1 Introduction
Suppose that
$\Omega $
is a bounded domain of
$\mathbb {R}^{N}$
with Lipschitz boundary
$\partial \Omega $
and T is a positive number. Denote
$\Omega _T=(0,T] \times \Omega $
,
$\Sigma =(0,T]\times \partial \Omega $
. This manuscript establishes the existence and uniqueness of both renormalized and entropy solutions for a general parabolic equation with merely integrable data in time-dependent spaces. Namely, we consider the parabolic problem
$$ \begin{align} \begin{cases} \partial_{t} u- \text{div} {\mathcal{A}}(t,x,\nabla u)= f(t,x) &\mbox{in } \Omega_{T},\\ u(t,x) = 0 &\mbox{on } \Sigma ,\\ u(0, \cdot)=u_{0}(\cdot) &\mbox{in } \Omega, \end{cases} \end{align} $$
where
$f\in L^1(\Omega _{T})$
and
$u_{0}\in L^{1}(\Omega )$
. We assume the following hypotheses.
-
(A1)
${\mathcal {A}}(t,x,\xi ):(0, T)\times \Omega \times \mathbb {R}^{N}\rightarrow \mathbb {R}^{N}$
is a Carathéodory function (that is, measurable in t and x for fixed
$\xi $
and continuous with respect to
$\xi $
for fixed
$(t,x)$
). -
(A2)
${\mathcal {A}}(t,x,0)=0$
for almost every
$(t,x)\in \Omega _T$
and there exist an N-function M (see Definition 2.1) and constants
$c_1\in (0,1)$
such that, for all
$\xi \in \mathbb {R}^N$
, where
$$ \begin{align*} c_1 ( M(t,x,|\xi|)+M^{*}(t,x,|{\mathcal{A}}(t,x,\xi)|)) \leq{\mathcal{A}}(t,x,\xi)\cdot\xi, \end{align*} $$
$M^*$
is the complementary function of M (see Definition 2.2).
-
(A3)
$ [ {\mathcal {A}}(t,x,\xi )-{\mathcal {A}}(t,x,\eta )]\cdot (\xi -\eta )> 0$
for all
$\xi ,\eta $
in
$\mathbb {R}^{N}$
,
$\xi \neq \eta $
and almost every
$(t,x)\in \Omega _T$
.
Motivated by fluids of nonstandard rheology, we focus on the general form of growth conditions for the operator’s leading term, making Musielak-Orlicz spaces a suitable function space for the considered problem. We do not assume any growth condition of the doubling type on the function M. Instead, we impose a condition that balances the behavior of M with respect to its variable, ensuring that smooth functions are modularly dense in the related Sobolev-type space.
Musielak–Orlicz spaces, which generalize Orlicz, variable exponent and double-phase spaces, have received significant attention since the seminal contributions of Marcellini [Reference Marcellini39, Reference Marcellini40]. There is a large amount of literature on partial differential equation (PDE) problems in the framework of Musielak–Orlicz spaces [Reference Colombo and Mingione19, Reference Diening, Harjulehto, Hästö and Ružička20, Reference Harjulehto and Hästö33]. We refer to [Reference Fan26, Reference Liu and Zhao37] for the existence of weak solutions in isotropic, separable and reflexive Musielak–Orlicz–Sobolev spaces. Donaldson was the first to study nonlinear elliptic boundary value problems in nonreflexive Musielak–Orlicz Sobolev spaces (see [Reference Donaldson22]). Subsequent advancements were made by Gossez [Reference Gossez27, Reference Gossez28]. The groundwork for studying PDE problems in anisotropic Musielak–Orlicz spaces was established in the works of [Reference Cianchi17, Reference Cianchi18, Reference Klimov35]. For more recent results concerning PDEs in Musielak–Orlicz spaces, the reader may refer to the monograph [Reference Chlebicka, Gwiazda, Świerczewska-Gwiazda and Wróblewska-Kamińska13] and the review paper [Reference Chlebicka12].
As we consider problems with data of low integrability, it is reasonable to work with renormalized solutions and entropy solutions, as they require less regularity in the data than standard weak solutions. The concept of renormalized solutions was initially introduced by DiPerna and Lions in [Reference DiPerna and Lions21] for analyzing the Boltzmann equation. At the same time, Bénilan et al. proposed the notion of entropy solutions in [Reference Bénilan, Boccardo, Gallouët, Gariepy, Pierre and Vázquez8] for the study of nonlinear elliptic problems. The existence of renormalized solutions within the context of variable exponents has been explored in [Reference Bendahmane, Wittbold and Zimmermann7, Reference Wittbold and Zimmermann47]. We refer to [Reference Nassar, Moussa and Rhoudaf41, Reference Redwane45] for this issue in the nonreflexive Orlicz–Sobolev space. Also, there have been a large number of papers devoted to the study of renormalized solutions for PDEs in Musielak–Orlicz space (see [Reference Chlebicka, Gwiazda and Zatorska-Goldstein14, Reference Chlebicka, Gwiazda and Zatorska-Goldstein15, Reference Gwiazda, Skrzypczak and Zatorska-Goldstein29–Reference Gwiazda, Wittbold, Wróblewska-Kamińska and Zimmermann32]). Entropy solutions for the
$p(x)$
-Laplace equation were investigated in [Reference Sanchón and Urbano46], and further research on entropy solutions exhibiting Orlicz growth is available in [Reference Zhang and Zhou49, Reference Zhang and Zhou51]. For studies on the existence of entropy solutions in Musielak–Orlicz spaces, see [Reference Benkirane, EL Haji and EL Moumni9, Reference Elarabi, Rhoudaf and Sabiki24]. Moreover, Droniou and Prignet established the equivalence between entropy solutions and renormalized solutions for parabolic problems with polynomial growth in [Reference Droniou and Prignet23], which was subsequently studied for variable-exponent and Orlicz growth in [Reference Zhang and Zhou48, Reference Zhang and Zhou50, Reference Zhang and Zhou52], and for
$(p(x),q(x))$
growth in parabolic equations in [Reference Alaa, Bendahmane and Charkaoui6]. In addition, the first author et al. [Reference Li, Yao and Zhou36] demonstrated this equivalence for elliptic equations with general growth in Musielak–Orlicz spaces.
Recently, Chlebicka et al. [Reference Chlebicka, Gwiazda and Zatorska-Goldstein16] showed the existence and uniqueness of renormalized solutions of the equation
$$ \begin{align*} \begin{cases} \partial_{t} u- \text{div} \mathcal{A}(t,x,\nabla u)= f(t,x) &\mbox{in } \Omega_{T},\\ u(t,x) = 0 &\mbox{on } \Sigma,\\ u(0,x)=u_{0}(x) &\mbox{in } \Omega, \end{cases} \end{align*} $$
where
$f\in L^1(\Omega _T)$
,
$u_0\in L^1(\Omega )$
and
$\mathcal {A}$
was assumed to be controlled by an N-function. The authors therein employed a delicate time-approximation method to achieve smoothness in the time direction. Unlike the proof in [Reference Chlebicka, Gwiazda and Zatorska-Goldstein16], we do not require separate approximations for time and space variables. We assume that the regularity of the modular function is strong enough to ensure the density of smooth functions in the related Sobolev-type space. In fact, this density can be guaranteed by the balance condition (B) in the isotropic Musielak–Orlicz space (see Lemma 2.6). Utilizing this density result, we provide direct proof of the existence and uniqueness of renormalized solutions to (1-1), demonstrate that the renormalized solution is also an entropy solution, and establish the uniqueness of the entropy solution, thereby showing the equivalence between entropy and renormalized solutions. It is worth noting that when
$M(t,x,\xi )=|\xi |^{p(t,x)}$
, Bulíček and Woźnicki [Reference Bulíček and Woźnicki11] studied the existence and uniqueness of entropy solutions to (1-1). For
$M(t, x, \xi )=|\xi |^p$
, the existence and uniqueness of renormalized solutions and entropy solutions to (1-1) were established in [Reference Blanchard and Murat10, Reference Prignet44]. Renormalized solutions of p-growth nonlinear parabolic equations with general measure data were investigated in [Reference Petitta42, Reference Petitta, Ponce and Porretta43]. We remark here that the method we have developed can be applied to measure data problems with slight modifications. The results of this work extend the results presented in [Reference Blanchard and Murat10, Reference Bulíček and Woźnicki11, Reference Chlebicka, Gwiazda and Zatorska-Goldstein16, Reference Li, Yao and Zhou36, Reference Prignet44, Reference Zhang and Zhou50].
We rely on the density of smooth functions in a relevant function space to study problem (1-1), namely,
$$ \begin{align*} \textbf{W}(\Omega_T):=\{u\in W^{1,x}_0L_M(\Omega_T)\cap L^2(\Omega_T),\partial_t u\in W^{-1,x}L_{M^{*}}(\Omega_T)+L^2(\Omega_T)\}, \end{align*} $$
where
$W^{1,x}_0 L_M(\Omega _T)$
and
$W^{-1,x}L_{M^{*}}(\Omega _T)$
are defined in Section 2 (see Definition 2.4 and Lemma 2.6).
To ensure the density, one may assume the regularity of M. Note that the smooth functions are dense in
$\textbf {W}(\Omega _T)$
in the modular topology if the following balance condition holds (see Lemma 2.6).
Balance condition
$\mathsf {(B)}$
. There exists a function
$\varrho :[0,\infty )\times \mathbb {R}^+ \to \mathbb {R}^+$
that is nondecreasing with respect to each of the variables such that, for
$(t,x)\in \Omega _T$
and
$(\tau ,y)\in \Omega _{T}$
,
$$ \begin{align*} M(t,x,s)\leq \varrho(|t-\tau|+c|x-y|,s) M(\tau,y,s)\quad \text{with } \limsup_{\epsilon\to 0^+}\varrho(\epsilon,\epsilon^{-N})<\infty. \end{align*} $$
We point out that the balance condition is only used to ensure the density of smooth functions in our proof. In addition, throughout the paper, we assume that the N-functions
$M(t,x,s)$
satisfy the following
$\mathsf {(Y)}$
-condition.
$\mathsf {(Y)}$
-condition. An N-function M is said to satisfy the
$\mathsf {(Y)}$
-condition on a segment
$[a,b]$
of the real line
$\mathbb {R}$
, if either:
-
(Y1): there exist
$q_0\in \mathbb {R}^+$
and
$1\leq i\leq N$
such that
$x_i\in [a,b]\to M(t,x,s)$
is increasing when
$x_i\geq q_0$
and decreasing when
$x_i\leq q_0$
, or vice versa;
or
-
(Y2): there exists
$1\leq i\leq N$
such that, for all
$ s\geq 0$
, the partial function
$x_i \in [a,b]\to M(t,x,s)$
is monotone on
$[a,b]$
.
Here
$x_i$
stands for the
$i\,\mathrm {th}$
component of
$x\in \Omega $
.
In Musielak–Orlicz spaces, the norm Poincaré inequality is no longer true in general. We remark that the
$\mathsf {(Y)}$
-condition is only used as a sufficient condition to obtain the norm Poincaré inequality (see Lemma 2.5 below), which is crucial in the proof of Lemma 2.8. This condition covers the assumption given by Maeda [Reference Maeda38] to provide the Poincaré integral form for variable exponents. See [Reference Ahmida and Youssfi3] for more information about this condition.
The structural conditions imposed on the Carathéodory vector field in our manuscript are not weaker, and in some aspects are even slightly stronger, than those considered in [Reference Chlebicka, Gwiazda and Zatorska-Goldstein16]. Our main goal, however, is not to relax these assumptions, but rather to extend the existing theory to a broader functional framework that covers time-dependent Musielak–Orlicz growth conditions
$M(t,x,\xi )$
.
In [Reference Chlebicka, Gwiazda and Zatorska-Goldstein16], the authors proved only the existence of a renormalized solution. To use the truncations of the solution as test functions, one needs to regularize them both in time and in space. However, when the modular function depends on t, the classical Landes-type time regularization cannot be applied, since it is no longer a self-mapping in
$L_M$
. Although [Reference Chlebicka, Gwiazda and Zatorska-Goldstein16, Theorem 3.1] introduces a new approximation technique, it does not allow the class of admissible test functions to include the truncations of the solution itself. Consequently, the almost everywhere convergence of the gradient, which is crucial for establishing the existence of entropy solutions, cannot be obtained. By contrast, our approach provides a framework that allows us to prove the existence and uniqueness of both renormalized and entropy solutions, and to establish their equivalence for the general class of parabolic equations with time-dependent modular functions
$M(t,x,\xi )$
. To the best of our knowledge, the existence of entropy solutions in this setting was previously known only for the special case
$M(t,x,\xi )=|\xi |^{p(t,x)}$
studied in [Reference Bulíček and Woźnicki11].
Before we proceed to define the renormalized and entropy solution to (1-1), we first introduce the truncation operator
$T_{k}(r)$
as
$$ \begin{align*} T_{k}(r)= \begin{cases} r \quad &\text{if } |r|\leq k,\\ k\dfrac{r}{|r|} \quad &\text{if } |r|> k. \end{cases} \end{align*} $$
Its primitive
$\Theta _{k}:\mathbb {R}\rightarrow \mathbb {R}^{+}$
defined by
$$ \begin{align*} \Theta_{k}(r):= \int^{r}_{0}T_{k}(s)\,ds = \begin{cases} \dfrac{r^{2}}{2} & \mbox{if } |r|\leq k,\\ k|r|-\dfrac{k^{2}}{2} & \mbox{if } |r|> k. \end{cases} \end{align*} $$
It is obvious that
$\Theta _{k}(r)\geq 0$
and
$\Theta _{k}(r)\le k|r|$
.
