The nonnegative weak solution of a degenerate parabolic equation with variable exponent growth order

Abstract

A degenerate parabolic equation of the form

$$\bigl( \vert v \vert ^{\beta-1}v\bigr)_{t}= \operatorname{div} \bigl(b(x,t) \vert \nabla v \vert ^{p(x,t)-2}\nabla v \bigr)+\nabla\vec{g}\cdot\nabla\vec{\gamma}(v) $$

is considered, where $\vec{g}=\{g^{i}(x,t)\}$, $\vec{\gamma}(v)=\{ \gamma_{i}(v)\}$. If the diffusion coefficient $b(x,t)\geq0$ is degenerate on the boundary, by adding some restrictions on $b(x,t)$ and g⃗, the existence and uniqueness of weak solutions are proved. Based on the uniqueness, the stability of weak solutions can be proved without any boundary condition.

1 Introduction and the main results

Consider the degenerate parabolic problem with exponent variable growth order

$$ \bigl( \vert v \vert ^{\beta-1}v\bigr)_{t}= \operatorname{div} \bigl(b(x,t) \vert \nabla v \vert ^{p(x,t)-2}\nabla v \bigr)+ \nabla\vec{g}\cdot\nabla\vec{\gamma}(v),\quad (x,t)\in Q_{T}=\varOmega \times(0,T), $$

(1.1)

where $b(x,t)$ and $p(x,t)$ are $C(\overline{Q_{T}})$ nonnegative functions, $\vec{g}=\{g^{i}(x,t)\}$, $\vec{\gamma}(v)=\{\gamma _{i}(v)\}$, $\beta>0$. We denote that $p^{-}=\min_{(x,t)\in\overline {Q_{T}}}p(x,t)>1$ and $p^{+}=\max_{(x,t)\in\overline{Q_{T}}}p(x,t)$ in this paper. The initial value matching up to equation (1.1) is

$$ \vert v \vert ^{\beta-1}v(x,0)= \bigl\vert v_{0}(x) \bigr\vert ^{\beta-1}v_{0}(x),\quad x\in\varOmega. $$

(1.2)

While the Dirichlet boundary value condition

$$ v(x,t)=0,\quad (x,t)\in\partial\varOmega\times(0,T) $$

(1.3)

is dispensable.

If $g=0$, equation (1.1) arises from the branches of flows of electro-rheological or thermo-rheological fluids (see [1–3]), and the processing of digital images [4–15]. If the variable exponent $p(x,t)$ is replaced by a constant p, equation (1.1) becomes the well-known non-Newtonian polytropic filtration equation with orientated convection [16], as well as the convection-diffusion-reaction equation in which the variable can be interpreted as temperature for heat transfer problems, concentration for dispersion problems, etc. [17]. Now, let us give some details in part of the above references. Ye and Yin studied the propagation profile for the equation

$$u_{t}=\operatorname{div}\bigl( \bigl\vert \nabla u^{m} \bigr\vert ^{p-2}\nabla u^{m}\bigr)-\vec{\beta}(x)\cdot \nabla u^{q}, $$

in which the orientation of the convection was specified to be either the convection with counteracting diffusion or the convection with promoting diffusion, that is, $\vec{\beta}(x)\cdot(-x)\geq0$ or $\vec{\beta}(x)\cdot x\geq0$, respectively [16]. Guo, Li, and Gao considered the following evolutionary $p(x)$-Laplacian equation:

$$v_{t} = \operatorname{div}\bigl( \vert \nabla v \vert ^{p(x)-2} \nabla v\bigr) + \vert v \vert ^{r-2}v,\quad (x,t)\in Q_{T}, $$

subject to homogeneous Dirichlet boundary condition, where $r > 1$ is a constant. By using the energy estimate method, the regularity of weak solutions and blow-up in finite time were revealed in [7]. Antontsev and Shmarev have published a series of papers [8–13] on the homogeneous Dirichlet problem for the doubly nonlinear parabolic equation

$$ v_{t} = \operatorname{div}\bigl(b(x,t,v) \vert v \vert ^{\alpha(x,t)} \vert \nabla v \vert ^{p(x,t)-2}\nabla v\bigr) + f(x,t),\quad (x,t)\in Q_{T}, $$

provided that $a(x,t,v)\geq a>0$. They established conditions on the data that guarantee the comparison principle and uniqueness of bounded weak solutions in suitable Orlicz–Sobolev spaces subject to some additional restrictions [12]. Gao, Chu, and Gao in [14] studied the nonlinear diffusion equation

$$v_{t} = \operatorname{div}\bigl( \vert \nabla v \vert ^{p(x,t)-2} \nabla v+b(x,t)\nabla v\bigr) +f(v),\quad (x,t)\in Q_{T}, $$

with the homogeneous Dirichlet boundary condition (1.3), where f is a continuous function satisfying

$$\bigl\vert f(v) \bigr\vert \leq a_{0} \vert v \vert ^{\alpha-1} , $$

with $a_{0}>0$ and $\alpha>1$. They constructed suitable function spaces and used Galerkin’s method to obtain the existence of weak solutions. It is worth pointing out that the requirement on $p_{t} (x,t)$ is only negative and integrable, which is a weaker condition than the corresponding conditions appearing in other papers. Recently, Liu and Dong [15] generalized [14]’s result to a more general equation

$$v_{t} = \operatorname{div}\bigl( \bigl\vert \nabla v^{m} \bigr\vert ^{p(x,t)-2}\nabla v^{m}+b(x,t)\nabla v^{m}\bigr) +v^{q(x,t)},\quad (x,t)\in Q_{T} $$

and gave a classification of the weak solutions. In addition, the equation arising from the double phase obstacle problems of the type

$$v_{t} = \operatorname{div}\bigl(a(x) \vert \nabla v \vert ^{p-2} \nabla v+b(x) \vert \nabla v \vert ^{q-2}\nabla v\bigr) +f(x,t,v, \nabla v),\quad (x,t)\in Q_{T} $$

has gained a wide attention [18, 19] etc., where $a(x)+b(x)>0$.

In recent years, we have been interested in the well-posedness of weak solutions to the nonlinear equation

$$ v_{t} = \operatorname{div}\bigl(b(x) \vert \nabla v \vert ^{p(x)-2}\nabla v\bigr) + f(x,t, v, \nabla v),\quad (x,t)\in Q_{T}, $$

(1.4)

with some restrictions in $f(x,t,v, \nabla v)$. Different from other researchers’ works [7–15], in which $b(x)=1$ or $b(x)>b^{-}>0$, where $b^{-}=\min_{x\in\overline{\varOmega}}b(x)$, we only assumed that

$$ b(x)>0,\quad x\in\varOmega,\qquad b(x)=0,\quad x\in\partial\varOmega, $$

and proved that the stability of weak solutions may be independent of the Dirichlet boundary value condition (1.3). One might refer to [20–24] for the details.

In this paper, for any $t\in[0,T]$, we assume

$$ b(x,t)>0,\quad x\in\varOmega,\qquad b(x,t)=0,\quad x\in\partial\varOmega. $$

(1.5)

Comparing with equation (1.4), equation (1.1) is with the nonlinearity of $|v|^{\beta-1}v$, with the diffusion coefficient $b(x,t)$ and the variable exponent $p(x,t)$ depending on time variable t, and with a more complicate convection term $\nabla\vec{g}\cdot \nabla\vec{\gamma}(v)$. These nonlinearities not only bring some essential changes to the proof of the existence, but also add difficulties to proving the stability of weak solutions. The readers will see that, in order to overcome these difficulties, a new technique based on the mean value theorem is posed to prove the uniqueness of weak solution, another new technique based on the proof by contradiction is introduced. Both of them supply a new method to prove uniqueness of weak solution for the nonlinear degenerate parabolic equations.

