1. Introduction
Significant research efforts have been devoted to characterising the conditions under which parabolic systems can accurately model pattern formation in chemotaxis processes, ensuring consistency with experimental observations. A cornerstone of this research is the Keller–Segel model, in which the chemotactic-diffusion mechanism enables the description of collective microbial behaviour driven by chemical gradients [Reference Bellomo, Bellouquid, Tao and Winkler1, Reference Keller and Segel8, Reference Ni and Takagi20].
Recent studies have demonstrated the crucial influence of chemotactic sensitivity function on the dynamic prospects of chemotaxis system
\begin{equation} \left \{ \begin{aligned} &u_{t}= \nabla \cdot (D(u)\nabla u)-\nabla \cdot (S(u)\nabla v),&x\in \Omega ,\, t\gt 0,\\[5pt] & v_{t}=\Delta v -v+u,&x\in \Omega ,\, t\gt 0,\\ \end{aligned} \right . \end{equation}
posed on a smoothly bounded n-dimensional domain. Specifically, the ratio
$\frac {S(u)}{D(u)}$
critically determines whether solutions to the zero-flux initial-boundary value problem of (1.1) are globally bounded or posses finite-time singularities. Indeed, when the diffusivity and sensitivity functions
$D,S\in C^2(\mathbb R_+)$
satisfy
$S(0)=0$
and the growth condition
$\frac {S(u)}{D(u)}\leq Cu^{\alpha }$
for all
$u\geq 1$
with
$C\gt 0$
and
$\alpha \lt \frac {2}{n}$
, the system admits a global bounded classical solution [Reference Ishida, Seki and Yokota6, Reference Tao and Winkler21]. In contrast, when
$\frac {S(u)}{D(u)}\geq Cu^{\alpha }$
for some
$C\gt 0$
and
$\alpha \gt \frac {2}{n}$
at large values, the solution exhibits finite-time blow-up through concentration phenomena [Reference Ciéslak and Stinner2, Reference Ciéslak and Stinner3].
When chemotaxis directs motion of bacteria (with density
$u$
) towards higher concentrations of nutrient (of concentration
$v$
), the resulting dynamics are mathematically described by chemotaxis-consumption models of the form
\begin{equation} \left \{ \begin{aligned} &u_{t}= \nabla \cdot (D(u)\nabla u)-\nabla \cdot (S(u)\nabla v),&x\in \Omega ,\, t\gt 0,\\[5pt] & v_{t}=\Delta v -vu,&x\in \Omega ,\, t\gt 0.\\ \end{aligned} \right . \end{equation}
In sharp contrast to the situation in (1.1), due to the essentially dissipative character of
$v$
-equation in (1.2), the latter system should exhibit some considerably stronger relaxation properties. Notably, when considering the specific case with constant diffusivity
$D(u)=1$
and linear sensitivity
$S(u)=u$
, subject to no-flux boundary conditions in bounded convex domains, the available literature reveals no evidence of non-trivial large-time dynamics for solutions of (1.2). At the level of rigorous mathematical analysis, it is confirmed that for reasonably regular but arbitrarily large initial data, two-dimensional versions of (1.2) are known to possess global bounded classical solutions (see [Reference Jin and Wang7, Reference Tao and Winkler22] for example); whereas when
$n =3$
, the certain global weak solutions emanating from large initial data at least eventually become bounded and smooth and particularly approach the unique relevant constant steady state in the large time limit [Reference Tao and Winkler22, Reference Winkler24].
Recent experimental studies indicate that motility constraints in Bacillus subtilis strains under nutrient-poor environments play a crucial role in the formation of diverse morphological aggregation patterns, including densely accumulating multiply branches structures [Reference Fujikawa4, Reference Fujikawa and Matsushita5, Reference Matsushita and Fujikawa19]. To gain a comprehensive understanding of the underlying mechanism, particularly the hypothesis that bacterial motility decreases under nutrient deprivation, a doubly degenerate nutrient taxis system of the form
\begin{equation} \left \{ \begin{aligned} &u_{t}=\nabla \cdot (u^{m-1}v\nabla u)-\chi \nabla \cdot (S(u)v\nabla v)+f(u,v),&x\in \Omega ,\, t\gt 0,\\[5pt] & v_{t}=\Delta v-uv,&x\in \Omega ,\, t\gt 0,\\ \end{aligned} \right . \end{equation}
with
$m=2, S(u)=u^2$
, was introduced in [Reference Kawasaki, Mochizuki, Umeda and Shigesada9, Reference Leyva, Málaga and Plaza12, Reference Plaza17]. A remarkable feature of (1.3) is the doubly degenerate structure of the first equation, arising from two distinct mechanisms: (i) porous-medium-type degeneracy as
$u$
approaches zero and (ii) cross-degeneracy induced by the factor
$ v$
, which vanishes when
$v$
tends to zero (as governed by the second equation). This peculiarity brings about noticeable mathematical challenges beyond classical porous medium-type diffusion theory. Hence, current understanding of even fundamental existence questions is still largely restricted to the global solvability in low-dimensional settings.
It is shown in [Reference Winkler26] that when
$\chi =0$
,
$f(u,v)=0$
and
$\Omega \subset \mathbb{R}^n$
is a bounded convex domain, the associated initial-boundary value problem for (1.3) possesses a global weak solution for the reasonably regular initial data. Moreover, within an appropriate topological setting, the solution can approach the non-homogeneous steady-state
$(u_\infty ,0)$
in the large time limit. It is noticed that due to the absence of taxis effects in (1.3), the comparison principle allows for deriving local bounds for
$\|u(\cdot ,t)\|_{L^\infty (\Omega )}$
, and the boundedness of
$\int _0^\infty \int _\Omega u^{q-1}v|\nabla u|^2$
for
$ q\in (0,1)$
becomes the basic regularity property of (1.3). These estimates provide the essential foundation for establishing the global existence of weak solution. However, when
$\chi \gt 0$
, the chemoattraction mechanisms in (1.3) may exert considerably destabilising effect, as evidenced by blow-up phenomena observed in analogous modelling contexts of (1.1) [Reference Bellomo, Bellouquid, Tao and Winkler1, Reference Lankeit and Winkler10]. To the best of our knowledge, the available analytical results for (1.3) with
$\chi \gt 0, f(u,v)=\ell uv$
so far remain restricted to low-dimensional settings:
-
i. In the one-dimensional case, when
$m=2$
and
$S(u)=u^2$
, global existence of continuous weak solutions and their asymptotic stabilisation towards non-homogeneous equilibria was established in [Reference Li and Winkler14, Reference Winkler29]. These results were later extended to cases where either
$2\leq m\lt 3$
and
$S(u)\leq Cu^{\alpha }$
for
$\alpha \in [m-1,\frac {m}{2}+1]$
or
$3\leq m\lt 4$
[Reference Wu32]. -
ii. In two-dimensional counterparts, global existence of weak solutions was first established for
$m=2$
and
$S(u)=u^{\alpha }$
with
$1\lt \alpha \lt \frac {3}{2}$
[Reference Li13]. Subsequent work in [Reference Winkler28] extended this result to all
$\alpha \lt 2$
, and in particular, proved
$L^{\infty }$
-bounds for the solutions. Notably, for the critical case
$\alpha =2$
, global existence of weak solutions was obtained under a smallness condition solely on the initial data
$v_0$
in [Reference Winkler27]. This restriction was later relaxed in [Reference Zhang and Li34]. Furthermore, the large-time behavior of solutions to (1.3) has been studied in [Reference Winkler31, Reference Wu33] using techniques based on Harnack inequalities.
In striking contrast to lower-dimensional cases where effective embedding theorems facilitate the analysis, the study of the three-dimensional version of (1.3) becomes significantly more subtle. To date, even fundamental questions regarding global solvability and boundedness, including the large-time behaviour, remain largely unestablished in the literature. For instance, the global solvability of (1.3) was established in [Reference Li13] when
$\frac {7}{6}\lt \alpha \lt \frac {13}{9}$
, which may be relaxed to
$1\lt \alpha \lt \frac {3}{2}$
[Reference Wu32]. Apart from that, the stabilising effect of logistic-type source terms
$f(u,v)=\rho u-\mu u^k$
on (1.3) is also investigated. For example, global existence of weak solutions was established in arbitrary dimensions
$n\geq 2$
under the condition
$k\gt \frac {n+2}{2}$
[Reference Pan16], and particularly the continuity of weak solutions was achieved when
$n=k=2$
[Reference Li and Winkler15].
The intention of the present work is to develop an analytical approach that not only establishes the global solvability of system (1.3) but also characterise the non-trivial long-time dynamics in three-dimensional domains. Specially, we shall consider the initial-boundary value problem
\begin{equation} \left \{ \begin{aligned} &u_{t}=\nabla \cdot (uv\nabla u)-\chi \nabla \cdot (u^{\alpha }v\nabla v)+\ell uv,&x\in \Omega ,\, t\gt 0,\\[5pt] & v_{t}=\Delta v-uv,&x\in \Omega ,\, t\gt 0,\\[5pt] &(uv\nabla u-\chi u^{\alpha }v\nabla v)\cdot \nu =\nabla v\cdot \nu =0,&x\in \partial \Omega ,\, t\gt 0,\\[5pt] &u(x,0)=u_0(x), v(x,0)=v_0(x), &x\in \Omega , \end{aligned} \right . \end{equation}
posed in a smoothly bounded domain
$\Omega \subset \mathbb{R}^3$
. Here, the parameters satisfy
$\chi \gt 0$
,
$\ell \gt 0$
and
$\alpha \in (\frac {3}{2},\frac {19}{12})$
. The initial data
$(u_0,v_0)$
are assumed to satisfy
\begin{equation} \left \{ \begin{aligned} &u_0\in W^{1,\infty }(\Omega )\; \hbox{is such that}\; u_0\geq 0\; \hbox{and}\; u_0\not \equiv 0, \quad \text{and} \\[5pt] &v_0\in W^{1,\infty }(\Omega )\; \hbox{is such that}\; v_0\gt 0\,\, \text{in} \,\,\Omega \\ \end{aligned} \right . \end{equation}
as well as
\begin{align} \|u_0\|_{L^{\infty }(\Omega )}+\|v_0\|_{L^{\infty }(\Omega )}+\|\nabla \ln v_0\|_{L^{\infty }(\Omega )}\leq K\quad \text{and}\quad \int _{\Omega }\ln {u_0}\geq -K \end{align}
for some
$K\gt 0$
.
In this framework, we will first address the basic issue of the global existence and boundedness of continuous weak solutions to (1.4) when the convexity of
$\Omega$
is not necessary.
Theorem 1.1.
Let
$\Omega \subset \mathbb R^3$
be a bounded domain with smooth boundary, and suppose that
$\alpha \in \left (\frac {3}{2}, \frac {19}{12} \right )$
as well as
$\chi \gt 0$
and
$\ell \gt 0$
. Then, for all
$p\gt 1$
, one can find
$C=C(p,K)\gt 0$
with the property that whenever
$u_0$
and
$v_0$
fulfill (1.5) and (1.6), there exist
\begin{equation} \left \{ \begin{aligned} &u\in C^{0}_{loc}(\overline {\Omega }\times [0,\infty ))\cap L^{\infty }_{loc}(\overline {\Omega }\times [0,\infty )) \quad \text{and} \\[5pt] &v\in C^{0}_{loc}(\overline {\Omega }\times [0,\infty ))\cap C^{2,1}_{loc}(\overline {\Omega }\times (0,\infty ))\cap L^{\infty }_{loc}([0,\infty );\;W^{1,\infty }(\Omega ))\\ \end{aligned} \right . \end{equation}
such that
$u\geq 0$
and
$v\gt 0$
a.e. in
$\Omega \times (0,\infty )$
, and that
$(u,v)$
forms a continuous global weak solution of (1.4) in the sense of Definition
2.1
below. Moreover, we have
\begin{align} \|u(\cdot ,t)\|_{L^{p}(\Omega )}+\|v(\cdot ,t)\|_{W^{1,\infty }(\Omega )}\leq C(p,K)\qquad for \;a.e.\; t\gt 0. \end{align}
By carefully tracking respective dependence on time, the estimates gained in the proof of Theorem1.1 already pave the way for our subsequent asymptotic analysis. Indeed, by means of a duality-based argument, we can achieve the following result on stability property enjoyed by function pairs
$(u_0,0)$
for all suitably regular
$u_0$
.
Theorem 1.2.
Let
$\Omega \subset \mathbb R^3$
be a bounded domain with smooth boundary, and suppose that
$\alpha \in \left (\frac {3}{2}, \frac {19}{12} \right )$
as well as
$\chi \gt 0$
and
$\ell \gt 0$
. Then for each
$\eta \gt 0$
, there exists
$\delta _1=\delta _1(\eta , K)\gt 0$
whenever
$u_0$
and
$v_0$
fulfill (1.5) and (1.6), as well as
\begin{equation*}\int _{\Omega }v_0\leq \delta _1,\end{equation*}
the solution
$(u,v)$
of (1.4) obtained in Theorem
1.1
satisfies
\begin{align} \|u(\cdot ,t)-u_0\|_{(W^{1,\infty }(\Omega ))^*}\leq \eta \qquad for \;all\; t\gt 0. \end{align}
Beyond that, we undertake a deeper qualitative analysis of the asymptotic properties of solutions. Although Theorem1.2 guarantees that the solution component
$u$
remains close to its initial data
$u_0$
in the weak-
$\ast$
topology of
$(W^{1,\infty }(\Omega ))^*$
for small
$v_0$
, the fundamental question of whether the limiting profile
$u_\infty$
must be constant remains open. We resolve this question by proving that when
$u_0\not \equiv const$
, sufficiently small initial concentrations of
$v_0$
necessarily yield nonconstant limiting profiles
$u_\infty$
.
Theorem 1.3.
Let
$\Omega \subset \mathbb R^3$
be a bounded domain with smooth boundary, and suppose that
$\alpha \in \left (\frac {3}{2}, \frac {19}{12} \right )$
as well as
$\chi \gt 0$
and
$\ell \gt 0$
. Then, there exists a nonnegative function
$u_{\infty }\in \bigcap _{p\geq 1}L^p(\Omega )$
such that as
$t\to \infty$
, the solution
$(u,v)$
of (1.4) from Theorem
1.1
satisfies
\begin{align} u(\cdot ,t)\rightharpoonup u_{\infty }\qquad in\;L^p(\Omega ),\qquad v(\cdot ,t)\rightarrow 0\qquad in\;W^{1,p}(\Omega )\;\; for \;all\; p\geq 1. \end{align}
Moreover, for the given nonnegative function
$u_0\not \equiv$
const., one can find
$\delta _2=\delta _2(K, u_0)\gt 0$
such that whenever
$u_0$
and
$v_0$
fulfill (1.5) and (1.6), as well as
\begin{equation*}\int _{\Omega }v_0\leq \delta _2,\end{equation*}
the limit function satisfies
$ u_{\infty }\not \equiv \mathrm{const}$
.
Main ideas. The main challenge in developing a mathematical theory for system (1.4) lies in effectively quantifying its dissipative mechanisms, which are substantially weakened by the signal-dependent degeneracy in the
$u$
-equation. Accordingly, existing literature has been limited to (1.4) posed on the bounded convex domain
$\Omega \subset \mathbb{R}^n, n\leq 2$
[Reference Li13, Reference Li and Winkler14, Reference Winkler26–Reference Winkler29]. In non-convex three-dimensional domains, however, the analysis becomes considerably more intricate due to the loss of favourable embedding properties.
Our approach is to improve the regularity of (1.4) via the bootstrap argument. The most crucial step is to obtain the global boundedness of
$\|u(\cdot ,t)\|_{L^{p_0}(\Omega )}$
for some
$p_0\gt \frac {3}{2}$
. To this end, we shall appropriately utilise the diffusion-induced contributions of the form
$\int _{\Omega }u^{k}v|\nabla u|^2$
(for suitable
$k$
) to counteract the unfavourable terms typified by
$\int _{\Omega }u^{\beta }v$
. As the starting point of our analysis, we certify that the integrals
$\int _{\Omega }u^{1-\alpha }v|\nabla u|^2$
and
$\int _{\Omega }u\frac {|\nabla v|^{2}}{v}$
are dissipated by tracking the time evolution of
$\int _{\Omega }u^{2-\alpha }$
and
${F}(t)\;:\!=\;a\int _{\Omega } \frac {|\nabla v(\cdot ,t)|^2}{v(\cdot ,t)}-\int _{\Omega }\ln {u(\cdot ,t)}$
(with some
$a\gt 0$
) along trajectories of (1.4), as detailed in Lemmas2.3 and 2.5, respectively, which distinguishes our approach from that in [Reference Li13]. Moreover, differing from [Reference Li13, Reference Wu32], our derivation of bounds for quantities such as
\begin{align} \int _0^{T}\int _\Omega u^{\frac {5}{3}}v \end{align}
is to make use of the aforementioned dissipation terms and an interpolation inequality (Lemma2.6). Building upon (1.11), we can estimate the unfavourable terms arising from the time evolution of the functional
\begin{equation*} \mathcal{ G}(t)\;:\!=\;\int _{\Omega }u(\cdot ,t)\ln {u}(\cdot ,t)+b\int _{\Omega }\frac {|\nabla v(\cdot ,t)|^4}{v^3(\cdot ,t)}\end{equation*}
for some
$b\gt 0$
, and thereby derive the dissipated quantity of the form
\begin{align} \int _{\Omega }u^{p_*}v|\nabla u|^2 \quad \text{and}\quad \int _{\Omega }u\frac {|\nabla v|^4}{v^3} \end{align}
for all
$p_*\in [1-\alpha , 0]$
(Lemmas3.3 and 3.4). A further crucial ingredient in our analysis is the novel observation: Assume that
$\Omega \subset \mathbb{R}^3$
is a smoothly bounded domain,
$L\gt 0, k\in (\!-1,-\frac {1}{3})$
and
$\beta \in [1,k+\frac {8}{3})$
, then there exists
$C(L,k, \beta )\gt 0$
such that
\begin{align} \int _{\Omega }\varphi ^{\beta }\psi \leq \int _{\Omega }\varphi ^{k}\psi |\nabla \varphi |^2+\int _{\Omega }\varphi \frac {|\nabla \psi |^4}{\psi ^3}+C(L,k, \beta )\int _{\Omega }\varphi \psi \end{align}
holds for all sufficiently regular positive functions
$\varphi$
and
$\psi$
fulfilling
$\int _{\Omega }\varphi \leq L$
(see Lemma3.5), which is somewhat different from that in [Reference Wu32]. Thanks to the estimates (1.12), we then employ inequality (1.13) to achieve the desired
$L^{p_0}$
bounds for
$u$
. This improved integrability of
$u$
allows us to derive a refined version of (1.13) (see Lemma3.7) and accordingly certify an energy-like property for the coupled quantity
\begin{align*} \mathcal{ H}(t)\;:\!=\;\int _{\Omega }\frac {|\nabla v(\cdot ,t)|^q}{v^{q-1}(\cdot ,t)}+\int _{\Omega }u^{p}(\cdot ,t) \end{align*}
for any fixed
$p\gt 1$
. This property thereby yields the bounds for
$\|u(\cdot ,t)\|_{L^{p}(\Omega )}$
as detailed in Lemma3.8.