The definitions of entropy and renormalized solutions for Problem (1-1) are as follows.
Definition 1.1. A function
$u \in C([0,T]; L^{1}(\Omega )) $
is an entropy solution to Problem (1-1) if u satisfies the following two conditions.
-
(E1) u is a measurable function satisfying
$$ \begin{align*}T_{k}(u)\in W_0^{1,x}L_M(\Omega_T)\quad \text{for each } k>0 \text{ and } {\mathcal{A}}(t,x,\nabla T_{k}(u))\in L_{M^*}(\Omega_{T}).\end{align*} $$
-
(E2) For every
$k>0$
and every
$\phi \in C^{1}(\overline {\Omega }_{T})$
with
$\phi |_{\Sigma }=0$
, the inequality (1-2)holds.
$$ \begin{align} &\int_{\Omega}\Theta_{k}(u-\phi)(T)\,dx -\int_{\Omega}\Theta_{k}(u_{0}-\phi(0))\,dx +\int^{T}_{0}\langle\phi_{t},T_{k}(u-\phi)\rangle \,dt \nonumber \\ & \quad+\int^{T}_{0}\int_{\Omega}{\mathcal{A}}(t,x,\nabla u)\cdot\nabla T_{k}(u-\phi)\,dx\,dt= \int^{T}_{0}\int_{\Omega}fT_{k}(u-\phi)\,dx\,dt \end{align} $$
Definition 1.2. A function
$u\in C([0,T];L^1(\Omega ))$
is a renormalized solution to problem (1-1) if u satisfies the following two conditions.
-
(R1) u is a measurable function satisfying
and
$$ \begin{align*}T_{k}(u)\in W_0^{1,x}L_M(\Omega_T)\quad \text{for each } k>0, \quad {\mathcal{A}}(t,x,\nabla T_{k}(u))\in L_{M^*}(\Omega_{T}) \end{align*} $$
$$ \begin{align*} \int_{\{l<|u|<l+1\}}{\mathcal{A}}(t,x,\nabla u)\cdot \nabla u\,dx\,dt \rightarrow 0 \quad \mbox{as } l \rightarrow +\infty. \end{align*} $$
-
(R2) For every
$ \phi \in C^{1}(\overline {\Omega }_{T})$
with
$\phi (\cdot ,T)=0$
and
$S\in W^{2,\infty }(\mathbb {R})$
with
$S'$
having compact support,
$$ \begin{align*} &-\int_{\Omega}S(u_0)\phi(x,0)\,dx-\int^{T}_{0}\int_{\Omega}S(u)\partial_{t}\phi\,dx\,dt\\ &\quad +\int^{T}_{0}\int_{\Omega}{\mathcal{A}}(t,x,\nabla u)\cdot \nabla(S'(u)\phi)\,dx\,dt =\int^{T}_{0}\int_{\Omega}fS'(u)\phi\,dx\,dt. \end{align*} $$
Now, we state the main results of this work.
Theorem 1.3. Assume that
$ f\in L^{1}(\Omega _{T})$
,
$u_{0}\in L^{1}(\Omega )$
,
${\mathcal {A}}$
satisfies the conditions
$(A1)$
–
$(A3)$
, N-function M is regular enough so that the set of smooth functions is dense in
$\textbf {W}(\Omega _T)$
in the modular topology, and M satisfies
$\mathsf {(Y)}$
-condition. Then there exists a unique renormalized solution to Problem (1-1).
Theorem 1.4. Assume that
$ f\in L^{1}(\Omega _{T})$
,
$u_{0}\in L^{1}(\Omega )$
,
${\mathcal {A}}$
satisfies the conditions
$(A1)$
–
$(A3)$
, N-function M is regular enough so that the set of smooth functions is dense in
$\textbf {W}(\Omega _T)$
in the modular topology, and M satisfies
$\mathsf {(Y)}$
-condition. Then the renormalized solution for Problem (1-1) is also an entropy solution to Problem (1-1), and the entropy solution is unique.
Remark 1.5. The entropy solution obtained in Theorem 1.4 is equivalent to the renormalized solution of (1-1).
Example 1.6. We provide examples of time- and space-dependent Musielak-type spaces that are admissible in our investigation.
-
(1) If
$M(t,x,s)=|s|^p$
with
$1<p<\infty $
, then the space
$W^{1,x}L_M(\Omega _{T})$
is the classical Bochner space
$L^p(0,T;W^{1,p}(\Omega ))$
defined as
$$ \begin{align*} L^{p}(0,T;W^{1,p}(\Omega)):=\bigg\{u:(0,T)\to W^{1,p}(\Omega):\int^{T}_0\|u(t)\|^p_{W^{1,p}(\Omega)}\,dt<+\infty\bigg\}. \end{align*} $$
-
(2) If
$M(t,x,s)=M(s)$
, then we obtain the Orlicz–Sobolev space. -
(3) If
$M(t,x,s)=s^{p(t,x)}$
,
$1<p^-\leq p(t,x)\leq p^+<+\infty $
, where
$p(t,x)$
is log-Hölder continuous and
$x_i\to p(t,x)$
is monotone on a compact subset of the real line
$\mathbb {R}$
, then we obtain the variable Sobolev space. Here
$x_i$
stands for the
$i\,\mathrm {th}$
component of
$x\in \Omega $
. -
(4) If
$M(t,x,s)=s^p+a(t,x)s^q$
, with
$0\leq a\in C^{0,\alpha }$
,
$\alpha \in (0,1]$
,
$1<p\leq q<+\infty $
,
${q}/{p}\leq 1+({\alpha }/{n})$
, and
$x_i\to a(t,x)$
is monotone on a compact subset of the real line
$\mathbb {R}$
, then we obtain the double-phase spaces. Here
$x_i$
stands for the
$i\,\mathrm {th}$
component of
$x\in \Omega $
.
We organize this paper as follows. In Section 2, we state some basic results that will be used later. We prove the main results in Section 3. In the following statement, C stands for a constant, which may vary even within the same inequality.
2 Functional setting and main tools
In this section, we give some functional settings that will be used below. For a more complete discussion on this subject, we refer the reader to [Reference Ahmida, Chlebicka, Gwiazda and Youssfi1, Reference Chlebicka, Gwiazda, Świerczewska-Gwiazda and Wróblewska-Kamińska13]. We begin with the definition of N-functions.
Definition 2.1. A function
$M(\cdot ,s):\Omega _T\times \mathbb {R}^+ \to \mathbb {R}^+$
is called an N-function if
$M(\cdot ,s)$
is a measurable function for every
$s\geq 0$
,
$M(t,x,\xi )$
is continuous with respect to its last variable, and
$M(t,x,\cdot )$
is a convex function for almost every
$(t,x)\in \Omega _T$
with
$M(t,x,0)=0, M(t,x,s)\to +\infty $
as
$s\to +\infty $
and
$$ \begin{align*} \lim\limits_{s\to 0}\frac{M(t,x,s)}{s}=0 \quad \text{and}\quad \lim\limits_{s\to +\infty}\frac{M(t,x,s)}{s}=+\infty. \end{align*} $$
Definition 2.2. The complementary function
$M^*$
to an N-function M in the sense of Young is defined by
$$ \begin{align*} M^*(t,x, \xi_1):=\sup_{ \xi_2\geq 0} [\xi_1\xi_2 -M(t,x,\xi_2)] \end{align*} $$
for any
$\xi _1 \geq 0 $
and almost every
$(t,x)\in \Omega _T$
.
For an N-function, we define the general Musielak–Orlicz class
$\mathcal {L}_{M}(\Omega _T)$
as the set of all measurable functions
$u(t,x):\Omega _T\to \mathbb {R}$
such that
$$ \begin{align*} \int^{T}_{0}\int_{\Omega}M(t,x,|u(t,x)|)\,dx\,dt < \infty. \end{align*} $$
The Musielak–Orlicz space
$L_{M}(\Omega _T)$
(respectively,
$E_M(\Omega _{T})$
) is defined as the set of all measurable functions
$u:\Omega _{T}\to \mathbb {R}$
such that
$$ \begin{align*} \int^T_0\int_{\Omega}M\bigg(t,x,\frac{|u(t,x)|}{\lambda}\bigg)\,dx\,dt<+\infty \end{align*} $$
for some
$\lambda>0$
(respectively, for all
$\lambda>0$
) equipped with Luxemburg norm
$$ \begin{align*} \|u\|_{L_{M}(\Omega_T)}=\inf \bigg\{\lambda>0:\int^{T}_{0}\int_{\Omega}M\bigg(t,x,\frac{|u(t,x)|}{\lambda}\bigg)\,dx\,dt\leq 1\bigg\}. \end{align*} $$
Then
$L_M(\Omega _{T})$
is a Banach space and
$E_M(\Omega _T)$
is its closed subset.
An N-function M is called locally integrable on
$\Omega _T$
if, for any constant number
$c>0$
and for any compact
$\Omega _T'$
of
$\Omega _T$
,
$$ \begin{align*}\int_{\Omega_T'}M(t,x,c)\,dx\,dt<+\infty.\end{align*} $$
We remark here that if an N-function M satisfies the balance condition (B), then the function is naturally locally integrable (see [Reference Ahmida, Fiorenza and Youssfi2, Reference Ahmida and Youssfi4]). It is shown in [Reference Ahmida and Youssfi5, Lemma 2.1] that the continuous embedding
$L_M(\Omega _T)\hookrightarrow L^1(\Omega _T)$
holds if either
$M^*$
is locally integrable or M satisfies
$ \operatorname *{\mbox {ess inf}}_{(t,x)\in \Omega _T} M(t,x,1)\geq c>0$
.
Definition 2.3. Suppose that the complementary N-function
$M^*$
of M is locally integrable on
$\Omega _T$
. We define
$$ \begin{align*} W^{1,x}L_{M}(\Omega_T):=\{u:\Omega_T\to \mathbb{R}:u\in L_{M}(\Omega_T),|\nabla u|\in L_{M}(\Omega_T)\} \end{align*} $$
and
$$ \begin{align*} W^{1,x}E_{M}(\Omega_T):=\{u:\Omega_T\to \mathbb{R}:u\in E_{M}(\Omega_T),|\nabla u|\in E_{M}(\Omega_T)\}. \end{align*} $$
We denote by
$\nabla u$
the vector gradient with respect to the space variable. These spaces are normed by
$\|u\|_{W^{1,x}L_{M}(\Omega _T)}:=\|u\|_{L_M(\Omega _T)}+\|\nabla u\|_{L_M(\Omega _T)}$
. Then
$W^{1,x}L_{M}(\Omega _{T})$
is a Banach space.
Let X and Y be subsets of
$L^1(\Omega _T)$
not necessarily related by duality. We say that
$f_n \to f$
for
$\sigma (X,Y)$
if
$$ \begin{align*} \int^{T}_0\int_\Omega f_n g \, dx\,dt \xrightarrow{n\to +\infty}\int^{T}_0 \int_\Omega fg \, dx\,dt \end{align*} $$
for all
$g \in Y$
. If
$X=L_M(\Omega _T)$
and
$Y=E_{M^\ast }(\Omega _T)$
, we recover the weak-
$\ast $
convergence and can also denote
$f_n \stackrel {\ast }\rightharpoonup f$
.
We define
$W^1L_M(\Omega )$
(respectively,
$W^1E_M(\Omega )$
) as the set of all measurable functions
$u:\Omega \to \mathbb {R}$
, such that, for all
$|\alpha |\leq 1$
, the function
$|D^{\alpha } u|$
belongs to
$L_{M}(\Omega )$
(respectively,
$E_{M}(\Omega )$
): that is,
$$ \begin{align*} \int_{\Omega}M\bigg( x,\frac{|D^\alpha u|}{\lambda}\bigg)\,dx< +\infty \quad \text{for some } \lambda>0 \ (\text{respectively, for all } \lambda>0). \end{align*} $$
Note that if an N-function M is locally integrable, then the set of
$C_0^{\infty }(\Omega )$
-functions is contained in
$W^{1}E_{M}(\Omega )$
. Therefore, the norm closure of the
$C_0^{\infty }(\Omega )$
-functions in
$W^{1}E_{M}(\Omega )$
, denoted by
$W_{0}^{1}E_{M}(\Omega )$
, is well defined. Moreover, if the pair of complementary N-functions
$(M, M^*) $
are both locally integrable, then the space
$ W_{0}^{1}L_{M}(\Omega )$
, defined as the closure of the
$C_0^{\infty }(\Omega )$
-functions with respect to the weak-* topology σ(L
M
, E
M
*
), is also well defined.
Definition 2.4. Suppose that the complementary N-function
$M^*$
of M is locally integrable on
$\Omega _T$
. We define
$$ \begin{align*} W_0^{1,x}L_{M}(\Omega_T):=\{u:(0,T)\to W_0^{1}L_M(\Omega):u\in L_{M}(\Omega_T),|\nabla u|\in L_{M}(\Omega_T)\} \end{align*} $$
and
$$ \begin{align*} W_0^{1,x}E_{M}(\Omega_T):=\{u:(0,T)\to W_0^{1}E_M(\Omega):u\in E_{M}(\Omega_T),|\nabla u|\in E_{M}(\Omega_T)\}. \end{align*} $$
These spaces are equipped with the norm
$\|u\|_{W^{1,x}L_{M}(\Omega _T)}$
.
