1. Introduction
This paper investigates the well-posedness and boundedness of the solution
$(\rho, A) = (\rho (\mathbf{x}, t), A(\mathbf{x}, t))$
to a reaction–advection–diffusion system, where
$\mathbf{x}$
represents location and
$t$
denotes time, in the following form:
\begin{equation} \left \{ \begin{array}{l@{\quad}l} \partial _tA=\eta \Delta A-A+A\rho +\bar{A},&\mathbf{x}\in \Omega, t\gt 0, \\[5pt] \partial _t\rho =\nabla \cdot (\nabla \rho - \phi (\rho ) \nabla \ln A) -A\rho +\bar{B},&\mathbf{x}\in \Omega, t\gt 0,\\[5pt] \dfrac{\partial A}{\partial \mathbf{n}}=\dfrac{\partial \rho }{\partial \mathbf{n}}=0, & \mathbf{x}\in \partial \Omega, t\gt 0,\\[9pt] A(\mathbf{x},0)=A_0(\mathbf{x})\gt 0, \rho (\mathbf{x},0)=\rho _0(\mathbf{x})\geq, \not\equiv 0, & \mathbf{x}\in \Omega . \end{array} \right . \end{equation}
Here,
$\Omega$
is a bounded domain in
$\mathbb{R}^2$
with a piecewise smooth boundary
$\partial \Omega$
endowed with unit outer normal
$\mathbf{n}$
.
$\phi (\rho )$
is a continuously differentiable function, and
$\eta, \bar{A}$
and
$\bar{B}$
are arbitrary positive constants. The main result of our paper is Theorem1.1 which proves that system (1.1) is globally well-posed, and both
$A$
and
$\rho$
are bounded in
$L^\infty (\mathbb{R}^+;\; L^\infty (\Omega ))$
.
1.1. Urban criminal activities
It is widely believed that human behaviour is influenced by a multitude of factors and is therefore too complex to obey the same laws that govern physics and other natural sciences. In particular, individual behaviour is difficult to describe with mechanistic models or to predict with quantitative methods. At the group level, however, human behaviour can exhibit spatiotemporal patterns that are more amenable to mathematical analysis. A prime example of this is the urban crime hotspot phenomenon, in which certain neighbourhoods experience higher crime rates compared to surrounding regions with lower rates. In addition, criminal activity may also fluctuate over time, with certain periods characterised by increased disorder. Such spatiotemporal clustering suggests that while individual actions are unpredictable, collective behaviour follows discernible patterns that can be studied mathematically (see [Reference Brantingham and Brantingham4, Reference Eck and Weisburd14]).
1.2. Mathematical models of residential urban crimes
In 2008, a group at UCLA [Reference Short, D’Orsogna, Pasour, Tita, Brantingham and Bertozzi55] developed a mathematical model to explore the spatiotemporal dynamics of urban crime hotspots. Their model is based on two key criminological assumptions: the broken-windows effect ([Reference Sampson and Raudenbush51, Reference Wilson and Kelling61]), which suggests that visible signs of disorder promote further crime, and the repeat and near-repeat victimisation effect ([Reference Anselin, Cohen, Cook, Gorr and Tita3, Reference Bowers and Johnson5, Reference Felsen15, Reference Johnson, Bowers and Hirschfield27, Reference Short, D’Orsogna, Brantingham and Tita54]), which argues that areas previously victimised are more likely to experience subsequent crimes. Consequently, they constructed a 2D agent-based lattice model that not only simulates the movement of criminals but also incorporates the dynamics of attractiveness values, representing locations more prone to criminal activity. The spatial continuum of the lattice model is governed by strongly coupled partial differential equations of parabolic type, expressed in the following form
\begin{equation} \left \{ \begin{array}{l@{\quad}l} A_t=\eta \Delta A-A+A\rho +\bar{A},&\mathbf{x}\in \Omega, t\gt 0,\\[5pt] \rho _t=\nabla \cdot (\nabla \rho - \chi \rho \nabla \ln A) -A\rho +\bar{B},&\mathbf{x}\in \Omega, t\gt 0. \end{array} \right . \end{equation}
where functions
$ A(\mathbf{x}, t)$
and
$ \rho (\mathbf{x}, t)$
represent the area attractiveness and the population density of criminal agents at the space-time location
$ (\mathbf{x}, t)$
, respectively;
$ \Delta$
and
$ \nabla$
denote the Laplace and gradient operators, while
$ \eta$
and
$ \bar{B}$
are positive constants. The term
$ \bar{A}$
represents a time-independent positive function of location
$ \mathbf{x}$
, capturing the static attractiveness of an area. The attractiveness
$A(\mathbf{x},t)$
consists of two components with
$A(\mathbf{x},t)= \bar{A}+B(\mathbf{x},t)$
,
$\bar{A}$
representing the static part and
$B$
the dynamic part. Diffusion rate
$\eta$
reflects the intensity of the near-repeat victimisation effect, whereas the broken-window effect is incorporated through the dynamics of
$ A\rho$
in the first equation. The positive coefficient
$\chi$
represents the strength of the migration rate of criminal agents towards attractive sites, driven by the gradient of the logarithm of perceived house attractiveness. We refer the reader to [Reference Gu, Wang and Yi20, Reference Short, D’Orsogna, Pasour, Tita, Brantingham and Bertozzi55] for the derivation, justification and extension of (1.2) and to [Reference D’Orsogna and Perc13, Reference Groff, Johnson and Thornton18, Reference Lindstrom and Bertozzi33] for a review of agent-based urban crime modelling.
Over the last 15 years, a growing interest in mathematical analysis of the urban crime model (1.2) has been driven by the work of [Reference Short, D’Orsogna, Pasour, Tita, Brantingham and Bertozzi55]. Weak linear analysis by [Reference Short, Bertozzi and Brantingham52, Reference Short, Brantingham, Bertozzi and Tita53] reveals the emergence of stable hotspots, while rigorous bifurcation analysis by [Reference Cantrell, Cosner and Manásevich11, Reference Gu, Wang and Yi20] provides deeper quantitative insights into these dynamics. Furthermore, stationary large-amplitude peaks in the continuum model (1.2) are investigated by [Reference Berestycki, Rodriguez and Ryzhik8, Reference Berestycki, Wei and Winter9, Reference Kolokolnikov, Ward and Wei28, Reference Lloyd and O’Farrell34, Reference Mei and Wei38]. The 2D lattice model and its continuum not only capture aggregation phenomena in urban criminal activity but also exhibit rich spatiotemporal dynamics and complex regular and irregular spatial patterns that urge further analytical and numerical studies. It seems necessary to mention that the migration of criminal agents towards attractive targets, as modelled here, is analogous to the movement of chemotactic organisms in response to uneven chemical distribution. See [Reference Hillen and Painter22, Reference Horstmann23, Reference Wang60] for some surveys on the mathematical modelling of chemotaxis.