By suitably taking into account respective dependence on time, we intend to characterise the large-time behaviour of solution
$(u,v)$
to system (1.4). To achieve this, we employ a duality-based estimate for the time derivative
$u_t$
, establishing that
\begin{align} \int _0^{\infty }\|u_t(\cdot ,t)\|_{(W^{1,\infty }(\Omega ))^*}dt\leq C(K)\cdot \left \{ \int _{\Omega } v_0\right \}^\sigma , \end{align}
for some
$C(K)\gt 0$
and
$\sigma \gt 0$
(see Lemma4.1). Thanks to (1.14), we prove that the solution of (1.4) converges to equilibrium state
$(u_{\infty },0)$
as
$t\to \infty$
(see Lemmas4.3 and 4.6). Moreover, we establish that for sufficiently small initial data
$v_0$
, the long-time limit
$u_{\infty }$
exhibits spatial heterogeneity (Lemma4.7).
The structure of this paper is as follows: In Section 2, we specify the weak solutions of system (1.4) and establishes fundamental a priori estimates for the regularised systems. In Section 3, crucial
$L^p$
-estimates for
$u_\varepsilon$
are derived through careful analysis of energy functionals and bootstrap arguments, which needs to effectively quantifying the interplay between the weaken smoothing effects of random diffusion and the potentially destabilisng influence of taxis-diffusion. Building upon these estimates and further higher regularity properties of the regularised systems, we then prove Theorem1.1 via compactness arguments. In Section 4, we investigates the stability properties of
$(u,v)$
by means of a duality-based argument.
2. Preliminaries
In view of the considered diffusion degeneracy in (1.4), the notation of weak solutions specified below seems fairly natural [Reference Winkler28], but it is somewhat stronger than those in [Reference Li13, Reference Winkler26], as it requires the integrability of
$\nabla u$
.
Definition 2.1.
Assume that (1.5) and (1.6) hold, suppose that
$\chi \gt 0$
and
$\ell \gt 0$
, and that
$\alpha \geq 1$
. Then, a pair of nonnegative functions
\begin{equation} \left \{ \begin{aligned} &u\in C^{0}(\overline {\Omega }\times [0,\infty ))\quad and \\[5pt] &v\in C^{0}(\overline {\Omega }\times [0,\infty ))\cap C^{2,1}(\overline {\Omega }\times (0,\infty ))\\ \end{aligned} \right . \end{equation}
such that
\begin{align} u^2 \in L^{1}_{loc}([0,\infty );\; W^{1,1}(\Omega ))\quad \hbox{and}\quad u^\alpha \nabla v \in L^{1}_{loc}(\overline {\Omega }\times [0,\infty );\;\mathbb{R}^3) \end{align}
will be called a continuous weak solution of (1.4) if
\begin{align} -\int _{0}^{\infty }\!\int _{\Omega }u\varphi _t-\int _{\Omega }u_0\varphi (\cdot ,0)= &-\frac {1}{2}\int _{0}^{\infty } \!\int _{\Omega } v \nabla u^2 \cdot \nabla \varphi +\chi \int _{0}^{\infty }\!\int _{\Omega }u^{\alpha }v\nabla v\cdot \nabla \varphi +\ell \int _{0}^{\infty }\!\int _{\Omega }uv\varphi \end{align}
for all
$\varphi \in C_0^{\infty }(\overline {\Omega }\times [0,\infty ))$
fulfilling
$\frac {\partial \varphi }{\partial \nu }=0$
on
$\partial \Omega \times (0,\infty )$
, and if
\begin{align} \int _{0}^{\infty }\int _{\Omega }v\varphi _t+\int _{\Omega }v_0\varphi (\cdot ,0)=\int _{0}^{\infty }\int _{\Omega }\nabla v\cdot \nabla \varphi +\int _{0}^{\infty }\int _{\Omega }uv\varphi . \end{align}
for any
$\varphi \in C_0^{\infty }(\overline {\Omega }\times [0,\infty ))$
.
To construct the defined weak solutions above through approximation by the classical solutions to the conveniently regularised problems, we consider the the regularised variants of (1.4) given by
\begin{equation} \left \{ \begin{aligned} &u_{\varepsilon t}=\nabla \cdot \big(u_{\varepsilon }v_{\varepsilon }\nabla u_{\varepsilon }\big)-\chi \nabla \cdot (u^{\alpha }_{\varepsilon }v_{\varepsilon }\nabla v_{\varepsilon })+\ell u_{\varepsilon }v_{\varepsilon },&x\in \Omega ,\, t\gt 0,\\[5pt] & v_{\varepsilon t}=\Delta v_{\varepsilon }-u_{\varepsilon }v_{\varepsilon },&x\in \Omega ,\, t\gt 0,\\[5pt] & \frac {\partial u_{\varepsilon }}{\partial \nu }=\frac {\partial v_{\varepsilon }}{\partial \nu }=0,&x\in \partial \Omega ,\, t\gt 0,\\[5pt] & u_{\varepsilon }(x,0)=u_0(x)+\varepsilon ,\;v_{\varepsilon }(x,0)=v_0(x),&x\in \Omega ,\, \end{aligned} \right . \end{equation}
for
$\varepsilon \in (0,1)$
. According to Lemma2.1 of [Reference Winkler27], we have the following statement regarding the existence, extensibility and basic properties of solutions to (2.5).
Lemma 2.1.
Let
$\Omega \subset \mathbb R^3$
be a bounded domain with smooth boundary,
$\chi \gt 0$
,
$\ell \gt 0$
and
$\alpha \geq 1$
, and assume that (1.5) and (1.6) hold. Then, for each
$\varepsilon \in (0,1)$
, there exist
$T_{max,\varepsilon }\in (0,\infty ]$
and functions
\begin{equation} \left \{ \begin{aligned} &u_{\varepsilon }\in C^0(\overline {\Omega }\times [0,T_{max,\varepsilon })\cap C^{2,1}(\overline {\Omega }\times (0,T_{max,\varepsilon })) \\[5pt] &v_{\varepsilon }\in C^0(\overline {\Omega }\times [0,T_{max,\varepsilon }))\cap C^{2,1}(\overline {\Omega }\times (0,T_{max,\varepsilon })) \\ \end{aligned} \right . \end{equation}
such that
$u_ \varepsilon \gt 0$
and
$v_{\varepsilon }\gt 0$
in
$\overline {\Omega } \times [0,T_{max,\varepsilon })$
, that
$(u_{\varepsilon },v_{\varepsilon })$
solves (2.5) classically in
$\Omega \times (0,T_{max,\varepsilon })$
, and that
\begin{equation} \hbox{if}\,\, T_{max,\varepsilon }\lt \infty , \,\, \hbox{then}\, \limsup _{t\nearrow T_{max,\varepsilon }}\|u(\cdot , t)\|_{L^{\infty }(\Omega )}=\infty . \end{equation}
Furthermore, this solution satisfies
\begin{align} \|v_{\varepsilon }(\cdot , t)\|_{L^{\infty }(\Omega )}\leq \|v_{0 }\|_{L^{\infty }(\Omega )}\qquad for\; all\; t\in (0,T_{max,\varepsilon }) \end{align}
and
\begin{align} \int _{\Omega }u_{0}\leq \int _{\Omega }u_{\varepsilon }(\cdot ,t)\leq \int _{\Omega }u_{0}+\ell \int _{\Omega }v_{0}+|\Omega |\qquad for\; all\; t\in (0,T_{max,\varepsilon }) \end{align}
as well as
\begin{align} \int _{0}^{T_{max,\varepsilon }}\int _{\Omega }u_{\varepsilon }v_{\varepsilon }\leq \int _{\Omega }v_{0}. \end{align}
Throughout the sequel, unless explicitly stated otherwise, we shall assume that
$\Omega \subset \mathbb R^3$
is a smoothly bounded domain (the convexity of
$\Omega$
is not required),
$\chi \gt 0$
,
$\ell \gt 0$
and
$\alpha \geq 1$
. Moreover, whenever functions
$u_0$
and
$v_0$
fulfilling (1.5)–(1.6) have been selected, we shall denote by
$(u_{\varepsilon },v_{\varepsilon })$
the solutions given by Lemma2.1, with
$T_{max,\varepsilon }$
representing their maximal existence time.
Due to the nonnegativity of
$u_{\varepsilon }$
and
$v_{\varepsilon }$
, the evolution of
$v_{\varepsilon }$
yields the following estimate.
Lemma 2.2.
For any
$K\gt 0$
with the property that (1.6) is valid, one can find
$C=C(K)\gt 0$
such that
\begin{align} \int _0^{T_{max,\varepsilon }}\int _{\Omega }v_{\varepsilon }|\nabla v_{\varepsilon }|^2\leq C. \end{align}
Proof. Multiplying the second equation in (2.5) by
$v_{\varepsilon }^2$
, we have
\begin{align} \frac {d}{dt}\int _{\Omega }v_{\varepsilon }^3+6\int _{\Omega }v_{\varepsilon }|\nabla v_{\varepsilon }|^2=-3\int _{\Omega }u_{\varepsilon }v_{\varepsilon }^3\leq 0\qquad for\; all\; t\in (0,T_{max,\varepsilon }). \end{align}
Integrating (2.12) over
$(0,T_{max,\varepsilon })$
and using (1.5), we then obtain (2.11).
It is noticed that in [Reference Li13], the bounds of
$\int _0^{T_{\max , \varepsilon }}\int _{\Omega } u^{2-2\alpha }_{\varepsilon } v_{\varepsilon }|\nabla u_{\varepsilon }|^2$
were utilised as a basic piece of information in deriving the bounds for
$\int _\Omega u^{p_0}_\varepsilon (\cdot ,t)$
with some
$p_0\gt 1$
. In contrast, our approach in this paper takes
$ \int _0^{T_{\max , \varepsilon }}\int _{\Omega } u_{\varepsilon }^{1-\alpha }v_{\varepsilon }|\nabla u_{\varepsilon }|^2$
as the starting point for the bootstrap argument, which is achieved by analysing the temporal evolution of
$ -\frac {1}{2-\alpha }\int _{\Omega } u_{\epsilon }^{2-\alpha }+ \frac {\chi (\alpha -1)}{2} \int _{\Omega } |\nabla v_{\varepsilon }|^2$
rather
$-\frac {1}{3-2\alpha }\int _{\Omega }u_{\varepsilon }^{3-2\alpha }$
.
Lemma 2.3.
Let
$\alpha \in \left (1,2\right ]$
. Then, for all
$K\gt 0$
with the property that (1.6) is valid, one can find
$C=C(K)\gt 0$
such that
\begin{align} \int _0^{T_{\max , \varepsilon }}\int _{\Omega } u_{\varepsilon }^{1-\alpha }v_{\varepsilon }|\nabla u_{\varepsilon }|^2 \leq C . \end{align}
Proof. Multiplying the equations in (2.5) by
$u_{\epsilon }^{1-\alpha }$
and
$\Delta v_{\varepsilon }$
, respectively, we obtain
\begin{align} -\frac {1}{2-\alpha }\frac {d}{dt}\int _{\Omega }u_{\varepsilon }^{2-\alpha }+(\alpha -1)\int _{\Omega } u_{\varepsilon }^{1-\alpha }v_{\varepsilon }|\nabla u_{\varepsilon }|^2=(\alpha -1)\chi \int _{\Omega } v_{\varepsilon }\nabla u_{\varepsilon }\cdot \nabla v_{\varepsilon }-\ell \int _{\Omega } u_{\varepsilon }^{2-\alpha }v_{\varepsilon } \end{align}
and
\begin{align} \frac {1}{2}\frac {d}{dt}\int _{\Omega }|\nabla v_{\varepsilon }|^2+\int _{\Omega }|\Delta v_{\varepsilon }|^2=\int _{\Omega }u_{\varepsilon }v_{\varepsilon }\Delta v_{\varepsilon }=-\int _{\Omega } v_{\varepsilon }\nabla u_{\varepsilon }\cdot \nabla v_{\varepsilon }-\int _{\Omega }u_{\varepsilon }|\nabla v_{\varepsilon }|^2 \end{align}
for all
$t \in (0, T_{\max , \varepsilon })$
. From (2.14) and (2.15), we have
\begin{align*} &\frac {d}{dt}\left \{-\frac {1}{2-\alpha }\int _{\Omega } u_{\epsilon }^{2-\alpha }+ \frac {\chi (\alpha -1)}{2} \int _{\Omega } |\nabla v_{\varepsilon }|^2\right \}+(\alpha -1)\int _{\Omega } u_{\varepsilon }^{1-\alpha }v_{\varepsilon }|\nabla u_{\varepsilon }|^2\\[5pt] &\leq -\ell \int _{\Omega } u_{\varepsilon }^{2-\alpha }v_{\varepsilon }-\chi (\alpha -1)\int _{\Omega }u_{\varepsilon }|\nabla v_{\varepsilon }|^2 \end{align*}
for all
$t \in (0, T_{\max , \varepsilon })$
. Integrating the above differential inequality on
$(0,t)$
, along with (1.6) and (2.9), we then get
\begin{align*} & \frac {1}{2-\alpha }\int _{\Omega } u_{0\epsilon }^{2-\alpha }+ \frac {\chi (\alpha -1)}{2} \int _{\Omega } |\nabla v_{\varepsilon }|^2 +(\alpha -1)\int _0^{t}\int _{\Omega } u_{\varepsilon }^{1-\alpha }v_{\varepsilon }|\nabla u_{\varepsilon }|^2\\[5pt] &+\ell \int _0^{t}\int _{\Omega } u_{\varepsilon }^{2-\alpha }v_{\varepsilon }+\chi (\alpha -1)\int _0^{t}\int _{\Omega }u_{\varepsilon }|\nabla v_{\varepsilon }|^2\\[5pt] ={} &\frac {1}{2-\alpha }\int _{\Omega } u_{\varepsilon }^{2-\alpha }+ \frac {\chi (\alpha -1)}{2} \int _{\Omega } |\nabla v_{0 }|^2\\[5pt] \leq{} &\frac {1}{2-\alpha }\int _{\Omega }\left (u_{\varepsilon }+1\right )+ \frac {\chi (\alpha -1)}{2} \int _{\Omega } |\nabla v_{0 }|^2\\[5pt] \leq{} &C(K) \end{align*}
for all
$t \in (0, T_{\max , \varepsilon })$
, due to
$2-\alpha \in [0,1)$
, which implies (2.13) immediately.
When the convexity of
$\Omega$
is not assumed, the following estimate indicates how the related boundary integral over
$\partial \Omega$
can be controlled by the corresponding higher-order integrals on
$\Omega$
involving singular weights.
Lemma 2.4. [Reference Winkler25] Let
$q\geq 2$
and
$\eta \gt 0$
. Then, there exists
$C=C(\eta ,q)\gt 0$
with the property that
\begin{align} \int _{\partial \Omega }\psi ^{-q+1}|\nabla \psi |^{q-2}\cdot \frac {\partial |\nabla \psi |}{\partial \nu }\leq \eta \int _{\Omega }\psi ^{-q+1}|\nabla \psi |^{q-2}|D^2\psi |^2+\eta \int _{\Omega }\psi ^{-q-1}|\nabla \psi |^{q+2}+C\int _{\Omega }\psi \end{align}
for any
$\psi \in C^2(\overline {\Omega })$
, which is such that
$\psi \gt 0$
in
$\overline {\Omega }$
and
$\frac {\partial \psi }{\partial \nu }=0$
on
$\partial \Omega$
.
By analysing the temporal evolution of
\begin{equation*}\|v_0\|_{L^{\infty }(\Omega )}^2\int _{\Omega } \frac {|\nabla v_{\varepsilon }|^2}{v_{\varepsilon }}-\int _{\Omega }\ln {u_{\varepsilon }},\end{equation*}
we establish spatio-temporal integral bounds for both
$v_{\varepsilon }$
and
$\frac {u_{\varepsilon }}{v_{\varepsilon }}|\nabla v_{\varepsilon }|^2$
, thanks to the estimates (2.13) and (2.16).
Lemma 2.5.