Definition 1.1

A function $v(x,t)$ is said to be a weak solution of equation (1.1) with initial value (1.2), provided that $v(x,t)$ satisfies

$$ \begin{gathered}v \in L_{\mathrm{loc}}^{\infty}\bigl(0,T; W^{1,p(x,t)}( \varOmega)\bigr),\qquad v\in W_{\mathrm{loc}}^{1,2}\bigl((0,T), L^{2}( \varOmega)\bigr),\\ b(x,t) \vert \nabla v \vert ^{p(x,t)} \in L^{1}(Q_{T}),\end{gathered} $$

(1.6)

and for $\forall\phi(x,t) \in{C_{0}^{1}}(Q_{T})$,

$$ \begin{aligned}[b] & \iint_{Q_{T}} \bigl(- \vert v \vert ^{\beta-1}v \phi_{t}\bigr) \,dx\,dt + \iint_{Q_{T}} b(x,t) \vert \nabla v \vert ^{p(x,t) - 2} \nabla v \cdot\nabla\phi \,dx\,dt \\ &\qquad+ \sum_{i=1}^{N} \iint_{Q_{T}}g^{i}(x,t)\gamma_{i}(v) \phi_{x_{i}}\,dx\,dt \\ &\quad=\sum_{i=1}^{N} \iint_{Q_{T}}\gamma_{i}(v)g^{i}(x,t) \phi(x,t)\,dx\,dt. \end{aligned} $$

(1.7)

Initial value (1.2) is true in the sense

$$ \lim_{t\rightarrow0} \int_{\varOmega} \vert v \vert ^{\beta-1}v(x,t)\varphi(x) \,dx= \int_{\varOmega} \vert v \vert ^{\beta-1}v_{0}(x) \varphi(x) \,dx,\quad\forall\varphi (x)\in C_{0}^{\infty}(\varOmega). $$

(1.8)

The main results are the following theorems.

Theorem 1.2

If$p^{-}\geq2$, $b(x,t)$satisfying (1.5), $v_{0}(x)\in L^{\infty}(\varOmega)$is nonnegative for$i\in\{ 1, 2, \ldots, N\}$, $\gamma_{i}(s)$is a$C^{1}$function satisfying$|\gamma_{i}'(s)|^{2}|s|^{1-\beta}\leq c$for$i=1,2,\ldots, N$, $g^{i}(x,t)$satisfies

$$ \int_{\varOmega}\frac{p(x,t)-2}{p(x,t)} \Biggl(\sum _{i=1}^{N}g^{i}(x,t)b(x,t)^{-\frac{2}{p(x,t)}} \Biggr)^{\frac {p(x,t)}{p(x,t)-2}}\,dx\leq c(T), $$

(1.9)

then there is a nonnegative weak solution of equation (1.1) with initial value (1.2) in the sense of Definition1.1.

Theorem 1.3

Let$u(x,t)$and$v(x,t)$be two weak solutions of equation (1.1) with the different initial values$u_{0}(x)$and$v_{0}(x)$respectively, $0< m\leq\|u\|_{L^{\infty}(Q_{T})}\leq M$, $0< m\leq\|v\|_{L^{\infty}(Q_{T})}\leq M$. If$p^{-}>1$, $\gamma_{i}(s)$is a Lipschitz function, $b(x,t)$satisfies (1.5), and

$$ \Biggl\vert \sum_{i=1}^{N} \frac{\partial g^{i}(x,t)}{\partial x_{i}} \Biggr\vert \leq cb(x,t)^{\frac{\alpha_{1}}{p(x,t)}}, $$

(1.10)

then there exists a constant$\alpha_{1}\geq2p^{+}$such that

$$ \begin{aligned}[b] &\int_{\varOmega}b(x,t)^{\frac{\alpha_{1}}{p(x,t)}} \bigl\vert \vert u \vert ^{\beta -1}u(x,t) - \vert v \vert ^{\beta-1}v(x,t) \bigr\vert ^{2}\,dx \\ &\quad\leq \int_{\varOmega}b(x,0)^{\frac{\alpha_{1}}{p(x,0)}} \bigl\vert \vert u_{0} \vert ^{\beta-1}u_{0}(x) - \vert v_{0} \vert ^{\beta-1}v_{0}(x) \bigr\vert ^{2}\,dx,\quad \forall t \in[0,T). \end{aligned} $$

(1.11)

Theorem 1.4

Let$u(x,t)$and$v(x,t)$be two weak solutions of equation (1.1) with the different initial values$u_{0}(x)$and$v_{0}(x)$respectively, $\gamma_{i}(s)$be a Lipschitz function. Suppose that$g^{i}(x,t)$satisfies (1.10) and

$$ \Biggl\vert \sum_{i=1}^{N}g^{i}(x,t) \Biggr\vert \leq cb(x,t)^{\frac{1}{p(x,t)}} $$

(1.12)

and one of the following conditions is true:

• (i) $\beta\leq1$.

• (ii) For$1\leq i\leq N$, $\gamma_{i}(s)$satisfies

  $$ \bigl\vert \gamma_{i}(t_{1})-\gamma_{i}(t_{2}) \bigr\vert \leq c \bigl\vert \vert t_{1} \vert ^{\beta-1}t_{1}- \vert t_{2} \vert ^{\beta -1}t_{2} \bigr\vert . $$

  (1.13)

Then

$$ \begin{aligned}[b] &\int_{\varOmega}\bigl\vert \vert u \vert ^{\beta-1}u(x,t) - \vert v \vert ^{\beta-1}v(x,t) \bigr\vert \,dx \\&\quad\leq \int_{\varOmega}\bigl| \vert u_{0} \vert ^{\beta-1}u_{0}(x) - \vert v_{0} \vert ^{\beta -1}v_{0}(x) \bigr|\,dx,\quad\forall t \in[0,T).\end{aligned} $$

(1.14)

Conditions (1.9), (1.10), and (1.12) all reflect the internal mutually dependent relationships between the diffusion coefficient $b(x,t)$ and the convective coefficients $g^{i}(x,t)$. Such an internal mutually dependent relationship that can affect the finite propagation has been studied in [16], while the internal mutually dependent relationships between the diffusion coefficient and the convection term arise in mathematics finance model for studying the agent’s decision under the risk [25].

At the end of introduction, it might be advisable to summarize briefly. First, as a classical work on the well-posed results of the solution of a nonlinear parabolic equation, there are many papers devoted to this problem (one can refer to [26–28] and the references therein). Secondly, the model studied in this paper is a parabolic equation with variable exponential term; we would like to point out that more details on the structural characteristics and the physical background of the variable exponential term have been described in [29–33], etc. Thirdly, one can see that the new method to prove uniqueness of weak solution can be generalized to study the double phase obstacle problems.

2 The existence of weak solutions

Let us consider the approximate initial-boundary value problem

$$\begin{aligned}& \bigl( \vert v \vert ^{\beta-1}v\bigr)_{t}= \operatorname{div} \bigl(\bigl(b(x,t)+\varepsilon\bigr) \vert \nabla v \vert ^{p(x,t)-2}\nabla v \bigr)+\nabla\vec{g}\cdot\nabla\vec{\gamma}(v), \end{aligned}$$

(2.1)

$$\begin{aligned}& \vert v \vert ^{\beta-1}v(x,0)= \vert v_{0} \vert ^{\beta-1}v_{0}(x), \quad x\in\varOmega, \end{aligned}$$

(2.2)

$$\begin{aligned}& v(x,t)=0,\quad (x,t)\in\partial\varOmega\times(0,T). \end{aligned}$$

(2.3)

Definition 2.1

If $u(x,t)$ satisfies

$$ v \in L_{\mathrm{loc}}^{\infty}\bigl(0,T; W_{0}^{1,p(x,t)}(\varOmega)\bigr), \qquad v\in W_{\mathrm{loc}}^{1,2} \bigl(0,T; L^{2}(\varOmega)\bigr), $$

(2.4)

and for any $\phi(x,t) \in{C_{0}^{1}}(Q_{T})$, there holds

$$ \begin{aligned}[b] &{-} \iint_{Q_{T}} \vert v \vert ^{\beta-1}v \phi_{t} \,dx\,dt+ \iint_{Q_{T}} \bigl(b(x,t)+\varepsilon\bigr) \vert \nabla v \vert ^{p(x,t) - 2}\nabla v \cdot\nabla\phi \,dx\,dt \\ &\qquad+ \sum_{i=1}^{N} \iint_{Q_{T}}\gamma_{i}(v)\phi_{x_{i}}\,dx\,dt \\ &\quad=\sum_{i=1}^{N} \iint_{Q_{T}}\gamma_{i}(v)g^{i}(x,t) \phi(x,t)\,dx\,dt. \end{aligned} $$

(2.5)

Then we say that $v(x,t)$ is said to be the weak solution of problem (2.1)–(2.3).