Mathematical modelling of system (1.2) has been extended in several directions. For instance, Gu, Wang, and Yi in [Reference Gu, Wang and Yi20] explored spatial heterogeneity in both the near-repeat victimisation effect and the dispersal strategy of criminal agents, proposing a class of reaction–advection–diffusion systems with nonlinear diffusion. Additionally, works such as [Reference Chaturapruek, Breslau, Yazdi, Kolokolnikov and McCalla10, Reference Levajkovic, Mena and Zarfl29, Reference Pan, Li and Wang41] model the dispersal of criminal agents using the Lévy process, which leads to a fractional Laplacian diffusion problem in the continuum limit of the agent-based model. Other modelling developments can be seen in the consideration of age-structured populations ([Reference Saldana, Aguareles and Avinyó50]) and geographic profiling ([Reference Mohler and Short39]), and we refer to [Reference Berestycki and Nadal7, Reference Berestycki, Wei and Winter9, Reference D’Orsogna and Perc13, Reference Mohler, Short, Brantingham, Schoenberg and Tita40] for further developments in this direction. It is also noteworthy that the authors of [Reference Jones, Brantingham and Chayes26, Reference Pitcher42] independently introduced an additional equation to (1.2) to describe police deployment and focused patrols, examining the effects of law enforcement on the dynamics of criminal activity. This three-component reaction-diffusion system has been further analysed and developed by [Reference Buttenschoen, Kolokolnikov, Ward and Wei6, Reference Pitcher42, Reference Ricketson44, Reference Rodríguez, Wang and Zhang47, Reference Short, Bertozzi and Brantingham52, Reference Short, Brantingham, Bertozzi and Tita53, Reference Tse and Ward58, Reference Zipkin, Short and Bertozzi66].
1.3. Global well-posedness of the PDE models and main result
Theoretical and numerical studies of system (1.2) have attracted considerable academic attention, as seen in works such as [Reference Buttenschoen, Kolokolnikov, Ward and Wei6–Reference Berestycki, Wei and Winter9, Reference Cantrell, Cosner and Manásevich11, Reference García-Huidobro, Manásevich and Mawhin19, Reference Kolokolnikov, Ward and Wei28, Reference Lloyd and O’Farrell34–Reference Manásevich, Garcia-Huidobro and Mawhin36, Reference Mei and Wei38, Reference Short, Bertozzi and Brantingham52, Reference Short, D’Orsogna, Pasour, Tita, Brantingham and Bertozzi55, Reference Tse and Ward57]. These works demonstrate that this system exhibits rich and complex spatiotemporal dynamics, with nontrivial patterns that successfully capture the characteristic crime hotspots associated with urban residential burglary. Depending on the choice of parameters, the models can produce either dynamic or static concentrating profiles, representing the spatial hotspots of criminal activity. Rodríguez and Bertozzi [Reference Rodríguez and Bertozzi46] established the local well-posedness theory for system (1.2), proving both the existence and the uniqueness of its solution. They also highlighted the possibility of finite time blow-up in a modified version of (1.2). In the 1D case, Wang, Wang and Feng [Reference Wang, Wang and Feng59] extended this well-posedness theory to global time and prove the uniform boundedness of the solution for any
$\chi \gt 0$
(see also [Reference Rodríguez and Winkler49]). However, in high dimensions, the global existence and boundedness of (1.2) remain open. Successful attempts have been made in special cases, including settings with small parameters or small data ([Reference Freitag16, Reference Jiang and Yang25, Reference Li, Wang and Xie30, Reference Tao and Winkler56]), in a weak sense ([Reference Fuest and Heihoff17, Reference Heihoff21, Reference Li and Xie31, Reference Winkler63]), or in variants of (1.2) with superlinear diffusion and/or sublinear advection enhancement of criminal agents ([Reference Rodríguez and Winkler48, Reference Yang and Yang64, Reference Zhao65]) as well as superlinear dissipation of criminal agents ([Reference Heihoff21, Reference Li, Xiao and Xie32, Reference Manásevich, Phan and Souplet37, Reference Qiu and Li43, Reference Rodríguez45]).
This paper extends the existing literature by proving global existence and uniform boundedness for the 2D urban crime model. We assume that there are some positive constants
$M$
and
$\theta$
such that
\begin{equation} 0\leq \phi (\rho )\leq M\rho ^{\theta }, \theta \lt \frac{3}{4} \quad \quad \forall \rho \geq 0. \end{equation}
Then our main result is as follows.
Theorem 1.1.
Let
$\Omega$
be a bounded domain in
$\mathbb{R}^2$
with piecewise smooth boundary
$\partial \Omega$
. Suppose that
$\phi \in C^1(\mathbb{R})$
and (1.3) holds, then for any nonnegative initial data
$(\rho _0,A_0)\in C(\bar{\Omega }) \times W^{1,\infty }(\Omega )$
, there exists a unique pair
$(\rho, A)$
of functions each belonging to
$C^0(\bar{\Omega }\times [0,\infty ))\cap C^{2,1}(\bar{\Omega }\times (0,\infty ))$
that solves
(1.1)
classically. Moreover, this classical solution is strictly positive in
$\bar{\Omega }$
, and it is uniformly bounded such that
$\Vert \rho (\cdot, t) \Vert _{L^\infty (\Omega )}+\Vert A(\cdot, t) \Vert _{L^\infty (\Omega )}\leq C$
for all
$t\gt 0$
.
This paper proves the well-posedness of the original system (1.2), for which we assume the sublinearity (1.3) of the sensitivity function for technical reasons. While this assumption deviates from linearity, it reflects the diminishing effect where criminal agents are less inclined to move to attractive regions that are already densely populated by other agents. Throughout the paper, we use
$C$
to represent a generic positive constant that may vary from line to line. For clarity and simplicity, we omit the differential “
$d\mathbf{x}$
” in calculations whenever possible.
2. Local existence and preliminary results
The mathematical analysis of the global well-posedness of system (1.1) is delicate since the maximum principle does not apply to the
$\rho$
equation. However, the local well-posedness can be easily established using the fundamental theory developed by Amann [Reference Amann2] (see also [Reference Rodríguez and Bertozzi46]) and standard parabolic regularity theory.
Proposition 1.