Lemma 2.5 [Reference Ahmida and Youssfi3, Theorem 1.1].
Assume that the pair of complementary N-functions M and
$M^*$
both satisfy Balance Condition
$(B)$
, and that M satisfies
$\mathsf {(Y)}$
-condition. Then there exists a constant C depending only on
$\Omega _{T}$
such that, for every
$u\in W^{1,x}_{0}L_M(\Omega _{T})$
,
$$ \begin{align*}\|u\|_{L_M(\Omega_{T})}\leq C\|\nabla u\|_{L_M(\Omega_{T})}.\end{align*} $$
From Lemma 2.5, the two norms
$\|\cdot \|_{L_M(\Omega _T)}+\|\nabla \cdot \|_{L_M(\Omega _T)}$
and
$\|\nabla \cdot \|_{L_M(\Omega _T)}$
are equivalent on
$W^{1,x}_0 L_{M}(\Omega _{T})$
. In addition, it follows from Definition 2.2 that
$$ \begin{align*} \int^{T}_{0}\int_{\Omega}|u_1 u_2|\,dx\,dt\leq 2\|u_1\|_{L_{M}(\Omega_T)}\|u_2\|_{L_{M^{*}}(\Omega_T)} \end{align*} $$
for all
$u_1\in L_{M}(\Omega _T)$
and
$u_2\in L_{M^*}(\Omega _T)$
. We say that a sequence
$\{v_n\}_{n=1}^{\infty }$
converges modularly to v in
$L_M(\Omega _T)$
if there exists
$\lambda>0$
such that
$$ \begin{align*} \int^{T}_{0}\int_{\Omega}M\bigg(t,x,\frac{|v_n-v|}{\lambda}\bigg)\,dx\,dt\to 0 \quad \text{as } n\to +\infty. \end{align*} $$
For the notion of this convergence, we write
$v_n \xrightarrow []{M}v$
.
Lemma 2.6 [Reference Ahmida and Youssfi5, Theorem 5.2].
Assume that
${\mathcal {A}}$
satisfies Conditions
$(A1)$
–
$(A3)$
, and that the pair of complementary N-functions M and
$M^*$
both satisfy Balance Condition
$(B)$
. Then, for every
$\phi \in \textbf {W}(\Omega _T)$
, there exists a sequence
$\{\phi _{\delta }\}_{\delta }\subset C_0^{\infty }((0,T];C^{\infty }_{0}(\Omega ))$
such that
$$ \begin{align*} \phi_{\delta}\xrightarrow[]{}\phi & \quad\text{in } L^2(\Omega_T), \\ \partial_t \phi_{\delta}\xrightarrow[]{M}\partial_t\phi & \quad\text{in } W^{-1,x}L_{M^*}(\Omega_T)+L^2(\Omega_T),\\ D^{\alpha} \phi_{\delta}\xrightarrow[]{M} D^{\alpha} \phi, |\alpha|\leq 1 & \quad\text{in }L_M(\Omega_T), \end{align*} $$
where
$W^{-1,x}L_{M^*}(\Omega _T)$
is defined as
$$ \begin{align*} W^{-1,x}L_{M^*}(\Omega_T):&=\{u:(0,T)\to W^{-1}L_{M^*}(\Omega):u=\tilde{u}-\mathrm{div}U, \\ &\qquad\textrm{with }\tilde{u}\in L_{M^*}(\Omega_T)\text{ and } U \in L_{M^*}(\Omega_T)\}. \end{align*} $$
The following fact is a consequence of modular topology.
Lemma 2.7 [Reference Ahmida, Chlebicka, Gwiazda and Youssfi1, Lemma 2].
Let M be an N-function and let
$u_n, u\in L_M(\Omega _{T})$
. If
$u_n\xrightarrow []{M} u$
modularly, then
$u_n\to u$
in
$\sigma (L_M,L_{M^*})$
.
Using a proof strategy analogous to that in [Reference Elmahi and Meskine25], we establish the following lemma.
Lemma 2.8. Suppose that
$f\in C_0^\infty (\Omega _T)$
and
$u_0\in C_0^\infty (\Omega )$
,
${\mathcal {A}}$
satisfies the conditions
$(A1)$
–
$(A3)$
, N-function M is regular enough so that the set of smooth functions is dense in
$\textbf {W}(\Omega _T)$
in the modular topology, and M satisfies
$\mathsf {(Y)}$
-condition. Then there exists at least one distributional solution
$u\in \textbf {W}(\Omega _T)$
of Problem (1-1) satisfying
$u(x,0)=u_0(x)$
for almost every
$x\in \Omega $
. Furthermore, for all
$\tau \in (0,T]$
,
$$ \begin{align*} -\int^{\tau}_0\int_{\Omega}\partial_t \phi u\,dx\,dt+\int_{\Omega}u\phi\,dx \bigg|^{\tau}_0+\int^{\tau}_0\int_{\Omega}{\mathcal{A}}(t,x,\nabla u)\cdot \nabla \phi \,dx\,dt =\int^{\tau}_0\int_{\Omega}f\phi\, dx\,dt \end{align*} $$
for every
$\phi \in \textbf {W}(\Omega _T)$
.
We present some preliminary lemmas that will be used later.
Lemma 2.9 [Reference Zhang and Zhou49, Lemma 2.7].
Let
$\Omega _T$
be a measurable with finite Lebesgue measure, and let
${f_{n}}$
be a sequence of functions in
$L^{p}(\Omega _T) (p \ge 1)$
such that
$$ \begin{align*} & f_{n} \rightharpoonup f \quad \mbox{weakly in } L^{p}(\Omega_T),\\ & f_{n} \rightarrow g \quad \mbox{almost everywhere in }\Omega_T. \end{align*} $$
Then
$f=g$
almost everywhere in
$\Omega _T$
.
Lemma 2.10 ([Reference Hewitt and Stromberg34, Theorem 13.47]).
Let
$f_n, f\in L^1(\Omega _T)$
such that
$ f_{n}\geq 0$
almost everywhere in
$\Omega _T$
,
$f_{n} \rightarrow f$
almost everywhere in
$\Omega _{T}$
and
$$ \begin{align*} \int^{T}_0\int_{\Omega} f_n\,dx\,dt \to \int^{T}_0\int_{\Omega}f\,dx\,dt \quad \text{as } n\to+\infty. \end{align*} $$
Then
$f_n\to f$
strongly in
$L^1(\Omega _T)$
.
3 The proofs of main results
We are now in a position to give the proofs of our main results. Some of the reasoning is based on the ideas developed in [Reference Chlebicka, Gwiazda and Zatorska-Goldstein14, Reference Gwiazda, Skrzypczak and Zatorska-Goldstein29, Reference Zhang and Zhou50].
We first consider the approximate problems
$$ \begin{align} \begin{cases} \partial_t u_{n}- \text{div}{\mathcal{A}}(t,x,\nabla u_{n})= f_{n} &\mbox{in } \Omega_{T},\\ u_{n} = 0 &\mbox{on } \Sigma,\\ u_{n}(0,x) = u_{0n} &\mbox{in } \Omega, \end{cases} \end{align} $$
where the two sequences of functions
$\{f_{n}\}\subset C^{\infty }_{0}(\Omega _{T})$
and
$\{u_{0n}\}\subset C^{\infty }_{0}(\Omega )$
are strongly convergent to f in
$L^{1}(\Omega _{T})$
and
$u_{0}$
in
$L^{1}(\Omega )$
, respectively, such that
$$ \begin{align} \|f_{n}\|_{L^{1}(\Omega_{T})} \leq \|f\|_{L^{1}(\Omega_{T})}, \quad \|u_{0n}\|_{L^{1}(\Omega)}\leq \|u_{0}\|_{L^{1}(\Omega)}. \end{align} $$
It follows from Lemma 2.8 that there exists a distributional solution
$u_n\in \textbf {W}(\Omega _T)$
for Problem (3-1), such that
$$ \begin{align} \int^{\tau}_0\int_{\Omega}\partial_tu_n\phi\,dx\,dt+\int^{\tau}_0\int_{\Omega}{\mathcal{A}}(t,x,\nabla u_n)\cdot \nabla \phi \,dx\,dt =\int^{\tau}_0\int_{\Omega}f_n\phi\, dx\,dt \end{align} $$
for every
$\phi \in \textbf {W}(\Omega _T)$
.
Taking the test function as
$T_{k}(u_{n})\chi _{(0,\tau )}$
with
$\tau \in (0,T]$
in (3-3), we have
$$ \begin{align*} &\int_{\Omega}\Theta_{k}(u_{n})(\tau)\,dx -\int_{\Omega}\Theta_{k}(u_{0n})\,dx \\ & \quad+\int^{\tau}_{0}\int_{\Omega}{\mathcal{A}}(t,x,\nabla T_{k}(u_{n}))\cdot \nabla T_{k}(u_{n})\,dx\,dt = \int_{0}^{\tau}\int_{\Omega}f_{n}T_{k}(u_{n})\,dx\,dt. \end{align*} $$
According to the definition of
$\Theta _{k}(r)$
and (3-2), we deduce that
$$ \begin{align} &\int^{\tau}_{0}\int_{\Omega}{\mathcal{A}}(t,x,\nabla T_{k}(u_{n}))\cdot \nabla T_{k}(u_{n})\,dx\,dt + \int_{\Omega}\Theta_{k}(u_{n})(\tau)\,dx\nonumber\\ &\quad \leq k ( \|f_{n}\|_{L^{1}(\Omega_{T})}+\|u_{0n}\|_{L^{1}(\Omega)} ) \leq k( \|f\|_{L^{1}(\Omega_{T})}+\|u_{0}\|_{L^{1}(\Omega)}). \end{align} $$
Recalling Condition (A2), we have
$$ \begin{align} c_1\int^{\tau}_{0}\int_{\Omega}M(t,x,|\nabla T_{k}(u_{n})|)\,dx\,dt \leq & \int^{\tau}_{0}\int_{\Omega}{\mathcal{A}}(t,x,\nabla T_{k}(u_{n}))\cdot \nabla T_{k}(u_{n})\,dx\,dt\leq Ck \end{align} $$
and
$$ \begin{align} c_1\int^{\tau}_{0}\int_{\Omega}M^*(t,x,|{\mathcal{A}}(t,x,\nabla T_{k}(u_{n}))|)\,dx\,dt \leq Ck. \end{align} $$
Since
$L_M(\Omega _T)\hookrightarrow L^1(\Omega _T)$
, we know that
$$ \begin{align} \int^{\tau}_{0}\int_{\Omega}|\nabla T_{k}(u_{n})|\,dx\,dt\le C(k+1), \end{align} $$
that is,
$T_{k}(u_{n})$
is bounded in
$L^{1}(0,T; W^{1,1}_{0}(\Omega ))$
.
Choosing
$k=1$
in the inequality (3-4), we find that
$$ \begin{align*} \int_{\Omega}\Theta_{1}(u_{n}(\tau))\,dx \leq \|f\|_{L^{1}(\Omega_{T})}+\|u_{0}\|_{L^{1}(\Omega)} \end{align*} $$
for almost every
$\tau \in (0.T]$
. Moreover,
$$ \begin{align*} \int_{\Omega}|u_{n}(\tau)|\,dx \le \text{meas}(\Omega)+\|f\|_{L^{1}(\Omega_{T})}+\|u_{0}\|_{L^{1}(\Omega)}. \end{align*} $$
Therefore, we obtain
$$ \begin{align} \|u_{n}\|_{L^{\infty}(0,T;L^{1}(\Omega))}\le C. \end{align} $$
Now, we are ready to prove the existence and uniqueness of renormalized solutions.
Proof of Theorem 1.3.
(1). Existence of renormalized solutions. We divide our proof into several steps.
Step 1: Prove the convergence of
$\{u_n\}$
in
$C([0,T]; L^1(\Omega ))$
and find its subsequence which is almost everywhere convergent in
$\Omega _T$
.