For any $k>0$, let

$$a_{k}=k^{2-\beta},\qquad b_{k}=k^{1-\beta} \frac{3-\beta}{2},\quad k=1,2,\ldots. $$

$\varphi_{k}(v)$ is an even function and is defined as

$$\varphi_{k}(v)=\left \{ \textstyle\begin{array}{l@{\quad}l} \beta v^{\beta-1}, & v\geq k^{-1}, \\ \beta(a_{k}v^{2}+b_{k} v),& 0\leq v< k^{-1}. \end{array}\displaystyle \right . $$

Then $\varphi_{k}(v)\in C^{1}(\mathbb{R})$, $\varphi_{k}(v)\rightarrow \beta v^{\beta-1}$ as $k\rightarrow\infty$. Instead of (2.1)–(2.3), we now consider the following problem:

$$\begin{aligned}& \varphi_{k}(v)v_{t}=\operatorname{div} \biggl( \bigl(b(x,t)+\varepsilon\bigr) \biggl( \vert \nabla v \vert ^{2}+ \frac{1}{k}\biggr)^{\frac{p(x,t)-2}{2}}\nabla v \biggr)+\nabla\vec {g}\cdot\nabla \vec{\gamma}(v), \end{aligned}$$

(2.6)

$$\begin{aligned}& {v_{k}}(x,t) = 0,\quad (x,t) \in\partial\varOmega \times(0,T), \end{aligned}$$

(2.7)

$$\begin{aligned}& v_{k}(x,0) = v_{0k}(x),\quad x\in\varOmega, \end{aligned}$$

(2.8)

where $\|v_{0k}(x)-v_{0}(x)\|_{p^{+}(0)}\rightarrow0$ as $k\rightarrow 0$, and $p^{+}(0)=\max_{x\in\overline{\varOmega}}p(x,0)$. From [8–14], there is a unique solution $v_{k\varepsilon}$ of initial-boundary value problem (2.6)–(2.8). Let $k\rightarrow\infty$. If $v_{0}(x)\in L^{\infty}(\varOmega)$ is nonnegative, similar to the process subject to the existence of weak solutions in [12](also[34]), one can prove that there is a nonnegative weak solution $v_{\varepsilon}\in L^{1}(0,T;W_{0}^{1,p(x,t)}(\varOmega))$ to initial-boundary value problem (2.1)–(2.3) in the sense of Definition 2.1. Moreover,

$$ \Vert v_{k\varepsilon} \Vert _{L^{\infty}(Q_{T})}\leq c,\qquad \Vert v_{\varepsilon} \Vert _{L^{\infty}(Q_{T})}\leq c. $$

(2.9)

Proof of Theorem 1.2

Let us choose $v_{\varepsilon}$ as a test function. Then

$$ \begin{aligned}[b] &\frac{\beta}{\beta+1} \int_{\varOmega}v_{\varepsilon}^{\beta+1}\,dx + \iint_{Q_{T}} \bigl(b(x,t)+\varepsilon\bigr) \vert \nabla v_{\varepsilon} \vert ^{p(x,t)}\,dx\,dt \\ &\qquad- \iint_{Q_{T}} v_{\varepsilon}\nabla\vec{g}\cdot\nabla\vec { \gamma}(v_{\varepsilon}) \,dx\,dt \\ &\quad= \frac{\beta}{\beta+1} \int_{\varOmega}v_{0}(x)^{\beta+1}\,dx. \end{aligned} $$

(2.10)

Since

$$ \begin{aligned}[b] &{-} \iint_{Q_{T}} v_{\varepsilon}\nabla\vec{g}\cdot\nabla\vec { \gamma}(v_{\varepsilon}) \,dx\,dt \\ &\quad=\sum_{i = 1}^{N} \iint_{Q_{T}} v_{\varepsilon}g^{i}(x,t) \frac {\partial\gamma_{i}(v_{\varepsilon})}{\partial x_{i}} \,dx\,dt \\ &\quad=\sum_{i = 1}^{N} \biggl[ \iint_{Q_{T}} v_{\varepsilon}\frac{\partial (\gamma_{i}(v_{\varepsilon})g^{i}(x,t))}{\partial x_{i}} \,dx\,dt- \iint _{Q_{T}} v_{\varepsilon}a_{i}(v_{\varepsilon}) \frac{\partial g^{i}(x,t)}{\partial x_{i}} \,dx\,dt \biggr] \\ &\quad= - \sum_{i = 1}^{N} \iint_{Q_{T}} \frac{\partial v_{\varepsilon}}{\partial x_{i}}g^{i}(x,t) \gamma_{i}(v_{\varepsilon})\,dx\,dt-\sum_{i=1}^{N} \iint_{Q_{T}} v_{\varepsilon}\gamma_{i}(v_{\varepsilon}) \frac {\partial g^{i}(x,t)}{\partial x_{i}} \,dx\,dt \\ &\quad= - \sum_{i = 1}^{N} \int_{\varOmega}\frac{\partial}{\partial x_{i}} \int _{0}^{v_{\varepsilon}} \gamma_{i}(v_{\varepsilon})\,ds\,dx- \sum_{i=1}^{N} \iint _{Q_{T}} v_{\varepsilon}\gamma_{i}(v_{\varepsilon}) \frac{\partial g^{i}(x,t)}{\partial x_{i}} \,dx\,dt \\ &\quad= -\sum_{i=1}^{N} \iint_{Q_{T}} v_{\varepsilon}\gamma_{i}(v_{\varepsilon}) \frac{\partial g^{i}(x,t)}{\partial x_{i}} \,dx\,dt, \end{aligned} $$

(2.11)

we have

$$ \frac{\beta}{\beta+1} \int_{\varOmega}v_{\varepsilon}^{\beta+1}\,dx + \iint_{Q_{T}} \bigl(b(x,t)+\varepsilon\bigr) \vert \nabla v_{\varepsilon} \vert ^{p(x,t)}\,dx\,dt \leqslant c, $$

(2.12)

which implies

$$ \iint_{Q_{T}} b(x,t) \vert \nabla v_{\varepsilon} \vert ^{p(x,t)}\,dx\,dt\leqslant c. $$

(2.13)

Moreover, let us multiply (2.1) with $v_{\varepsilon t}$, and obtain

$$ \begin{aligned}[b] &\beta \int_{\varOmega}v_{\varepsilon}^{\beta-1}(v_{{\varepsilon}t})^{2}\,dx \\ &\quad= \int_{\varOmega}\operatorname{div}\bigl(\bigl(\rho(x,t)+\varepsilon\bigr) \vert \nabla v_{\varepsilon} \vert ^{p(x,t)-2}\nabla v_{\varepsilon} \bigr)v_{{\varepsilon}t}\,dx+ \int_{\varOmega}\nabla\vec {g}\cdot\nabla\vec{\gamma}(v_{\varepsilon})v_{\varepsilon t}\,dx. \end{aligned} $$

(2.14)