Let all assumptions in Theorem
1.1
hold. Then there exist
$T_{\max }\in (0,\infty ]$
and a unique couple
$(\rho, A)$
of nonnegative functions from
$C^0(\bar{\Omega }\times [0,T_{\max }))\cap C^{2,1}(\bar{\Omega }\times (0,T_{\max }))$
solving (
1.1
) classically in
$\Omega \times (0,T_{\max })$
. Moreover,
$\rho (\mathbf{x},t)\gt 0$
and
$A(\mathbf{x},t)\gt 0$
in
$\Omega \times (0,T_{\max })$
and the following dichotomy holds:
\begin{equation*} \text {either }\quad T_{\max }=\infty \qquad \text {or}\qquad T_{\max }\lt \infty \,\text {and}\, \limsup _{t\nearrow T_{\max }^-} \|\rho (\cdot, t)\|_{L^{\infty }(\Omega )}=\infty . \end{equation*}
Next, we collect some properties of the local solution obtained in Proposition1. To begin with, for any space dimension, Lemma2.1 and Lemma2.2 in [Reference Wang, Wang and Feng59] ensure the existence of positive constants
$\delta$
and
$C$
, which depend on
$A_0(\mathbf{x})$
, such that
\begin{equation} \min _{\mathbf{x}\in \bar \Omega }A(\mathbf{x},t)\geq \delta \gt 0 \quad \quad \forall t\in (0,T_{\max }) \end{equation}
and
\begin{equation} \int _\Omega A(\mathbf{x},t) d\mathbf{x}+\int _\Omega \rho (\mathbf{x},t) d\mathbf{x}\leq C\quad \quad \forall t\in (0,T_{\max }). \end{equation}
Lemma 2.1.
Let
$(\rho, A)$
be a nonnegative classical solution of
(1.1)
in
$\Omega \times (0,T_{\max })$
. If
$A\rho \in L^p(\Omega )$
for some
$p\in [1,\infty )$
, then there exists a positive constant
$C$
dependent on
$\Vert A_0 \Vert _{L^{p}(\Omega )}$
and
$\vert \Omega \vert$
such that
\begin{equation} \Vert A(\cdot, t) \Vert _{W^{1,q}(\Omega )} \le C\Big (1+\sup _{s\in (0,t)} \Vert (A\rho )(\cdot, s)\Vert _{L^p(\Omega )}\Big )\quad \quad \forall t\geq 0, \end{equation}
where
$q\in [1,\frac{2p}{2-p})$
if
$p\in [1,2)$
,
$q\in [1,\infty )$
if
$p=2$
and
$q=\infty$
if
$p\gt 2$
.
Proof. We first write the following abstract formula of
$A$
\begin{equation*} A(\cdot, t)=e^{(\eta \Delta -1)t}A_0+\int _0^t e^{(\eta \Delta -1)(t-s)} \big ( (A\rho )(\cdot, s)+\bar {A} \big )ds, \end{equation*}
Thanks to the
$L^p$
–
$L^q$
estimates between semigroups
$\{e^{t\Delta }\}_{t\geq 0}$
(cf. Lemma 1.3 of [Reference Winkler62] with
$N=2$
), we can find positive constants
$C_{21}$
,
$C_{22}$
and
$C_{23}$
such that
\begin{align} &\Vert A(\cdot, t) \Vert _{W^{1,q}}\nonumber \\[5pt] =&\Big \Vert e^{(\eta \Delta -1)t}A_0+\int _0^t e^{(\eta \Delta -1)(t-s)} \big ((A\rho )(\cdot, s)+\bar{A} \big )ds\Big \Vert _{W^{1,q}} \nonumber \\[5pt] \leq & C_{21} \Vert A_0\Vert _{L^p}+C_{21}\int _0^t e^{-\nu (t-s)}(t-s)^{-\frac{1}{2}-(\frac{1}{p}-\frac{1}{q})}\Vert (A\rho )(\cdot, s)+\bar{A} \Vert _{L^p} ds \nonumber \\[5pt] \leq & C_{22} +C_{23}\int _0^t e^{-\nu (t-s)}(t-s)^{-\frac{1}{2}-(\frac{1}{p}-\frac{1}{q})} \Vert (A\rho )(\cdot, s)+1\Vert _{L^p} ds \nonumber \\[5pt] \leq & C_{22}+C_{23} \Big (\int _0^t e^{-\nu (t-s)}(t-s)^{-\frac{1}{2}-(\frac{1}{p}-\frac{1}{q})}ds\Big )\sup _{s\in (0,t)} \Vert (A\rho )(\cdot, s)+1\Vert _{L^p}, \end{align}
where
$\nu$
is the first Neumann eigenvalue of
$-\eta \Delta$
. On the other hand, under the conditions on
$q$
after (2.3), we have that
\begin{equation*} \sup _{t\in (0,\infty )}\int _0^t e^{-\nu (t-s)}(t-s)^{-\frac {1}{2}-(\frac {1}{p}-\frac {1}{q})} ds\lt \infty, \end{equation*}
2.1. Boundedness of
$\Vert \nabla A(\cdot, t) \Vert _{L^2(\Omega )}$
As we will see, establishing the boundedness of
$\|\nabla A(\cdot, t)\|_{L^\infty (\Omega )}$
is essential for proving the boundedness of
$\|\rho (\cdot, t)\|_{L^\infty (\Omega )}$
. To achieve this, we first focus on establishing a weaker regularity condition for the boundedness of
$\|\nabla A(\cdot, t)\|_{L^2(\Omega )}$
.