Owing to (3-3), we can write the weak form as
$$ \begin{align} &\int^{T}_{0} \langle \partial_t(u_{n}-u_{m}),\phi\rangle\,dt +\int^{T}_{0}\int_{\Omega}[{\mathcal{A}}(t,x,\nabla u_{n})-{\mathcal{A}}(t,x,\nabla u_{m})]\cdot \nabla \phi\,dx\,dt \notag\\ &\quad=\int^{T}_{0}\int_{\Omega}(f_{n}-f_{m})\phi \,dx\,dt \end{align} $$
for all
$m,n \in \mathbb {Z}$
and
$\phi \in \textbf {W}(\Omega _T)$
. It follows from the Fenchel–Young inequality and (3-4) that
$$ \begin{align*} |{\mathcal{A}}(t,x,\nabla u_{n})\cdot \nabla u_{m}|\le (M(t,x,|\nabla u_{n}|)+M^{*}(t,x,|{\mathcal{A}}(t,x,\nabla u_{m})|)) \in L^{1}(\Omega_{T}). \end{align*} $$
We define
$$ \begin{align*} \alpha_{n,m}:=\int^{T}_{0}\int_{\Omega}|f_{n}-f_{m}|\,dx\,dt +\int_{\Omega}|u_{0n}-u_{0m}|\,dx. \end{align*} $$
Since
$\{f_{n}\}$
and
$\{u_{0n}\}$
are convergent in
$L^{1}$
,
$$ \begin{align*} \lim_{n,m \to+ \infty}\alpha_{n,m}=0. \end{align*} $$
By taking
$w=T_{1}(u_{n}-u_{m})\chi _{(0,\tau )}$
with
$\tau \le T$
as a test function in (3-9) and discarding the positive term we obtain
$$ \begin{align*} \int_{\Omega}\Theta_{1}(u_{n}-u_{m})(\tau)\,dx&\le\int_{\Omega}\Theta_{1}(u_{0n}-u_{0m})\,dx+\|f_{n}-f_{m}\|_{L^{1}(\Omega_{T})}\nonumber\\ &\le\|u_{0n}-u_{0m}\|_{L^{1}(\Omega)}+\|f_{n}-f_{m}\|_{L^{1}(\Omega_{T})}=\alpha_{n,m}. \end{align*} $$
Therefore, we conclude that
$$ \begin{align*} & \int_{\{|u_{n}-u_{m}|<1\}}\frac{|u_{n}-u_{m}|^{2}(\tau)}{2}\,dx +\int_{\{|u_{n}-u_{m}|\ge1 \}}\frac{|u_{n}-u_{m}|(\tau)}{2}\,dx \\ & \quad \le \int_{\Omega}[\Theta_{1}(u_{n}-u_{m})](\tau)\,dx \le \alpha_{n,m}. \end{align*} $$
Moreover,
$$ \begin{align*} \int_{\Omega}|u_{n}-u_{m}|(\tau)\,dx &=\int_{\{|u_{n}-u_{m}|<1\}}|u_{n}-u_{m}|(\tau)\,dx +\int_{\{|u_{n}-u_{m}|\ge 1\}}|u_{n}-u_{m}|(\tau)\,dx\nonumber\\ &\le \bigg(\!\int_{\{|u_{n}-u_{m}|<1\}}|u_{n}-u_{m}|^{2}(\tau)\,dx\bigg)^{{1}/{2}}\text{meas}(\Omega)^{{1}/{2}}+2\alpha_{n,m}\nonumber\\ &\le ( 2\text{meas}(\Omega))^{{1}/{2}}\alpha^{{1}/{2}}_{n,m}+2\alpha_{n,m}. \end{align*} $$
Thus, we deduce that
$$ \begin{align*} \|u_{n}-u_{m}\|_{C([0,T];L^{1}(\Omega))}\rightarrow 0 \quad \mbox{as } n, \ m \rightarrow +\infty, \end{align*} $$
which implies that
$\{u_{n}\}$
is a Cauchy sequence in
$C([0,T];L^{1}(\Omega ))$
. Then
$u_{n}$
converges to u in
$C([0,T];L^{1}(\Omega ))$
. We find an almost everywhere convergent subsequence (still denoted by
$\{u_{n}\})$
in
$\Omega _{T}$
such that
$$ \begin{align} u_{n}\rightarrow u \quad \mbox{almost everywhere in } \Omega_{T} \quad \mbox{as } n\rightarrow +\infty. \end{align} $$
Step 2: Show that the sequence
$\{\nabla u_{n}\}$
converges almost everywhere in
$\Omega _{T}$
to
$\nabla u$
(up to a subsequence). We first set
$\delta>0$
and denote
$$ \begin{align*} &A_{1}:=\{ (t,x)\in \Omega_{T}:|\nabla u_{n}|>h\}\cup \{(t,x)\in \Omega_{T}:|\nabla u_{m}|>h\},\\ &A_{2}:=\{ (t,x)\in \Omega_{T}:|u_{n}-u_{m}|>1\} \end{align*} $$
and
$$ \begin{align*} A_{3}:=\{ (t,x)\in\Omega_{T}:|\nabla u_{n}|\le h,|\nabla u_{m}|\le h,|u_{n}-u_{m}|\le1,|\nabla u_{n}-\nabla u_{m}|>\delta\}, \end{align*} $$
where h will be chosen later. Next, we show that
$\{\nabla u_{n}\}$
is a Cauchy sequence in measure. It is easy to check that
$$ \begin{align*} \{ (t,x)\in\Omega_{T}:|\nabla u_{n}-\nabla u_{m}|>\delta \}\subset A_{1}\cup A_{2}\cup A_{3}. \end{align*} $$
First, we notice that
$$ \begin{align*} \{ (t,x)\in\Omega_{T}:|\nabla u_{n}|\ge h\}\subset \{ (t,x)\in\Omega_{T}:|u_{n}|\ge k\}\cup\{ (t,x)\in\Omega_{T}:|\nabla T_{k}(u_{n})|\ge h\} \end{align*} $$
for all
$k>0$
. Thus, using (3-8) and (3-7), we know that there exist constants
$C>0$
such that
$$ \begin{align*} \mbox{meas}\{ (t,x)\in \Omega_{T}:|\nabla u_{n}|\ge h\} \le \frac{C}{k}+\frac{C(k+1)}{h}, \end{align*} $$
when h is appropriately large. By choosing
$k=Ch^{1/2}$
, we deduce that
$$ \begin{align*} \mbox{meas}\{ (t,x)\in \Omega_{T}:|\nabla u_{n}|\ge h\} \le Ch^{-{1}/{2}}. \end{align*} $$
Let
$\varepsilon>0$
. We let
$h=h(\varepsilon )$
be large enough such that
$$ \begin{align} \mbox{meas}(A_{1})\le \frac{\varepsilon}{3} \quad \mbox{for all } n,m>0. \end{align} $$
Second, by Step 1 we know that
$\{u_{n}\}$
is a Cauchy sequence in measure. Then there exists
$N_{1}(\varepsilon )\in \mathbb {N}$
such that
$$ \begin{align} \mbox{meas}(A_{2}) \le \frac{\varepsilon}{3} \quad \mbox{for all } n,m\ge N_{1}(\varepsilon). \end{align} $$
Finally, from Condition
$(A3)$
, we know that there exists a real-valued function
$m(h,\delta )>0$
such that
$$ \begin{align*} [{\mathcal{A}}(t,x,\nabla \eta)-{\mathcal{A}}(t,x,\nabla \zeta)]\cdot({\nabla\eta-\nabla \zeta})\ge m(h,\delta)>0 \end{align*} $$
for all
$\eta ,\zeta \in \mathbb {R}^{N}$
with
$|\eta |,|\zeta |\le h,\delta \le |\eta -\zeta |$
. By taking
$T_{1}(u_{n}-u_{m})$
as a test function in (3-9) and integrating on
$A_{3}$
, we obtain
$$ \begin{align*} m(h,\delta) \mbox{meas} (A_{3})&\le \int_{A_{3}}[{\mathcal{A}}(t,x,\nabla u_{n})-{\mathcal{A}}(t,x,\nabla u_{m})]\cdot(\nabla u_{n}-\nabla u_{m})\,dx\,dt\\ &\le\int^{T}_{0}\int_{\Omega}[{\mathcal{A}}(t,x,\nabla u_{n})-{\mathcal{A}}(t,x,\nabla u_{m})]\cdot(\nabla u_{n}-\nabla u_{m})\,dx\,dt\\ &\le \int^{T}_{0}\int_{\Omega}|f_{n}-f_{m}|\,dx\,dt+\int_{\Omega}|u_{0n}-u_{0m}|\,dx =\alpha(n,m), \end{align*} $$
which implies that
$$ \begin{align} \mbox{meas}(A_{3}) \le \frac{\alpha(n,m)}{m(h,\delta)}\le \frac{\varepsilon}{3} \quad \mbox{for all } n,m \ge N_{2}(\varepsilon,\delta). \end{align} $$
By combining the estimates (3-11)–(3-13), we obtain
$$ \begin{align*} \mbox{meas}\{ (t,x)\in \Omega_{T}:|\nabla u_{n}-\nabla u_{m}|> \delta \}\le \varepsilon \quad \mbox{for all } n,m \ge \mathrm{max}\{N_{1},N_{2}\}, \end{align*} $$
that is,
$\{\nabla u_{n}\}$
is a Cauchy sequence in measure. Therefore, we obtain a subsequence of
$\{\nabla u_{n}\}$
that is almost everywhere convergent in
$\Omega _{T}$
. Moreover, the a priori estimate (3-5) and weak lower semicontinuity of a convex functional give that
$$ \begin{align*} T_{k}(u_{n}) \rightharpoonup T_{k}(u) \quad \mbox{weakly in } L^{1}(0,T; W^{1,1}_{0}(\Omega)). \end{align*} $$
Therefore, we deduce from Lemma 2.9 that
$$ \begin{align*} \nabla u_{n}\rightarrow \nabla u \quad \mbox{almost everywhere in }\Omega_{T} \quad \mbox{as } n\rightarrow +\infty. \end{align*} $$
Step 3: Prove a decay condition for
$u_n$
. In this step, we prove that
$$ \begin{align*} \lim\limits_{l\to +\infty}\lim\limits_{n\to +\infty}\int_{\{l\leq |u_n|\leq l+1\}}{\mathcal{A}}(t,x,\nabla u_n)\cdot \nabla u_n\,dx\,dt=0. \end{align*} $$
We define the function
$T_{l,a}(s)=T_{a}(s-T_{l}(s))$
as
$$ \begin{align*} T_{l,a}(s):= \begin{cases}{ccc} s-l\textrm{sign}(s) & \mbox{if } l \leq |s| < l+a,\\ a\textrm{sign}(s)& \mbox{if } l+a \leq |s|,\\ 0 & \mbox{if } |s|< l. \end{cases} \end{align*} $$
Using
$T_{l,a}(u_{n})=T_{a}(u_{n}-T_{l}(u_{n}))$
as a test function in (3-3), we find that
$$ \begin{align*} &\int_{\{|u_{n}|>l\}}\Theta_{a}(u_{n}\mp l)(T)\,dx -\int_{\{|u_{0n}|>l\}}\Theta_{a}(u_{0n}\mp l)\,dx\\ &\quad\quad +\int_{\{l \leq |u_{n}|\leq l+a\}}{\mathcal{A}}(t,x,\nabla u_{n})\cdot \nabla u_{n}\,dx\,dt \leq \int^{T}_0\int_{\Omega}f_{n}T_{l,a}(u_{n})\,dx\,dt, \end{align*} $$
which yields that
$$ \begin{align} \int_{\{l\leq |u_{n}|\leq l+a\}} {\mathcal{A}}(t,x,\nabla u_{n})\cdot \nabla u_{n}\,dx\,dt \le a \bigg(\!\int_{\{|u_{n}|>l\}}|f_{n}|\,dx\,dt+\int_{\{|u_{0n}|>l\}}|u_{0n}|\,dx\bigg). \end{align} $$
Recalling the convergence of
$\{u_{n}\}$
in
$C([0,T];L^{1}(\Omega ))$
, we have
$$ \begin{align*} \lim\limits_{l\rightarrow +\infty} \text{meas}\{ (t,x)\in \Omega_T:|u_{n}|>l\}=0 \quad \mbox{uniformly with respect to } n. \end{align*} $$
Therefore, passing to the limit first in n then in l, we conclude from (3-14) that
$$ \begin{align*} \lim_{l \to+ \infty}\lim\limits_{n\rightarrow +\infty} \int_{\{ l \leq |u_n| \leq l+a\}} {\mathcal{A}}(t,x,\nabla u_n)\cdot \nabla u_n \,dx\,dt=0. \end{align*} $$
Choosing
$a=1$
gives
$$ \begin{align} \lim_{l\to+\infty}\lim\limits_{n\rightarrow +\infty} \int_{\{ l \leq |u_n| \leq l+1\}} {\mathcal{A}}(t,x,\nabla u_n)\cdot \nabla u_n \,dx\,dt=0. \end{align} $$
Step 4: Establish the convergence of
${\mathcal {A}}(t,x,\nabla T_{k}(u_n))\cdot \nabla T_{k}(u_n)$
.