Since

$$ \begin{aligned}[b] & \int_{\varOmega}\operatorname{div}\bigl(\bigl(b(x,t)+\varepsilon\bigr) \vert \nabla v_{\varepsilon} \vert ^{p(x,t)-2}\nabla v_{\varepsilon} \bigr)v_{\varepsilon t}\,dx \\ &\quad=-\frac{1}{2} \int_{\varOmega}\bigl(b(x,t)+\varepsilon\bigr) \vert \nabla v_{\varepsilon} \vert ^{p(x,t)-2} \vert \nabla v_{\varepsilon} \vert ^{2}_{t}\,dx \\ &\quad=-\frac{1}{2} \int_{\varOmega}\bigl(b(x,t)+\varepsilon\bigr)\frac{d}{dt} \int _{0}^{ \vert \nabla v_{\varepsilon} \vert ^{2}}s^{\frac{p(x,t)-2}{2}}\,ds\,dx\\ &\qquad+ \frac{1}{2} \int _{\varOmega}\bigl(b(x,t)+\varepsilon\bigr) \int_{0}^{ \vert \nabla v_{\varepsilon} \vert ^{2}}\frac{d}{dt}s^{\frac{p(x,t)-2}{2}}\,ds\,dx, \end{aligned} $$

(2.15)

and by $|\gamma_{i}'(s)|^{2}|s|^{1-\beta}\leq c$, $p^{-}\geq2$ and by (1.9), using the Young inequality, we have

$$ \begin{aligned}[b] & \Biggl\vert \int_{\varOmega}\sum_{i=1}^{N} \frac{\partial \gamma_{i}(v_{\varepsilon})}{\partial x_{i}}g^{i}(x,t)v_{\varepsilon t}\,dx \Biggr\vert \\ &\quad\leq \int_{\varOmega}\sum_{i=1}^{N} \bigl\vert \gamma_{i}'(v_{\varepsilon }) \bigr\vert \bigl\vert g^{i}(x,t)v_{\varepsilon x_{i}} \bigr\vert \vert v_{\varepsilon t} \vert \,dx \\ &\quad\leq\frac{\beta}{2} \int_{\varOmega}v_{\varepsilon}^{\beta -1}(v_{\varepsilon t})^{2}\,dx+ \frac{2}{\beta} \int_{\varOmega}\sum_{i=1}^{N} \bigl\vert \gamma_{i}'(v_{\varepsilon})g^{i}(x,t)v_{\varepsilon x_{i}} \bigr\vert ^{2} \vert v_{\varepsilon} \vert ^{1-\beta} \,dx \\ &\quad\leq\frac{\beta}{2} \int_{\varOmega}v_{\varepsilon}^{\beta -1}(v_{\varepsilon t})^{2}\,dx \\ &\qquad+c \int_{\varOmega} \Biggl[\frac{p(x,t)-2}{p(x,t)} \Biggl(\sum _{i=1}^{N}g^{i}(x,t)b(x,t)^{-\frac{2}{p(x,t)}} \Biggr)^{\frac {p(x,t)}{p(x,t)-2}}+\frac{2}{p(x,t)}b(x,t) \vert \nabla v_{\varepsilon } \vert ^{p(x,t)} \Biggr]\,dx. \end{aligned} $$

(2.16)

From (2.14)–(2.16), we extrapolate that

$$\begin{aligned}& \frac{\beta}{2} \int_{\varOmega}v^{\beta-1}(u_{\varepsilon t})^{2}\,dx+ \frac{1}{2} \int_{\varOmega}\bigl(b(x,t)+\varepsilon\bigr)\frac {d}{dt} \int_{0}^{ \vert \nabla v_{\varepsilon} \vert ^{2}}s^{\frac{p(x,t)-2}{2}}\,ds\,dx \\ & \quad\leq c \int_{\varOmega} \biggl[\frac{p(x,t)-2}{p(x,t)} \bigl(g^{i}(x,t)b(x,t)^{-\frac{2}{p(x,t)}} \bigr)^{\frac {p(x,t)}{p(x,t)-2}}+\frac{2}{p(x,t)}b(x,t) \vert \nabla v_{\varepsilon } \vert ^{p(x,t)} \biggr]\,dx \\& \qquad{}+ \biggl\vert \frac{1}{2} \int_{\varOmega}\bigl(b(x,t)+\varepsilon\bigr) \int _{0}^{ \vert \nabla v_{\varepsilon} \vert ^{2}}\frac{d}{dt}s^{\frac{p(x,t)-2}{2}}\,ds\,dx \biggr\vert \\& \quad\leq c. \end{aligned}$$

Then

$$ \bigl\Vert \bigl(v_{\varepsilon}^{\frac{\beta+1}{2}} \bigr)_{t} \bigr\Vert _{L^{2}(Q_{T})}=\frac{\beta+1}{2} \bigl\Vert v_{\varepsilon}^{\frac{\beta -1}{2}}v_{\varepsilon t} \bigr\Vert _{L^{2}(Q_{T})}\leq c, $$

(2.17)

and

$$ \begin{aligned}[b] \iint_{Q_{T}} \vert v_{\varepsilon t} \vert ^{2}\,dx\,dt&\leq \int_{0}^{T} \int_{\varOmega }v_{\varepsilon}^{\beta-1} \vert v_{\varepsilon t} \vert ^{2}v_{\varepsilon}^{1-\beta}\,dx\,dt\leq \Vert v_{\varepsilon} \Vert _{L^{\infty}(Q_{T})}^{1-\beta } \int_{0}^{T} \int_{\varOmega}v_{\varepsilon}^{\beta-1} \vert v_{\varepsilon t} \vert ^{2}\,dx\,dt\hspace{-24pt}\\&\leq c.\end{aligned} $$

(2.18)

From (2.12), (2.18), we are able to extrapolate that $v_{\varepsilon}\rightarrow v$ a.e. in $Q_{T}$. Accordingly, $\gamma _{i}(v_{\varepsilon})\rightarrow\gamma_{i}(v)$ a.e. in $Q_{T}$.

Let $\varepsilon\rightarrow0$ in (2.10). Similar to that in [35], which is subject to the evolutionary p-Laplacian equation, it is not difficult to deduce that

$$\bigl(b(x,t)+\varepsilon\bigr) \vert \nabla v_{\varepsilon} \vert ^{p(x,t)-2}\nabla v_{\varepsilon}\rightharpoonup b(x,t) \vert \nabla v \vert ^{p(x,t)-2}\nabla v, \quad\text{in } L^{1}\bigl(0,T; L^{\frac{p(x,t)}{p(x.t)-1}}(\varOmega)\bigr). $$

Also, we can show that initial value (1.2) is true in the sense of (1.8) as in [12]. Theorem 1.2 is proved. □

3 Proof of Theorem 1.3

Lemma 3.1

([36, 37])

• (i) The space$(L^{p(x)}(\varOmega), \|\cdot\| _{L^{p(x)}(\varOmega)} )$, $(W^{1,p(x)}(\varOmega), \|\cdot\|_{W^{1,p(x)}(\varOmega)} )$and$W^{1,p(x)}_{0}(\varOmega)$are reflexive Banach spaces.