First of all, let us calculate using the
$A$
-equation as follows:
\begin{align*} \frac{d}{dt} \int _{\Omega } \frac{|\nabla A|^2}{A} =&2 \int _{\Omega } \frac{1}{A} \nabla A \cdot \nabla A_t-\int _{\Omega } \frac{|\nabla A|^2}{A^2} \cdot A_t \\[5pt] =&2\int _{\Omega } \frac{\nabla A}{A} \cdot \nabla (\eta \Delta A-A+A\rho +\bar{A})-\int _{\Omega }\frac{|\nabla A|^2}{A^2} \cdot (\eta \Delta A-A+A\rho +\bar{A}) \\[5pt] =&2\eta \int _{\Omega } \frac{\nabla A}{A}\cdot \nabla \Delta A +2\int _{\Omega } \frac{\nabla A}{A}\cdot \nabla (-A+A\rho )\\[5pt] &-\eta \int _{\Omega } \frac{|\nabla A|^2}{A^2} \Delta A+\int _{\Omega } \frac{|\nabla A|^2}{A} -\int _{\Omega } \frac{\rho }{A} |\nabla A|^2-\bar{A} \int _{\Omega } \frac{|\nabla A|^2}{A^2}. \end{align*}
Then, we apply the identity
$\nabla A\cdot \nabla \Delta A=\frac{1}{2}\Delta |\nabla A|^2-|D^2A|^2$
and integrate by parts to obtain that
\begin{align} \frac{d}{dt} \int _{\Omega } \frac{|\nabla A|^2}{A}=&\eta \int _{\Omega } \frac{1}{A} \Delta |\nabla A|^2-2\eta \int _{\Omega } \frac{|D^2A|^2}{A}-2\int _{\Omega } \frac{|\nabla A|^2}{A}+2 \int _{\Omega } \frac{\rho }{A} |\nabla A|^2 +2\int _{\Omega } \nabla A \cdot \nabla \rho \nonumber \\[5pt] &-\eta \int _{\Omega }\frac{|\nabla A|^2}{A^2} \Delta A+\int _{\Omega }\frac{|\nabla A|^2}{A}- \int _{\Omega } \frac{\rho }{A} |\nabla A|^2-\bar{A} \int _{\Omega } \frac{|\nabla A|^2}{A^2} \nonumber \\[5pt] =&\eta \int _{\partial \Omega } \frac{1}{A}\frac{\partial |\nabla A|^2}{\partial \mathbf{n}} +\eta \int _{\Omega } \frac{1}{A^2} \nabla A \cdot \nabla |\nabla A|^2 -2\eta \int _{\Omega } \frac{|D^2A|^2}{A} -\eta \int _{\Omega }\frac{|\nabla A|^2}{A^2} \Delta A \nonumber \\[5pt] &-\int _{\Omega } \frac{|\nabla A|^2}{A}+\int _{\Omega } \frac{\rho }{A} |\nabla A|^2 +2\int _{\Omega } \nabla A \cdot \nabla \rho -\bar{A} \int _{\Omega } \frac{|\nabla A|^2}{A^2} \nonumber \\[5pt] =& \eta \overbrace{\int _{\partial \Omega } \frac{1}{A}\frac{\partial |\nabla A|^2}{\partial \mathbf{n}}}^{ I_{\partial }} + \eta \overbrace{\Big (\int _{\Omega } \frac{1}{A^2} \nabla A \cdot \nabla |\nabla A|^2 -2\int _{\Omega } \frac{|D^2A|^2}{A} -\int _{\Omega }\frac{|\nabla A|^2}{A^2} \Delta A\Big )}^{I_{01}} \nonumber \\[5pt] &-\int _{\Omega } \frac{|\nabla A|^2}{A}-\bar{A} \int _{\Omega } \frac{|\nabla A|^2}{A^2} \overbrace{-2 \int _{\Omega } A\rho \Delta \ln A}^{I_{02}}-\int _{\Omega } \frac{\rho }{A} |\nabla A|^2. \end{align}
To estimate the boundary integral
$I_\partial$
in (2.5), we apply the uniform lower bound of
$A$
in (2.1) along with the pointwise inequality
$\frac{\partial |\nabla A|^2}{\partial \mathbf{n}}\leq C_{\Omega }|\nabla A|^2$
(cf. inequality (2.4) in [Reference Ishida, Seki and Yokota24]) to find that, with
$C_{\Omega }$
being a positive constant depending only on the curvatures of
$\partial \Omega$
, the following holds
\begin{equation} \int _{\partial \Omega } \frac{1}{A}\frac{\partial |\nabla A|^2}{\partial \mathbf{n}} \leq C_{\Omega } \int _{\partial \Omega } \frac{|\nabla A|^{2}}{A}\;:\!=\;4C_{\Omega }\|\nabla A^{\frac{1}{2}}\|^2_{L^2(\partial \Omega )}. \end{equation}
The Sobolev trace embedding (cf. (1.9), Lemma 2.3 and Lemma 2.4 in [Reference Ishida, Seki and Yokota24]) implies that there exists a positive constant
$C$
such that
\begin{align} 4C_{\Omega }\|\nabla A^{\frac{1}{2}}\|_{L^2(\partial \Omega )}\leq C_{24} \|\nabla A^{\frac{1}{2}}\|_{W^{\frac{3}{4},2}(\Omega )}; \end{align}
moreover, applying the fractional Gagliardo–Nirenberg interpolation inequality to (2.7) finds that
\begin{align*} &C_{24}\|\nabla A^{\frac{1}{2}}\|_{W^{\frac{3}{4},2}(\Omega )}^2 \\[5pt] \leq & C_{25} \|\nabla \cdot \nabla A^{\frac{1}{2}}\|^{\frac{3}{2}}_{L^2(\Omega )}\|\nabla A^{\frac{1}{2}}\|^{\frac{1}{2}}_{L^{2}(\Omega )}+C_{25} \|\nabla A^{\frac{1}{2}}\|^{2}_{L^{2}(\Omega )} \\[5pt] = & C_{25} \Big \|\frac{\Delta A}{A}-\frac{|\nabla A|^2}{A^{\frac{3}{2}}}\Big \|^{\frac{3}{2}}_{L^2(\Omega )}\|\nabla A^{\frac{1}{2}}\|^{\frac{1}{2}}_{L^{2}(\Omega )}+C_{25}\|\nabla A^{\frac{1}{2}}\|^{2}_{L^{2}(\Omega )} \\[5pt] \leq & \sqrt{2} C_{25} \Big \|\frac{\Delta A}{A}\Big \|^{\frac{3}{2}}_{L^2(\Omega )}\|\nabla A^{\frac{1}{2}}\|^{\frac{1}{2}}_{L^{2}(\Omega )}+\sqrt{2} C_{25}\Big \|\frac{|\nabla A|^2}{A^{\frac{3}{2}}}\Big \|^{\frac{3}{2}}_{L^2(\Omega )}\|\nabla A^{\frac{1}{2}}\|^{\frac{1}{2}}_{L^{2}(\Omega )}+C_{25} \|\nabla A^{\frac{1}{2}}\|^{2}_{L^{2}(\Omega )}, \end{align*}
where to derive the second inequality we have used the inequality that