We show that
$$ \begin{align} {\mathcal{A}}(t,x,\nabla T_{k}(u_n)) \nabla T_k(u_n) \rightarrow {\mathcal{A}}(t,x,\nabla T_{k}(u))\nabla T_{k}(u) \quad \mbox{strongly in } L^{1}(\Omega_T). \end{align} $$
In fact, according to (3-6) and Condition (A2), we obtain
$$ \begin{align*} \int^{T}_{0}\int_{\Omega}M^*(t,x,|{\mathcal{A}}(t,x,\nabla T_{k}(u_n))|)\,dx\,dt \leq C. \end{align*} $$
Therefore, there exists a subsequence of
${\mathcal {A}}(t,x,\nabla T_{k}(u_n)$
such that, as
$n\to +\infty $
,
$$ \begin{align} {\mathcal{A}}(t,x,\nabla T_{k}(u_n)) \rightharpoonup {\mathcal{A}}_{k} \quad \text{weakly}-* \text{ in } L_{M^*}(\Omega_T). \end{align} $$
Recalling the fact that
$\nabla u_n \to \nabla u$
almost everywhere in
$\Omega _T$
and that
${\mathcal {A}}(t,x,\nabla T_{k}(u_n))$
is continuous with respect to
$\nabla T_{k}(u_n)$
, we deduce that
$$ \begin{align*} {\mathcal{A}}_{k}={\mathcal{A}}(t,x,\nabla T_{k}(u)). \end{align*} $$
To prove (3-16), we first show that
$$ \begin{align*} \lim_{n\to+\infty}\int^{T}_{0}\int_{\Omega}{\mathcal{A}}(t,x,\nabla T_{k}(u_n))\cdot\nabla T_{k}(u_n)\,dx\,dt=\int^{T}_{0}\int_{\Omega}{\mathcal{A}}(t,x,\nabla T_{k}(u))\cdot\nabla T_{k}(u)\,dx\,dt. \end{align*} $$
Since
$T_{k}(u)\in \textbf {W}(\Omega _T)$
, it follows from Lemma 2.6 that there exists a sequence
$\{T_{k}(u)\}_{\delta }\subset C_0^{\infty }((0,T];C^{\infty }_{0}(\Omega ))$
such that
$$ \begin{align*} \{T_{k}(u)\}_{\delta}\xrightarrow[]{M}T_{k}(u)& \quad \text{in } L_M( \Omega_T), \\ \nabla \{T_{k}(u_n)\}_{\delta}\xrightarrow[]{M}\nabla T_{k}(u)& \quad \text{in } L_M( \Omega_T),\\ \partial_t \{T_{k}(u)\}_{\delta}\xrightarrow[]{M} \partial_t T_{k}(u)& \quad \text{in } W^{-1,x}L_{M^*}(\Omega_T)+L^2(\Omega_T). \end{align*} $$
We define
$\psi _l(r)=\min \{(l+1-|r|)^+,1\}$
. Taking
$\psi _{l}(u_n)\{T_{k}(u)\}_{\delta }\chi _{(0,\tau )}$
as a test function for Problem (3-1), we have
$$ \begin{align} &\int^{\tau}_0\int_{\Omega}\partial_tu_n \psi_{l}(u_n)(T_{k}(u))_{\delta}\,dx\,dt +\int^{\tau}_0\int_{\Omega}{\mathcal{A}}(t,x,u_{n})\cdot \nabla(\psi_{l}(u_n)(T_{k}(u))_{\delta})\,dx\,dt \notag\\ &\quad=\int^{\tau}_0\int_{\Omega}f_{n}\psi_{l}(u_n)(T_{k}(u))_{\delta}\,dx\,dt. \end{align} $$
For the first term on the left-hand side of (3-18), and recalling the fact that
$u_n\to u$
almost everywhere in
$\Omega _T$
as
$n\to +\infty $
, we deduce from the Lebesgue dominated convergence theorem that
$$ \begin{align*} \begin{aligned} &\lim\limits_{\delta\to 0}\lim\limits_{n\to +\infty}\int^{\tau}_0\int_{\Omega}\partial_tu_n \psi_{l}(u_n)(T_{k}(u))_{\delta}\,dx\,dt \\ &\quad=\lim\limits_{\delta\to 0}\lim\limits_{n\to +\infty}\int^{\tau}_0\int_{\Omega} \partial_t \int^{u_n}_0\psi_{l}(r)\,dr (T_{k}(u))_{\delta}\,dx\,dt\\ &\quad=\lim\limits_{\delta\to 0}\lim\limits_{n\to +\infty}\bigg[\int_{\Omega}\int^{u_n}_0\psi_{l}(r)\,dr (T_{k}(u))_{\delta}\,dx\bigg|^{\tau}_0-\int^\tau_0\int_{\Omega}\int^{u_n}_0\psi_{l}(r)\,dr \partial_t (T_{k}(u))_{\delta}\,dx\,dt\bigg]\\ &\quad=\int^{\tau}_0\int_{\Omega}\partial_tu\psi_{l}(u)T_{k}(u)\,dx\,dt. \end{aligned} \end{align*} $$
Taking
$l\to +\infty $
, it is obvious that
$$ \begin{align} \lim\limits_{l\to +\infty}\lim\limits_{\delta\to 0}\lim\limits_{n\to +\infty}\int^{\tau}_0\int_{\Omega}\partial_tu_n \psi_{l}(u_n)(T_{k}(u))_{\delta}\,dx\,dt =\int^{\tau}_0\int_{\Omega}\partial_tu T_{k}(u)\,dx\,dt. \end{align} $$
For the second term on the left-hand side of (3-18), we know that
$$ \begin{align*} \begin{aligned} &\int^{\tau}_0\int_{\Omega}{\mathcal{A}}(t,x, \nabla u_{n})\cdot \nabla(\psi_{l}(u_n)(T_{k}(u))_{\delta})\,dx\,dt\\ &\quad= \int^{\tau}_0\int_{\Omega}{\mathcal{A}}(t,x, \nabla u_{n})\cdot \nabla T_{l+1}(u_n) \psi_l'(u_n)(T_{k}(u))_{\delta}\,dx\,dt\\ &\quad\quad +\int^{\tau}_0\int_{\Omega}{\mathcal{A}}(t,x, \nabla u_{n})\cdot \nabla (T_{k}(u))_{\delta} \psi_l(u_n)\,dx\,dt\\ &\quad:=Z_1+Z_2. \end{aligned} \end{align*} $$
For
$Z_1$
, it follows from (3-15) that
$$ \begin{align*} \begin{aligned} &\lim\limits_{l\to +\infty}\lim\limits_{\delta\to 0}\lim\limits_{n\to +\infty}\int^{\tau}_0\int_{\Omega}{\mathcal{A}}(t,x, \nabla u_{n})\cdot \nabla T_{l+1}(u_n) \psi_l'(u_n)(T_{k}(u))_{\delta}\,dx\,dt\\ &\quad \leq \lim\limits_{l\to +\infty}\lim\limits_{n\to +\infty}C\int_{\{l\leq |u_n|\leq l+1\}}{\mathcal{A}}(t,x, \nabla u_{n})\cdot \nabla T_{l+1}(u_n)\,dx\,dt\to 0. \end{aligned} \end{align*} $$
For
$Z_2$
, we deduce from (3-17) that
$$ \begin{align} {\mathcal{A}}(t,x, \nabla T_{l+1}(u_{n}))\psi_l(u_n)\rightharpoonup {\mathcal{A}}(t,x, T_{l+1}(\nabla u))\psi_l(u) \quad\text{weakly in } L^1(\Omega_T) \quad \text{as } n\to +\infty, \end{align} $$
which gives that
$$ \begin{align*} \begin{aligned} &\lim\limits_{l\to +\infty}\lim\limits_{\delta\to 0}\lim\limits_{n\to +\infty}\int^{\tau}_0\int_{\Omega}{\mathcal{A}}(t,x, \nabla u_{n})\cdot \nabla (T_{k}(u))_{\delta} \psi_l(u_n)\,dx\,dt\\ &\quad=\lim\limits_{l\to +\infty}\lim\limits_{\delta\to 0}\lim\limits_{n\to +\infty}\int^{\tau}_0\int_{\Omega}{\mathcal{A}}(t,x, \nabla T_{l+1}(u_{n}))\cdot \nabla (T_{k}(u))_{\delta} \psi_l(u_n)\,dx\,dt\\ &\quad=\int^{\tau}_0\int_{\Omega}{\mathcal{A}}(t,x, \nabla u)\cdot \nabla T_{k}(u)\,dx\,dt. \end{aligned} \end{align*} $$
The limit as
$\delta \to 0$
results from Lemma 2.7. Therefore,
$$ \begin{align} &\lim\limits_{l\to +\infty}\lim\limits_{\delta\to 0}\lim\limits_{n\to +\infty}\int^{\tau}_0\int_{\Omega}{\mathcal{A}}(t,x, \nabla u_{n})\cdot \nabla(\psi_{l}(u_n)(T_{k}(u))_{\delta})\,dx\,dt\notag \\ &\quad=\int^{\tau}_0\int_{\Omega}{\mathcal{A}}(t,x, \nabla u)\cdot \nabla T_{k}(u)\,dx\,dt. \end{align} $$
For the term on the right-hand side of (3-18),
$$ \begin{align} \lim\limits_{l\to +\infty}\lim\limits_{\delta\to 0}\lim\limits_{n\to +\infty}\int^{\tau}_0\int_{\Omega}f_{n}\psi_{l}(u_n)(T_{k}(u))_{\delta}\,dx\,dt=\int^{\tau}_0\int_{\Omega}fT_{k}(u)\,dx\,dt. \end{align} $$
Thus, it follows from (3-18)–(3-22) that
$$ \begin{align} \int^{\tau}_0\int_{\Omega}\partial_tu T_{k}(u)\,dx\,dt+\int^{\tau}_0\int_{\Omega}{\mathcal{A}}(\tau,x, \nabla u)\cdot \nabla T_{k}(u)\,dx\,dt=\int^{\tau}_0\int_{\Omega}fT_{k}(u)\,dx\,dt. \end{align} $$
In addition, testing the approximate problem (3-1) by
$T_{k}(u_{n})\chi _{(0,\tau )}$
,
$$ \begin{align} & \lim_{n\to+\infty}\bigg(\!\int_{\Omega}\Theta_{k}(u_{n})(\tau)\,dx -\int_{\Omega}\Theta_{k}(u_{0n})\,dx+ \int^{\tau}_{0}\int_{\Omega}{\mathcal{A}}(t,x,\nabla T_{k}(u_{n}))\cdot \nabla T_{k}(u_{n})\,dx\,dt\bigg)\notag\\ &\quad = \lim_{n\to+\infty}\int^{\tau}_{0}\int_{\Omega}f_{n}T_{k}(u_{n})\,dx\,dt=\int^{\tau}_{0}\int_{\Omega}fT_{k}(u)\,dx\,dt. \end{align} $$
By combining (3-23) and (3-24), we obtain
$$ \begin{align*} \lim_{n\to+\infty} \int^{\tau}_{0}\int_{\Omega}{\mathcal{A}}(t,x,\nabla T_{k}(u_n))\cdot\nabla T_k(u_n)\,dx\,dt= \int^{\tau}_{0}\int_{\Omega}{\mathcal{A}}(t,x,\nabla T_{k}(u))\cdot\nabla T_{k}(u)\,dx\,dt. \end{align*} $$
Thus, Lemma 2.10 gives that
$$ \begin{align*} {\mathcal{A}}(t,x,\nabla T_{k}(u_n)) \cdot \nabla T_k(u_n) \rightarrow {\mathcal{A}}(t,x,\nabla T_{k}(u))\cdot \nabla T_{k}(u) \quad \mbox{strongly in } L^{1}(\Omega_T). \end{align*} $$
Step 5: We prove that u is a renormalized solution.
Condition (R1). First, we show that the u satisfies Condition (R1). By the properties of truncations, for every
$l\in \mathbb {N}$
,
$$ \begin{align*} \begin{aligned} \nabla u_n=0 \quad \text{almost everywhere in } \{(t,x) \in \Omega_T : |u_n|\in \{l,l+1\}\}. \end{aligned} \end{align*} $$
From this and (3-15), it follows that
$$ \begin{align} \lim_{l \to+ \infty}\lim\limits_{n\rightarrow +\infty} \int_{\{ l-1 \leq |u_n| \leq l+2\}} {\mathcal{A}}(t,x,\nabla u_n)\cdot \nabla u_n \,dx\,dt=0. \end{align} $$
Now, we define the function
$G_l:\mathbb {R}\to \mathbb {R}$
by
$$ \begin{align*}G_l(r)= \begin{cases} 1 & \mbox{if } l\leq |r|\leq l+1, \\ 0 & \mbox{if } |r|<l-1 \ \text{or } |r|>l+2,\\ \text{affine} & \text{otherwise}. \end{cases} \end{align*} $$
Using this, we write
$$ \begin{align*} \begin{aligned} \int_{\{l<|u|<l+1\}}{\mathcal{A}}(t,x,\nabla u)\cdot \nabla u \,dx\,dt \leq \int^{T}_{0}\int_{\Omega}G_l(u){\mathcal{A}}(t,x,\nabla T_{l+2}(u))\cdot \nabla T_{l+2}(u)\,dx\,dt. \end{aligned} \end{align*} $$
Taking the limit in the inequality above gives
$$ \begin{align} 0 &\leq \lim\limits_{l\to +\infty} \int_{\{l<|u|<l+1\}}{\mathcal{A}}(t,x,\nabla u)\cdot \nabla u \,dx\,dt \nonumber\\&\leq \lim\limits_{l\to +\infty} \int^{T}_{0}\int_{\Omega}G_l(u){\mathcal{A}}(t,x,\nabla T_{l+2}(u))\cdot \nabla T_{l+2}(u)\,dx\,dt. \end{align} $$
Using (3-16), (3-25) and the fact that
$G_l$
is bounded and continuous, we get
$$ \begin{align} &\lim\limits_{l\to +\infty}\int^{T}_{0}\int_{\Omega}G_l(u){\mathcal{A}}(t,x,\nabla T_{l+2}(u))\cdot \nabla T_{l+2}(u)\,dx\,dt\notag\\ &\quad=\lim\limits_{l\to +\infty}\lim\limits_{n\to +\infty}\int^{T}_{0}\int_{\Omega}G_l(u){\mathcal{A}}(t,x,\nabla T_{l+2}(u_n))\cdot \nabla T_{l+2}(u_n)\,dx\,dt\notag\\ &\quad \leq \lim\limits_{l\to +\infty}\lim_{n\to +\infty}\int_{\{l-1<|u_n|<l+2\}}{\mathcal{A}}(t,x,\nabla u_n)\cdot \nabla u_n \,dx\,dt \notag\\ &\quad=0. \end{align} $$
From (3-26) and (3-27), we conclude that
$$ \begin{align*} \lim\limits_{l\to +\infty}\int_{\{l<|u|<l+1\}}{\mathcal{A}}(t,x,\nabla u)\cdot \nabla u \,dx\,dt=0, \end{align*} $$
which gives Condition (R1).