• (ii) Let$p(x)$and$q(x)$be two functions with$\frac{1}{p(x)}+\frac{1}{q(x)} = 1$. The conjugate space of$L^{p(x)}(\varOmega)$is$L^{q(x)}(\varOmega)$. For any$u \in L^{p(x)}(\varOmega)$and$v \in L^{q(x)}(\varOmega)$,

  $$\biggl\vert \int_{\varOmega}uv \,dx \biggr\vert \leq2 \Vert u \Vert _{L^{p(x)}(\varOmega)} \Vert v \Vert _{L^{q(x)}(\varOmega)}. $$
• (iii)
  $$ \begin{gathered} \textit{If } \Vert u \Vert _{L^{p(x)}(\varOmega)} = 1,\quad \textit{then } \int_{\varOmega} \vert u \vert ^{p(x)} \,dx = 1. \\ \textit{If } \Vert u \Vert _{L^{p(x)}(\varOmega) }> 1,\quad \textit{then } \vert u \vert ^{p^{-}}_{L^{p(x)}(\varOmega ) }\leq \int_{\varOmega} \vert u \vert ^{p(x)} \,dx\leq \vert u \vert ^{p^{+}}_{L^{p(x)}(\varOmega) }. \\ \textit{If } \Vert u \Vert _{L^{p(x)}(\varOmega)} < 1,\quad \textit{then } \vert u \vert ^{p^{+}}_{L^{p(x)}(\varOmega ) }\leq \int_{\varOmega} \vert u \vert ^{p(x)} \,dx\leq \vert u \vert ^{p^{-}}_{L^{p(x)}(\varOmega) }. \end{gathered} $$

Proof of Theorem 1.3

For any given $t\in(0,T)$ and small enough $\lambda>0$, we denote $\varOmega_{\lambda t}=\{x\in\varOmega: b(x,t)>\lambda\}$ and define

$$\xi_{\lambda}(x,t)=\bigl[b(x,t)-\lambda\bigr]_{+}^{\frac{\alpha_{1}}{p(x,t)}}, $$

where $\alpha_{1}\geq2p^{+}$.

We choose $\chi_{[\tau,s]}(t)[u(x,t)-v(x,t)]\xi_{\lambda}(x,t)$ as a test function, where $\chi_{[\tau,s]}$ is the characteristic function on $[\tau,s]\subset(0,t)$. Then

$$ \begin{aligned}[b] & \iint_{Q_{\tau s}}(u-v)\xi_{\lambda}(x,t)\frac{\partial ( \vert u \vert ^{\beta-1}u - \vert v \vert ^{\beta-1}v)}{\partial t}\,dx\,dt \\ &\quad=- \iint_{Q_{\tau s}} \bigl(b(x,t) \vert \nabla u \vert ^{p(x,t)-2}\nabla u- \vert \nabla v \vert ^{p(x,t)-2}\nabla v \bigr) \nabla\bigl[(u-v)\xi_{\lambda }(x,t)\bigr] \,dx\,dt \\ &\qquad-\sum_{i=1}^{N} \iint_{Q_{\tau s}}g^{i}(x,t)\bigl[\gamma_{i}(u)- \gamma _{i}(v)\bigr] \bigl[u-v)\xi_{\lambda}(x,t) \bigr]_{x_{i}} \,dx\,dt \\ &\qquad-\sum_{i=1}^{N} \iint_{Q_{\tau s}}\bigl[\gamma_{i}(u)-\gamma _{i}(v)\bigr] \bigl[(u-v)\xi_{\lambda}(x,t)\bigr] \frac{\partial g^{i}(x,t)}{\partial x_{i}}\,dx\,dt, \end{aligned} $$

(3.1)

where $Q_{\tau s}=\varOmega\times[\tau, s]$ as usual.

In the first place,

$$ \begin{aligned}[b] & \iint_{Q_{\tau s}}b(x,t) \bigl( \vert \nabla u \vert ^{p(x,t)-2}\nabla u- \vert \nabla v \vert ^{p(x,t)-2}\nabla v \bigr) \nabla \bigl[(u-v)\xi_{\lambda} \bigr] \,dx\,dt \\ &\quad= \iint_{Q_{\tau s}}b(x,t)\xi_{\lambda} \bigl( \vert \nabla u \vert ^{p(x,t)-2}\nabla u- \vert \nabla v \vert ^{p(x,t)-2}\nabla v \bigr)\nabla(u-v) \,dx\,dt \\ &\qquad+ \iint_{Q_{\tau s}}b(x,t) \bigl( \vert \nabla u \vert ^{p(x,t)-2}\nabla u- \vert \nabla v \vert ^{p(x,t)-2}\nabla v \bigr) (u-v)\nabla\xi_{\lambda} \,dx\,dt, \end{aligned} $$

(3.2)

we have

$$ \iint_{Q_{\tau s}}b(x,t)\xi_{\lambda} \bigl( \vert \nabla u \vert ^{p(x,t)-2}\nabla u- \vert \nabla v \vert ^{p(x,t)-2}\nabla v \bigr)\nabla(u-v) \,dx\,dt\geq0 $$

(3.3)

and

$$ \begin{aligned}[b] & \biggl\vert \iint_{Q_{\tau s}}(u-v)b(x,t) \bigl( \vert \nabla u \vert ^{p(x,t)-2}\nabla u- \vert \nabla v \vert ^{p(x,t)-2}\nabla v \bigr) \nabla\xi _{\lambda}\,dx\,dt \biggr\vert \\ &\quad\leq c \biggl( \int_{\tau}^{s} \int_{\varOmega_{\lambda t}}b(x,t) \bigl( \vert \nabla u \vert ^{p(x,t)}+ \vert \nabla v \vert ^{p(x,t)} \bigr)\,dx\,dt \biggr)^{\frac{1}{q_{1}}} \\ &\qquad\cdot \biggl( \int_{\tau}^{s} \int_{\varOmega_{\lambda t}}b(x,t) \vert \nabla\xi_{\lambda} \vert ^{p(x,t)} \vert u-v \vert ^{p(x,t)}\,dx\,dt \biggr)^{\frac{1}{p_{1}}} \\ &\quad\leq c \biggl( \int_{\tau}^{s} \int_{\varOmega}b(x,t) \bigl( \vert \nabla u \vert ^{p(x,t)}+ \vert \nabla v \vert ^{p(x,t)} \bigr)\,dx\,dt \biggr)^{\frac{1}{q_{1}}} \\ &\qquad\cdot \biggl( \int_{\tau}^{s} \int_{\varOmega_{\lambda t}}b(x,t)\bigl[b(x,t)-\lambda\bigr]^{p(x,t) (\frac{\alpha _{1}}{p(x,t)}-1 )} \vert \nabla b \vert ^{p(x,t)} \vert u-v \vert ^{p(x,t)}\,dx\,dt \biggr)^{\frac{1}{p_{1}}} \\ &\quad\leq c \biggl( \int_{\tau}^{s} \int_{\varOmega_{\lambda t}}b(x,t)\bigl[b(x,t)-\lambda\bigr]^{p(x,t) (\frac{\alpha _{1}}{p(x,t)}-1 )} \vert u-v \vert ^{p(x,t)}\,dx\,dt \biggr)^{\frac{1}{p_{1}}}. \end{aligned} $$

(3.4)

Here, $q(x,t)=\frac{p(x,t)}{p(x,t)-1}$, from (iii) of Lemma 3.1, $q_{1}=q^{+}$ or $q^{-}$ according to

$$\int_{\tau}^{s} \int_{\varOmega_{\lambda t}}b(x,t) \bigl( \vert \nabla u \vert ^{p(x,t)}+ \vert \nabla v \vert ^{p(x,t)} \bigr)\,dx\,dt< 1, $$

or

$$\int_{\tau}^{s} \int_{\varOmega_{\lambda t}}b(x,t) \bigl( \vert \nabla u \vert ^{p(x,t)}+ \vert \nabla v \vert ^{p(x,t)} \bigr)\,dx\,dt\geq1, $$

$p_{1}$ has a similar meaning, and we have used the fact that $|\nabla b|\leq c$ in (3.4).