Assume
$\alpha \in \left (\frac {3}{2},2\right ]$
and
$K\gt 0$
such that (1.6) holds. Then, there exists
$C\gt 0$
independent of
$T_{\max , \varepsilon }$
such that
\begin{align} \int _0^{T_{\max , \varepsilon }}\int _{\Omega }v_{\varepsilon }\leq C \end{align}
and
\begin{align} \int _0^{T_{\max , \varepsilon }}\int _{\Omega } \frac {u_{\varepsilon }}{v_{\varepsilon }}|\nabla v_{\varepsilon }|^2 \leq C \end{align}
as well as
\begin{align} \int _0^{T_{\max , \varepsilon }}\int _{\Omega } \frac {v_{\varepsilon }}{u_{\varepsilon }}|\nabla u_{\varepsilon }|^2 \leq C\qquad and \qquad \int _0^{T_{\max , \varepsilon }}\int _{\Omega } \frac {|\nabla v_{\varepsilon }|^4}{v_{\varepsilon }^3} \leq C. \end{align}
Proof. According to the second equation in (2.5), several integrations by parts show that for all
$t \in \left (0, T_{\max , \varepsilon }\right )$
,
\begin{align*} \frac {1}{2} \frac {d}{dt} \int _{\Omega } \frac {\left |\nabla v_{\varepsilon }\right |^2}{v_{\varepsilon }} ={} & \int _{\Omega } \frac {1}{v_{\varepsilon }} \nabla v_{\varepsilon } \cdot \nabla \left \{\Delta v_{\varepsilon } -u_{\varepsilon } v_{\varepsilon }\right \} -\frac {1}{2} \int _{\Omega } \frac {1}{v_{\varepsilon }^2}\left |\nabla v_{\varepsilon }\right |^2 \cdot \left \{\Delta v_{\varepsilon }-u_{\varepsilon } v_{\varepsilon }\right \} \\[5pt] ={} & \int _{\Omega } \frac {1}{v_{\varepsilon }} \nabla v_{\varepsilon } \cdot \nabla \Delta v_{\varepsilon } -\int _{\Omega } \frac {u_{\varepsilon }}{v_{\varepsilon }}\left |\nabla v_{\varepsilon }\right |^2 -\int _{\Omega } \nabla u_{\varepsilon } \cdot \nabla v_{\varepsilon }\\[5pt] & -\frac {1}{2} \int _{\Omega } \frac {1}{v_{\varepsilon }^2}\left |\nabla v_{\varepsilon }\right |^2 \cdot \Delta v_{\varepsilon } +\frac {1}{2} \int _{\Omega } \frac {u_{\varepsilon }}{v_{\varepsilon }}\left |\nabla v_{\varepsilon }\right |^2\\[5pt] ={} & \int _{\Omega } \frac {1}{v_{\varepsilon }} \nabla v_{\varepsilon } \cdot \nabla \Delta v_{\varepsilon } -\frac {1}{2} \int _{\Omega } \frac {1}{v_{\varepsilon }^2}\left |\nabla v_{\varepsilon }\right |^2 \cdot \Delta v_{\varepsilon }-\frac {1}{2} \int _{\Omega } \frac {u_{\varepsilon }}{v_{\varepsilon }}\left |\nabla v_{\varepsilon }\right |^2 -\int _{\Omega } \nabla u_{\varepsilon } \cdot \nabla v_{\varepsilon }\\[5pt] ={} & \frac {1}{2}\int _{\Omega } \frac {1}{v_{\varepsilon }}\Delta |\nabla v_{\varepsilon }|^2-\int _{\Omega } \frac {1}{v_{\varepsilon }}|D^2 v_{\varepsilon }|^2+ \frac {1}{2}\int _{\Omega } \nabla \bigg(\frac {|\nabla v_{\varepsilon }|^2}{v_{\varepsilon }^2} \bigg) \cdot \nabla v_{\varepsilon }\\[5pt] &-\frac {1}{2} \int _{\Omega } \frac {u_{\varepsilon }}{v_{\varepsilon }}\left |\nabla v_{\varepsilon }\right |^2 -\int _{\Omega } \nabla u_{\varepsilon } \cdot \nabla v_{\varepsilon }\\[5pt] ={} &\frac {1}{2} \int _{\partial \Omega } \frac {1}{v_{\varepsilon }} \frac {\partial |\nabla v_{\varepsilon }|^2}{\partial \nu }+\int _{\Omega }\frac {1}{v_{\varepsilon }^2} \nabla v_{\varepsilon }\cdot \nabla |\nabla v_{\varepsilon }|^2 -\int _{\Omega } \frac {1}{v_{\varepsilon }}|D^2 v_{\varepsilon }|^2-\int _{\Omega } \frac {|\nabla v_{\varepsilon }|^4}{ v_{\varepsilon }^3}\\[5pt] &-\frac {1}{2} \int _{\Omega } \frac {u_{\varepsilon }}{v_{\varepsilon }}\left |\nabla v_{\varepsilon }\right |^2 -\int _{\Omega } \nabla u_{\varepsilon } \cdot \nabla v_{\varepsilon }, \end{align*}
where
$\nabla v_{\varepsilon } \cdot \nabla \Delta v_{\varepsilon }=\frac {1}{2}\Delta |\nabla v_{\varepsilon }|^2-|D^2 v_{\varepsilon }|^2$
is used. Thanks to the identity [Reference Winkler25, Lemma 3.2]
\begin{align} |D^2 v_{\varepsilon }|^2=v_{\varepsilon }^2|D^2 \ln v_{\varepsilon }|^2+\frac {1}{v_{\varepsilon }}\nabla v_{\varepsilon }\cdot \nabla |\nabla v_{\varepsilon }|^2-\frac {1}{v_{\varepsilon }^2}|\nabla v_{\varepsilon }|^4, \end{align}
we then get
\begin{align} \frac {1}{2} \frac {d}{dt}\int _{\Omega } \frac {\left |\nabla v_{\varepsilon }\right |^2}{v_{\varepsilon }} +\int _{\Omega } v_{\varepsilon }\left |D^2 \ln v_{\varepsilon }\right |^2 +\frac {1}{2} \int _{\Omega } \frac {u_{\varepsilon }}{v_{\varepsilon }}\left |\nabla v_{\varepsilon }\right |^2 = \frac {1}{2} \int _{\partial \Omega } \frac {1}{v_{\varepsilon }} \frac {\partial |\nabla v_{\varepsilon }|^2}{\partial \nu } -\int _{\Omega } \nabla u_{\varepsilon } \cdot \nabla v_{\varepsilon } \end{align}
for all
$t \in \left (0, T_{\max , \varepsilon }\right )$
. From Lemma3.4 in [Reference Winkler25], it follows that there exists
$c_1\gt 0$
such that
\begin{equation} \int _{\Omega } v_{\varepsilon }\left |D^2 \ln v_{\varepsilon }\right |^2 \geq c_1 \int _{\Omega } \frac {\left |D^2 v_{\varepsilon }\right |^2}{v_{\varepsilon }} +c_1 \int _{\Omega } \frac {\left |\nabla v_{\varepsilon }\right |^4}{v_{\varepsilon }^3}. \end{equation}
As an application of Lemma2.4 with
$\eta \;:\!=\;c_1, q\;:\!=\;2$
, we arrive at
\begin{align} \int _{\partial \Omega } \frac {1}{v_{\varepsilon }} \cdot \frac {\partial \left |\nabla v_{\varepsilon }\right |^2}{\partial \nu } \leq c_1 \int _{\Omega } \frac {\left |D^2 v_{\varepsilon }\right |^2}{v_{\varepsilon }} +c_1 \int _{\Omega } \frac {\left |\nabla v_{\varepsilon }\right |^4}{v_{\varepsilon }^3} +C(c_1,2) \int _{\Omega } v_{\varepsilon }. \end{align}
Apart from that, by Young’s inequality, we infer that for all
$t \in (0, T_{\max , \varepsilon })$
\begin{align} -\int _{\Omega } \nabla u_{\varepsilon } \cdot \nabla v_{\varepsilon } &\leq \int _{\Omega }u_{\varepsilon }^{1-\alpha }v_{\varepsilon }|\nabla u_{\varepsilon }|^2 +\frac {1}{4}\int _{\Omega } \frac {u_{\varepsilon }^{\alpha -1}}{v_{\varepsilon }} |\nabla v_{\varepsilon }|^2\nonumber \\[5pt] &\leq \int _{\Omega }u_{\varepsilon }^{1-\alpha }v_{\varepsilon }|\nabla u_{\varepsilon }|^2 +\frac {1}{4}\int _{\Omega } \left (u_{\varepsilon }+1\right )\frac {|\nabla v_{\varepsilon }|^2}{v_{\varepsilon }}.\nonumber \\[5pt] &\leq \int _{\Omega }u_{\varepsilon }^{1-\alpha }v_{\varepsilon }|\nabla u_{\varepsilon }|^2 +\frac {1}{4}\int _{\Omega } \frac {u_{\varepsilon }}{v_{\varepsilon }}|\nabla v_{\varepsilon }|^2+\frac {c_1}{4}\int _{\Omega } \frac {|\nabla v_{\varepsilon }|^4}{v_{\varepsilon }^3}+ \frac {1}{2c_1} \int _{\Omega } v_{\varepsilon }. \end{align}
Summing up (2.21)–(2.24), we conclude that
\begin{align} \frac {d}{dt}\int _{\Omega } \frac {|\nabla v_{\varepsilon }|^2}{v_{\varepsilon }} +\frac {1}{2} \int _{\Omega } \frac {u_{\varepsilon }}{v_{\varepsilon }} |\nabla v_{\varepsilon }|^2+\frac {c_1}{2}\int _{\Omega } \frac {|\nabla v_{\varepsilon }|^4}{v_{\varepsilon }^3} \leq 2 \int _{\Omega }u_{\varepsilon }^{1-\alpha }v_{\varepsilon }|\nabla u_{\varepsilon }|^2 +c_2 \int _{\Omega } v_{\varepsilon } \end{align}
with
$ c_2 \;:\!=\;C(c_1,2)+\frac {1}{c_1}$
for all
$t \in (0, T_{\max , \varepsilon })$
. On the other hand, multiplying the first equation in (2.5) by
$-\frac {1}{u_{\varepsilon }}$
and using (2.8) and Young’s inequality, we have
\begin{align} &-\frac {d}{dt}\int _{\Omega }\ln {u_{\varepsilon }}+\ell \int _{\Omega } v_{\varepsilon }+\int _{\Omega } \frac {v_{\varepsilon }}{u_{\varepsilon }} |\nabla u_{\varepsilon }|^2\nonumber \\[5pt] &=\chi \int _{\Omega }u_{\varepsilon }^{\alpha -2}v_{\varepsilon }\nabla u_{\varepsilon }\cdot \nabla v_{\varepsilon }\nonumber \\[5pt] &\leq \frac {1}{2}\int _{\Omega } \frac {v_{\varepsilon }}{u_{\varepsilon }} |\nabla u_{\varepsilon }|^2 +\frac {\chi ^2}{2}\int _{\Omega } u_{\varepsilon }^{2\alpha -3}v_{\varepsilon } |\nabla v_{\varepsilon }|^2\nonumber \\[5pt] &\leq \frac {1}{2}\int _{\Omega } \frac {v_{\varepsilon }}{u_{\varepsilon }} |\nabla u_{\varepsilon }|^2 +\frac {\chi ^2}{2}\int _{\{u_{\varepsilon }\geq 1\}} u_{\varepsilon }^{2\alpha -3}v_{\varepsilon } |\nabla v_{\varepsilon }|^2+\frac {\chi ^2}{2}\int _{\{u_{\varepsilon }\lt 1\}} v_{\varepsilon } |\nabla v_{\varepsilon }|^2\nonumber \\[5pt] &\leq \frac {1}{2}\int _{\Omega } \frac {v_{\varepsilon }}{u_{\varepsilon }} |\nabla u_{\varepsilon }|^2 +\frac {\chi ^2\eta }{2}\int _{\Omega } u_{\varepsilon }v_{\varepsilon } |\nabla v_{\varepsilon }|^2+\frac {\chi ^2}{2}(\eta ^{-1})^{\frac {2\alpha -3}{4-2\alpha }}\int _{\Omega } v_{\varepsilon } |\nabla v_{\varepsilon }|^2+\frac {\chi ^2}{2}\int _{\{u_{\varepsilon }\lt 1\}} v_{\varepsilon } |\nabla v_{\varepsilon }|^2\nonumber \\[5pt] &\leq \frac {1}{2}\int _{\Omega } \frac {v_{\varepsilon }}{u_{\varepsilon }} |\nabla u_{\varepsilon }|^2 +\frac {\chi ^2\eta }{2}\|v_0\|_{L^{\infty }(\Omega )}^2\int _{\Omega } \frac {u_{\varepsilon }}{v_{\varepsilon }} |\nabla v_{\varepsilon }|^2+\frac {\chi ^2}{2}(\eta ^{\frac {3-2\alpha }{4-2\alpha }}+1)\int _{\Omega } v_{\varepsilon } |\nabla v_{\varepsilon }|^2 \end{align}
for all
$t \in (0, T_{\max , \varepsilon })$
, due to
$\alpha \in \left (\frac {3}{2},2\right ]$
. Letting
\begin{equation*} \eta \;:\!=\;\frac {\ell }{4\chi ^2c_2\|v_0\|_{L^{\infty }(\Omega )}^2} \quad \text{and} \quad C(\alpha )\;:\!=\;\frac {\chi ^2}{2}\big(\eta ^{\frac {3-2\alpha }{4-2\alpha }}+1\big). \end{equation*}
Then, combining (2.25) with (2.26) yields
\begin{align*} &\frac {d}{dt}\left \{\frac {\ell }{2c_2}\int _{\Omega } \frac {|\nabla v_{\varepsilon }|^2}{v_{\varepsilon }}-\int _{\Omega }\ln {u_{\varepsilon }} \right \}+\frac {\ell }{4c_2} \int _{\Omega } \frac {u_{\varepsilon }}{v_{\varepsilon }} |\nabla v_{\varepsilon }|^2+\frac {\ell c_1}{4c_2}\int _{\Omega } \frac {|\nabla v_{\varepsilon }|^4}{v_{\varepsilon }^3}+\frac {\ell }{2}\int _{\Omega } v_{\varepsilon }+\frac {1}{2}\int _{\Omega } \frac {v_{\varepsilon }}{u_{\varepsilon }} |\nabla u_{\varepsilon }|^2\\[5pt] &\leq \frac {\chi ^2\eta }{2}\|v_0\|_{L^{\infty }(\Omega )}^2\int _{\Omega } \frac {u_{\varepsilon }}{v_{\varepsilon }} |\nabla v_{\varepsilon }|^2 +\frac {\ell }{c_2} \int _{\Omega }u_{\varepsilon }^{1-\alpha }v_{\varepsilon }|\nabla u_{\varepsilon }|^2+C(\alpha )\int _{\Omega } v_{\varepsilon } |\nabla v_{\varepsilon }|^2\\[5pt] &= \frac {\ell }{8c_2} \int _{\Omega } \frac {u_{\varepsilon }}{v_{\varepsilon }} |\nabla v_{\varepsilon }|^2 +\frac {\ell }{c_2} \int _{\Omega }u_{\varepsilon }^{1-\alpha }v_{\varepsilon }|\nabla u_{\varepsilon }|^2+C(\alpha )\int _{\Omega } v_{\varepsilon } |\nabla v_{\varepsilon }|^2 \end{align*}
for all
$t \in (0, T_{\max , \varepsilon })$
. On integration in time, according to
$\ln {\xi }\leq \xi$
for all
$\xi \gt 0$
entails that
\begin{align} &\frac {\ell }{2c_2}\int _{\Omega } \frac {|\nabla v_{\varepsilon }|^2}{v_{\varepsilon }}+\frac {\ell }{8c_2} \int _0^t\int _{\Omega } \frac {u_{\varepsilon }}{v_{\varepsilon }} |\nabla v_{\varepsilon }|^2+\frac {\ell c_1}{4c_2}\int _0^t\int _{\Omega } \frac {|\nabla v_{\varepsilon }|^4}{v_{\varepsilon }^3}+\frac {\ell }{2}\int _0^t\int _{\Omega } v_{\varepsilon }+\frac {1}{2}\int _0^t\int _{\Omega } \frac {v_{\varepsilon }}{u_{\varepsilon }} |\nabla u_{\varepsilon }|^2\nonumber \\[5pt] &\leq -\int _{\Omega }\ln {u_{0\varepsilon }}+\int _{\Omega }\ln {u_{\varepsilon }}+\frac {\ell }{2c_2}\int _{\Omega } \frac {|\nabla v_{0\varepsilon }|^2}{v_{0\varepsilon }}+ \frac {\ell }{c_2} \int _0^t\int _{\Omega }u_{\varepsilon }^{1-\alpha }v_{\varepsilon }|\nabla u_{\varepsilon }|^2+C(\alpha )\int _0^t\int _{\Omega } v_{\varepsilon } |\nabla v_{\varepsilon }|^2\nonumber \\[5pt] &\leq -\int _{\Omega }\ln {u_0}+\int _{\Omega }u_{\varepsilon }+\frac {\ell }{2c_2}\int _{\Omega } \frac {|\nabla v_{0\varepsilon }|^2}{v_{0\varepsilon }}+ \frac {\ell }{c_2} \int _0^t\int _{\Omega }u_{\varepsilon }^{1-\alpha }v_{\varepsilon }|\nabla u_{\varepsilon }|^2+C(\alpha )\int _0^t\int _{\Omega } v_{\varepsilon } |\nabla v_{\varepsilon }|^2 \end{align}
for all
$t \in (0, T_{\max , \varepsilon })$
. Therefore, (2.27) implies (2.17)–(2.19) in view of (1.6), (2.9), (2.11) and (2.13).
With the help of the following class of interpolation inequalities, the results derived from (2.13) and (2.18) can be used to estimate
$\int _0^{T_{\max }}\int _\Omega u_{\varepsilon }^{\frac {5}{3}}v_{\varepsilon }$
in the next section.
Lemma 2.6. [Reference Winkler26] Let
$p^*\geq 1$
and
$p\;:\!=\;\frac {2p^*+3}{3}$
. Then, for all
$M\gt 0$
, one can find
$C(M,p^*)\gt 0$
such that for any
$q\in [0,\frac {2p^*}{3}]$
and functions
$\phi ,\psi \in C^1(\overline {\Omega })$
satisfying
$\phi \gt 0,\psi \gt 0$
in
$\overline {\Omega }$
as well as
\begin{align*} \int _{\Omega }\phi ^{p^*}\leq M, \end{align*}
we have
\begin{align} \int _{\Omega }\phi ^{p}\psi \leq C(M,p^*) \left \{\int _{\Omega }\phi ^{q-1}\psi |\nabla \phi |^2+\int _{\Omega }\frac {\phi }{\psi }|\nabla \psi |^2+\int _{\Omega }\phi \psi \right \}\!. \end{align}
Since the convexity requirement for the domain
$\Omega$
is not assumed, we establish a refined results stated in Lemma2.3 of [Reference Winkler28].
Lemma 2.7.