Condition (R2). Let
$S\in W^{2,\infty }(\mathbb {R})$
be such that
$\text {supp} S'\in [-M,M]$
for some
$M>0$
. For every
$ \phi \in C^{1}(\overline {\Omega }_{T})$
with
$\phi (\cdot , T)=0$
, taking
$S'(u_n)\phi $
as a test function in (3-1), we obtain
$$ \begin{align*} \begin{aligned} &\int^{T}_{0}\int_{\Omega} \frac{\partial S(u_n)}{\partial t}\phi\,dx\,dt+\int^{T}_{0}\int_{\Omega}{\mathcal{A}}(t,x,\nabla u_n)\cdot\nabla\phi S'(u_n)\\ &\quad+{\mathcal{A}}(t,x,\nabla u_n)\cdot\nabla u_n S"(u_n)\phi\,dx\,dt =\int^{T}_{0}\int_{\Omega}f_n S'(u_n)\phi\,dx\,dt. \end{aligned} \end{align*} $$
Recalling the fact that
$u_n\to u$
almost everywhere in
$\Omega _T$
, we obtain that
$$ \begin{align*} \int^{T}_{0}\int_{\Omega} \frac{\partial S(u_n)}{\partial t}\phi\,dx\,dt \to \int^{T}_{0}\int_{\Omega} \frac{\partial S(u)}{\partial t}\phi\,dx\,dt \end{align*} $$
and
$$ \begin{align*} \int^{T}_{0}\int_{\Omega}f_n S'(u_n)\phi\,dx\,dt\to \int^{T}_{0}\int_{\Omega}f S'(u)\phi\,dx\,dt \end{align*} $$
as
$n\to +\infty $
.
In addition, since
$\text {supp} S'\in [-M,M]$
for
$M>0$
, we know that
$$ \begin{align*}S'(u_n) {\mathcal{A}}(t,x,\nabla u_n)= S'(u_n){\mathcal{A}}(t,x,\nabla T_{M}(u_n)).\end{align*} $$
Note that
$S'(u_n)$
is bounded and satisfies
$$ \begin{align*}S'(u_n)\to S'(u) \quad \text{almost everywhere in } \Omega_T.\end{align*} $$
Thus, it follows from
$$ \begin{align*}{\mathcal{A}}(t,x,\nabla T_{M}(u_n))\rightharpoonup{\mathcal{A}}(t,x,\nabla T_{M}(u)) \quad \text{weakly in } L^1(\Omega_T)\end{align*} $$
that
$$ \begin{align*}S'(u_n){\mathcal{A}}(t,x,\nabla T_{M}(u_n))\rightharpoonup S'(u){\mathcal{A}}(t,x,\nabla T_{M}(u))\quad\text{weakly in } L^1(\Omega_T),\end{align*} $$
which implies that
$$ \begin{align*} \int^{T}_{0}\int_{\Omega}{\mathcal{A}}(t,x,\nabla u_n) S'(u_n)\cdot\nabla\phi\,dx\,dt \to \int^{T}_{0}\int_{\Omega}{\mathcal{A}}(t,x,\nabla u) S'(u)\cdot \nabla\phi\,dx\,dt \end{align*} $$
as
$n\to +\infty $
. Moreover, it follows from (3-16) that
$$ \begin{align*} \int^{T}_{0}\int_{\Omega}{\mathcal{A}}(t,x,\nabla u_n)\cdot\nabla u_n S"(u_n)\phi\,dx\,dt\to \int^{T}_{0}\int_{\Omega}{\mathcal{A}}(t,x,\nabla u)\cdot\nabla u S"(u)\phi\,dx\,dt \end{align*} $$
as
$n\to +\infty $
. Thus, we conclude that
$$ \begin{align*} \begin{aligned} &\int^{T}_{0}\int_{\Omega} \frac{\partial S(u)}{\partial t}\phi\,dx\,dt+\int^{T}_{0}\int_{\Omega}({\mathcal{A}}(t,x,\nabla u)\cdot\nabla\phi S'(u)+{\mathcal{A}}(t,x,\nabla u)\cdot\nabla u S"(u)\phi)\,dx\,dt\\ &\quad=\int^{T}_{0}\int_{\Omega}f S'(u)\phi\,dx\,dt. \end{aligned} \end{align*} $$
We have proved the existence of the renormalized solution. Now, we proceed to demonstrate its uniqueness.
(2) Uniqueness of renormalized solutions. Let u and v be two renormalized solutions to Problem (1-1). For
$\sigma>0$
, we define the function
$S_{\sigma }$
as
$$ \begin{align*} S_\sigma(r):= \begin{cases} r & \text{if }|r|<\sigma,\\ ( \sigma+\tfrac{1}{2})-\tfrac{1}{2}( r-(\sigma+1))^2 &\text{if }\sigma\le r\le\sigma+1,\\ -( \sigma+\tfrac{1}{2})+\tfrac{1}{2}( r+(\sigma+1))^2 &\text{if }-\sigma-1\le r\le-\sigma,\\ ( \sigma+\tfrac{1}{2}) &\text{if }r>\sigma+1,\\ -( \sigma+\tfrac{1}{2}) &\text{if }r<-\sigma-1. \end{cases} \end{align*} $$
It is obvious that
$$ \begin{align*} S_\sigma'(r)= \begin{cases} 1 &\text{if }|r|< \sigma,\\ \sigma+1-|r| &\text{if }\sigma\le |r|\le \sigma+1,\\ 0 &\text{if }|r|>\sigma+1, \end{cases} \end{align*} $$
$S_\sigma \in W^{2,\infty }(\mathbb {R})$
with
$\text {supp}\,S_\sigma '\subset [-\sigma -1,\sigma +1]$
and
$$ \begin{align*}\text{supp}\,S_\sigma"\subset [\sigma,\sigma+1]\cup[-\sigma-1,-\sigma]. \end{align*} $$
We take
$S=S_{\sigma }$
, to obtain
$$ \begin{align*} \int^{T}_{0}\int_{\Omega} \frac{\partial S_{\sigma}(u)}{\partial t}\phi \,dx\,dt +\int^{T}_{0}\int_{\Omega}{\mathcal{A}}(t,x,\nabla u)\cdot \nabla (S'(u)\phi)\,dx\,dt=\int^{T}_{0}\int_{\Omega}fS^{\prime}_{\sigma}(u)\phi\,dx\,dt \end{align*} $$
and
$$ \begin{align*} \int^{T}_{0}\int_{\Omega} \frac{\partial S_{\sigma}(v)}{\partial t}\phi\,dx\,dt +\int^{T}_{0}\int_{\Omega}{\mathcal{A}}(t,x,\nabla v)\cdot \nabla (S'(v)\phi)\,dx\,dt=\int^{T}_{0}\int_{\Omega}fS^{\prime}_{\sigma}(v)\phi\,dx\,dt. \end{align*} $$
For fixed
$k>0$
, taking
$\phi =T_{k}(S_{\sigma }(u)-S_{\sigma }(v))$
and subtracting the two equations given above, we deduce that
$$ \begin{align*} &\int^{T}_{0}\int_{\Omega} \bigg(\frac{\partial S_{\sigma}(u)}{\partial t}-\frac{\partial S_{\sigma}(v)}{\partial t}\bigg)T_{k}(S_{\sigma}(u)-S_{\sigma}(v))\,dx\,dt \\ &\qquad +\int^{T}_{0}\int_{\Omega}{\mathcal{A}}(t,x,\nabla u)\cdot\nabla u S"(u)T _{k}(S_{\sigma}(u) -S_{\sigma}(v))\\ &\qquad-{\mathcal{A}}(t,x,\nabla v)\cdot\nabla v S"(v)T_{k}(S_{\sigma}(u)-S_{\sigma}(v))\,dx\,dt\\ &\qquad+\int^{T}_{0}\int_{\Omega}S'(u){\mathcal{A}}(t,x,\nabla u)\cdot\nabla T_{k}(S_{\sigma}(u)-S_{\sigma}(v)) \\ &\qquad -S'(v){\mathcal{A}}(t,x,\nabla v)\cdot\nabla T_{k}(S_{\sigma}(u)-S_{\sigma}(v))\,dx\,dt\\ &\quad=\int^{T}_{0}\int_{\Omega}(fS^{\prime}_{\sigma}(u)-fS^{\prime}_{\sigma}(v))T_{k}(S_{\sigma}(u)-S_{\sigma}(v))\,dx\,dt. \end{align*} $$
We set
$$ \begin{align*} J_0&:=\int^{T}_{0}\int_{\Omega}\bigg(\frac{\partial S_{\sigma}(u)}{\partial t}-\frac{\partial S_{\sigma}(v)}{\partial t}\bigg)T_{k}(S_{\sigma}(u)-S_{\sigma}(v))\,dx\,dt, \\ J_1&:=\int^{T}_{0}\int_{\Omega}{\mathcal{A}}(t,x,\nabla u)\cdot\nabla u S_{\sigma}"(u)T _{k}(S_{\sigma}(u) -S_{\sigma}(v))\\ &\quad -{\mathcal{A}}(t,x,\nabla v)\cdot\nabla v S_{\sigma}"(v)T_{k}(S_{\sigma}(u)-S_{\sigma}(v))\,dx\,dt,\\ J_2&:=\int^{T}_{0}\int_{\Omega}S_{\sigma}'(u){\mathcal{A}}(t,x,\nabla u)\cdot\nabla T_{k}(S_{\sigma}(u) -S_{\sigma}(v))\\ & \quad-S_{\sigma}'(v){\mathcal{A}}(t,x,\nabla v)\cdot\nabla T_{k}(S_{\sigma}(u)-S_{\sigma}(v))\,dx\,dt,\\ J_3&:=\int^{T}_{0}\int_{\Omega}(fS^{\prime}_{\sigma}(u)-fS^{\prime}_{\sigma}(v))T_{k}(S_{\sigma}(u)-S_{\sigma}(v))\,dx\,dt. \end{align*} $$
Thus,
$$ \begin{align*} J_0+J_1+J_2=J_3. \end{align*} $$
Next, we estimate
$J_i(i=0,1,2,3)$
one by one.
Estimates of
$J_0$
. Owing to the same initial condition for u and v, and the properties of
$\Theta _k$
, we get
$$ \begin{align*} J_0=\int_\Omega\Theta_k( S_\sigma(u)-S_\sigma(v))(T)\,dx\ge 0. \end{align*} $$
Estimates of
$J_1$
. It follows from the definition of
$S_{\sigma }$
that
$$ \begin{align*} |J_1|\leq Ck\bigg(\!\int_{\{\sigma<|u|<\sigma+1\}} {\mathcal{A}}(t,x,\nabla u)\cdot\nabla u\,dx\,dt+\int_{\{\sigma<|v|<\sigma+1\}} {\mathcal{A}}(t,x,\nabla v)\cdot\nabla v\,dx\,dt\bigg). \end{align*} $$
According to the definition of renormalized solutions,
$$ \begin{align*} \lim_{\sigma\to+\infty} J_1=0. \end{align*} $$
Estimates of
$J_2$
. We write
$$ \begin{align*} \begin{aligned} J_2&:=\int^{T}_{0}\int_{\Omega} {\mathcal{A}}(t,x,\nabla u) (S^{\prime}_{\sigma}(u)-1)\cdot\nabla T_{k}(S_{\sigma}(u)-S_{\sigma}(v))\,dx\,dt\\ &\quad +\int^{T}_{0}\int_{\Omega}({\mathcal{A}}(t,x,\nabla u)-{\mathcal{A}}(t,x,\nabla v))\cdot\nabla T_{k}(S_{\sigma}(u)-S_{\sigma}(v))\,dx\,dt\\ &\quad+\int^{T}_{0}\int_{\Omega} {\mathcal{A}}(t,x,\nabla v) (1-S^{\prime}_{\sigma}(v))\cdot\nabla T_{k}(S_{\sigma}(u)-S_{\sigma}(v))\,dx\,dt\\ &:= J^1_2+J^2_2+J^2_3. \end{aligned} \end{align*} $$
Setting
$\sigma>k$
, we obtain
$$ \begin{align*} J^2_2 \geq \int_{\{|u-v|\leq k\cap |u|\leq k,|v|\leq k\}}({\mathcal{A}}(t,x,\nabla u)-{\mathcal{A}}(t,x,\nabla v))\cdot\nabla (u-v)\,dx\,dt. \end{align*} $$
Moreover, since meas(
$\{u=\infty \}=0$
) and meas(
$\{v=\infty \}=0$
), we deduce that
$$ \begin{align*} \begin{aligned} &\lim_{\sigma\to+\infty}(|J^1_2|+|J^3_2|)\\ &\quad \leq \lim_{\sigma\to+\infty} \int_{\{\sigma\leq |u|\}} {\mathcal{A}}(t,x,\nabla u) \cdot\nabla T_{k}(S_{\sigma}(u)-S_{\sigma}(v))\,dx\,dt\\ &\qquad+ \lim_{\sigma\to+\infty} \int_{\{\sigma\leq |v|\}} {\mathcal{A}}(t,x,\nabla v)\cdot \nabla T_{k}(S_{\sigma}(u)-S_{\sigma}(v))\,dx\,dt\\ &\quad =0. \end{aligned} \end{align*} $$
Thus, we conclude that
$$ \begin{align*} \lim_{\sigma\to+\infty} J_2\geq \int_{\{|u-v|\leq k\cap |u|\leq k,|v|\leq k\}}({\mathcal{A}}(t,x,\nabla u)-{\mathcal{A}}(t,x,\nabla v))\cdot \nabla (u-v)\,dx\,dt. \end{align*} $$
Estimates of
$J_3$
. It is easy to check that
$$ \begin{align*} |J_3|\leq k\int_{\{\max\{|u|,|v|\}>\sigma\}}|f|\,dx\,dt, \end{align*} $$
which implies that
$$ \begin{align*} \lim_{\sigma\to+\infty} |J_3|=0. \end{align*} $$
To summarize, by sending
$\sigma \to +\infty $
, we obtain
$$ \begin{align*} \begin{aligned} 0&\leq \int_{\{|u|\leq {k}/{2}, |v|\leq{k}/{2}\}}({\mathcal{A}}(t,x,\nabla u)-{\mathcal{A}}(t,x,\nabla v))\cdot\nabla (u-v)\,dx\,dt\\ &\leq 0, \end{aligned} \end{align*} $$
which implies that
$\nabla u=\nabla v$
almost everywhere on the set
$\{|u|\leq {k}/{2}, |v|\leq {k}/{2}\}$
. Since k is arbitrary, we have
$\nabla u=\nabla v$
almost everywhere in
$\Omega _T$
. Thus, from the Poincaré inequality, we have
$u=v$
almost everywhere in
$\Omega _T$
.