$(a+b)^{\frac{3}{2}}\leq \sqrt{2}(a^{\frac{3}{2}}+b^{\frac{3}{2}})$
for any positive real numbers
$a,b\gt 0$
. We then further apply Cauchy’s inequality to find that for any arbitrary positive constant
$\epsilon$
, there exists a positive constant
$C_\epsilon$
such that
\begin{align} & C_{24}\|\nabla A^{\frac{1}{2}}\|_{W^{\frac{3}{4},2}(\Omega )}^2 \nonumber \\[5pt] \leq & \frac{\epsilon }{4} \Big \|\frac{\Delta A}{A}\Big \|^{2}_{L^2(\Omega )} +\frac{\epsilon }{2} \Big \|\frac{|\nabla A|^2}{A^{\frac{3}{2}}}\Big \|^{2}_{L^2(\Omega )} +2C_{\epsilon } \|\nabla A^{\frac{1}{2}}\|^{2}_{L^{2}(\Omega )} +C_{25} \|\nabla A^{\frac{1}{2}}\|^{2}_{L^{2}(\Omega )} \nonumber \\[5pt] = & \frac{\epsilon }{4} \int _{\Omega } \frac{|\Delta A|^2}{A^2} +\frac{\epsilon }{2} \int _{\Omega } \frac{|\nabla A|^4}{A^3} +C_{26} \int _{\Omega } |\nabla A^{\frac{1}{2}}|^2 \nonumber \\[5pt] \leq & \frac{\epsilon }{2} \int _{\Omega } \frac{|D^2A|^2}{A^2}+\frac{\epsilon }{2} \int _{\Omega } \frac{|\nabla A|^4}{A^3}+C_{26} \int _{\Omega } |\nabla A^{\frac{1}{2}}|^2. \end{align}
To estimate
$C_{26} \int _{\Omega } |\nabla A^{\frac{1}{2}}|^2$
in (2.8), we have from Young’s inequality and the Gagliardo–Nirenberg interpolation inequality
\begin{align} & C_{26} \int _{\Omega } |\nabla A^{\frac{1}{2}}|^2 \nonumber \\[5pt] = & 4C_{26}\int _{\Omega } |\nabla A^{\frac{1}{4}}|^2 A^{\frac{1}{2}} \nonumber \\[5pt] \leq & \frac{\epsilon }{4} \int _{\Omega } \frac{|\nabla A|^4}{A^3}+C_{\epsilon } \int _{\Omega } A^2\nonumber \\[5pt] \leq & \frac{\epsilon }{4} \int _{\Omega } \frac{|\nabla A|^4}{A^3}+C_{27}\|\nabla A^{\frac{1}{4}}\|^2_{L^4(\Omega )} \|A^{\frac{1}{4}}\|^6_{L^4(\Omega )}+ C_{27}\|A^{\frac{1}{4}}\|^8_{L^4(\Omega )}\nonumber \\[5pt] \leq & \frac{\epsilon }{2} \int _{\Omega } \frac{|\nabla A|^4}{A^3}+C_{28}. \end{align}
Therefore, we have from (2.6)-(2.9) that
\begin{equation} I_\partial \leq \frac{\epsilon \eta }{2} \int _{\Omega } \frac{|D^2A|^2}{A^2}+\epsilon \eta \int _{\Omega } \frac{|\nabla A|^4}{A^3}+\eta C_{28} \leq \frac{\varepsilon }{2} \int _{\Omega } \frac{|D^2A|^2}{A^2}+\varepsilon \int _{\Omega } \frac{|\nabla A|^4}{A^3}+C_{29}, \end{equation}
where
$\varepsilon \;:\!=\;\epsilon \eta$
is an arbitrary positive constant. To estimate
$I_{01}$
in (2.5), we first integrate by parts to have that
\begin{align} I_{01}=&-\int _{\Omega } \nabla \frac{\nabla A}{A^2} \cdot |\nabla A|^2 -2\int _{\Omega } \frac{|D^2A|^2}{A} -\int _{\Omega }\frac{|\nabla A|^2}{A^2} \Delta A \nonumber \\[5pt] =&-2 \int _{\Omega } \frac{|D^2A|^2}{A} -2\int _{\Omega } \frac{1}{A} |\nabla A|^2 \Delta A+2\int _{\Omega }\frac{|\nabla A|^4}{A^3}. \end{align}
We then use the identity
$\nabla |\nabla A|^2=2D^2A\cdot \nabla A$
and integrate by parts again to find that
\begin{align} \int _{\Omega }A|D^2 \ln A|^2 =&\int _{\Omega }\frac{|D^2 A|^2}{A}-2\int _{\Omega } \frac{\nabla A}{A^2} \cdot (D^2A \cdot \nabla A)+\int _{\Omega } \frac{|\nabla A|^4}{A^3} \nonumber \\[5pt] =&\int _{\Omega }\frac{|D^2 A|^2}{A}-\int _{\Omega } \frac{1}{A} \nabla A\cdot \nabla |\nabla A|^2+\int _{\Omega } \frac{|\nabla A|^4}{A^3} \nonumber \\[5pt] =&\int _{\Omega }\frac{|D^2 A|^2}{A}+\int _{\Omega } \frac{1}{A^2}|\nabla A|^2 \Delta A-\int _{\Omega } \frac{|\nabla A|^4}{A^3}. \end{align}
Thanks to (2.11) and (2.12), we find that
$I_{01}=-2 \int _{\Omega }A|D^2 \ln A|^2$
. Moreover, we have that for the same arbitrarily small
$\varepsilon \gt 0$
as in (2.10)
\begin{align*} I_{01} =&-(2-\varepsilon )\int _{\Omega } A|D^2 \ln A|^2 +\frac{\varepsilon }{2}I_{01} \\[5pt] =&-(2-\varepsilon )\int _{\Omega } A|D^2 \ln A|^2 +\frac{\varepsilon }{2}\Big (\int _{\Omega } \frac{1}{A^2} \nabla A \cdot \nabla |\nabla A|^2-2\int _{\Omega } \frac{|D^2A|^2}{A}-\int _{\Omega } \frac{|\nabla A|^2}{A^2} \Delta A \Big )\\[5pt] =&-(2-\varepsilon )\int _{\Omega } A|D^2 \ln A|^2 +\frac{\varepsilon }{2}\Big ( 2\int _\Omega \frac{|\nabla A|^4}{A^3}-2\int _\Omega \frac{|\nabla A|^2}{A^2}\Delta A - 2 \int _{\Omega } \frac{|D^2A|^2}{A} \Big )\\[5pt] =&-(2-\varepsilon )\int _{\Omega } A|D^2 \ln A|^2 -\varepsilon \int _{\Omega } \frac{|D^2A|^2}{A}-\varepsilon \int _{\Omega } \frac{|\nabla A|^2}{A^2} \Delta A+\varepsilon \int _{\Omega } \frac{|\nabla A|^4}{A^3}. \end{align*}
Moreover, in the light of the following inequality due to Young’s inequality
\begin{align*} -\varepsilon \int _{\Omega } \frac{|\nabla A|^2}{A^2} \Delta A \leq \varepsilon \int _{\Omega } \frac{|\nabla A|^4}{A^3}+ \frac{\varepsilon }{4} \int _{\Omega } \frac{|\Delta A|^2}{A} \leq \varepsilon \int _{\Omega } \frac{|\nabla A|^4}{A^3}+\frac{\varepsilon }{2} \int _{\Omega } \frac{|D^2A|^2}{A}, \end{align*}