Then

$$ \begin{aligned}[b] &\lim_{\lambda\rightarrow0} \biggl\vert \iint_{Q_{\tau s}}(u-v)b(x,t) \bigl( \vert \nabla u \vert ^{p(x,t)-2}\nabla u- \vert \nabla v \vert ^{p(x,t)-2}\nabla v \bigr) \nabla\xi_{\lambda}\,dx\,dt \biggr\vert \\ &\quad\leq\lim_{\lambda\rightarrow0}c \biggl( \int_{\tau}^{s} \int _{\varOmega_{\lambda t}}b(x,t) \bigl(b(x,t)-\lambda\bigr)^{p(x,t)(\frac{\alpha _{1}}{p(x,t)}-1)} \vert u-v \vert ^{p(x,t)}\,dx\,dt \biggr)^{\frac{1}{p_{1}}} \\ &\quad\leq c \biggl( \int_{\tau}^{s} \int_{\varOmega}b(x,t)^{1+p(x,t) (\frac{\alpha_{1}}{p(x,t)}-1 )} \vert u-v \vert ^{p(x,t)}\,dx\,dt \biggr)^{\frac {1}{p_{1}}}. \end{aligned} $$

(3.5)

If we denote that

$$\varOmega_{1t}=\bigl\{ x\in\varOmega: p(x,t)\geq2\bigr\} ,\qquad \varOmega_{2t}=\bigl\{ x\in\varOmega : p(x,t)< 2\bigr\} , $$

by $u,v\in L^{\infty}$, we have

$$ \begin{aligned}[b] & \biggl( \int_{\tau}^{s} \int_{\varOmega_{1t}}b(x,t)^{1+p(x,t) (\frac{\alpha_{1}}{p(x,t)}-1 )} \vert u-v \vert ^{p(x,t)}\,dx\,dt \biggr)^{\frac{1}{p_{1}}} \\ &\quad\leq c \biggl( \iint_{Q_{\tau s}}b(x,t)^{\frac{\alpha _{1}}{p(x,t)}} \vert u-v \vert ^{2}\,dx\,dt \biggr)^{\frac{1}{p_{1}}}, \end{aligned} $$

(3.6)

and using the Hölder inequality, we get

$$ \begin{aligned} [b]& \biggl( \int_{\tau}^{s} \int_{\varOmega_{2t}}b(x,t)^{1+p(x,t) (\frac{\alpha_{1}}{p(x,t)}-1 )} \vert u-v \vert ^{p(x,t)}\,dx\,dt \biggr)^{\frac{1}{p_{1}}} \\ &\quad\leq \biggl( \int_{\tau}^{s} \int_{\varOmega_{2t}} \bigl[b(x,t)^{1+p(x,t) (\frac{\alpha_{1}}{p(x,t)}-1)-\frac{\alpha_{1}}{2}} \bigr]^{\frac {2}{2-p(x,t)}}\,dx\,dt \biggr)^{\frac{1}{p_{12}}\frac{1}{p_{1}}} \\ &\qquad\cdot \biggl( \int_{\tau}^{s} \int_{\varOmega_{2t}}b(x,t)^{\frac {\alpha_{1}}{p(x,t)}} \vert u-v \vert ^{2}\,dx\,dt \biggr)^{\frac{1}{2}} \\ &\quad\leq c \biggl( \iint_{Q_{\tau s}}b(x,t)^{\frac{\alpha _{1}}{p(x,t)}} \vert u-v \vert ^{2}\,dx\,dt \biggr)^{\frac{1}{2}}, \end{aligned} $$

(3.7)

where $p_{12}(x,t)=\frac{2}{2-p(x,t)}$, from (iii) of Lemma 3.1, $p_{12}=p_{12}^{+}$ or $p_{12}^{-}$ according to

$$\int_{\tau}^{s} \int_{\varOmega_{2t}} \bigl[b(x,t)^{1+p(x,t) (\frac{\alpha_{1}}{p(x,t)}-1)-\frac{\alpha_{1}}{2}} \bigr]^{\frac {2}{2-p(x,t)}}\,dx\,dt< 1, $$

or

$$\int_{\tau}^{s} \int_{\varOmega_{2t}} \bigl[b(x,t)^{1+p(x,t) (\frac{\alpha_{1}}{p(x,t)}-1)-\frac{\alpha_{1}}{2}} \bigr]^{\frac {2}{2-p(x,t)}}\,dx\,dt \geq1. $$

In the second place,

$$ \begin{aligned}[b] &\sum_{i=1}^{N} \iint_{Q_{\tau s}}\bigl[\gamma_{i}(u)-\gamma _{i}(v)\bigr] \bigl[(u-v)\xi_{\lambda}\bigr]_{x_{i}} \,dx\,dt \\ &\quad=\sum_{i=1}^{N} \iint_{Q_{\tau s}}\bigl[\gamma_{i}(u)-\gamma _{i}(v)\bigr](u-v)\xi_{\lambda x_{i}} \,dx\,dt \\ &\qquad+\sum_{i=1}^{N} \iint_{Q_{\tau s}}\bigl[\gamma_{i}(u)-\gamma _{i}(v)\bigr](u-v)_{x_{i}}\xi_{\lambda} \,dx\,dt. \end{aligned} $$

(3.8)

Since $|\nabla b|\leq c$, $\alpha_{1}\geq2p^{+}$, there hold

$$ \begin{aligned}[b] &\lim_{\lambda\rightarrow0}\sum _{i=1}^{N} \iint_{Q_{\tau s}}\bigl[\gamma _{i}(u)- \gamma_{i}(v)\bigr](u-v)\xi_{\lambda x_{i}} \,dx\,dt \\ &\quad\leq c\lim_{\lambda\rightarrow0}\sum_{i=1}^{N} \int_{\tau}^{s} \int _{\varOmega_{\lambda t}}\bigl[\gamma_{i}(u)-\gamma _{i}(v)\bigr](u-v)\bigl[b(x,t)-\lambda\bigr]^{\frac{\alpha_{1}}{p(x,t)}-1} \vert b_{ x_{i}} \vert \,dx \\ &\quad\leq c \biggl( \iint_{Q_{\tau s}}b(x,t)^{\frac{\alpha _{1}}{p(x,t)}} \vert u-v \vert ^{2}\,dx \biggr)^{\frac{1}{2}} \end{aligned} $$

(3.9)

and

$$ \begin{aligned} [b]&\lim_{\lambda\rightarrow0}\sum _{i=1}^{N} \iint_{Q_{\tau s}}\bigl[\gamma _{i}(u)- \gamma_{i}(v)\bigr](u-v)_{x_{i}}\xi_{\lambda} \,dx\,dt \\ &\quad=\lim_{\lambda\rightarrow0}\sum_{i=1}^{N} \iint_{Q_{\tau s}}\bigl[\gamma _{i}(u)- \gamma_{i}(v)\bigr](u-v)_{x_{i}}\bigl[b(x,t)-\lambda \bigr]_{+}^{\frac{\alpha _{1}}{p(x,t)}} \,dx\,dt \\ &\quad\leq\sum_{i=1}^{N} \biggl( \iint_{Q_{\tau s}} \bigl(b(x,t)^{\frac {\alpha_{1}-1}{p(x,t)}} \bigr)^{q(x,t)} \bigl\vert \gamma_{i}(u)-\gamma _{i}(v) \bigr\vert ^{p'(x,t)}\,dx\,dt \biggr)^{\frac{1}{q_{1}}} \\ &\qquad\cdot \biggl( \iint_{Q_{\tau s}}b(x,t) \bigl( \vert \nabla u \vert ^{p(x,t)}+ \vert \nabla v \vert ^{p(x,t)} \bigr) \,dx\,dt \biggr)^{\frac{1}{p_{1}}} \\ &\quad\leq c\sum_{i=1}^{N} \biggl( \iint_{Q_{\tau s}}b(x,t)^{\frac{\alpha _{1}-1}{p(x,t)-1}} \bigl\vert \gamma_{i}(u)-\gamma_{i}(v) \bigr\vert ^{q(x,t)}\,dx\,dt \biggr)^{\frac{1}{q_{1}}}. \end{aligned} $$

(3.10)