We have proved the existence and uniqueness of the renormalized solution to (1-1). Next, we demonstrate that this renormalized solution is also its entropy solution and prove the uniqueness of the entropy solution. Consequently, we obtain the equivalence between the entropy and renormalized solutions for this equation.
Proof of Theorem 1.4.
First, we prove that the renormalized solution of (1-1) is also its entropy solution.
(1) Existence of entropy solutions. Now we choose
$T_{k}(u_{n}-\phi )$
as a test function in (3-1) for
$k\in \mathbb {N}^{+}$
and
$\phi \in C^{1}(\overline {\Omega }_{T})$
with
$\phi |_{\Sigma }=0$
. Set
$L=k+\|\phi \|_{L^{\infty }(\Omega _{T})}$
. We deduce that
$$ \begin{align*} & \int^{T}_{0}\int_{\Omega}{\mathcal{A}}(t,x,\nabla u_{n})\cdot \nabla T_{k}(u_{n}-\phi)\,dx\,dt\\ &\quad =\int^{T}_{0}\int_{\Omega}{\mathcal{A}}(t,x,\nabla T_{L}(u_{n}))\cdot \nabla T_{k}(T_{L}(u_{n})-\phi)\,dx\,dt \end{align*} $$
and
$$ \begin{align*} &\int^{T}_{0}\langle \partial_t u_{n},T_{k}(u_{n}-\phi)\rangle \,dt+\int^{T}_{0}\int_{\Omega}{\mathcal{A}}(t,x,\nabla T_{L}(u_{n}))\cdot \nabla T_{k}(T_{L}(u_{n})-\phi)\,dx\,dt \nonumber\\ &\quad =\int^{T}_{0}\int_{\Omega}f_{n}T_{k}(u_{n}-\phi)\,dx\,dt. \end{align*} $$
Since
$\partial _t u_{n}=\partial _t(u_{n}-\phi )+\partial _t\phi $
,
$$ \begin{align*} \int^{T}_{0}\langle \partial_t u_{n},T_{k}(u_{n}-\phi)\rangle \,dt &= \int_{\Omega}\Theta_{k}(u_{n}-\phi)(T)\,dx- \int_{\Omega}\Theta_{k}(u_{n}-\phi)(0)\,dx \\ &\quad +\int^{T}_{0}\langle \partial_t\phi,T_{k}(u_{n}-\phi)\rangle dt, \end{align*} $$
which yields that
$$ \begin{align} &\int_{\Omega}\Theta_{k}(u_{n}-\phi)(T)\,dx -\int_{\Omega}\Theta_{k}(u_{n}-\phi)(0)\,dx +\int^{T}_{0}\langle\phi_{t},T_{k}(u_{n}-\phi)\rangle\,dt \nonumber\\ & \quad +\int^{T}_{0}\int_{\Omega}{\mathcal{A}}(t,x,\nabla T_{L}(u_{n}))\cdot \nabla T_{k}(T_{L}(u_{n})-\phi)\,dx\,dt =\int^{T}_{0}\int_{\Omega}f_{n}T_{k}(u_{n}-\phi)\,dx\,dt. \end{align} $$
Recalling that
$u_{n}$
converges to u in
$C([0,T];L^{1}(\Omega ))$
, we have
$u_{n}(t)\rightarrow u(t)$
in
$L^{1}(\Omega )$
as
$n \to + \infty $
for all
$t \le T$
. Since
$\Theta _{k}$
is Lipschitz continuous, we get
$$ \begin{align*} \int_{\Omega}\Theta_{k}(u_{n}-\phi)(T)\,dx \rightarrow \int_{\Omega}\Theta_{k}(u-\phi)(T)\,dx \end{align*} $$
and
$$ \begin{align*} \int_{\Omega}\Theta_{k}(u_{n}-\phi)(0)\,dx \rightarrow \int_{\Omega}\Theta_{k}(u_{0}-\phi(0))\,dx \end{align*} $$
as
$n\rightarrow +\infty $
.
The fourth term on the left-hand side of (3-28) can be written as
$$ \begin{align*} \begin{aligned} &\int^{T}_{0}\int_{\Omega}{\mathcal{A}}(t,x,\nabla T_{L}(u_{n})) \cdot \nabla T_{k}(T_{L}(u_{n})-\phi)\,dx\,dt\\ &\quad=\int_{\{|T_{L}(u_{n})-\phi|\le k\}}{\mathcal{A}}(t,x,\nabla T_{L}(u_{n})) \cdot \nabla T_{L}(u_{n})\,dx\,dt\\ &\quad \quad -\int_{\{|T_{L}(u_{n})-\phi|\le k\}}{\mathcal{A}}(t,x,\nabla T_{L}(u_{n})) \cdot \nabla \phi \,dx\,dt\\ &\quad =:I_{1}+I_{2}, \end{aligned} \end{align*} $$
where
$$ \begin{align*} I_{1}:=\int_{\{|T_{L}(u_{n})-\phi|\le k\}}{\mathcal{A}}(t,x,\nabla T_{L}(u_{n})) \cdot \nabla T_{L}(u_{n})\,dx\,dt \end{align*} $$
and
$$ \begin{align*} I_{2}:=-\int_{\{|T_{L}(u_{n})-\phi|\le k\}}{\mathcal{A}}(t,x,\nabla T_{L}(u_{n})) \cdot \nabla \phi\,dx\,dt. \end{align*} $$
Estimate of
$I_{1}$
. We derive from (3-16) that
$$ \begin{align} &\int_{\{|T_{L}(u)-\phi|\le k\}}{\mathcal{A}}(t,x,\nabla T_{L}(u))\cdot\nabla T_{L}(u)\,dx\,dt \nonumber\\&\quad = \lim_{n \to+ \infty}\int_{\{|T_{L}(u_{n})-\phi|\le k\}}{\mathcal{A}}(t,x,\nabla T_{L}(u_{n})) \cdot \nabla T_{L}(u_{n})\,dx\,dt \nonumber\\&\quad =:\lim_{n \to+ \infty} I_{1}. \end{align} $$
Estimate of
$I_{2}$
. For convenience, we define
$$ \begin{align*} \eta_{n}:= -{\mathcal{A}}(t,x,\nabla T_{L}(u_{n})),\quad E_{n}:=\{ (t,x)\in\Omega_{T}:|T_{L}(u_{n})-\phi|\leq k\} \end{align*} $$
and
$$ \begin{align*}E:=\{ (t,x)\in \Omega_{T}:|T_{L}(u)-\phi|\leq k\}.\end{align*} $$
We write
$$ \begin{align*} I_{2}=\int_{E_{n}}\eta_{n}\cdot \nabla \phi \,dx\,dt= \int_{E}\eta_{n}\cdot \nabla \phi dz +\int_{E_{n}\setminus E}\eta_{n}\cdot \nabla \phi \,dx\,dt:= I_{21}+I_{22}. \end{align*} $$
Recalling the fact that, for
$n \to + \infty $
,
$$ \begin{align*} {\mathcal{A}}(t,x,\nabla T_{L}(u_{n})) \rightharpoonup {\mathcal{A}}(t,x,\nabla T_{L}(u)) \quad\mbox{weakly in } L^{1}(\Omega_{T}), \end{align*} $$
we have
$$ \begin{align*} \lim_{n \to+ \infty}I_{21}=-\int_{\{|T_{L}(u)-\phi|\leq k\}}{\mathcal{A}}(t,x,\nabla T_{L}(u))\cdot \nabla \phi \,dx\,dt. \end{align*} $$
Moreover,
$M^{*}(t,x,\eta )$
satisfies the condition
$ \lim _{|\eta |\rightarrow +\infty } ({M^{*}(t,x, |\eta |)}/{|\eta |})=+\infty $
. Then, for every
$\varepsilon>0$
, there exists a constant
$\Lambda>0$
such that
$$ \begin{align*} |\eta| \leq \varepsilon M^{*}(t,x,|\eta|) \quad \mbox{for all } |\eta|>\Lambda. \end{align*} $$
It follows from (3-5) that
$$ \begin{align*} |I_{22}| &\leq \|\nabla\phi\|_{L^{\infty}(\Omega_{T})}\int^{T}_{0}\int_{\Omega}|\eta_{n}| \chi_{E_{n}\setminus E} \,dx\,dt \\ &= C\bigg(\!\int_{\{|\eta_{n}|\leq \Lambda\}}|\eta_{n}|\chi_{E_{n}\setminus E}\,dx\,dt+\int_{\{|\eta_{n}|> \Lambda \}}|\eta_{n}|\chi_{E_{n}\setminus E}\,dx\,dt\bigg)\\ &\leq C\bigg(\Lambda\mbox{meas}(E_{n}\setminus E)+\varepsilon \int^T_0\int_{\Omega}M^{*}(t,x, |\eta_{n}|)\,dx\,dt\bigg)\\ &\leq C\Lambda \mbox{meas}(E_{n}\setminus E)+C\varepsilon. \end{align*} $$
By the arbitrariness of
$\varepsilon $
, we get
$$ \begin{align*} \lim_{n\rightarrow+\infty}|I_{22}|=0. \end{align*} $$
Therefore,
$$ \begin{align} \lim_{n\rightarrow+\infty} I_{2} =-\int_{\{|T_{L}(u)-\phi|\le k\}}{\mathcal{A}}(t,x,\nabla T_{L}(u))\cdot\nabla \phi\,dx\,dt. \end{align} $$
According to (3-29) and (3-30), we obtain
$$ \begin{align*} \int^{T}_{0}\int_{\Omega}{\mathcal{A}}(t,x,\nabla u)\cdot \nabla T_{k}(u-\phi)\,dx\,dt &= \int^{T}_{0}\int_{\Omega}{\mathcal{A}}(t,x,\nabla T_{L}(u))\cdot \nabla T_{k}(T_{L}(u)-\phi)\,dx\,dt\nonumber\\&= \int_{\{|T_{L}(u)-\phi|<k\}}{\mathcal{A}}(t,x,\nabla T_{L}(u))\cdot\nabla T_{L}(u)\,dx\,dt \nonumber\\& \quad -\int_{\{|T_{L}(u)-\phi|<k\}}{\mathcal{A}}(t,x,\nabla T_{L}(u))\cdot\nabla \phi\,dx\,dt\nonumber \\&= \lim_{n\rightarrow+\infty}(I_{1}+I_{2}). \end{align*} $$
Using the strong convergence of
$f_{n}$
, (3-10) and the Lebesgue dominated convergence theorem, we pass to the limits as
$n\rightarrow +\infty $
in the other term of (3-28) to conclude that
$$ \begin{align*} &&\int_{\Omega}\Theta_{k}(u-\phi)(T)\,dx -\int_{\Omega}\Theta_{k}(u_{0}-\phi(0))\,dx +\int^{T}_{0}\langle \partial_t\phi,T_{k}(u-\phi)\rangle \,dt \nonumber \\ & & \quad +\int^{T}_{0}\int_{\Omega}{\mathcal{A}}(t,x,\nabla u)\cdot\nabla T_{k}(u-\phi)\,dx\,dt = \int^{T}_{0}\int_{\Omega}fT_{k}(u-\phi)\,dx\,dt \end{align*} $$
for all
$k>0$
and
$\phi \in C^{1}(\overline {\Omega }_{T})$
with
$\phi |_{\Sigma }=0$
. Hence, our solution u satisfies Condition (E2). Since
$u_{n}$
is the distributional solution of Problem (3-1), then its limit u satisfies Condition (E1) naturally. This completes the proof.
We have established the existence of the entropy solution. Now, we further show its uniqueness.