Let
$\Omega \subset \mathbb R^3$
be a bounded domain with smooth boundary and
$q\geq 2$
. Then, for all
$t\in (0,T_{max,\epsilon })$
, there exist
$\Gamma (q)\gt 0$
and
$\gamma (q)\gt 0$
such that
\begin{align} \frac {d}{dt}\int _{\Omega }\frac {|\nabla v_{\epsilon }|^q}{v_{\epsilon }^{q-1}}+\Gamma (q)\int _{\Omega }\frac {|\nabla v_{\epsilon }|^{q-2}}{v_{\epsilon }^{q-3}}|D^2\ln {v_{\epsilon }}|^2+\Gamma (q)\int _{\Omega }u_{\epsilon }\frac {|\nabla v_{\epsilon }|^q}{v_{\epsilon }^{q-1}}\leq \frac {1}{\Gamma (q)}\int _{\Omega }v_{\epsilon }\big(u_{\epsilon }^{\frac {q+2}{2}}+1\big) \end{align}
and
\begin{align} \frac {d}{dt}\int _{\Omega }\frac {|\nabla v_{\varepsilon }|^q}{v_{\varepsilon }^{q-1}}+\gamma (q)\int _{\Omega }\frac {|\nabla v_{\varepsilon }|^{q-2}}{v_{\varepsilon }^{q-3}}|D^2\ln {v_{\varepsilon }}|^2+\gamma (q)\int _{\Omega }u_{\varepsilon }\frac {|\nabla v_{\varepsilon }|^q}{v_{\varepsilon }^{q-1}}\leq \frac {1}{\gamma (q)}\int _{\Omega }v_{\varepsilon }\big(|\nabla u_{\varepsilon }|^{\frac {q+2}{3}}+1\big). \end{align}
Proof. According to the second equation in (2.5) and the identity
$\nabla v_{\varepsilon } \cdot \nabla \Delta v_{\varepsilon }=\frac {1}{2}\Delta |\nabla v_{\varepsilon }|^2-|D^2 v_{\varepsilon }|^2$
, we can obtain that
\begin{align} \frac {d}{dt}\int _{\Omega }\frac {|\nabla v_{\varepsilon }|^q}{v_{\varepsilon }^{q-1}}={}&q\int _{\Omega }v_{\varepsilon }^{-q+1}|\nabla v_{\varepsilon }|^{q-2}\nabla v_{\varepsilon }\cdot \left \{\nabla \Delta v_{\varepsilon }-\nabla (u_{\varepsilon } v_{\varepsilon })\right \}-(q-1)\int _{\Omega }v_{\varepsilon }^{-q}|\nabla v_{\varepsilon }|^q \cdot \left \{\Delta v_{\varepsilon }-u_{\varepsilon } v_{\varepsilon }\right \}\nonumber \\ ={}&q\int _{\Omega }v_{\varepsilon }^{-q+1}|\nabla v_{\varepsilon }|^{q-2}\nabla v_{\varepsilon }\cdot \nabla \Delta v_{\varepsilon }-q\int _{\Omega }v_{\varepsilon }^{-q+1}|\nabla v_{\varepsilon }|^{q-2}\nabla v_{\varepsilon }\cdot \nabla (u_{\varepsilon } v_{\varepsilon })\nonumber \\ &-(q-1)\int _{\Omega }v_{\varepsilon }^{-q}|\nabla v_{\varepsilon }|^q \cdot \Delta v_{\varepsilon }+(q-1)\int _{\Omega }u_{\varepsilon }v_{\epsilon }^{-q+1}|\nabla v_{\varepsilon }|^{q}\nonumber \\ ={}&\frac {q}{2}\int _{\Omega }v_{\varepsilon }^{-q+1}|\nabla v_{\varepsilon }|^{q-2}\Delta |\nabla v_{\varepsilon }|^2-q\int _{\Omega }v_{\varepsilon }^{-q+1}|\nabla v_{\varepsilon }|^{q-2}|D^2 v_{\varepsilon }|^2-(q-1)\int _{\Omega }v_{\varepsilon }^{-q}|\nabla v_{\varepsilon }|^{q} \Delta v_{\varepsilon }\nonumber \\ &-q\int _{\Omega }v_{\varepsilon }^{-q+1}|\nabla v_{\varepsilon }|^{q-2}\nabla v_{\varepsilon }\cdot \nabla (u_{\varepsilon } v_{\varepsilon })+(q-1)\int _{\Omega }u_{\varepsilon }v_{\varepsilon }^{-q+1}|\nabla v_{\varepsilon }|^{q}\nonumber \\ ={}&\frac {q}{2}\int _{\Omega }v_{\varepsilon }^{-q+1}|\nabla v_{\varepsilon }|^{q-2}\cdot \frac {\partial |\nabla v_{\varepsilon }|}{\partial \nu }-\frac {q(q-2)}{4}\int _{\Omega }v_{\varepsilon }^{-q+1}|\nabla v_{\varepsilon }|^{q-4}|\nabla |\nabla v_{\varepsilon }|^2|^2\nonumber \\ &-q\int _{\Omega }v_{\varepsilon }^{-q+1}|\nabla v_{\varepsilon }|^{q-2}|D^2 v_{\varepsilon }|^2+q(q-1)\int _{\Omega }v_{\varepsilon }^{-q}|\nabla v_{\varepsilon }|^{q-2}\nabla |\nabla v_{\varepsilon }|^2\cdot \nabla v_{\varepsilon }\nonumber \\ &-q(q-1)\int _{\Omega }v_{\varepsilon }^{-q-1}|\nabla v_{\varepsilon }|^{q+2}-q\int _{\Omega }v_{\varepsilon }^{-q+1}|\nabla v_{\varepsilon }|^{q-2}\nabla v_{\varepsilon }\cdot \nabla (u_{\varepsilon } v_{\epsilon })\nonumber \\ &+(q-1)\int _{\Omega }u_{\varepsilon }v_{\varepsilon }^{-q+1}|\nabla v_{\varepsilon }|^{q} \end{align}
for all
$t \in (0, T_{\max , \varepsilon })$
. Thanks to (2.16), one can see that for any
$\eta \gt 0$
, there exists
$c_1=C(\eta ,q)\gt 0$
such that
\begin{align} \int _{\partial \Omega }v_{\varepsilon }^{-q+1}|\nabla v_{\varepsilon }|^{q-2}\cdot \frac {\partial |\nabla v_{\varepsilon }|}{\partial \nu }\leq \eta \int _{\Omega }v_{\varepsilon }^{-q+1}|\nabla v_{\varepsilon }|^{q-2}|D^2v_{\varepsilon }|^2+\eta \int _{\Omega }v_{\varepsilon }^{-q-1}|\nabla v_{\varepsilon }|^{q+2}+c_1\int _{\Omega }v_{\varepsilon }. \end{align}
Furthermore, by two well-known inequalities (Lemma3.4 of [Reference Winkler25])
\begin{align} \int _{\Omega }v_{\varepsilon }^{-q-1}|\nabla v_{\varepsilon }|^{q+2}\leq (q+\sqrt {3})^2\int _{\Omega }v_{\varepsilon }^{-q+3}|\nabla v_{\varepsilon }|^{q-2}|D^2 \ln v_{\varepsilon }|^2 \end{align}
and
\begin{align} \int _{\Omega }v_{\varepsilon }^{-q+1}|\nabla v_{\varepsilon }|^{q-2}|D^2 v_{\varepsilon }|^2\leq (q+\sqrt {3}+1)^2\int _{\Omega }v_{\varepsilon }^{-q+3}|\nabla v_{\varepsilon }|^{q-2}|D^2 \ln v_{\varepsilon }|^2, \end{align}
we can conclude that
\begin{align} &-\frac {q(q-2)}{4}\int _{\Omega }v_{\varepsilon }^{-q+1}|\nabla v_{\varepsilon }|^{q-4}|\nabla |\nabla v_{\varepsilon }|^2|^2-q\int _{\Omega }v_{\varepsilon }^{-q+1}|\nabla v_{\varepsilon }|^{q-2}|D^2 v_{\varepsilon }|^2\nonumber \\ &+q(q-1)\int _{\Omega }v_{\varepsilon }^{-q}|\nabla v_{\varepsilon }|^{q-2}\nabla |\nabla v_{\varepsilon }|^2\cdot \nabla v_{\varepsilon }-q(q-1)\int _{\Omega }v_{\varepsilon }^{-q-1}|\nabla v_{\varepsilon }|^{q+2}\nonumber \\ &=-\frac {q(q-2)}{4}\int _{\Omega }v_{\varepsilon }^{-q+1}|\nabla v_{\varepsilon }|^{q-4}|\nabla |\nabla v_{\varepsilon }|^2|^2-q\int _{\Omega }v_{\varepsilon }^{-q+3}|\nabla v_{\varepsilon }|^{q-2}|D^2 \ln v_{\varepsilon }|^2\nonumber \\ &+q(q-2)\int _{\Omega }v_{\varepsilon }^{-q}|\nabla v_{\varepsilon }|^{q-2}\nabla |\nabla v_{\varepsilon }|^2\cdot \nabla v_{\varepsilon }-q(q-2)\int _{\Omega }v_{\varepsilon }^{-q-1}|\nabla v_{\varepsilon }|^{q+2}\nonumber \\ &\leq -q\int _{\Omega }v_{\varepsilon }^{-q+3}|\nabla v_{\varepsilon }|^{q-2}|D^2 \ln v_{\varepsilon }|^2\nonumber \\ &-\frac {q(q-2)}{4}\int _{\Omega }v_{\varepsilon }^{-q+1}|\nabla v_{\varepsilon }|^{q-4}| \nabla |\nabla v_{\varepsilon }|^2 - \frac {2}{v_{\varepsilon }}|\nabla v_{\varepsilon }|^2\nabla v_{\varepsilon } |^2\nonumber \\ &\leq -q\int _{\Omega }v_{\varepsilon }^{-q+3}|\nabla v_{\varepsilon }|^{q-2}|D^2 \ln v_{\varepsilon }|^2. \end{align}
By the Young inequality, there exist
$c_2\gt 0$
and
$c_3\gt 0$
such that
\begin{align} &-q\int _{\Omega }v_{\varepsilon }^{-q+1}|\nabla v_{\varepsilon }|^{q-2}\nabla v_{\varepsilon }\cdot \nabla (u_{\varepsilon } v_{\varepsilon })+(q-1)\int _{\Omega }u_{\varepsilon }v_{\varepsilon }^{-q+1}|\nabla v_{\varepsilon }|^{q}\nonumber \\ &=-q\int _{\Omega }v_{\varepsilon }^{-q+2}|\nabla v_{\varepsilon }|^{q-2}\nabla v_{\varepsilon }\cdot \nabla u_{\varepsilon } -\int _{\Omega }u_{\varepsilon }v_{\varepsilon }^{-q+1}|\nabla v_{\varepsilon }|^{q}\nonumber \\ &\leq q\int _{\Omega }v_{\varepsilon }^{-q+2}|\nabla v_{\varepsilon }|^{q-2}\nabla v_{\varepsilon }\cdot \nabla u_{\varepsilon } -\int _{\Omega }u_{\varepsilon }v_{\varepsilon }^{-q+1}|\nabla v_{\varepsilon }|^{q}\nonumber \\ &\leq c_3\int _{\Omega }v_{\varepsilon }^{-q-1}|\nabla v_{\varepsilon }|^{q+2}+c_2\int _{\Omega }v_{\varepsilon }|\nabla u_{\varepsilon }|^{\frac {q+2}{3}} -\int _{\Omega }u_{\varepsilon }v_{\varepsilon }^{-q+1}|\nabla v_{\varepsilon }|^{q}. \end{align}
On the other hand, since the identities
$\nabla |\nabla v_{\varepsilon }|^2=2D^2v_{\varepsilon }\cdot \nabla v_{\varepsilon }$
and
$|\Delta v_{\varepsilon }|=\sqrt {3}|D^2v_{\varepsilon }|$
, it follows that for all
$t \in (0, T_{\max , \varepsilon })$
\begin{align} &-q\int _{\Omega }v_{\varepsilon }^{-q+1}|\nabla v_{\varepsilon }|^{q-2}\nabla v_{\varepsilon }\cdot \nabla (u_{\varepsilon } v_{\varepsilon })+(q-1)\int _{\Omega }u_{\varepsilon }v_{\varepsilon }^{-q+1}|\nabla v_{\varepsilon }|^{q}\nonumber \\[5pt] &=q(q-2)\int _{\Omega }u_{\varepsilon }v_{\varepsilon }^{-q+2}|\nabla v_{\varepsilon }|^{q-4}\nabla v_{\varepsilon }\cdot \big(D^2v_{\varepsilon }\cdot \nabla v_{\varepsilon }\big)\nonumber \\[5pt] &+q\int _{\Omega }u_{\varepsilon }v_{\varepsilon }^{-q+2}|\nabla v_{\varepsilon }|^{q-2}\Delta v_{\varepsilon }-(q-1)^2\int _{\Omega }u_{\varepsilon }v_{\epsilon }^{-q+1}|\nabla v_{\varepsilon }|^{q}\nonumber \\[5pt] &\leq q(q-2+\sqrt {n})\int _{\Omega }u_{\varepsilon }v_{\varepsilon }^{-q+2}|\nabla v_{\varepsilon }|^{q-2}|D^2v_{\varepsilon }| -(q-1)^2\int _{\Omega }u_{\varepsilon }v_{\varepsilon }^{-q+1}|\nabla v_{\varepsilon }|^{q}\nonumber \\[5pt] &\leq c_4\int _{\Omega }v_{\varepsilon }^{-q+1}|\nabla v_{\varepsilon }|^{q-2}|D^2v_{\varepsilon }|^2+c_5\int _{\Omega }u_{\varepsilon }^2v_{\varepsilon }^{-q+3}|\nabla v_{\varepsilon }|^{q-2}-(q-1)^2\int _{\Omega }u_{\varepsilon }v_{\varepsilon }^{-q+1}|\nabla v_{\varepsilon }|^{q}\nonumber \\[5pt] &\leq c_4\int _{\Omega }v_{\varepsilon }^{-q+1}|\nabla v_{\varepsilon }|^{q-2}|D^2v_{\varepsilon }|^2+c_6\int _{\Omega }v_{\varepsilon }^{-q-1}|\nabla v_{\varepsilon }|^{q+2}\nonumber \\[5pt] &+c_7\int _{\Omega }u_{\varepsilon }^{\frac {q+2}{2}}v_{\varepsilon }-(q-1)^2\int _{\Omega }u_{\varepsilon }v_{\varepsilon }^{-q+1}|\nabla v_{\varepsilon }|^{q}, \end{align}
where
$c_i\gt 0, i=4,\cdots , 7$
, are constants. Therefore, (2.29) follows from (2.31)–(2.36) and (2.30) results from (2.31)–(2.35) and (2.37).
By means of the Gagliardo–Nirenberg inequality and the Ehrling lemma, we deduce the following functional inequality, which plays an important role in estimating an integral of the type
$\int _{\Omega }\phi ^{\beta }\psi$
below.
Lemma 2.8.
Let
$\Omega \subset \mathbb R^3$
, and supposed that
$p\gt 0$
and
$1\lt r\lt 6$
. Then, for any
$\eta \gt 0$
, one can find
$C(\eta , p)\gt 0$
such that
\begin{align} \|\phi ^{\frac {p+1}{2}}\sqrt {\psi }\|^2_{L^{r}(\Omega )}\leq \eta \int _{\Omega }\phi ^{p-1}\psi |\nabla \phi |^2+\eta \int _{\Omega }\phi ^{p+1}\psi ^{-1}|\nabla \psi |^2+C(\eta , p)\cdot \left \{\int _{\Omega }\phi \right \}^{p}\left \{\int _{\Omega }\phi \psi \right \} \end{align}
for all
$\phi \in C^1(\bar {\Omega })$
and
$\psi \in C^1(\overline {\Omega })$
with
$\phi \gt 0$
and
$\psi \gt 0$
in
$\overline {\Omega }$
.
Proof. From the Gagliardo–Nirenberg inequality, it follows that for
$1\lt r\lt 6$
, there exist constants
$a\in (0,1)$
and
$c_1\gt 0$
such that
\begin{align*} \|\phi ^{\frac {p+1}{2}}\sqrt {\psi }\|^2_{L^{r}(\Omega )}&\leq c_1\|\nabla (\phi ^{\frac {p+1}{2}}\sqrt {\psi })\|^{2a}_{L^{2}(\Omega )}\|\phi ^{\frac {p+1}{2}}\sqrt {\psi }\|^{2(1-a)}_{L^{1}(\Omega )} +c_1\|\phi ^{\frac {p+1}{2}}\sqrt {\psi }\|_{L^{1}(\Omega )}^2. \end{align*}
Furthermore, by Young’s inequality, one can see that for all
$\eta \gt 0$
there is
$ c_2(\eta )\gt 0$
such that
\begin{align} c_1\|\nabla (\phi ^{\frac {p+1}{2}}\sqrt {\psi })\|^{2a}_{L^{2}(\Omega )}\|\phi ^{\frac {p+1}{2}}\sqrt {\psi }\|^{2(1-a)}_{L^{1}(\Omega )}&\leq \eta \|\nabla (\phi ^{\frac {p+1}{2}}\sqrt {\psi })\|_{L^{2}(\Omega )}^2+c_2(\eta )\|\phi ^{\frac {p+1}{2}}\sqrt {\psi }\|_{L^{1}(\Omega )}^2. \end{align}
For
$p\geq 1$
, the compact embedding
$L^{r}\hookrightarrow L^{1}$
and continuous embedding
$L^{1}\hookrightarrow L^{\frac {2}{p+1}}$
allow us to apply Ehrling lemma, which yields
\begin{equation*}c_2(\eta )\|\phi ^{\frac {p+1}{2}}\sqrt {\psi }\|_{L^{1}(\Omega )}^2\leq \frac {1}{2}\|\phi ^{\frac {p+1}{2}}\sqrt {\psi }\|^2_{L^{r}(\Omega )}+c_3(\eta )\|\phi ^{\frac {p+1}{2}}\sqrt {\psi }\|^2_{L^{\frac 2{p+1}}(\Omega )}\end{equation*}
with some
$c_3(\eta )\gt 0$
. Therefore, for all
$p\geq 1$
, we have
\begin{equation*}\|\phi ^{\frac {p+1}{2}}\sqrt {\psi }\|^2_{L^{r}(\Omega )}\leq \eta \|\nabla \big(\phi ^{\frac {p+1}{2}}\sqrt {\psi }\big)\|_{L^{2}(\Omega )}^2+c_3(\eta )\|\phi ^{\frac {p+1}{2}}\sqrt {\psi }\|^2_{L^{\frac {2}{p+1}}(\Omega )}.\end{equation*}
On the other hand, for
$p\in (0,1)$
, one can see that
\begin{align*} \|\phi ^{\frac {p+1}{2}}\sqrt {\psi }\|^2_{L^{r}(\Omega )} &\leq \eta \|\nabla (\phi ^{\frac {p+1}{2}}\sqrt {\psi })\|_{L^{2}(\Omega )}^2+c_2(\eta )\|\phi ^{\frac {p+1}{2}}\sqrt {\psi }\|_{L^{1}(\Omega )}^2\\[5pt] &\leq \eta \|\nabla (\phi ^{\frac {p+1}{2}}\sqrt {\psi })\|_{L^{2}(\Omega )}^2+c_4(\eta )\|\phi ^{\frac {p+1}{2}}\sqrt {\psi }\|_{L^{\frac 2{p+1}}(\Omega )}^2. \end{align*}
Hence, from the Hölder inequality, it follows that
\begin{align} &\eta \|\nabla (\phi ^{\frac {p+1}{2}}\sqrt {\psi })\|_{L^{2}(\Omega )}^2+ (c_3(\eta )+ c_4(\eta ))\|\phi ^{\frac {p+1}2}\sqrt {\psi }\|_{L^{\frac 2{p+1}}(\Omega )}^2\nonumber \\[5pt] &\leq \eta \int _{\Omega }\phi ^{p-1}\psi |\nabla \phi |^2+\eta \int _{\Omega }\phi ^{p+1}\psi ^{-1}|\nabla \psi |^2+C(\eta ,p)\cdot \left \{\int _{\Omega }\phi \right \}^{p}\cdot \left \{\int _{\Omega }\phi \psi \right \}\!. \end{align}
3. Estimates of
$ \|u_{\epsilon }(\cdot ,t)\|_{L^p(\Omega )}$
3.1.
Space-time estimate of
$u_{\varepsilon }^{p_*-1}v_{\varepsilon }|\nabla u_{\varepsilon }|^2$
The purpose of this subsection is to derive the space-time estimate of
$u_{\varepsilon }^{p_*}v_{\varepsilon }|\nabla u_{\varepsilon }|^2$
with
$p^*\in [1-\alpha ,0]$
. This refines the space-time estimate of
$u_{\varepsilon }^{1-\alpha }v_{\varepsilon }|\nabla u_{\varepsilon }|^2$
provided in Lemma2.3 and allows us to obtain the bounds for
$\|u_{\varepsilon }(\cdot ,t)\|_{L^{p_0}(\Omega )}$
with
$p_0\gt \frac {3}{2}$
in the subsequent subsection. To this end, as outlined in the introduction, we shall make sure that
$\mathcal{ G}_{\varepsilon }(t)$
enjoys the feature of the energy-like quantity.
Lemma 3.1.