When $1< p(x,t)<2$, we know $q(x,t)>2$. Since $\alpha_{1}\geq p^{+}$, if $b(x,t)<1$, then $b(x,t)^{\frac{\alpha_{1}-1}{p(x,t)-1}}\leq b(x,t)^{\frac{\alpha_{1}}{p(x,t)}}$. If $1\leq b(x,t)\leq D=\max_{(x,t)\in\overline{\varOmega}\times[0,T]}b(x,t)$, then

$$b(x,t)^{\frac{\alpha_{1}-1}{p(x,t)-1}-\frac{\alpha _{1}}{p(x,t)}}=b(x,t)^{\frac{\alpha_{1}-p(x,t)}{p(x,t)(p(x,t)-1)}}\leq D^{\frac{\alpha_{1}-p(x,t)}{p(x,t)(p(x,t)-1)}}\leq c, $$

which implies that $b(x,t)^{\frac{\alpha_{1}-1}{p(x,t)-1}}\leq cb(x,t)^{\frac{\alpha_{1}}{p(x,t)}}$ is always true. Thus, we extrapolate that

$$ \begin{aligned}[b] &\sum_{i=1}^{N} \biggl( \int_{\tau}^{s} \int_{\varOmega_{1t}}b(x,t)^{\frac {\alpha_{1}-1}{p(x,t)-1}} \bigl\vert \gamma_{i}(u)-\gamma _{i}(v) \bigr\vert ^{q(x,t)}\,dx\,dt \biggr)^{\frac{1}{q_{1}}} \\ &\quad\leq c \biggl( \iint_{Q_{\tau s}}\rho(x,t)^{\frac{\alpha _{1}}{p(x,t)}} \vert u-v \vert ^{2}\,dx\,dt \biggr)^{\frac{1}{q_{1}}}. \end{aligned} $$

(3.11)

When $p(x,t)\geq2$, we know $q(x,t)<2$. By $\alpha_{1}\geq2$, using the Hölder inequality, we have

$$ \begin{aligned}[b] &\sum_{i=1}^{N} \biggl( \int_{\tau}^{s} \int_{\varOmega_{2t}}b(x,t)^{\frac {\alpha_{1}-1}{p(x,t)-1}} \bigl\vert \gamma_{i}(u)-\gamma _{i}(v) \bigr\vert ^{p'(x,t)}\,dx\,dt \biggr)^{\frac{1}{q_{1}}} \\ &\quad\leq c \biggl( \iint_{Q_{\tau s}} \bigl[b(x,t)^{\frac{\alpha_{1}-1}{p(x,t)-1}-\frac{\alpha _{1}}{2(p(x,t)-1)}} \bigr]^{\frac{2}{2-q(x,t)}}\,dx\,dt \biggr)^{\frac {1}{q_{22}}\frac{1}{q_{1}}} \\ &\qquad\cdot \biggl( \iint_{Q_{\tau s}}b(x,t)^{\frac{\alpha _{1}}{p(x,t)}} \vert u-v \vert ^{2}\,dx\,dt \biggr)^{\frac{1}{2}} \\ &\quad\leq c \biggl( \iint_{Q_{\tau s}}b(x,t)^{\frac{\alpha _{1}}{p(x,t)}} \vert u-v \vert ^{2}\,dx\,dt \biggr)^{\frac{1}{2}}, \end{aligned} $$

(3.12)

where $q_{22}(x,t)=\frac{2-q(x,t)}{2}$, $q_{22}={q}_{22}^{+}$, or ${q}_{22}^{-}$.

In the third place, since $\vert \sum_{i=1}^{N}\frac{\partial g^{i}(x,t)}{\partial x_{i}} \vert \leq cb(x,t)^{\frac{\alpha_{1}}{p(x,t)}}$

$$ \begin{aligned}[b] & \Biggl\vert -\lim _{\lambda\rightarrow0}\sum_{i=1}^{N} \iint_{Q_{\tau s}}\bigl[\gamma_{i}(u)-\gamma_{i}(v) \bigr] \bigl[(u-v)\xi_{\lambda}(x,t)\bigr]\frac {\partial g^{i}(x,t)}{\partial x_{i}}\,dx\,dt \Biggr\vert \\ &\quad= \Biggl\vert -\sum_{i=1}^{N} \iint_{Q_{\tau s}}\bigl[\gamma_{i}(u)-\gamma _{i}(v)\bigr](u-v)\frac{\partial g^{i}(x,t)}{\partial x_{i}}\,dx\,dt \Biggr\vert \\ &\quad\leq c \iint_{Q_{\tau s}} \vert u-v \vert ^{2} \Biggl\vert \sum_{i=1}^{N}\frac{\partial g^{i}(x,t)}{\partial x_{i}} \Biggr\vert \,dx\,dt \\ &\quad\leq c \iint_{Q_{\tau s}}b(x,t)^{\frac{\alpha_{1}}{p(x,t)}} \vert u-v \vert ^{2}\,dx\,dt. \end{aligned} $$

(3.13)

From (3.4)–(3.13), letting $\lambda\rightarrow0$ in (3.1), we deduce that

$$ \begin{aligned}[b] & \iint_{Q_{\tau s}}b(x,t)^{\frac{\alpha_{1}}{p(x,t)}}(u-v)\frac {\partial ( \vert u \vert ^{\beta-1}u - \vert v \vert ^{\beta-1}v )}{\partial t}\,dx\,dt \\ &\quad\leq c \biggl( \iint_{Q_{\tau s}}b(x,t)^{\frac{\alpha _{1}}{p(x,t)}} \bigl\vert u(x,t)-v(x,t) \bigr\vert ^{2}\,dx\,dt \biggr)^{l}, \end{aligned} $$

(3.14)

where $l\leq1$.

Last but not least, by the mean value theorem,

$$ \begin{aligned}[b] & \iint_{Q_{\tau s}}b(x,t)^{\frac{\alpha_{1}}{p(x,t)}}(u-v)\frac {\partial ( \vert u \vert ^{\beta-1}u - \vert v \vert ^{\beta-1}v )}{\partial t}\,dx\,dt \\ &\quad= \iint_{Q_{\tau s}}b(x,t)^{\frac{\alpha_{1}}{p(x,t)}}\frac {u-v}{ \vert u \vert ^{\beta-1}u - \vert v \vert ^{\beta-1}v} \bigl( \vert u \vert ^{\beta-1}u - \vert v \vert ^{\beta-1}v \bigr) \frac{\partial( \vert u \vert ^{\beta-1}u - \vert v \vert ^{\beta -1}v)}{\partial t}\,dx\,dt\hspace{-18pt} \\ &\quad=\frac{1}{2} \iint_{Q_{\tau s}}\frac{b(x,t)^{\frac{\alpha _{1}}{p(x,t)}}}{\beta \vert \zeta \vert ^{\beta-1}}\frac{\partial ( \vert u \vert ^{\beta-1}u - \vert v \vert ^{\beta-1}v )^{2}}{\partial t}\,dx\,dt, \end{aligned} $$

(3.15)

where $\zeta\in(v,u)$.

One of the possibilities of (3.15) is that, for any $s\geq \tau$,

$$ \frac{d}{dt} \bigl\Vert b(x,t)^{\frac{\alpha_{1}}{2p(x,t)}}\bigl( \vert u \vert ^{\beta -1}u - \vert v \vert ^{\beta-1}v\bigr) \bigr\Vert _{L^{2}(\varOmega)}\leq0,\quad t\in[\tau,s], $$

(3.16)

is true, then

$$\begin{gathered} \int_{\varOmega}b(x,t)^{\frac{\alpha_{1}}{p(x,t)}} \bigl\vert \vert u \vert ^{\beta -1}u(x,s) - \vert v \vert ^{\beta-1}v(x,s) \bigr\vert ^{2}\,dx\\\quad\leq \int_{\varOmega}b(x,t)^{\frac{\alpha_{1}}{p(x,t)}} \bigl\vert \vert u \vert ^{\beta-1}u(x,\tau) - \vert v \vert ^{\beta-1}v(x, \tau) \bigr\vert ^{2}\,dx\end{gathered} $$

is clear.