we have that
\begin{equation} I_{01}\leq -(2-\varepsilon )\int _{\Omega } A|D^2 \ln A|^2 -\frac{\varepsilon }{2} \int _{\Omega } \frac{|D^2A|^2}{A} +2\varepsilon \int _{\Omega } \frac{|\nabla A|^4}{A^3}. \end{equation}
To estimate
$I_{02}$
, we apply the pointwise identity
$|\Delta \ln A|^2\leq 2|D^2\ln A|^2$
and use Young’s inequality
\begin{align} I_{02} \leq 2 \sqrt{2} \int _{\Omega } A\rho |D^2\ln A| \leq \int _{\Omega } A|D^2\ln A|^2+2\int _{\Omega }A \rho ^2. \end{align}
We then combine (2.10), (2.13) and (2.14) with (2.5) to obtain that
\begin{align} \frac{d}{dt} \int _{\Omega } \frac{|\nabla A|^2}{A} \leq & -(1-\varepsilon )\int _{\Omega } A |D^2 \ln A |^2+3\varepsilon \int _{\Omega } \frac{|\nabla A|^4}{A^3}-\int _{\Omega } \frac{|\nabla A|^2}{A} \nonumber \\[5pt] &-\bar{A}\int _{\Omega } \frac{|\nabla A|^2}{A^2}-\int _{\Omega } \frac{\rho }{A} |\nabla A|^2+2\int _{\Omega } A \rho ^2+C_{29}\nonumber \\[5pt] \leq & -(1-4\varepsilon )\int _{\Omega } \frac{|\nabla A|^4}{A^3}-\int _{\Omega } \frac{|\nabla A|^2}{A}-\bar{A}\int _{\Omega } \frac{|\nabla A|^2}{A^2} \nonumber \\[5pt] &-\int _{\Omega } \frac{\rho }{A} |\nabla A|^2+2 \int _{\Omega } A \rho ^2 +C_{29}\nonumber \\[5pt] \leq & -(1-4\varepsilon )\int _{\Omega } \frac{|\nabla A|^4}{A^3}-\int _{\Omega } \frac{|\nabla A|^2}{A} +2 \int _{\Omega } A \rho ^2+C_{29}, \end{align}
where the second inequality follows from the inequality
$\int _{\Omega } \frac{|\nabla A |^4}{A^3}\leq \int _{\Omega } A |D^2 \ln A |^2$
, and the last inequality skips anything redundant for our future estimates.
On the other hand, we first apply the
$\rho$
-equation and then use Young’s inequality with the same
$\varepsilon$
as (2.10) to find that for a positive constant
$C_\varepsilon$
we have
\begin{align} \frac{1}{2} \frac{d}{dt} \int _{\Omega } \rho ^2=&\int _{\Omega } \rho \rho _t =-\int _{\Omega } |\nabla \rho |^2+\int _{\Omega } \phi (\rho )\nabla \rho \cdot \nabla \ln A-\int _{\Omega } A\rho ^2+\bar{B} \int _{\Omega } \rho \nonumber \\[5pt] \leq &-\frac{1}{2}\int _{\Omega } |\nabla \rho |^2+\varepsilon \int _{\Omega } |\nabla \ln A|^4+C_{\varepsilon } \int _{\Omega } \rho ^{4\theta }-\int _{\Omega }A\rho ^2+C_{210}\nonumber \\[5pt] \leq & -\frac{1}{2}\int _{\Omega } |\nabla \rho |^2+\frac{\varepsilon }{\delta } \int _{\Omega } \frac{|\nabla A|^4}{A^3}+\varepsilon \int _{\Omega } \rho ^3-\int _{\Omega }A\rho ^2+C_{211}, \end{align}
where the last inequality is obtained by (1.3) and Lemma 2.1. By Gagliardo–Nirenberg interpolation inequality, there exists a positive constant
$C$
such that
\begin{align} \int _{\Omega } \rho ^3\leq C(\| \rho \|_{L^1}) \int _{\Omega } |\nabla \rho |^2+C (\| \rho \|_{L^1}). \end{align}
We are now ready to present the following uniform-in-time boundedness of
$\nabla A$
and
$\rho$
in their
$L^2$
norms.
Lemma 2.2.
There exists a positive constant
$C$
such that
\begin{equation} \Vert \nabla A(\cdot, t) \Vert _{L^2(\Omega )} +\Vert \rho (\cdot, t) \Vert _{L^2(\Omega )} \leq C\quad \quad \forall t\in (0,T_{\max }). \end{equation}
Proof. Substituting (2.17) into (2.16) gives us
\begin{align} \frac{1}{2} \frac{d}{dt} \int _{\Omega } \rho ^2 \leq -\frac{1}{4} \int _{\Omega } |\nabla \rho |^2+\frac{\varepsilon }{\delta } \int _{\Omega } \frac{|\nabla A|^4}{A^3}-\int _{\Omega } A\rho ^2+C_{212}. \end{align}
We use Gagliardo–Nirenberg interpolation inequality to estimate
$\int _{\Omega } \rho ^2$
\begin{align*} \Vert \rho \Vert _{L^2(\Omega )}^2 \leq & C\Vert \nabla \rho \Vert _{L^2(\Omega )} \Vert \rho \Vert _{L^1(\Omega )}+C(\Vert \rho \Vert _{L^1(\Omega )}) \\[5pt] \leq & \varepsilon \Vert \nabla \rho \Vert _{L^2(\Omega )}^2 +C_{\varepsilon } \Vert \rho \Vert _{L^1(\Omega )}^2+C(\Vert \rho \Vert _{L^1(\Omega )}) \\[5pt] \leq & \varepsilon \Vert \nabla \rho \Vert _{L^2(\Omega )}^2 +C_{213} \end{align*}
This, together with (2.15) and (2.19), gives us that
\begin{align} \frac{d}{dt}\Big (\int _{\Omega } \frac{|\nabla A|^2}{A}+ \int _{\Omega } \rho ^2\Big ) \leq & -\Big (\int _{\Omega } \frac{|\nabla A|^2}{A}+ \frac{1}{2}\int _{\Omega } |\nabla \rho |^2\Big )+C_{214} \nonumber \\[5pt] \leq & -\Big (\int _{\Omega } \frac{|\nabla A|^2}{A}+ \int _{\Omega } \rho ^2\Big )+C_{214} \end{align}
where
$\varepsilon$
is chosen small so that
$1-4\varepsilon \geq \frac{2\varepsilon }{\delta }$
. Then solving (2.20) easily implies the boundedness of
$\int _\Omega \frac{|\nabla A|^2}{A}$
and
$\int _\Omega \rho ^2$
.