Let
$\Omega \subset \mathbb R^3$
. Then, one can find constants
$a\gt 0$
and
$b\gt 0$
such that for all
$t\in (0,T_{max,\varepsilon })$
the functional
\begin{equation*}\mathcal{ G}_{\varepsilon } (t)\;:\!=\;\int _{\Omega }u_{\varepsilon }(\cdot ,t)\ln {u_{\varepsilon }(\cdot ,t)}+a\int _{\Omega }\frac {|\nabla v_{\varepsilon }(\cdot ,t)|^4}{v_{\varepsilon }^{3}(\cdot ,t)}\end{equation*}
satisfies
\begin{align} \mathcal{ G}_{\varepsilon }'(t)&+b\int _{\Omega }\frac {|\nabla v_{\varepsilon }|^{2}}{v_{\varepsilon }}|D^2\ln {v_{\varepsilon }}|^2+b\int _{\Omega }u_{\varepsilon }\frac {|\nabla v_{\varepsilon }|^4}{v_{\varepsilon }^3}+\frac {1}{2}\int _{\Omega }v_{\varepsilon }|\nabla u_{\varepsilon }|^2 \nonumber \\[5pt] &\leq \chi ^2\int _{\Omega }u_{\varepsilon }^{2\alpha -2}v_{\varepsilon }|\nabla v_{\varepsilon }|^2+\ell \int _{\Omega } u_{\varepsilon } v_{\varepsilon } \ln u_{\varepsilon }+\ell \int _{\Omega }u_{\varepsilon }v_{\varepsilon }+\frac {1}{4}\int _{\Omega }v_{\varepsilon }. \end{align}
Proof. Multiplying the first equation in (2.5) by
$1+\ln u_{\varepsilon }$
, we use the Young inequality and
$\ln \xi \leqslant \xi$
for all
$\xi \gt 0$
to infer that
\begin{align} & \frac {d}{dt} \int _{\Omega } u_{\varepsilon } \ln u_{\varepsilon }+\int _{\Omega } v_{\varepsilon }\left |\nabla u_{\varepsilon }\right |^2\nonumber \\[5pt] ={}& \chi \int _{\Omega } u_{\varepsilon }^{\alpha -1} v_{\varepsilon } \nabla u_{\varepsilon } \cdot \nabla v_{\varepsilon }+ \ell \int _{\Omega } u_{\varepsilon } v_{\varepsilon } \ln u_{\varepsilon } + \ell \int _{\Omega } u_{\varepsilon } v_{\varepsilon }\nonumber \\[5pt] \leq{} & \frac {1}{4} \int _{\Omega } v_{\varepsilon }\left |\nabla u_{\varepsilon }\right |^2+ \chi ^2 \int _{\Omega } u_{\varepsilon }^{2\alpha -2} v_{\varepsilon }\left |\nabla v_{\varepsilon }\right |^2 +\ell \int _{\Omega } u_{\varepsilon } v_{\varepsilon } \ln u_{\varepsilon }+\ell \int _{\Omega } u_{\varepsilon } v_{\varepsilon } \end{align}
for all
$t \in (0, T_{max, \varepsilon })$
. On the other hand, applying (2.30) to
$q\;:\!=\;4$
, we have
\begin{align} \frac {d}{dt}\int _{\Omega }\frac {|\nabla v_{\varepsilon }|^4}{v_{\varepsilon }^{3}}+\gamma (4)\int _{\Omega }\frac {|\nabla v_{\varepsilon }|^{2}}{v_{\varepsilon }}|D^2\ln {v_{\varepsilon }}|^2+\gamma (4)\int _{\Omega }u_{\varepsilon }\frac {|\nabla v_{\varepsilon }|^4}{v_{\varepsilon }^{3}}\leq \frac {1}{\gamma (4)}\int _{\Omega }v_{\varepsilon }(|\nabla u_{\varepsilon }|^{2}+1). \end{align}
Combining (3.2) with (3.3), we arrive at
\begin{align*} &\frac {d}{dt}\left \{\int _{\Omega } u_{\varepsilon } \ln u_{\varepsilon }+\frac {\gamma (4)}{4}\int _{\Omega } \frac {\left |\nabla v_{\varepsilon }\right |^4}{v_{\varepsilon }^3}\right \}\\[5pt] &+\frac {\gamma ^2(4)}{4}\int _{\Omega }\frac {|\nabla v_{\varepsilon }|^{2}}{v_{\varepsilon }}|D^2\ln {v_{\varepsilon }}|^2+\frac {\gamma ^2(4)}{4}\int _{\Omega }u_{\varepsilon }\frac {|\nabla v_{\varepsilon }|^4}{v_{\varepsilon }^3}+\frac {1}{2}\int _{\Omega }v_{\varepsilon }|\nabla u_{\varepsilon }|^2 \\[5pt] &\leq \chi ^2\int _{\Omega }u_{\varepsilon }^{2\alpha -2}v_{\varepsilon }|\nabla v_{\varepsilon }|^2+\ell \int _{\Omega } u_{\varepsilon } v_{\varepsilon } \ln u_{\varepsilon }+\ell \int _{\Omega }u_{\varepsilon }v_{\varepsilon }+\frac {1}{4}\int _{\Omega }v_{\varepsilon } \end{align*}
for all
$t \in (0, T_{max, \varepsilon })$
. This readily leads to (3.1) with
$a=\frac {\gamma (4)}{4}$
and
$b=\frac {\gamma ^2(4)}{4}$
.
By Lemmas2.3, 2.5 and 2.6, we can obtain the estimate of
$\int _0^{T_{\max }}\int _\Omega u_{\varepsilon }^{\frac {5}{3}}v_{\varepsilon }$
, which allows us to address the unfavourable contributions on the right-hand side of (3.1) whenever the exponent
$\alpha \in \left (\frac {3}{2},2\right ]$
.
Lemma 3.2.
Let
$\Omega \subset \mathbb R^3$
, suppose that
$\alpha \in \left (\frac {3}{2},2\right ]$
and
$K\gt 0$
with the property (1.6) be valid. Then, there exists
$C=C(K)\gt 0$
such that
\begin{align} \int _0^{T_{max,\varepsilon }}\int _{\Omega }u_{\varepsilon }^{\frac {5}{3}}v_{\varepsilon }\leq C. \end{align}
Proof. Since (2.9) and (1.6), there exists
$c_1=c_1(K)\gt 0$
such that
\begin{align*} \int _{\Omega }u_{\varepsilon }(\cdot ,t)\leq c_1 \qquad for\; all\; t\in (0,T_{max,\varepsilon }). \end{align*}
Due to
$\alpha \in (\frac {3}{2},2]$
, then
$q\;:\!=\;2-\alpha \in [0,\frac {2}{3}]$
. As an application of Lemma2.6 to
$p^*\;:\!=\;1$
, there exists
$c_2\gt 0$
such that
\begin{align*} \int _0^{t}\int _{\Omega }u_{\varepsilon }^{\frac {5}{3}}v_{\varepsilon }\leq c_2 \left \{\int _0^{t}\int _{\Omega }u_{\varepsilon }^{1-\alpha }v_{\varepsilon }|\nabla u_{\varepsilon }|^2+\int _0^{t}\int _{\Omega }\frac {u_{\varepsilon }}{v_{\varepsilon }}|\nabla v_{\varepsilon }|^2+\int _0^{t}\int _{\Omega }u_{\varepsilon }v_{\varepsilon }\right \} \end{align*}
for all
$t \in (0, T_{max, \varepsilon })$
. Hence from (2.10), (2.13) and (2.18), there exists
$C(K)\gt 0$
such that (3.4) is valid.
With the help of two lemmas above, the quasi-energy feature of
$G_{\varepsilon }(t)$
can be verified and thereby space-time estimates of
$u_{\varepsilon }\frac {|\nabla v_{\varepsilon }|^4}{v_{\varepsilon }^{3}}$
and
$v_{\varepsilon }|\nabla u_{\varepsilon }|^2$
are established.
Lemma 3.3.
Let
$\Omega \subset \mathbb R^3$
and
$\alpha \in \left (\frac {3}{2},\frac {5}{3}\right ]$
, suppose that
$K\gt 0$
with the property (1.6) be valid. Then, there exists
$C=C(K)\gt 0$
such that
\begin{align} \int _0^{T_{max,\varepsilon }}\int _{\Omega }u_{\varepsilon }\frac {|\nabla v_{\varepsilon }|^4}{v_{\varepsilon }^{3}}\leq C \end{align}
and
\begin{align} \int _0^{T_{max,\varepsilon }}\int _{\Omega }v_{\varepsilon }|\nabla u_{\varepsilon }|^2\leq C. \end{align}
Proof. In light of condition
$\alpha \in \left (\frac {3}{2},\frac {5}{3}\right ]$
,
$4\alpha -5\in (0,\frac {5}{3}]$
. Therefore, from Lemma3.1, it follows that for all
$t\in (0,T_{max,\varepsilon })$
,
\begin{align} \mathcal{ G}_{\varepsilon }'(t)&+b\int _{\Omega }\frac {|\nabla v_{\varepsilon }|^{2}}{v_{\varepsilon }}|D^2\ln {v_{\varepsilon }}|^2+b\int _{\Omega }u_{\varepsilon }\frac {|\nabla v_{\varepsilon }|^4}{v_{\varepsilon }^3}+\frac {1}{2}\int _{\Omega }v_{\varepsilon }|\nabla u_{\varepsilon }|^2 \nonumber \\[5pt] &\leq \chi ^2\int _{\Omega }u_{\varepsilon }^{2\alpha -2}v_{\varepsilon }|\nabla v_{\varepsilon }|^2+\ell \int _{\Omega } u_{\varepsilon } v_{\varepsilon } \ln u_{\varepsilon }+\ell \int _{\Omega }u_{\varepsilon }v_{\varepsilon }+\frac {1}{4}\int _{\Omega }v_{\varepsilon }\nonumber \\[5pt] &\leq c_1\int _{\Omega }u_{\varepsilon }^{4\alpha -5}v_{\varepsilon }+\frac {b}{2}\int _{\Omega }u_{\varepsilon }\frac {|\nabla v_{\varepsilon }|^4}{v_{\varepsilon }^3}+\ell \int _{\Omega } u_{\varepsilon }^{\frac {5}{3}}v_{\varepsilon }+\ell \int _{\Omega }u_{\varepsilon }v_{\varepsilon }+\frac {1}{4}\int _{\Omega }v_{\varepsilon }\nonumber \\[5pt] &\leq (c_1+ \ell ) \int _{\Omega } u_{\varepsilon }^{\frac {5}{3}}v_{\varepsilon }+\frac {b}{2}\int _{\Omega }u_{\varepsilon }\frac {|\nabla v_{\varepsilon }|^4}{v_{\varepsilon }^3}+\ell \int _{\Omega }u_{\varepsilon }v_{\varepsilon }+(c_1+1)\int _{\Omega }v_{\varepsilon } \end{align}
with
$c_1\;:\!=\;\frac {\chi ^2\|v_0\|_{L^{\infty }}^4}{b}$
and hence
\begin{align} &\mathcal{ G}_{\varepsilon}'(t)+b\int _{\Omega }\frac {|\nabla v_{\varepsilon }|^{2}}{v_{\varepsilon }}|D^2\ln {v_{\varepsilon }}|^2+\frac {b}{2}\int _{\Omega }u_{\varepsilon }\frac {|\nabla v_{\varepsilon }|^4}{v_{\varepsilon }^3}+\frac {1}{2}\int _{\Omega }v_{\varepsilon }|\nabla u_{\varepsilon }|^2\nonumber \\[5pt] &\leq c_2\int _{\Omega } u_{\varepsilon }^{\frac {5}{3}}v_{\varepsilon }+ \ell \int _{\Omega }u_{\varepsilon }v_{\varepsilon }+c_2\int _{\Omega }v_{\varepsilon } \end{align}
with
$c_2\;:\!=\;c_1+1+\ell$
. After an integration over time, this together with (3.4), (2.10), (2.17) and (1.6) entails that
\begin{align*} & \frac {b}{2}\int _0^{t}\int _{\Omega }u_{\varepsilon }\frac {|\nabla v_{\varepsilon }|^4}{v_{\varepsilon }^3}+\frac {1}{2}\int _0^{t}\int _{\Omega }v_{\varepsilon }|\nabla u_{\varepsilon }|^2+b\int _0^{t}\int _{\Omega }\frac {|\nabla v_{\varepsilon }|^{2}}{v_{\varepsilon }}|D^2\ln {v_{\varepsilon }}|^2 \\[5pt] \leq{} & c_2\int _0^{t}\int _{\Omega } u_{\varepsilon }^{\frac {5}{3}}v_{\varepsilon }+\ell \int _0^{t}\int _{\Omega }u_{\varepsilon }v_{\varepsilon }+c_2\int _0^{t}\int _{\Omega }v_{\varepsilon } \\[5pt] &+\int _{\Omega } u_{0\varepsilon } \ln u_{0\varepsilon }+a \int _{\Omega } \frac {\left |\nabla v_{0\varepsilon }\right |^4}{v_{0\varepsilon }^3}-\int _{\Omega } u_{\varepsilon } \ln u_{\varepsilon } \\[5pt] \leq{} &c_2\int _0^{t}\int _{\Omega } u_{\varepsilon }^{\frac {5}{3}}v_{\varepsilon }+\ell \int _0^{t}\int _{\Omega }u_{\varepsilon }v_{\varepsilon }+c_2 \int _0^{t}\int _{\Omega }v_{\varepsilon } \\[5pt] &+\int _{\Omega } (u_0+1)^2+a\int _{\Omega } \frac {\left |\nabla v_{0}\right |^4}{v_{0}^3}+\frac {|\Omega |}{e} \\[5pt] \leq {}&C(K) \end{align*}
for all
$t\in (0,T_{max,\varepsilon })$
, and thereby completes the proof.
As a direct consequence of (3.6) and (2.13), we have
Lemma 3.4.
Let
$\alpha \in \left (\frac {3}{2},\frac {5}{3}\right ]$
and
$K\gt 0$
with the property (1.6) be valid. Then for any
$p_*\in [1-\alpha , 0]$
and
$C=C(K)\gt 0$
such that
\begin{align} \int _0^{T_{max,\varepsilon }}\int _{\Omega }u_{\varepsilon }^{p_*}v_{\varepsilon }|\nabla u_{\varepsilon }|^2\leq C. \end{align}
Proof. Since
$ u_{\varepsilon }^{p_*}\leq 1+u_{\varepsilon }^{1-\alpha }$
for any
$p_*\in [1-\alpha , 0]$
, we have
\begin{equation*} \int _0^{T_{max,\varepsilon }}\int _{\Omega } u_{\varepsilon }^{p_*}v_{\varepsilon }|\nabla u_{\varepsilon }|^2 \leq \int _0^{T_{max,\varepsilon }}\int _{\Omega } v_{\varepsilon }|\nabla u_{\varepsilon }|^2+\int _0^{T_{max,\varepsilon }}\int _{\Omega }u_{\varepsilon }^{1-\alpha }v_{\varepsilon }|\nabla u_{\varepsilon }|^2, \end{equation*}
3.2.
$L^{p_0}$
-estimates of
$u_{\varepsilon }$
with some
$p_0\gt \frac {3}{2}$
This subsection establishes the
$L^{p_0}$
-bounds for
$u_{\varepsilon }$
with some
$p_0\gt \frac {3}{2}$
, which serves as the foundation for a bootstrap argument to achieve the
$L^{p}$
bounds of
$u_{\varepsilon }$
for arbitrary
$p\gt 1$
. As a prerequisite for the derivation of estimate on
$L^{p_0}$
-bounds for
$u_{\varepsilon }$
, we apply Lemma2.8 to show that the integral of the form
$\int _{\Omega }\varphi ^{\beta }\psi$
can be dominated by
$\int _{\Omega }\varphi ^{k}\psi |\nabla \varphi |^2$
when added
$ \int _{\Omega }\varphi \frac {|\nabla \psi |^4}{\psi ^3}$
and term
$\int _{\Omega } \varphi \psi$
.
Lemma 3.5.
Let
$L\gt 0$
,
$k\in (\!-1,-\frac {1}{3})$
and
$\beta \in [1,k+\frac {8}{3})$
. There exists
$C=C(L,k, \beta )\gt 0$
such that if
$\int _{\Omega }\varphi \leq L$
, then
\begin{align} \int _{\Omega }\varphi ^{\beta }\psi \leq \int _{\Omega }\varphi ^{k}\psi |\nabla \varphi |^2+\int _{\Omega }\varphi \frac {|\nabla \psi |^4}{\psi ^3}+C\int _{\Omega }\varphi \psi \end{align}
is valid for all
$\varphi \in C^1(\overline \Omega )$
and
$\psi \in C^1(\overline \Omega )$
fulfilling
$\varphi \gt 0$
and
$\psi \gt 0$
in
$\overline \Omega$
.
Proof. Let
$\theta \;:\!=\;\frac {1}{k+3-\beta }$
and
$ \theta _*\;:\!=\;\frac {1}{\beta -k-2}.$
Then,
\begin{align} 1\leq \theta \lt 3 \end{align}
can be warranted by
$\beta \in [k+2,k+\frac {8}{3})$
. Consequently, by the Hölder inequality and applying Lemma2.8 to
$\eta =\frac 12$
, we conclude that there exists
$c_1=c_1(\beta ,k,L)$
such that for all
$t\in (0,T_{max,\varepsilon })$
,
\begin{align} \int _{\Omega }\varphi ^{\beta }\psi &=\int _{\Omega }\big(\varphi ^{\frac {k+2}{2}}\psi ^{\frac {1}{2}}\big)^2\varphi ^{\beta -k-2} \nonumber \\[5pt] &\leq \|\varphi ^{\frac {k+2}{2}}\psi ^{\frac {1}{2}}\|^2_{L^{2\theta }(\Omega )}\cdot \|\varphi ^{\beta -k-2}\|_{L^{\theta _*}(\Omega )}\nonumber \\[5pt] &\leq L^{\frac {1}{\theta _*}}\|\varphi ^{\frac {k+2}{2}}\psi ^{\frac {1}{2}}\|^2_{L^{2\theta }(\Omega )}\nonumber \\[5pt] &\leq \frac {1}{2}\int _{\Omega }\varphi ^{k}\psi |\nabla \varphi |^2+\frac {1}{2} \int _{\Omega }\varphi ^{k+2}\frac {|\nabla \psi |^2}{\psi }+c_1\int _{\Omega }\varphi \psi . \end{align}
Furthermore, let
$\lambda \;:\!=\;-\frac {1}{k}$
and
$ \lambda _*\;:\!=\;\frac {1}{k+1}$
. Then, the restriction
$k\in (\!-1,-\frac {1}{3})$
warrants that
$ 1\lt \lambda \lt 3$
. Applying Lemma2.8 to
$p\;:\!=\;k+1, r\;:\!=\;2\lambda , \eta \;:\!=\;\min \{L^{-\frac 1{\lambda _*}},1\}$
and Young’s inequality once more, we obtain that there exists positive constant
$c_2=c_2(k,\beta ,L)$
such that for all
$t\in (0,T_{max,\varepsilon })$
,
\begin{align} \int _{\Omega }\varphi ^{k+2}\frac {|\nabla \psi |^2}{\psi }&\leq \frac {1}{2}\int _{\Omega }\varphi \frac {|\nabla \psi |^4}{\psi ^3}+\frac 1{2}\int _{\Omega }\varphi ^{2k+3}\psi \nonumber \\[5pt] &=\frac {1}{2}\int _{\Omega }\varphi \frac {|\nabla \psi |^4}{\psi ^3}+\frac 1{2}\int _{\Omega }\big(\varphi ^{\frac {k+2}{2}}\psi ^{\frac {1}{2}}\big)^2\varphi ^{k+1}\nonumber \\[5pt] &\leq \frac {1}{2}\int _{\Omega }\varphi \frac {|\nabla \psi |^4}{\psi ^3}+\frac 1{2}\|\varphi ^{\frac {k+2}{2}}\psi ^{\frac {1}{2}}\|_{L^{2\lambda }(\Omega )}^2\|\varphi ^{k+1}\|_{L^{\lambda _*}(\Omega )}\nonumber \\[5pt] &\leq \frac {1}{2}\int _{\Omega }\varphi \frac {|\nabla \psi |^4}{\psi ^3}+\frac {L^{\frac 1{ \lambda _* }}}{2}\|\varphi ^{\frac {k+2}{2}}\psi ^{\frac {1}{2}}\|_{L^{2\lambda }(\Omega )}^2\nonumber \\[5pt] &\leq \frac {1}{2}\int _{\Omega }\varphi \frac {|\nabla \psi |^4}{\psi ^3}+\frac {1}{2}\int _{\Omega }\varphi ^{k+2}\frac {|\nabla \psi |^2}{\psi }+\frac {1}{2}\int _{\Omega }\varphi ^{k}\psi |\nabla \varphi |^2+c_2\int _{\Omega }\varphi \psi . \end{align}
Therefore, if
$\beta \in [k+2,k+\frac {8}{3})$
, it follows from (3.12) and (3.13) that
\begin{align} \int _{\Omega }\varphi ^{\beta }\psi \leq \eta \int _{\Omega }\varphi ^{k}\psi |\nabla \varphi |^2+ \int _{\Omega }\varphi \frac {|\nabla \psi |^4}{\psi ^3}+C(\beta , k, L)\int _{\Omega }\varphi \psi \end{align}
for all
$t\in (0,T_{max,\varepsilon })$
. Apart from that, it is observed that for
$\beta \in [1,k+2)$
,
\begin{equation*} \int _{\Omega }\varphi ^{\beta }\psi \leq \int _{\Omega }\varphi ^{k+2}\psi + \int _{\Omega }\varphi \psi \quad \hbox{for all} \,\, t\in (0,T_{max,\varepsilon }). \end{equation*}
The combination of Lemmas3.3, 3.4 and 3.5 yields the result of follows.