(2) Uniqueness of entropy solutions. Suppose that v is another entropy solution of Problem (1-1). We show that
$u=v$
almost everywhere in
$\Omega _{T}$
. For
$\sigma>0, 0<\varepsilon \le 1$
, we define the function
$S_{\sigma ,\varepsilon }$
in
$W^{2,\infty }(\mathbb {R})$
by
$$ \begin{align*} S_{\sigma,\varepsilon}(r):= \begin{cases} r &\mbox{if } |r|\le \sigma,\\ \bigg(\sigma+\dfrac{\varepsilon}{2}\bigg)- \dfrac{r}{|r|}\dfrac{1}{2\varepsilon}\bigg(r- \dfrac{r}{|r|}(\sigma +\varepsilon)\bigg)^{2} & \mbox{if } \sigma< |r|<\sigma+\varepsilon, \\[7pt] \dfrac{r}{|r|}\bigg(\sigma+\dfrac{\varepsilon}{2}\bigg) &\mbox{if } |r|\ge \sigma+\varepsilon. \end{cases} \end{align*} $$
Clearly,
$$ \begin{align*} S^{\prime}_{\sigma,\varepsilon}(r)= \begin{cases} 1 &\mbox{if } |r|\le \sigma,\\[4pt] \dfrac{1}{\varepsilon}(\sigma+\varepsilon -|r|) &\mbox{if } \sigma < |r| <\sigma +\varepsilon, \\[6pt] 0 &\mbox{if } |r|\ge \sigma+\varepsilon. \end{cases} \end{align*} $$
Choosing
$\phi =S_{\sigma ,\varepsilon }(u_{n})\chi _{(0,\tau )}$
as a test function in (1-2) for entropy solution v gives
$$ \begin{align} H_{0}+H_{1}+H_{2} = H_{3}, \end{align} $$
where
$$ \begin{align*} &H_{0}:=\int_{\Omega}\Theta_{k}(v- S_{\sigma,\varepsilon}(u_{n}))(\tau)\,dx -\int_{\Omega}\Theta_{k}(u_{0}-S_{\sigma,\varepsilon}(u_{0n}))\,dx, \\ &H_{1}:=\int^{\tau}_{0}\langle \partial_t u_{n}, S^{\prime}_{\sigma,\varepsilon}(u_{n})T_{k}(v-S_{\sigma,\varepsilon}(u_{n}))\rangle \,dt, \\ &H_{2}:=\int^{\tau}_{0}\int_{\Omega}{\mathcal{A}}(t,x,\nabla v)\cdot \nabla T_{k}(v-S_{\sigma,\varepsilon}(u_{n}))\,dx\,dt, \\ &H_{3}:=\int^{\tau}_{0}\int_{\Omega}fT_{k}(v-S_{\sigma,\varepsilon}(u_{n}))\,dx\,dt. \end{align*} $$
By taking
$S^{\prime }_{\sigma ,\varepsilon }(u_{n})T_{k}(v-S_{\sigma ,\varepsilon }(u_{n}))\chi _{(0,\tau )}$
as a test function for Problem (3-1), we get
$$ \begin{align} &H_{1}= \int^{\tau}_{0}\langle \partial_t u_{n},S^{\prime}_{\sigma,\varepsilon}(u_{n})T_{k}(v-S_{\sigma,\varepsilon}(u_{n}))\rangle\,dt \nonumber \\ &\quad =\int^{\tau}_{0}\int_{\Omega}f_{n}S^{\prime}_{\sigma,\varepsilon}(u_{n}) T_{k}( v-S_{\sigma,\varepsilon}(u_{n}))\,dx\,dt \nonumber \\ & \quad \quad -\int^{\tau}_{0}\int_{\Omega}S^{\prime\prime}_{\sigma,\varepsilon}(u_{n})T_{k}(v-S_{\sigma,\varepsilon}(u_{n})){\mathcal{A}}(t,x,\nabla u_{n})\cdot\nabla u_{n}\,dx\,dt \nonumber \\ & \quad \quad -\int^{\tau}_{0}\int_{\Omega}S^{\prime}_{\sigma,\varepsilon}(u_{n}){\mathcal{A}}(t,x,\nabla u_{n})\cdot \nabla T_{k}(v-S_{\sigma,\varepsilon}(u_{n}))\,dx\,dt \nonumber \\ &\quad =: H_{11}+H_{12}+H_{13}. \end{align} $$
Combining with (3-31) and (3-32) gives
$$ \begin{align*} &\int_{\Omega}\Theta_{k}(v-S_{\sigma,\varepsilon}(u_{n}))(\tau)\,dx -\int_{\Omega}\Theta_{k}(u_{0}-S_{\sigma,\varepsilon}(u_{0n}))\,dx \\ & \qquad-\int^{\tau}_{0}\int_{\Omega}S^{\prime}_{\sigma,\varepsilon}(u_{n}){\mathcal{A}}(t,x,\nabla u_{n})\cdot \nabla T_{k}(v-S_{\sigma,\varepsilon}(u_{n}))\,dx\,dt \\ & \qquad +\int^{\tau}_{0}\int_{\Omega}{\mathcal{A}}(t,x,\nabla v)\cdot \nabla T_{k}(v-S_{\sigma,\varepsilon}(u_{n}))\,dx\,dt \\ &\quad\le \int^{\tau}_{0}\int_{\Omega}fT_{k}(v-S_{\sigma,\varepsilon}(u_{n}))\,dx\,dt \\ &\qquad -\int^{\tau}_{0}\int_{\Omega}f_{n}S^{\prime}_{\sigma,\varepsilon}(u_{n}) T_{k}( v-S_{\sigma,\varepsilon}(u_{n}))\,dx\,dt\\ &\qquad+\bigg|\! \int^{\tau}_{0}\int_{\Omega}S^{\prime\prime}_{\sigma,\varepsilon}(u_{n})T_{k}(v-S_{\sigma,\varepsilon}(u_{n})) {\mathcal{A}}(t,x,\nabla u_{n})\cdot\nabla u_{n}\,dx\,dt\bigg|. \end{align*} $$
That is,
$$ \begin{align} H_{0}+H_{13}+H_{2}\le H_{3}-H_{11}+|H_{12}|. \end{align} $$
We pass to the limits as
$\varepsilon \rightarrow 0$
,
$n \to + \infty $
and
$\sigma \to + \infty $
, successively.
We begin with
$\varepsilon \rightarrow 0$
. Since
$|\Theta _{k}(v-S_{\sigma ,\varepsilon }(u_{n}))(\tau )|\le k(|v|+|T_{\sigma +1}(u_{n})|)(\tau )$
,
$S^{\prime }_{\sigma ,\varepsilon }(r)\le T^{\prime }_{\sigma +1}(r)$
and
$|\nabla T_{k}(v-S_{\sigma ,\varepsilon }(u_{n}))| \le |\nabla T_{\sigma +k+1}(v)|+|\nabla T_{\sigma +1}(u_{n})|$
, the three terms of the left-hand side and the two terms of the right-hand side in (3-33) pass to the limit for
$\varepsilon \rightarrow 0$
by the Lebesgue dominated convergence theorem. Now we estimate
$|H_{12}|$
. Let
$R_{\sigma ,\varepsilon }$
be an even function such that
$R_{\sigma ,\varepsilon }(r)=r-S_{\sigma ,\varepsilon }(r)$
for
$r \ge 0$
. Then we choose
$R^{\prime }_{\sigma ,\varepsilon }(u_{n})\chi _{(0,\tau )}$
as a test function in (3-1) to get
$$ \begin{align*} & \int_{\Omega}R_{\sigma,\varepsilon}(u_{n})(\tau)\,dx- \int_{\Omega}R_{\sigma,\varepsilon}(u_{0n})\,dx\\ & \quad + \int^{\tau}_{0}\int_{\Omega}R^{\prime\prime}_{\sigma,\varepsilon}(u_{n}){\mathcal{A}}(t, x,\nabla u_{n})\cdot \nabla u_{n}\,dx\,dt =\int^\tau_{0}\int_{\Omega}f_{n}R^{\prime}_{\sigma,\varepsilon}(u_{n})\,dx\,dt. \end{align*} $$
Since
$R_{\sigma ,\varepsilon }(r)\ge 0$
,
$R_{\sigma ,\varepsilon }(r)\le |r|$
on the set
$\{|r|>\sigma \}$
and
$|S^{\prime \prime }_{\sigma ,\varepsilon }(r)|=R^{\prime \prime }_{\sigma ,\varepsilon }(r)$
, we obtain that
$$ \begin{align*} \int^{\tau}_{0}\int_{\Omega}|S^{\prime\prime}_{\sigma,\varepsilon}(u_{n})|{\mathcal{A}}(t,x,\nabla u_{n})\cdot \nabla u_{n}\,dx\,dt &= \int^{\tau}_{0}\int_{\Omega}R^{\prime\prime}_{\sigma,\varepsilon}(u_{n}){\mathcal{A}}(t,x,\nabla u_{n})\cdot \nabla u_{n}\,dx\,dt \\ &\le \int_{\{|u_{n}|>\sigma\}}|f_{n}|\,dx\,dt+\int_{\{|u_{0n}|>\sigma\}}|u_{0n}|\,dx. \end{align*} $$
Therefore,
$$ \begin{align*} |H_{12}|\le k\bigg( \int_{\{|u_{n}|>\sigma\}}|f_{n}| \,dx\,dt+ \int_{\{|u_{0n}|>\sigma\}}|u_{0n}|\,dx\bigg). \end{align*} $$
Recalling that
${\mathcal {A}}(t,x,0)=0$
,
$T^{\prime }_{\sigma }(u_{n}){\mathcal {A}}(t,x,\nabla u_{n})={\mathcal {A}}(t,x,\nabla T_{\sigma }(u_{n}))$
, and letting
$\varepsilon \rightarrow 0$
in (3-33), we obtain
$$ \begin{align*} &\int_{\Omega} \Theta_{k}( v-T_{\sigma}(u_{n}))(\tau)\,dx -\int_{\Omega}\Theta_{k}(u_{0}-T_{\sigma}(u_{0n}))\,dx\\ &\quad \quad +\int^{\tau}_{0}\int_{\Omega}({\mathcal{A}}(t,x,\nabla v)-{\mathcal{A}}(t,x,\nabla T_{\sigma}(u_{n})))\cdot \nabla T_{k}(v-T_{\sigma}(u_{n}))\,dx\,dt \\ &\quad \le \int^{\tau}_{0}\int_{\Omega}(f-f_{n}T^{\prime}_{\sigma}(u_{n}))T_{k}(v-T_{\sigma}(u_{n}))\,dx\,dt \\ & \quad \quad +k \bigg(\!\int_{\{|u_{n}|>\sigma\}}|f_{n}|\,dx\,dt+\int_{\{|u_{0n}|>\sigma\}}|u_{0n}|\,dx\bigg). \end{align*} $$
Next, we take
$n\rightarrow +\infty $
. Owing to the fact that
$\nabla T_{\sigma }(u_{n})\rightarrow \nabla T_{\sigma }(u)$
almost everywhere in
$\Omega _{T}$
as
$n\rightarrow +\infty $
, by Fatou’s lemma and the Lebesgue dominated convergence theorem, sending
$n\rightarrow +\infty $
in the above inequality gives
$$ \begin{align*} &\int_{\Omega} \Theta_{k}( v-T_{\sigma}(u))(\tau)\,dx -\int_{\Omega}\Theta_{k}(u_{0}-T_{\sigma}(u_{0}))\,dx\\ &\quad \quad +\int^{\tau}_{0}\int_{\Omega}({\mathcal{A}}(t,x,\nabla v)-{\mathcal{A}}(t,x,\nabla T_{\sigma}(u)))\cdot \nabla T_{k}(v-T_{\sigma}(u))\,dx\,dt \\ &\quad \le \int^{\tau}_{0}\int_{\Omega}f(1-T^{\prime}_{\sigma}(u))T_{k}(v-T_{\sigma}(u))\,dx\,dt \\ & \quad \quad +k \bigg(\!\int_{\{|u|>\sigma\}}|f|\,dx\,dt +\int_{\{|u_{0}|>\sigma\}}|u_{0}|\,dx\bigg). \end{align*} $$
Now we let
$\sigma \to + \infty $
. Since
$$ \begin{align*} |\Theta_{k}(v-T_{\sigma}(u))(\tau)|\le k(|v(\tau)|+|u(\tau)|) \quad\mbox{and} \quad |\Theta_{k}(u_{0}-T_{\sigma}(u_{0}))|\le k|u_{0}|, \end{align*} $$
by the Lebesgue dominated convergence theorem, we obtain
$$ \begin{align*} \int_{\Omega}\Theta_{k}(v-T_{\sigma}(u))(\tau)\,dx\rightarrow \int_{\Omega}\Theta_{k}(v-u)(\tau)\,dx, \quad \int_{\Omega}\Theta_{k}(u_{0}-T_{\sigma}(u_{0}))\,dx \rightarrow 0 \end{align*} $$
and
$$ \begin{align*} \int_{\{|u|>\sigma\}}|f|\,dx\,dt +\int_{\{|u_{0}|>\sigma\}}|u_{0}|\,dx \rightarrow 0. \end{align*} $$
Therefore, we conclude that
$$ \begin{align*} \int_{\Omega}\Theta_{k}(v-u)(\tau)\,dx+\int^{\tau}_{0}\int_{\Omega}({\mathcal{A}}(t,x,\nabla v)-{\mathcal{A}}(t,x,\nabla u))\cdot \nabla T_{k}(v-u)\,dx\,dt \le 0, \end{align*} $$
which implies that
$$ \begin{align*} \int_{\Omega}\Theta_{k}(v-u)(\tau)\,dx +\int_{\{|u|\le {k}/{2},|v|\le {k}/{2}\}}({\mathcal{A}}(t,x,\nabla v)-{\mathcal{A}}(t,x,\nabla u))\cdot \nabla (v-u)\,dx\,dt \le 0. \end{align*} $$
Using the nonnegativity of the two terms in the above inequality, we conclude that
$u=v$
almost everywhere in
$\Omega _{T}$
. Therefore, we obtain the uniqueness of entropy solutions.
Acknowledgment
The authors thank the reviewer for carefully reading the manuscript and for their very useful suggestions.