To prove the boundedness of
$\int _\Omega |\nabla A|^2$
, we choose
$q=2$
and
$p=\frac{3}{2}$
in (2.3) to conclude from the Young’s inequality that
\begin{equation} \Vert A(\cdot, t) \Vert _{W^{1,2}(\Omega )} \leq C_{215}\Big (1+\sup _{s\in (0,t)} \Vert (A\rho )(\cdot, s)\Vert _{L^{\frac{3}{2}}(\Omega )}\Big ) \leq C_{216}\Bigg (1+\sup _{s\in (0,t)} \Big (\int _{\Omega } A^6 +\int _{\Omega } \rho ^2 \Big ) \Bigg ). \end{equation}
Due to the fact that
$A^{\frac{1}{2}} \in W^{1,2}$
, we readily have from the Sobolev embedding that
\begin{align*} \Vert A^{\frac{1}{2}} \Vert _{L^{12}(\Omega )}^{12} \leq C \Vert \nabla A^{\frac{1}{2}} \Vert _{L^2(\Omega )}^{10} \Vert A^{\frac{1}{2}} \Vert _{L^2{(\Omega )}}^{2}+C\Vert A^{\frac{1}{2}} \Vert _{L^2{(\Omega )}}^{12} \leq C, \end{align*}
We can iterate the process to find that
$\Vert A \Vert _{L^p}$
is bounded for any
$p\in (1,\infty )$
, which then leads us to (2.18) thanks to the GN inequality.
Remark 1. The boundedness of
$\Vert \nabla A \Vert _{L^2}$
is not enough and we need to prove that of
$\Vert \nabla A \Vert _{L^q}$
for
$q$
sufficiently large to show the boundedness of
$\rho$
in
$L^\infty$
. Indeed, we can show that
$\Vert \nabla A \Vert _{L^\infty }$
. To see this, in the light of (2.18), we can show that (2.3) holds for some
$p\gt 1$
, which then implies the boundedness of
$A$
in
$W^{1,p}$
for some
$p\gt 1$
, hence
$A\in L^\infty$
as claimed.
Corollary 2.1.
For any
$q\in (1,\infty )$
, there exists a constant
$C_q$
dependent on
$q$
such that
\begin{equation} \Vert \nabla A(\cdot, t) \Vert _{L^q(\Omega )} \leq C_q \quad \forall t\in (0,T_{\text{max}}); \end{equation}
moreover, there exists a positive constant
$C$
such that
\begin{equation} \Vert A(\cdot, t) \Vert _{L^\infty (\Omega )} \leq C \quad \forall t\in (0,T_{\text{max}}); \end{equation}
Proof. Let
$p=\frac{2}{2\varepsilon +1}$
in (2.3),
$\varepsilon \gt 0$
being arbitrary, then we have that for any
$q\in (1,\frac{1}{\varepsilon })$
\begin{align*} \Vert A(\cdot, t) \Vert _{W^{1,q}(\Omega )} &\leq C_{217}\Big (1+\sup _{s\in (0,t)} \Vert (A\rho )(\cdot, s)\Vert _{L^{\frac{2}{2\varepsilon +1}(\Omega )}}\Big ) \\[5pt] &\leq C_{217}\Big (1+\sup _{s\in (0,t)} \Vert A(\cdot, s)\Vert _{L^{\frac{1}{\varepsilon }}(\Omega )}+\sup _{s\in (0,t)} \Vert \rho (\cdot, s)\Vert _{L^2(\Omega )}\Big ) \\[5pt] &\leq C_q. \end{align*}
We infer from (2.22) the boundedness of
$A$
in
$W^{1,q}$
for any
$q$
, which eventually leads to the uniform boundedness of
$A$
.
3. Global well-posedness of the fully parabolic system
According to Proposition1, in order to prove Theorem1.1, it is sufficient to show that
$\Vert \rho (\cdot, t)\Vert _{L^\infty (\Omega )}$
is bounded for each
$t\in (0,T_{\max })$
and therefore
$T_{\max }=\infty$
and the solution is global. In fact, we will show that
$\Vert \rho (\cdot, t)\Vert _{L^\infty (\Omega )}$
is uniformly bounded in
$t\in (0,\infty )$
. To this end, it suffices to prove that
$\Vert \rho (\cdot, t)\Vert _{L^m(\Omega )}$
is bounded for some
$m$
large according to (2.3), now that we already have the boundedness of
$\rho$
in
$L^2$
. Choosing
$m\gt 2$
, one obtains from (2.3) the boundedness of
$\Vert A \Vert _{W^{1,q}}$
with
$q$
being sufficient large. One can then find that the boundedness of
$\Vert \rho \Vert _{L^{\infty }}$
follows from the standard Moser–Alikakos iteration in [Reference Alikakos1].