Lemma 3.6.
Assume
$\alpha \in \left (\frac {3}{2},\frac {19}{12}\right )$
and
$K\gt 0$
be parameter such that property (1.6) is valid. Then there exist
$p_0\gt \frac {3}{2}$
and
$C=C(K)\gt 0$
such that
\begin{align} \int _{\Omega }u_{\varepsilon }^{p_0}(\cdot ,t)\leq C\qquad for\;all\; t\in (0,T_{max,\varepsilon }). \end{align}
Proof. According to (2.9), there exists
$c_1=c_1(K)\gt 0$
such that
\begin{align*} \int _{\Omega }u_{\varepsilon }(\cdot ,t)\leq c_1\qquad for\; all\; t\in (0,T_{max,\varepsilon }). \end{align*}
Let
$ \delta \;:\!=\;\min \{\frac 12 (\alpha -\frac {4}{3}) ,\frac {19}{12}-\alpha ,\frac 12 \}\gt 0$
and
$ p_0\;:\!=\;\frac {3}{2}+ \frac {\delta }2$
. Then, multiplying the first equation in (2.5) by
$u_{\varepsilon }^{p_0-1}$
and by the Young inequality, we infer that
\begin{align} & \frac {1}{p_0}\frac {d}{dt}\int _{\Omega }u_{\varepsilon }^{p_0}+\frac {p_0-1}{2}\int _{\Omega }u_{\varepsilon }^{p_0-1}v_{\varepsilon }|\nabla u_{\varepsilon }|^2\nonumber \\ &\leq \frac {(1-p_0)\chi ^2}{2}\int _{\Omega }u_{\varepsilon }^{p_0+2\alpha -3}v_{\varepsilon }|\nabla v_{\varepsilon }|^2+\ell \int _{\Omega }u_{\varepsilon }^{p_0}v_{\varepsilon }\nonumber \\ &\leq \frac {(1-p_0)\chi ^2\|v_0\|_{L^{\infty }}^4}{2}\int _{\Omega }u_{\varepsilon }^{2p_0+4\alpha -7}v_{\varepsilon }+ \int _{\Omega }u_{\varepsilon }\frac {|\nabla v_{\varepsilon }|^4}{v_{\varepsilon }^3}+\ell \int _{\Omega }u_{\varepsilon }^{\frac 53}v_{\varepsilon } +\ell \int _{\Omega } u_{\varepsilon }v_{\varepsilon }. \end{align}
Let
$k\;:\!=\;-\frac 13-\delta$
. Then,
$k\in (\!-1,-\frac 13)$
due to the fact
$\delta \lt \frac 23$
. In addition, by the definition of
$\delta$
, we can see that
$k\in (1-\alpha ,0)$
. Therefore,
$k\in (1-\alpha ,0)\bigcap (\!-1,-\frac 13)$
, and thus
\begin{align} \int _0^{T_{max,\varepsilon }}\int _{\Omega }u_{\varepsilon }^{k}v_{\varepsilon }|\nabla u_{\varepsilon }|^2\leq c_1(K,T_{max,\varepsilon }) \end{align}
for some
$c_1(K,T_{max,\varepsilon })\gt 0$
by Lemma3.4.
On other hand, according to the definitions of
$\delta$
and
$p_0$
, we have
$\delta \lt \frac {19}{12}-\alpha$
and thereby
$2p_0+4\alpha -7\lt \frac 73-\delta =k+\frac 83$
. Apart from that, it is claimed that
\begin{align} 2p_0+4\alpha -7\gt 1. \end{align}
Indeed, according to the definition of
$p_0$
, it is easy to see that
$ 2p_0+4\alpha -8=\delta +4\alpha -5\gt 0$
due to
$\alpha \gt \frac 32$
.
Therefore, applying Lemma3.5 to
$\beta \;:\!=\;2p_0+4\alpha -7$
, we can conclude that there exists
$c_2 \gt 0$
such that
\begin{align} \frac {(1-p_0)\chi ^2\|v_0\|_{L^{\infty }}^4}{2}\int _{\Omega }u_{\varepsilon }^{2p_0+4\alpha -7}v_{\varepsilon }\leq \int _{\Omega }u_{\varepsilon }^{k}v_{\varepsilon }|\nabla u_{\varepsilon }|^2+\int _{\Omega }u_{\varepsilon }\frac {|\nabla v_{\varepsilon }|^4}{v_{\varepsilon }^3}+c_2\int _{\Omega }u_{\varepsilon }v_{\varepsilon } \end{align}
for
$t\in (0,T_{max,\varepsilon })$
. Combining (3.19), (3.16) with (3.17) and integrating the results over time, together with (3.5), (1.6), (2.17) and (2.10), we have
\begin{align} & \frac {1}{p_0}\int _{\Omega }u_{\varepsilon }^{p_0} +\frac {p_0-1}{2}\int _0^{t}\int _{\Omega }u_{\varepsilon }^{p_0-1}v_{\varepsilon }|\nabla u_{\varepsilon }|^2 \nonumber \\ &\leq \int _0^{t}\int _{\Omega }u_{\varepsilon }^{k}v_{\varepsilon }|\nabla u_{\varepsilon }|^2+2\int _0^{t}\int _{\Omega }u_{\varepsilon }\frac {|\nabla v_{\varepsilon }|^4}{v_{\varepsilon }^3}+c_2\int _0^{t}\int _{\Omega }u_{\varepsilon }v_{\varepsilon }+ \int _{\Omega }(u_{0}+1)^{p_0}+\int _0^{t}\int _{\Omega }v_{\varepsilon }\nonumber \\ &\leq C(K,T_{max,\varepsilon }) \end{align}
for all
$t\in (0,T_{max,\varepsilon })$
, and thus complete the proof.
3.3.
$L^{p}$
-estimates of
$u_{\varepsilon }$
for all
$p\gt 1$
The aim of this subsection is to achieve
$L^p$
-bounds of
$u_{\varepsilon }$
for all
$p\gt 1$
. Of crucial importance to our approach in this direction is to refine (3.10) in Lemma3.5, which involves the term
$\int _{\Omega }\varphi \frac {|\nabla \psi |^q}{\psi ^{q-1}}$
instead of
$\int _{\Omega }\varphi \frac {|\nabla \psi |^4}{\psi ^{3}}$
, when the enhanced integrability properties of
$\varphi$
are taken into account.
Lemma 3.7.
For
$L\gt 0,p_0\gt \frac {3}{2}$
,
$p\gt 1$
,
$q\gt 2+\frac {3p}{p_0}$
,
$\beta \in [1,\frac {2p_0}{3}+p+1)$
and any
$\eta \in (0,1)$
, there exists
$C=C(\eta , L,p,\beta )\gt 0$
such that whenever
$\int _{\Omega }\varphi ^{p_0}\leq L$
,
\begin{align} \int _{\Omega }\varphi ^{\beta } \psi \leq \eta \int _{\Omega }\varphi ^{p-1}\psi |\nabla \varphi |^2+\eta \int _{\Omega }\varphi \frac {|\nabla \psi |^q}{{\psi }^{q-1}}+C\int _{\Omega }\varphi \psi \end{align}
holds for all
$\varphi \in C^1(\overline \Omega )$
and
$\psi \in C^1(\overline \Omega )$
fulfilling
$\varphi \gt 0$
and
$\psi \gt 0$
in
$\overline \Omega$
.
Proof. It is observed that the restriction
$q\gt 2+\frac {3p}{p_0}$
ensures that
$ \frac {qp+q-2}{q-2}\lt \frac {2p_0}{3}+p+1.$
Now we first verify (3.21) in the case
\begin{align} \beta \in \left [\frac {qp+q-2}{q-2},\frac {2p_0}{3}+p+1\right )\!. \end{align}
To this end, let
\begin{equation*}\rho \;:\!=\;\frac {p_0}{p_0-\beta +p+1}\quad and \quad \rho _*\;:\!=\;\frac {p_0}{\beta -p-1}.\end{equation*}
Due to (3.22), it follows that
\begin{align} \rho \geq 1\quad and\quad 1\lt 2\rho \lt 6. \end{align}
Hence, thanks to
$\int _{\Omega }\varphi ^{p_0}\leq L$
for
$p_0\gt \frac {3}{2}$
, the Hölder inequality, the Young inequality and Lemma2.8, we deduce that for any
$\eta \gt 0$
, there exists positive constant
$c_1=c_1(\eta ,p,\beta ,L)$
such that
\begin{align*} \int _{\Omega }\varphi ^{\beta }\psi &=\int _{\Omega }(\varphi ^{\frac {p+1}{2}}\psi ^{\frac {1}{2}})^2\varphi ^{\beta -p-1} \\[5pt] &\leq \|\varphi ^{\frac {p+1}{2}}\psi ^{\frac {1}{2}}\|^2_{L^{2\rho }(\Omega )}\cdot \|\varphi ^{\beta -p-1}\|_{L^{\rho _*}(\Omega )} \\[5pt] &=\|\varphi ^{\frac {p+1}{2}}\psi ^{\frac {1}{2}}\|^2_{L^{2\rho }(\Omega )}\cdot \|\varphi \|_{L^{p_0}(\Omega )}^{\beta -p-1} \\[5pt] &\leq L^{\frac {\beta -p-1}{p_0}}\|\varphi ^{\frac {p+1}{2}}\psi ^{\frac {1}{2}}\|^2_{L^{2\rho }(\Omega )} \\[5pt] &\leq \frac {\eta }{2}\int _{\Omega }\varphi ^{p-1}\psi |\nabla \varphi |^2+\frac {\eta }{2}\int _{\Omega }\varphi ^{p+1}\frac {|\nabla \psi |^2}{\psi }+c_1\int _{\Omega }\varphi \psi \\[5pt] &\leq \frac {\eta }{2}\int _{\Omega }\varphi ^{p-1}\psi |\nabla \varphi |^2+\frac {\eta }{2}\int _{\Omega }\varphi \frac {|\nabla \psi |^q}{\psi ^{q-1}}+\frac 12\int _{\Omega }\varphi ^{\frac {qp+q-2}{q-2}}\psi +c_1\int _{\Omega }\varphi \psi \\[5pt] &\leq \frac {\eta }{2}\int _{\Omega }\varphi ^{p-1}\psi |\nabla \varphi |^2+\frac {\eta }{2}\int _{\Omega }\varphi \frac {|\nabla \psi |^q}{\psi ^{q-1}}+ \frac {1}{2}\int _{\Omega }\varphi ^{\beta }\psi + (c_1+1)\int _{\Omega }\varphi \psi . \end{align*}
and thus
\begin{align} \int _{\Omega }\varphi ^{\beta }\psi \leq \eta \int _{\Omega }\varphi ^{p-1}\psi |\nabla \varphi |^2+\eta \int _{\Omega }\varphi \frac {|\nabla \psi |^q}{\psi ^{q-1}}+2 (c_1+1) \int _{\Omega }\varphi \psi . \end{align}
Furthermore, it is observed that for
$\beta \in [1, \frac {qp+q-2}{q-2})$
, we have
\begin{equation*}\int _{\Omega }\varphi ^{\beta }\psi \leq \int _{\Omega }\varphi ^{\frac {qp+q-2}{q-2}}\psi +\int _{\Omega }\varphi \psi , \end{equation*}
which along with (3.24) complete the proof readily.
We are now in the position to achieve
$L^p$
-estimates for
$u_{\varepsilon }$
for any
$p\gt 1$
through tracing the evolution of
$ \mathcal{ H}(t)\;:\!=\; \int _{\Omega }u_{\varepsilon }^{p}+\int _{\Omega }\frac {|\nabla v_{\varepsilon }|^q}{v_{\varepsilon }^{q-1}}$
, thanks to the
$L^{p_0}$
-estimates of
$u_{\varepsilon }$
with some
$p_0\gt \frac {3}{2}$
established in Lemma3.6.
Lemma 3.8.
Let
$\Omega \subset \mathbb R^3$
and
$K\gt 0$
with the property that (1.6) holds. Supposed that
$\alpha \in \left (\frac {3}{2},\frac {19}{12}\right )$
and
$p\gt 1$
, there exists
$C=C(K, p)\gt 0$
such that for all
$ t\in (0,T_{max,\varepsilon })$
,
\begin{align} \int _{\Omega }u_{\varepsilon }^{p}\leq C\quad \hbox{and}\quad \int _0^{T_{max,\varepsilon }}\int _{\Omega }u_{\varepsilon }^{p-1}v_{\varepsilon }|\nabla u_{\varepsilon }|^2\leq C \end{align}
as well as
\begin{align} \int _0^{T_{max,\varepsilon }}\int _{\Omega }u_{\varepsilon }^{p+1}v_{\varepsilon }\leq C. \end{align}
Proof. From (2.29) in Lemma2.7, it follows that
\begin{align} \frac {d}{dt}\int _{\Omega }\frac {|\nabla v_{\varepsilon }|^q}{v_{\varepsilon }^{q-1}}+\Gamma (q)\int _{\Omega }\frac {|\nabla v_{\varepsilon }|^{q-2}}{v_{\varepsilon }^{q-3}}|D^2\ln {v_{\varepsilon }}|^2+\Gamma (q)\int _{\Omega }u_{\varepsilon }\frac {|\nabla v_{\varepsilon }|^q}{v_{\varepsilon }^{q-1}}\leq \frac {1}{\Gamma (q)}\int _{\Omega }v_{\varepsilon }\big(u_{\varepsilon }^{\frac {q+2}{2}}+1\big) \end{align}
for all
$t\in (0,T_{max,\varepsilon })$
.
Multiplying the first equation in (2.5) by
$u_{\varepsilon }^{p-1}$
, applying Young’s inequality and (1.6), there exists
$c_1\gt 0$
such that
\begin{align} &\frac {1}p \frac {d}{dt}\int _{\Omega }u_{\varepsilon }^{p}+\frac {p-1}{2}\int _{\Omega }u_{\varepsilon }^{p-1}v_{\varepsilon }|\nabla u_{\varepsilon }|^2 \\[5pt] &\leq \frac {p-1}{2}\int _{\Omega }u_{\varepsilon }^{p+2\alpha -3}v_{\varepsilon }|\nabla v_{\epsilon }|^2+\ell \int _{\Omega }u_{\varepsilon }^pv_{\varepsilon }\nonumber \\[5pt] &\leq (p-1)\int _{\Omega }u_{\varepsilon }^{p+1}v_{\varepsilon }|\nabla v_{\varepsilon }|^2+(p-1)\int _{\Omega }v_{\varepsilon }|\nabla v_{\epsilon }|^2+\ell \int _{\Omega }u_{\varepsilon }^{p+1}v_{\varepsilon }+\ell \int _{\Omega }u_{\varepsilon }v_{\varepsilon }\nonumber \\[5pt] &\leq \frac {\Gamma (q)} 2 \int _{\Omega }u_{\varepsilon }\frac {|\nabla v_{\varepsilon }|^q}{v_{\varepsilon }^{q-1}}+(p-1)\int _{\Omega }v_{\varepsilon }|\nabla v_{\varepsilon }|^2+\ell \int _{\Omega }u_{\varepsilon }^{p+1}v_{\varepsilon }+c_1\int _{\Omega }u_{\varepsilon }^{\frac {qp+q-2}{q-2}}v_{\varepsilon }^{\frac {3q-2}{q-2}}+\ell \int _{\Omega }u_{\varepsilon }v_{\varepsilon }\nonumber \\[5pt] &\leq \frac {\Gamma (q)} 2\int _{\Omega }u_{\varepsilon }\frac {|\nabla v_{\varepsilon }|^q}{v_{\varepsilon }^{q-1}}+(p-1)\int _{\Omega }v_{\varepsilon }|\nabla v_{\varepsilon }|^2+\ell \int _{\Omega }u_{\varepsilon }^{p+1}v_{\varepsilon }+ c_1K^{\frac {2q}{q-2}}\int _{\Omega }u_{\varepsilon }^{\frac {qp+q-2}{q-2}}v_{\varepsilon }+ \ell \int _{\Omega }u_{\varepsilon }v_{\varepsilon }\nonumber \end{align}
for all
$t\in (0,T_{max,\varepsilon })$
.
Noting that if
$p_0\gt \frac {3}{2}$
, then for any fixed
$p\gt 1$
, one can find
$q\gt 2$
satisfying
\begin{align} q\in \left (2+\frac {3p}{p_0},\frac {4p_0}{3}+2p \right ) \end{align}
which ensures that
\begin{align} \frac {qp+q-2}{q-2}\lt \frac {2p_0}{3}+p+1 \end{align}
and
\begin{align} \frac {q+2}{2}\lt \frac {2p_0}{3}+p+1. \end{align}
Therefore, thanks to (3.15), (3.30) and (3.31), the application of Lemma3.7 to
$\eta \;:\!=\;\min \{\frac {\Gamma (q)} 4, \frac {p-1}4\}$
yields
\begin{align} &\ell \int _{\Omega }u_{\varepsilon }^{p+1}v_{\varepsilon }+c_1K^{\frac {2q}{q-2}}\int _{\Omega }u_{\varepsilon }^{\frac {qp+q-2}{q-2}}v_{\varepsilon }+\frac {1}{\Gamma (q)}\int _{\Omega }u_{\varepsilon }^{\frac {q+2}{2}}v_{\varepsilon }\nonumber \\[5pt] &\leq \eta \int _{\Omega }u_{\varepsilon }^{p-1}v_{\varepsilon }|\nabla u_{\varepsilon }|^2+\eta \int _{\Omega }u_{\varepsilon }\frac {|\nabla v_{\varepsilon }|^q}{v_{\varepsilon }^{q-1}}+C(\eta ,K,p)\int _{\Omega }u_{\varepsilon }v_{\varepsilon } \end{align}
for all
$t\in (0,T_{max,\varepsilon })$
.