3.1. A priori estimates
For any
$m\gt 1$
, we multiply the
$\rho$
-equation in (1.1) by
$\rho ^{m-1}$
and then integrate it over
$\Omega$
by parts to have that
\begin{align} &\frac{1}{m}\frac{d}{dt}\int _{\Omega } \rho ^m\nonumber \\[5pt] =&\int _{\Omega } \rho ^{m-1} \nabla \cdot \nabla \rho -\int _{\Omega } \rho ^{m-1} \nabla \cdot ( \phi (\rho )\nabla \ln A)-\int _{\Omega } \rho ^m A +\bar{B}\int _{\Omega } \rho ^{m-1} \nonumber \\[5pt] =&-(m-1)\int _{\Omega }\rho ^{m-2}|\nabla \rho |^2+(m-1)\int _{\Omega } \phi (\rho )\rho ^{m-2}\nabla \rho \cdot \nabla \ln A -\int _{\Omega } \rho ^m A+\bar{B}\int _{\Omega } \rho ^{m-1}\nonumber \\[5pt] =&-\frac{4(m-1)}{m^2}\int _{\Omega }|\nabla \rho ^{\frac{m}{2}}|^2+(m-1)\int _{\Omega } \phi (\rho )\rho ^{m-2}\nabla \rho \cdot \nabla \ln A -\int _{\Omega } \rho ^m A+\bar{B}\int _{\Omega } \rho ^{m-1}, \end{align}
where the second identity holds as
$\rho ^{m-2}|\nabla \rho |^2=\frac{4}{m^2} |\nabla \rho ^{\frac{m}{2}}|^2$
. Moreover, Young’s inequality implies that
\begin{align} & (m-1)\int _{\Omega } \phi (\rho )\rho ^{m-2}\nabla \rho \cdot \nabla (\ln A) \nonumber \\[5pt] \leq &\frac{m-1}{2}\int _{\Omega } \rho ^{m-2}|\nabla \rho |^2+ \frac{m-1}{2} \int _{\Omega } \rho ^{m-2}\phi ^2(\rho )\Big |\frac{\nabla A}{A}\Big |^2 \nonumber \\[5pt] \leq &\frac{2(m-1)}{m^2}\int _{\Omega }|\nabla \rho ^{\frac{m}{2}}|^2+\frac{M^2(m-1)}{2} \int _{\Omega } \rho ^{m+2\theta -2} \Big |\frac{\nabla A}{A}\Big |^2, \end{align}
and
\begin{equation} -\int _{\Omega } \rho ^m A+\bar{B}\int _{\Omega } \rho ^{m-1} \leq -\delta \int _{\Omega } \rho ^m+\bar{B}\int _{\Omega } \rho ^{m-1} \leq -\frac{\delta }{2} \int _{\Omega } \rho ^m+C_{31}, \end{equation}
where
$C_{31}$
is a positive constant dependent on
$m$
. Thanks to (3.2) and (3.3), we have from (3.1) that
\begin{equation} \frac{1}{m}\frac{d}{dt}\int _{\Omega } \rho ^m +\frac{\delta }{2}\int _{\Omega } \rho ^m +\frac{2(m-1)}{m^2}\int _{\Omega }|\nabla \rho ^{\frac{m}{2}}|^2 \leq \frac{M^2 (m-1)}{2} \int _{\Omega } \rho ^{m+2\theta -2}\Big |\frac{\nabla A}{A}\Big |^2+C_{31}. \end{equation}
Since
$\rho$
is uniformly bounded in
$L^2$
, we infer from (3.4) that it is also bounded in
$L^4$
. This allows us to iterate the a priori estimates concerning the boundedness of
$\|\nabla A\|_{L^q}$
for arbitrary
$q$
. We will present this iteration in the final section.
3.2. Global existence and uniform boundedness
Proof of Theorem
1.1. First of all, we have the uniform boundedness of
$\Vert A(\cdot, t)\Vert _{L^{\infty }(\Omega )}$
from (2.23). The verification of the boundedness of
$\Vert \rho \Vert _{L^{\infty }}$
follows from the standard
$L^p-$
iteration sketches as follows. By (3.4), we have from the uniform lower bound of
$A$
given in (2.1) that
\begin{align} &\frac{1}{m}\frac{d}{dt} \int _{\Omega } \rho ^m+\frac{2(m-1)}{m^2}\int _{\Omega }|\nabla \rho ^{\frac{m}{2}}|^2 \nonumber \\[5pt] \leq &\frac{M^2 (m-1)}{2\delta ^2} \int _{\Omega } \rho ^{m+2\theta -2}|\nabla A|^2+C_{31} \nonumber \\[5pt] \leq &\frac{M^2 (m-1)}{2} \int _{\Omega } \rho ^{(m+2\theta -2)\cdot \frac{m}{m+2\theta -2}}+C_{\delta }\int _{\Omega } |\nabla A|^{2\cdot \frac{m}{2-2\theta }}+C_{31} \nonumber \\[5pt] \leq &\frac{M^2 (m-1)}{2} \int _{\Omega } \rho ^m+C_{32}, \end{align}
where apply (2.22) to derive the last inequality. Then we apply Gagliardo–Ladyzhenskaya–Nirenberg inequality (cf. Corollary 1 in [Reference Chertock, Kurganov, Wang and Wu12] with
$d=2$
) and the Young’s inequality to estimate (3.5) such that
\begin{align} &\frac{M^2 (m-1)}{m} \int _{\Omega } \rho ^2 \nonumber \\[5pt] =&\frac{2(m-1)}{m^2\varepsilon } \int _{\Omega } \rho ^m-\frac{M^2 (m-1)}{2} \int _{\Omega } \rho ^m \nonumber \\[5pt] \leq & \frac{2(m-1)}{m^2} \int _{\Omega }|\nabla \rho ^{\frac{m}{2}}|^2+\frac{2(m-1)\kappa (1+\varepsilon ^{-1})}{m^2\varepsilon }\Big (\int _{\Omega }\rho ^{\frac{m}{2}}\Big )^2-\frac{M^2 (m-1)}{2} \int _{\Omega } \rho ^m, \end{align}
where we choose
$\varepsilon \;:\!=\;\frac{2}{M^2m^2}$
and
$\frac{1+\varepsilon ^{-1}}{\varepsilon }=(1+\frac{M^2m^2}{2})\frac{M^2m^2}{2}$
. In the light of (3.6), (3.5) becomes
\begin{align} \frac{d}{dt} \int _{\Omega } \rho ^m &\leq -\frac{M^2m(m-1)}{2} \int _{\Omega } \rho ^m+M^2m(m-1)\kappa (1+\frac{M^2m^2}{2})\Big (\int _{\Omega }\rho ^{\frac{m}{2}}\Big )^2 \end{align}
Let us denote
$\lambda \;:\!=\;\frac{M^2m(m-1)}{2}$
, then for each
$T\in (0,\infty )$
we can solve the differential inequality (3.7) and obtain for all
$t\in (0,T)$
\begin{align} \int _{\Omega } \rho ^m&\leq e^{-\lambda t} \int _{\Omega } \rho _0^m+M^2m(m-1)\kappa (1+\frac{M^2m^2}{2}) \int _0^t e^{-\lambda (t-s)} (\int _{\Omega }\rho ^{\frac{m}{2}})^2 ds \nonumber \\[5pt] &\leq \int _{\Omega } \rho _0^m+2\lambda \kappa (1+\frac{M^2m^2}{2}) \sup \limits _{t\in (0,T)} \Big (\int _{\Omega }\rho ^{\frac{m}{2}}\Big )^2\quad \forall m\geq 2. \end{align}
Finally, the
$L^{\infty }$
boundedness of
$\rho$
follows from the Moser–Alikakos iteration following (3.8). This concludes the proof of Theorem1.1.
Acknowledgements
QW is partially supported by the Guanghua Talent Program at SWUFE. Both authors thank all the three referees for their helpful comments and suggestions.
Competing interest
The authors have no competing financial and/or non-financial interests in relation to this work.
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