From (3.27), (3.28) and (3.32), there exists
$C(p, K)\gt 0$
such that
\begin{align} &\frac {d}{dt}\left \{ \frac {1}{p}\int _{\Omega }u_{\varepsilon }^{p}+\int _{\Omega }\frac {|\nabla v_{\varepsilon }|^q}{v_{\varepsilon }^{q-1}}\right \} +\frac {\Gamma (q)}{4}\int _{\Omega }u_{\varepsilon }\frac {|\nabla v_{\varepsilon }|^q}{v_{\varepsilon }^{q-1}}+\frac {p-1}{4}\int _{\Omega }u_{\varepsilon }^{p-1}v_{\varepsilon }|\nabla u_{\varepsilon }|^2\nonumber \\[5pt] &\leq C(p,K)\int _{\Omega }\left (v_{\varepsilon }|\nabla v_{\varepsilon }|^2+u_{\varepsilon }v_{\varepsilon }+v_{\varepsilon } \right )\!. \end{align}
Integrating (3.33) with respect to time, we have
\begin{align*} &\int _{\Omega }u_{\varepsilon }^{p}+\int _{\Omega }\frac {|\nabla v_{\varepsilon }|^q}{v_{\varepsilon }^{q-1}}+\frac {\Gamma (q)}{4}\int _0^{t}\int _{\Omega }u_{\varepsilon }\frac {|\nabla v_{\varepsilon }|^q}{v_{\varepsilon }^{q-1}}+\frac {p-1}{4}\int _0^{t}\int _{\Omega }u_{\varepsilon }^{p-1}v_{\varepsilon }|\nabla u_{\varepsilon }|^2 \\[5pt] &\leq C(p,K)\int _0^{t}\int _{\Omega }\left (v_{\varepsilon }|\nabla v_{\varepsilon }|^2+u_{\varepsilon }v_{\varepsilon }+v_{\varepsilon }\right ) +\int _{\Omega }(u_{0}+1)^p+\int _{\Omega }\frac {|\nabla v_{0}|^q}{v_0^{q-1}} \end{align*}
for all
$t\in (0,T_{max,\varepsilon })$
. Hence, from (2.11), (2.17), (1.6) and (2.10), it follows that
\begin{align} \int _{\Omega }u_{\varepsilon }^{p}+\frac {p-1}{2}\int _0^{t}\int _{\Omega }u_{\varepsilon }^{p-1}v_{\varepsilon }|\nabla u_{\varepsilon }|^2\leq C(p, K). \end{align}
Furthermore, (3.26) results from (3.32), (3.34) and (2.10), and hence completes the proof.
As a natural consequence of (3.25), the application of standard smoothing estimates for the Neumann heat semigroup directly yields lower bounds for
$v_{\varepsilon }$
.
Lemma 3.9.
Let
$\Omega \subset \mathbb R^3$
,
$\alpha \in (\frac {3}{2},\frac {19}{12})$
and
$K\gt 0$
with the property that (1.6) holds. Assuming that
$T_{max,\varepsilon }\lt \infty$
, then there exists
$C=C(K,T_{max,\varepsilon })\gt 0$
such that
\begin{align} v_{\varepsilon }(x,t)\geq C\qquad for\; all\; (x,t)\in \Omega \times (0,T_{max,\varepsilon }). \end{align}
Proof. First, it is observed from Lemma2.1 that
$ v_\varepsilon \in C^{2,1}(\overline {\Omega } \times (0, T_{\max ,\varepsilon }))$
with
$v_\varepsilon \gt 0$
in
$\overline {\Omega } \times (0, T_{\max ,\varepsilon })$
. Defining the transformed variable
$w_\varepsilon \;:\!=\; \ln \frac {\|v_0\|_{L^\infty (\Omega )}}{v_\varepsilon } \in C^{2,1}(\overline {\Omega } \times [0, T_{\max ,\varepsilon }))$
and invoking (2.5), we derive
\begin{equation*}w_{\varepsilon t} = \Delta w_\varepsilon - |\nabla w_\varepsilon |^2 + u_\varepsilon \leq \Delta w_\varepsilon + u_\varepsilon \quad \text{in } \Omega \times (0, T_{\max ,\varepsilon }).\end{equation*}
Using the comparison principle along with known regularisation features of the Neumann heat semigroup
$(e^{t\Delta })_{t \geq 0}$
on
$ \Omega$
[Reference Winkler30], one can see that there is constant
$c_1\gt 0$
such that for all
$ t \in (0, T_{\max ,\varepsilon })$
,
\begin{align*} w_\varepsilon (\cdot , t) &\leq e^{t\Delta } w_{0\varepsilon } + \int _{0}^t e^{(t-s)\Delta } u_\varepsilon (\cdot , s) \, ds \\[5pt] &\leq \sup _{x \in \Omega } w_{0\varepsilon }(x) + c_1 \int _{0}^t (1+(t-s)^{-\frac {3}{4}} ) \|u_\varepsilon (\cdot , s)\|_{L^2(\Omega )} \, ds \\[5pt] &\leq \sup _{x \in \Omega } w_{0\varepsilon }(x) + c_1(T_{\max ,\varepsilon }+ 4T^{\frac 14}_{\max ,\varepsilon })\cdot \sup _{s \in (0, T_{\max ,\varepsilon })} \|u_\varepsilon (\cdot , s)\|_{L^2(\Omega )}. \end{align*}
This along with (3.25) implies that
$w_\varepsilon (\cdot , t)\leq c_2(K,T_{\max ,\varepsilon })$
for some
$c_2(K,T_{\max ,\varepsilon })$
and thus
$v_{\varepsilon }(x,t)\geq \|v_0\|_{L^\infty (\Omega )} e^{-c_2(K,T_{\max ,\varepsilon })}$
.
3.4. Proof of Theorem 1.1
On the basis of the result stated by Lemma3.8, we further derive the higher regularity properties of the solution components to (2.5) and then obtain a global continuous weak solution of (1.4).
As a consequence of (3.25), by leveraging standard heat semigroup estimates, one can verify that
$ \|v_{\varepsilon }(\cdot ,t)\|_{W^{1,\infty }(\Omega )}$
is locally bounded for
$t\in (0, T_{max,\varepsilon })$
.
Lemma 3.10.
Let
$\Omega \subset \mathbb R^3$
and
$K\gt 0$
with the property that (1.6) holds. Supposed that
$\alpha \in \left (\frac {3}{2},\frac {19}{12}\right )$
, there exists
$C=C(K,p)\gt 0$
such that for all
$\varepsilon \in (0,1)$
we have
\begin{align} \|v_{\varepsilon }(\cdot ,t)\|_{W^{1,\infty }(\Omega )}\leq C\qquad for\; all\; t\in (0,T_{max,\varepsilon }). \end{align}
Proof. According to Lemma3.8, we have the uniform boundedness of
$u_{\varepsilon }$
in
$L^{\infty }((0,T_{max,\varepsilon });\; L^p(\Omega ))$
for any
$p\gt 1$
. Therefore, by invoking the well-established smoothing properties of the Neumann heat semigroup documented in [Reference Winkler30], there exist
$c_1 \gt 0$
and
$c_2 \gt 0$
such that for any
$p\gt 3$
\begin{align} \|\nabla v_\varepsilon (t)\|_{L^\infty (\Omega )} &= \left \|\nabla e^{t(\Delta -1)}v_0 - \int _0^t \nabla e^{(t-s)(\Delta -1)} \big \{u_\varepsilon (s)v_\varepsilon (s) - v_\varepsilon (s)\big \} \, ds \right \|_{L^\infty (\Omega )} \nonumber \\[5pt] &\leq c_1 \|v_0\|_{W^{1,\infty }(\Omega )} + c_1 \int _0^t \left (1 + (t-s)^{-\frac 12 - \frac {3}{2p}} \right ) e^{-(t-s)} \|u_\varepsilon (\cdot ,s)v_\varepsilon (\cdot ,s) - v_\varepsilon (\cdot ,s)\|_{L^p(\Omega )} ds\nonumber \\[5pt] &\leq c_1 \|v_0\|_{W^{1,\infty }(\Omega )} + +c_1|\Omega |^{\frac 1p} \|v_0\|_{L^\infty (\Omega )}\int _0^t \left (1 + (t-s)^{-\frac 12 - \frac {3}{2p}} \right ) e^{-(t-s)} \, ds\nonumber \\[5pt] &+c_1 \|v_0\|_{L^\infty (\Omega )}\sup _{ t\in (0,T_{max,\varepsilon })} \|u_\varepsilon (\cdot ,t)\|_{L^p(\Omega )} \int _0^t \left (1 + (t-s)^{-\frac 12 - \frac {3}{2p}} \right ) e^{-(t-s)} \, ds\nonumber \\[5pt] &\leq c_2 \end{align}
for all
$t \in (0, T_{\max ,\varepsilon })$
and
$\varepsilon \in (0,1)$
. Then, we finish the proof.
At this position, we can proceed to prove the
$L^\infty$
-bounds for
$u_{\varepsilon }$
.
Lemma 3.11.
Let
$\alpha \in \left (\frac {3}{2},\frac {19}{12}\right )$
. Suppose
$T_{max,\varepsilon }\lt \infty$
, then there exists
$C(T_{max,\varepsilon })\gt 0$
such that for all
$\varepsilon \in (0,1)$
we have
\begin{align} \|u_{\varepsilon }(\cdot ,t)\|_{L^{\infty }(\Omega )}\leq C(T_{max,\varepsilon }) \,\qquad for\; all\; \,t\in (0,T_{max,\varepsilon }). \end{align}
Proof. We begin by reformulating the governing equation for
$u_\varepsilon$
as
\begin{equation*}u_{\varepsilon t}=\nabla \cdot \left (A_{\varepsilon }(x,t,u_{\varepsilon })\nabla u_{\varepsilon }\right )+\nabla \cdot B_{\varepsilon }(x,t)+D_{\varepsilon }(x,t) \qquad (x,t)\in \Omega \times (0,T_{max,\varepsilon }),\end{equation*}
where the nonlinear operators are defined by
\begin{equation*}A_{\varepsilon }(x,t,\xi )\;:\!=\;v_{\varepsilon }(x,t)\xi \qquad (x,t,\xi )\in \Omega \times (0,T_{max,\varepsilon })\times [0,\infty ),\end{equation*}
and
\begin{equation*}B_{\varepsilon }(x,t)\;:\!=\;-\chi u_{\varepsilon }^{\alpha }(x,t)v_{\varepsilon }(x,t)\nabla v_{\varepsilon }(x,t)\qquad (x,t)\in \Omega \times (0,T_{max,\varepsilon }),\end{equation*}
as well as
\begin{equation*} D_{\varepsilon }(x,t)\;:\!=\;\ell u_{\varepsilon }(x,t)v_{\varepsilon }(x,t) \qquad (x,t)\in \Omega \times (0,T_{max,\varepsilon }).\end{equation*}
Through applications of Lemmas3.8, 3.9 and 3.10, we derive the following estimates for any
$p \gt 1$
\begin{equation*}A_{\varepsilon }(x,t)\geq c_{1}(T_{max,\varepsilon })\xi \qquad \text{for all }(x,t,\xi )\in \Omega \times (0,T_{max,\varepsilon })\times [0,\infty ),\end{equation*}
and
\begin{equation*}\|B_{\varepsilon }(\cdot ,t)\|_{L^{p}(\Omega )}\leq c_{2}\quad \text{and}\quad \|D_{\varepsilon }(\cdot ,t)\|_{L^{p}(\Omega )}\leq c_2\qquad \text{for all }t\in (0,T_{max,\varepsilon }),\end{equation*}
with some
$c_1\gt 0,c_2\gt 0$
. Thereafter (3.38) can be derived from a Moser-type iteration, as recorded in ([Reference Tao and Winkler21], Lemma A.1).
As an application of (2.7) and Lemma3.11, one can show that the solutions to (2.5) are in fact global in time.
Lemma 3.12.
Under the assumptions of Theorem1.1, we have
$T_{max,\varepsilon }=\infty$
for all
$\varepsilon \in (0,1).$
Drawing on the a priori estimates from the previous lemmas, we now deduce the Hölder regularity for the global solution to (2.5) by applying standard parabolic regularity theory.
Lemma 3.13.
For all
$T\gt 0$
and
$\varepsilon \in (0,1)$
, there exist
$\theta _1=\theta _1(T)\in (0,1)$
and
$C(T)\gt 0$
such that
\begin{align} \|u_{\varepsilon }\|_{C^{\theta _1,\frac {\theta _1}{2}}(\overline {\Omega }\times [0,T])} \leq C(T) \end{align}
and
\begin{align} \|v_{\varepsilon }\|_{C^{\theta _1,\frac {\theta _1}{2}}(\overline {\Omega }\times [0,T])} \leq C(T). \end{align}
Moreover, for all
$\tau \gt 0$
and
$T\gt \tau$
, there exist
$\theta _2=\theta _2(\tau ,T)\in (0,1)$
and
$C(\tau ,T)\gt 0$
such that
\begin{align} \left \|v_{\varepsilon }\right \|_{C^{2+\theta _2,1+\frac {\theta _2}{2}}(\overline {\Omega }\times [\tau ,T])}\leq C(\tau ,T)\qquad \text{for all }\varepsilon \in (0,1). \end{align}
Proof. The Hölder estimates (3.39) and (3.40) are achieved through the combination of Lemmas3.9–3.11, (2.8) and an application of the parabolic Hölder regularity framework established in [Reference Porzio and Vespri18, Theorem 1.3; Remark 1.4]. The subsequent estimate (3.41) then follows as a direct consequence of classical Schauder theory for scalar parabolic equations [Reference Ladyzenskaja, Solonnikov and Uraleva11, Chapter IV], when coupled with the established Hölder continuity from (3.39).
Upon the above estimates, we can construct a continuous weak solutions to (1.4) through a standard extraction procedure.
Lemma 3.14.
Let
$\alpha \in (\frac {3}{2},\frac {19}{12})$
and suppose that
$(u_0,v_0)$
satisfies (1.5) and (1.6). Then there exist
$(\varepsilon _{j})_{j\in \mathbb{N}}\subset (0,1)$
and functions
$u$
and
$v$
fulfilling (1.7), as well as
$u\geq 0$
and
$v\gt 0$
in
$\overline {\Omega }\times (0,\infty )$
, such that
\begin{align} &u_{\varepsilon }\rightarrow {u}\qquad \text{in}\quad C_{loc}^{0}(\overline {\Omega }\times [0,\infty )) \end{align}
\begin{align} &v_{\varepsilon }\rightarrow {v}\qquad \text{in}\quad C_{loc}^{0}(\overline {\Omega }\times [0,\infty ))\quad \text{and}\quad \text{in}\quad C_{loc}^{2,1}(\overline {\Omega }\times (0,\infty ))\qquad \text{and} \end{align}
\begin{align} &\nabla v_{\varepsilon }\stackrel {*}{\rightharpoonup }\nabla v\qquad \text{in}\quad L^{\infty }(\Omega \times (0,\infty )) \end{align}
as
$\varepsilon =\varepsilon _{j}\searrow 0$
and that
$(u,v)$
forms a continuous global weak solutions of (1.4) in the sense of Definition
2.1
.
Proof. By virtue of Lemmas3.8, and 3.9 and Young’s inequality, there exists
$C=C(K,T)\gt 0$
such that for all
$T\gt 0$
,
\begin{align*} \int _0^{T}\int _{\Omega }|\nabla u_{\varepsilon }^2|\leq \int _0^{T}\int _{\Omega }u_{\varepsilon }v_{\varepsilon }|\nabla u_{\varepsilon }|^2+\int _0^{T}\int _{\Omega }u_{\varepsilon }v_{\varepsilon }^{-1}\leq C, \end{align*}
thereby establishing the boundedness of the gradient sequence
$(\nabla u_{\varepsilon }^2)_{\varepsilon \in (0,1)}$
in
$ L^1((0,T);\; W^{1,1}(\Omega ))$
. Subsequently, employing Lemmas3.10, and 3.13 and the Arzelá-Ascoli compactness theorem, a straightforward extraction procedure allows us to construct a vanishing subsequence
$(\varepsilon _{j})_{j\in \mathbb{N}}$
with
$\varepsilon _{j}\searrow 0$
. This yields nonnegative functions
$(u,v)$
satisfying (1.7), (2.2) and (3.42)–(3.44). Finally, (2.3) and (2.4) can be accomplished in the respective weak formulation associated with (2.5) on the basis of these convergence properties.
4. Large time behaviour
This section is devoted to establishing the stability properties and nontrivial stabilisation results stated in Theorems1.2 and 1.3, respectively. The proofs rely crucially on the temporal decay of
$u_{\varepsilon t}$
in suitable dual spaces, which is established via the integral estimates (2.10), (2.11) and (2.19).
Lemma 4.1.
Let
$\Omega \subset \mathbb R^3$
and
$K\gt 0$
with the property that (1.6) holds. Supposed that
$\alpha \in \left (\frac {3}{2},\frac {19}{12}\right )$
, there exist
$\sigma \gt 0$
and
$C=C(K)\gt 0$
such that for all
$\varepsilon \in (0,1)$
, we have
\begin{align} \int _0^{\infty }\|u_{\varepsilon t}(\cdot ,t)\|_{(W^{1,\infty }(\Omega ))^*}dt\leq C\cdot \left \{\int _{\Omega } v_{0}\right \}^{\sigma }. \end{align}
Proof. Using the first equation in (2.5), one can see that for all
$t\gt 0$
and any
$\psi \in W^{1,\infty }(\Omega )$
such that
$\|\psi \|_{W^{1,\infty }(\Omega )}\equiv \max \{\|\psi \|_{L^{\infty }(\Omega )},\|\nabla \psi \|_{L^{\infty }(\Omega )}\}\leq 1$
,
\begin{align*} \left |\int _{\Omega }u_{\varepsilon t}\psi \right |&=\left |-\int _{\Omega }u_{\varepsilon }v_{\varepsilon }\nabla u_{\varepsilon }\cdot \nabla \psi +\chi \int _{\Omega }u_{\varepsilon }^{\alpha }v_{\varepsilon }\nabla v_{\varepsilon }\cdot \nabla \psi +\ell \int _{\Omega }u_{\varepsilon }v_{\varepsilon }\right |\\[5pt] &\leq \int _{\Omega }u_{\varepsilon }v_{\varepsilon }|\nabla u_{\varepsilon }| +\chi \int _{\Omega }u_{\varepsilon }^{\alpha }v_{\varepsilon }|\nabla v_{\varepsilon }|+\ell \int _{\Omega }u_{\varepsilon }v_{\varepsilon }, \end{align*}
so that
\begin{align} \|u_{\varepsilon t}(\cdot ,t)\|_{(W^{1,\infty }(\Omega ))^*}\leq \int _{\Omega }u_{\varepsilon }v_{\varepsilon }|\nabla u_{\varepsilon }| +\chi \int _{\Omega }u_{\varepsilon }^{\alpha }v_{\varepsilon }|\nabla v_{\varepsilon }|+\ell \int _{\Omega }u_{\varepsilon }v_{\varepsilon }. \end{align}
Thanks to the Höder inequality and (2.10), for all
$T\gt 0$
, we infer that
\begin{align} \int _0^{T}\int _{\Omega }u_{\varepsilon }v_{\varepsilon }|\nabla u_{\varepsilon }| &\leq \left \{\int _0^{T}\int _{\Omega }\frac {v_{\varepsilon }}{u_{\varepsilon }}|\nabla u_{\varepsilon }|^2\right \}^{\frac {1}{2}}\cdot \left \{\int _0^{T}\int _{\Omega }u_{\varepsilon }^3v_{\varepsilon } \right \}^{\frac {1}{2}}\nonumber \\[5pt] &\leq \left \{\int _0^{T}\int _{\Omega }\frac {v_{\varepsilon }}{u_{\varepsilon }}|\nabla u_{\varepsilon }|^2\right \}^{\frac {1}{2}}\cdot \left \{\int _0^{T}\int _{\Omega }u_{\varepsilon }^{\frac {3-r}{1-r}}v_{\varepsilon } \right \}^{\frac {1-r}{2}}\cdot \left \{\int _0^{T}\int _{\Omega }u_{\varepsilon }v_{\varepsilon } \right \}^{\frac {r}{2}}\nonumber \\[5pt] &\leq \left \{\int _0^{T}\int _{\Omega }\frac {v_{\varepsilon }}{u_{\varepsilon }}|\nabla u_{\varepsilon }|^2\right \}^{\frac {1}{2}}\cdot \left \{\int _0^{T}\int _{\Omega }u_{\varepsilon }^{\frac {3-r}{1-r}}v_{\varepsilon } \right \}^{\frac {1-r}{2}}\cdot \left \{\int _{\Omega }v_{0} \right \}^{\frac {r}{2}} \end{align}
and
\begin{align} \int _0^{T}\int _{\Omega }u_{\varepsilon }^{\alpha }v_{\varepsilon }|\nabla v_{\varepsilon }| &\leq \left \{\int _0^{T}\int _{\Omega }v_{\varepsilon }|\nabla v_{\varepsilon }|^2\right \}^{\frac {1}{2}}\cdot \left \{\int _0^{T}\int _{\Omega }u_{\varepsilon }^{2\alpha }v_{\varepsilon } \right \}^{\frac {1}{2}}\nonumber \\[5pt] &\leq \left \{\int _0^{T}\int _{\Omega }v_{\varepsilon }|\nabla v_{\varepsilon }|^2\right \}^{\frac {1}{2}}\cdot \left \{\int _0^{T}\int _{\Omega }u_{\varepsilon }^{\frac {2\alpha -r}{1-r}}v_{\varepsilon } \right \}^{\frac {1-r}{2}}\cdot \left \{\int _0^{T}\int _{\Omega }u_{\varepsilon }v_{\varepsilon } \right \}^{\frac {r}{2}}\nonumber \\[5pt] &\leq \left \{\int _0^{T}\int _{\Omega }v_{\varepsilon }|\nabla v_{\varepsilon }|^2\right \}^{\frac {1}{2}}\cdot \left \{\int _0^{T}\int _{\Omega }u_{\varepsilon }^{\frac {2\alpha -r}{1-r}}v_{\varepsilon } \right \}^{\frac {1-r}{2}}\cdot \left \{\int _{\Omega }v_{0} \right \}^{\frac {r}{2}} \end{align}
as well as
\begin{align} \int _0^{T}\int _{\Omega }u_{\varepsilon }v_{\varepsilon }\leq \int _{\Omega }v_{0}\leq \left \{\int _{\Omega }v_{0} \right \}^{\frac {r}{2}}\cdot \left (\|v_{0}\|_{L^{\infty }(\Omega )}\cdot |\Omega |\right )^{\frac {1-r}{2}}, \end{align}
where
$r\in (0,1)$
and
$\min \{\frac {2\alpha -r}{1-r},\frac {3-r}{1-r}\}\gt 2$
. It follows from (4.2)–(4.5) that there exists
$C(K)\gt 0$
such that
\begin{align*} \int _0^T\|u_{\varepsilon t}(\cdot ,t)\|_{(W^{1,\infty }(\Omega ))^*}dt\leq C(K)\cdot \left \{\int _{\Omega }v_{0} \right \}^{\frac {r}{2}} \end{align*}
due to (1.6), (2.11), (2.19) and (3.26), and thereby we arrive at (4.1) with
$\sigma \;:\!=\;\frac {r}{2}$
.
Making use of Lemma4.1, we can derive the following consequence that exclusively involves the zero-order expression of
$u$
.
Lemma 4.2.
Let
$\alpha \in \left (\frac {3}{2},\frac {19}{12}\right )$
and
$K\gt 0$
with the property that (1.6) holds. Given
$\sigma \gt 0$
and
$C=C(K)\gt 0$
as in Lemma
4.1
, then for any nondecreasing
$(t_{k})_{k\in \mathbb{N}}\subset [0,\infty )$
, we have
\begin{align} \sum _{k=1}^{\infty }\|u(\cdot ,t_{k+1})-u(\cdot ,t_{k})\|_{(W^{1,\infty }(\Omega ))^*}dt\leq C\cdot \left \{\int _{\Omega } v_{0}\right \}^{\sigma }, \end{align}
where we set
$u(\cdot ,0)\;:\!=\;u_0$
.
Proof. Fixing any such nondecreasing sequence
$(t_k)_{k \in \mathbb{N}}$
, we infer from Lemma4.1 that
\begin{align*} \sum _{k\in \mathbb{N}} \| u_{\varepsilon }(\cdot , t_{k+1}) - u_{\varepsilon }(\cdot , t_k) \|_{(W^{1,\infty }(\Omega ))^*} &= \sum _{k\in \mathbb{N}} \left \| \int _{t_k}^{t_{k+1}} u_{\varepsilon t}(\cdot , t) \, dt \right \|_{(W^{1,\infty }(\Omega ))^*}\\[5pt] &\leq \sum _{k\in \mathbb{N}} \int _{t_k}^{t_{k+1}} \| u_{\varepsilon }(\cdot , t) \|_{(W^{1,\infty }(\Omega ))^*} \, dt\\[5pt] &\leq C(K) \cdot \left \{ \int _{\Omega } v_{0} \right \}^{\sigma }, \end{align*}
because
$(t_k, t_{k+1}) \cap (t_l, t_{l+1}) = \emptyset$
for
$k \in \mathbb{N}$
and
$l \in \mathbb{N}$
with
$k \neq l$
. This along with (3.42) implies (4.6) immediately.
The quantitative dependence on
$v_0$
not only establishes large-time stabilisation of individual trajectories in their first component but also quantifies the proximity between the limiting profile and initial data.
Lemma 4.3.
Let
$\alpha \in \left (\frac {3}{2},\frac {19}{12}\right )$
and
$K\gt 0$
with the property that (1.6) holds. Then, the function
$u$
obtained in Lemma
4.2
exhibits the convergence
\begin{align} u(\cdot ,t)\rightarrow u_{\infty }\qquad in\;(W^{1,\infty }(\Omega ))^*\qquad as\; t\to \infty \end{align}
with some
$u_{\infty }\in (W^{1,\infty }(\Omega ))^*$
that satisfies
\begin{align} \|u_{\infty }-u_0\|_{(W^{1,\infty }(\Omega ))^*}\leq C(K)\cdot \left \{\int _{\Omega } v_{0}\right \}^{\sigma } \end{align}
with
$C(K)\gt 0$
as in Lemma
4.1
.
Proof. Lemma4.2 implies that
$\{u(\cdot ,t_{k})\}_{k\in \mathbb{N}}$
forms a Cauchy sequence in
$(W^{1,\infty }(\Omega ))^*$
, which establishes (4.7) with some
$u_{\infty }\in (W^{1,\infty }(\Omega ))^*$
. Now select the sequence
$(t_{k})_{k\in \mathbb{N}}$
with
$t_1\;:\!=\;0$
and
$t_k\;:\!=\;t$
for
$k\geq 2$
in (4.6), yielding
\begin{align} \|u(\cdot ,t)-u_0\|_{(W^{1,\infty }(\Omega ))^*}\leq C(K)\cdot \left \{\int _{\Omega } v_{0}\right \}^{\sigma }\qquad for \;all\; t\gt 0, \end{align}
The quantitative forms of the right-hand side in (4.8) and (4.9) allow us to derive the following stability property of function pairs
$(u_0,0)$
.
Lemma 4.4.
Let
$\alpha \in \left (\frac {3}{2},\frac {19}{12}\right )$
and
$K\gt 0$
with the property that (1.6) holds. Then for each
$\eta \gt 0$
, there exists
$\delta _1=\delta _1(K,\eta )\gt 0$
whenever
$u_0$
and
$v_0$
fulfill (1.5), as well as
\begin{align} \int _{\Omega }v_0\leq \delta _1, \end{align}
the solution
$(u,v)$
of (1.4) obtained in Theorem
1.1
satisfies
\begin{align} \|u(\cdot ,t)-u_0\|_{(W^{1,\infty }(\Omega ))^*}\leq \eta \qquad for \;all\; t\gt 0. \end{align}
Moreover, the corresponding limit function from Lemma 4.3 admits
\begin{align} \|u_{\infty }-u_0\|_{(W^{1,\infty }(\Omega ))^*}\leq \eta . \end{align}
Proof. From (4.8) and (4.9), it follows that for any
$\eta \gt 0$
, there exists
$\delta _1=\delta _1(K,\eta )\gt 0$
such that (4.10) warrants (4.11) and (4.12).
The following lemma is used to obtain the decay estimate for
$v$
.
Lemma 4.5. [Reference Tao and Winkler23] Let
$y\in C^1([0,\infty ))$
and
$h\in L^1_{loc}([0,\infty ))$
both be nonnegative, and suppose that
\begin{align*} \int _t^{t-1}h(s)ds\rightarrow 0\qquad as\; t\rightarrow \infty , \end{align*}
and that there exists
$\lambda \gt 0$
such that
\begin{align*} y'(t)+\lambda y(t)\leq h(t) \qquad for \;all\;t\gt 0. \end{align*}
Then,
\begin{align*} y(t)\rightarrow 0\qquad as\; t\rightarrow \infty . \end{align*}
The estimates established in (2.17) and (2.19) imply the following asymptotic decay behaviour for the second solution component.
Lemma 4.6.
Let
$\alpha \in \left (\frac {3}{2},\frac {19}{12}\right )$
and
$K\gt 0$
with the property that (1.6) holds. Then, we have
\begin{align} v(\cdot ,t)\rightarrow 0\qquad in\;W^{1,p}(\Omega )\;\;for\;all\;p\geq 1\qquad as\; t\to \infty . \end{align}
Proof. Thanks to (2.10), (2.17), (3.42) and (3.43), the application of Fatou’s lemma then yields
\begin{equation*}\int _0^{\infty }\int _{\Omega }v\lt \infty \quad \text{and}\quad \int _0^{\infty }\int _{\Omega }uv\lt \infty .\end{equation*}
Consequently,
\begin{align} \int _{t-1}^{t}\int _{\Omega }v\rightarrow 0\quad \text{and}\quad \int _{t-1}^{t}\int _{\Omega }uv\rightarrow 0\qquad as\; t\to \infty . \end{align}
Testing the second equation in (1.4) by
$-\Delta v$
and applying Young’s inequality together with Hölder’s inequality, one can find
$c_1\gt 0$
such that
\begin{align} \frac {d}{dt}\int _{\Omega }|\nabla v|^2&=-2\int _{\Omega }|\Delta v|^2+2\int _{\Omega }uv\Delta v\nonumber \\[5pt] &\leq -\int _{\Omega }|\Delta v|^2+\int _{\Omega }u^2v^2\nonumber \\[5pt] &\leq -\int _{\Omega }|\Delta v|^2+\|v_0\|_{L^{\infty }(\Omega )}\left (\int _{\Omega }u^3\right )^{\frac {1}{2}}\left (\int _{\Omega }uv\right )^{\frac {1}{2}}\nonumber \\[5pt] &\leq -\int _{\Omega }|\Delta v|^2+c_1\left (\int _{\Omega }uv\right )^{\frac {1}{2}} \end{align}
for all
$t\gt 0$
, due to (2.8) and (3.25). Apart from that, integration by parts followed by Young’s inequality yields
\begin{align} \int _{\Omega }|\nabla v|^2=\int _{\Omega }v\Delta v\leq \int _{\Omega }|\Delta v|^2+\frac {1}{4}\int _{\Omega }v^2. \end{align}
Combining (2.8), (4.15) with (4.16), we get
\begin{align} \frac {d}{dt}\int _{\Omega }|\nabla v|^2+\int _{\Omega }|\nabla v|^2\leq c_1\left (\int _{\Omega }uv\right )^{\frac {1}{2}}+\frac {\|v_0\|_{L^{\infty }(\Omega )}}{4}\int _{\Omega }v. \end{align}
Define
\begin{equation*}y(t)\;:\!=\;\int _{\Omega }|\nabla v|^2\qquad \text{and}\qquad g(t)\;:\!=\;c_1\left (\int _{\Omega }uv\right )^{\frac {1}{2}}+\frac {\|v_0\|_{L^{\infty }(\Omega )}}{4}\int _{\Omega }v.\end{equation*}
Then it follows from (4.17) that
\begin{align*} y'(t)+y(t)\leq g(t). \end{align*}
Furthermore, by Hölder’s inequality and (4.14), we arrive at
\begin{align} \int _{t-1}^{t}g(t)\leq c_1\left (\int _{t-1}^{t}\int _{\Omega }uv\right )^{\frac {1}{2}}+\frac {\|v_0\|_{L^{\infty }(\Omega )}}{4}\int _{t-1}^{t}\int _{\Omega }v\rightarrow 0\qquad as\quad t\to \infty . \end{align}
As an application of Lemma4.5, we infer that
\begin{align} \int _{\Omega }|\nabla v|^2\rightarrow 0\qquad as\; t\to \infty . \end{align}
According to the interpolation inequality, (4.19) and (3.36), we obtain for any
$p\gt 2$
\begin{align} \|\nabla v(\cdot ,t)\|_{L^{p}(\Omega )}\leq \|\nabla v(\cdot ,t)\|_{L^{\infty }(\Omega )}^{\frac {p-2}{p}}\|\nabla v(\cdot ,t)\|_{L^{2}(\Omega )}^{\frac {2}{p}}\leq C\|\nabla v(\cdot ,t)\|_{L^{2}(\Omega )}^{\frac {2}{p}}\rightarrow 0 \quad \text{as} \quad t \to \infty . \end{align}
Beyond the stabilisation result established in Theorem1.2 for solutions to (1.4), we further demonstrate that the limiting profile
$u_{\infty }$
of the solution component
$u$
obtained in (4.7) becomes non-homogeneous when the initial signal concentration
$v_0$
is sufficiently small, provided that
$u_0$
is not identically constant.
Lemma 4.7.
Let
$\alpha \in \left (\frac {3}{2}, \frac {19}{12} \right )$
and
$K \gt 0$
be such that (1.6) holds and suppose
$u_0\not \equiv$
const. Then there exists
$\delta _2 = \delta _2(K, u_0)\gt 0$
such that if
$u_0$
and
$v_0$
satisfy (1.5) and (1.6), as well as
\begin{align} \int _{\Omega }v_0\lt \delta _2, \end{align}
the corresponding limit function
$u_{\infty } \in (W^{1,\infty }(\Omega ))^*$
from Lemma
4.3
satisfies
$u_{\infty } \not \equiv \mathrm{const}$
.
Proof. Since
$u_{0}$
is continuous and not constant, we can fix numbers
$c_{1}\gt 0$
,
$c_{2}\gt c_{1}$
, as well as open sets
$\Omega _1\subset \Omega$
and
$\Omega _2\subset \Omega$
, such that
$u_{0}\leq c_{1}$
in
$\Omega _1$
and
$u_{0}\geq c_{2}$
in
$\Omega _2$
. Furthermore, let functions
$0\leq \psi _{i}\in C^{\infty }_{0}(\Omega )$
,
$i\in \{1,2\}$
, such that
$\operatorname {supp}\psi _{i}\subset \Omega _i$
and
$\|\psi _{i}\|_{W^{1,\infty }(\Omega )}\leq 1$
. Choose
$\delta _2= \delta _2(K, u_0)\gt 0$
sufficiently small so that
\begin{align} C(K)\cdot \delta _2^{\sigma }\lt \kappa \cdot \min \left \{\int _{\Omega _1}\psi _1,\int _{\Omega _2}\psi _2\right \} \end{align}
with
$\kappa \;:\!=\;\frac {c_{2}-c_{1}}{4}$
,
$C(K)\gt 0$
and
$\sigma \gt 0$
given as in Lemma4.1.
Now if
$u_{\infty }\not \in L^\infty (\Omega )$
, then
$u_{\infty }$
is non-constant because any constant function is bounded, and the proof is thus complete. Therefore, we may assume
$u_{\infty } \in L^\infty (\Omega )$
. At this position, we claim that
\begin{align} \mathrm{ess\,sup}_{\Omega } u_{\infty }\geq c_{2}-\kappa \quad \text{and} \quad \mathrm{ess\,inf}_{\Omega } u_{\infty }\leq c_{1}+\kappa . \end{align}
To verify this, suppose that
$\mathrm{ess\,sup}_{\Omega } u_{\infty } \lt c_2 - \kappa$
, the properties of
$\psi _2$
imply
\begin{equation}\begin{aligned} \|u_{\infty }-u_{0}\|_{(W^{1,\infty }(\Omega ))^{*}} &\geq \left | \int _{\Omega } (u_{\infty } - u_0)\cdot \psi _2 \right | \\[5pt] &=\int _{\Omega _2}(u_{0}-u_{\infty })\cdot \psi _{2}\\[5pt] &\geq \int _{\Omega _2}\{c_2+(\kappa -c_{2})\}\cdot \psi _{2} \\[5pt] &=\kappa \int _{\Omega _2}\psi _2, \end{aligned} \end{equation}
On the other hand, Lemma4.3 and (4.22) yield
\begin{align} \|u_{\infty }-u_0\|_{(W^{1,\infty }(\Omega ))^*} \leq C(K)\cdot \left \{\int _{\Omega } v_{0}\right \}^{\sigma } \lt \kappa \cdot \min \left \{\int _{\Omega _1}\psi _1,\int _{\Omega _2}\psi _2\right \}\!, \end{align}
which contradicts (4.24). The second inequality of (4.23) follows analogously, and thereby
$u_{\infty }$
cannot coincide with any constant due to our choice of
$\kappa$
.
Data availability statement
All data that support the findings of this study are included within the article.
Funding statement
This work was supported by the National Natural Science Foundation of China (No. 12071030 and No. 12271186) and Beijing Key Laboratory on MCAACI.
Competing interests
No potential conflict of interest is reported by the authors.
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Ethics approval is not required for this research.
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