Another possibility of (3.15) is that there is $s_{0}\geq\tau$ such that

$$ \frac{d}{dt} \bigl\Vert b(x,t)^{\frac{\alpha_{1}}{2p(x,t)}} \bigl( \vert u \vert ^{\beta-1}u - \vert v \vert ^{\beta-1}v \bigr) \bigr\Vert _{L^{2}(\varOmega)}> 0,\quad t\in[\tau,s_{0}], $$

(3.17)

then

$$ \begin{aligned}[b] & \iint_{Q_{\tau s_{0}}}b(x,t)^{\frac{\alpha_{1}}{p(x,t)}}(u-v)\frac {\partial ( \vert u \vert ^{\beta-1}u - \vert v \vert ^{\beta-1}v )}{\partial t}\,dx\,dt \\ &\quad=\frac{1}{2} \iint_{Q_{\tau s_{0}}}\frac{b(x,t)^{\frac{\alpha _{1}}{p(x,t)}}}{\beta \vert \zeta \vert ^{\beta-1}}\frac{\partial ( \vert u \vert ^{\beta-1}u - \vert v \vert ^{\beta-1}v )^{2}}{\partial t}\,dx\,dt \\ &\quad\geq\frac{1}{2\beta M^{\beta-1}} \iint_{Q_{\tau s_{0}}}b(x,t)^{\frac {\alpha_{1}}{p(x,t)}}\frac{\partial ( \vert u \vert ^{\beta-1}u - \vert v \vert ^{\beta -1}v )^{2}}{\partial t}\,dx\,dt, \end{aligned} $$

(3.18)

where $\zeta\in(v,u)$, $M=\max \{\|u\|_{L^{\infty}(Q_{T})}, \|v\| _{L^{\infty}(Q_{T})} \}$.

Combining (3.14)–(3.15) with (3.18), we can extrapolate that

$$ \begin{aligned}[b] & \int_{\varOmega}b(x,s_{0})^{\frac{\alpha_{1}}{p(x,s_{0})}} \bigl\vert \vert u \vert ^{\beta-1}u(x,s_{0}) - \vert v \vert ^{\beta-1}v(x,s_{0}) \bigr\vert ^{2}\,dx \\ &\quad\leq \int_{\varOmega}b(x,\tau)^{\frac{\alpha_{1}}{p(x,\tau)}} \bigl\vert \vert u \vert ^{\beta-1}u(x,\tau) - \vert v \vert ^{\beta-1}v(x,\tau) \bigr\vert ^{2}\,dx \\ &\qquad+\frac{2 cM^{\beta-1}}{\beta m^{2(\beta-1)}} \biggl( \int_{\tau}^{s_{0}} \int_{\varOmega}b(x,t)^{\frac{\alpha_{1}}{p(x,t)}} \bigl\vert \vert u \vert ^{\beta -1}u(x,t) - \vert v \vert ^{\beta-1}v(x,t) \bigr\vert ^{2}\,dx\,dt \biggr)^{l}. \end{aligned} $$

(3.19)

Here $m=\min \{\|u\|_{L^{\infty}(Q_{T})}, \|v\|_{L^{\infty }(Q_{T})} \}$. From (3.19), we have

$$\begin{gathered} \int_{\varOmega}b(x,s_{0})^{\frac{\alpha_{1}}{p(x,s_{0}}} \bigl\vert \vert u \vert ^{\beta -1}u(x,s_{0}) - \vert v \vert ^{\beta-1}v(x,s_{0}) \bigr\vert ^{2}\,dx \\ \quad\leq \int_{\varOmega}b(x,\tau)^{\frac{\alpha_{1}}{p(x,\tau)}} \bigl\vert \vert u \vert ^{\beta-1}u(x,\tau) - \vert v \vert ^{\beta-1}v(x,\tau) \bigr\vert ^{2}\,dx, \end{gathered} $$

which contradicts assumption (3.17). In other words, (3.17) is impossible. This fact implies that, for any $s,\tau\in [0,T)$, inequality (3.16) is always true. By the arbitrariness of τ, we have

$$ \begin{aligned} & \int_{\varOmega}b(x,s)^{\frac{\alpha_{1}}{p(x,s)}} \bigl\vert \vert u \vert ^{\beta -1}u(x,s)- \vert v \vert ^{\beta-1}v(x,s) \bigr\vert ^{2} \,dx \\ &\quad\leq \int_{\varOmega}b(x,0)^{\frac{\alpha_{1}}{p(x,0)}} \bigl\vert \vert u_{0} \vert ^{\beta-1}u_{0}(x) - \vert v_{0} \vert ^{\beta-1}v_{0}(x) \bigr\vert ^{2}\,dx, \end{aligned} $$

Theorem 1.3 follows. □

4 The stability of weak solutions

Let ${h_{n}}(u)$ be an odd function defined as

$$h_{n}(u) = \left \{ \textstyle\begin{array}{l@{\quad}l} 1, & u > \frac{1}{n}, \\ n^{2}u^{2}e^{1 - n^{2}u^{2}},& 0\leq u \leqslant\frac{1}{n}. \end{array}\displaystyle \right . $$

Then

$$\begin{aligned}& \lim_{n\rightarrow\infty}h_{n}(u)=\operatorname{sign}(u),\quad u\in(-\infty, +\infty), \end{aligned}$$

(4.1)

$$\begin{aligned}& 0\leq h_{n}'(u)\leq\frac{c}{u},\qquad 0< u< \frac{1}{n}, \qquad\lim_{n\rightarrow \infty}h'_{n}(u)u=0. \end{aligned}$$

(4.2)

Proof of Theorem 1.4

Since $g^{i}(x,t)$ satisfies (1.10), then from Theorem 1.3 we know that the weak solution of equation (1.1) with initial value (1.2) is unique. Let $u(x,t)$ and $v(x,t)$ be two solutions of equation (1.1) with the different initial values $u_{0}(x)$ and $v_{0}(x)$ respectively. Since the weak solution of equation (1.1) with initial value (1.2) is unique, there are two asymptotic solutions of asymptotic problem (2.1)–(2.3), $u_{\varepsilon}$ and $v_{\varepsilon}$, satisfying

$$ \lim_{\varepsilon\rightarrow0}u_{\varepsilon}=u,\qquad \lim _{\varepsilon \rightarrow0}v_{\varepsilon}=v,\quad \text{a.e. }(x,t)\in Q_{T}, $$

(4.3)

and

$$ \begin{gathered} b(x,t)^{\frac{1}{p(x,t)}}\nabla u_{\varepsilon}\rightharpoonup b(x,t)^{\frac{1}{p(x,t)}}\nabla u,\\ b(x,t)^{\frac{1}{p(x,t)}}\nabla v_{\varepsilon}\rightharpoonup b(x,t)^{\frac{1}{p(x,t)}}\nabla v,\quad \text{in } L^{1} \bigl(0,T; L^{p(x,t)}(\varOmega)\bigr).\end{gathered} $$

(4.4)

We now choose $\chi_{[\tau,s]}(t){h_{n}}(u_{\varepsilon}(x,t) - v_{\varepsilon}(x,t))$ as a test function, and so

$$ \begin{aligned}[b] & \iint_{Q_{\tau s}} h_{n}(u_{\varepsilon} - v_{\varepsilon})\frac {\partial ( \vert u \vert ^{\beta-1}u - \vert v \vert ^{\beta-1}v )}{\partial t}\,dx\,dt \\ &\qquad+ \iint_{Q_{\tau s}}b(x,t) \bigl( \vert \nabla u \vert ^{p(x,t)- 2}\nabla u - \vert \nabla v \vert ^{p(x,t)- 2}\nabla v \bigr) \cdot\nabla(u_{\varepsilon}- v_{\varepsilon})h'_{n}(u_{\varepsilon} - v_{\varepsilon}) \,dx\,dt \\ &\qquad+ \sum_{i = 1}^{N} \iint_{Q_{\tau s}}g^{i}(x,t)\bigl[\gamma_{i}(u) - \gamma _{i}(v)\bigr] \cdot(u_{\varepsilon} - v_{\varepsilon})_{x_{i}}h'_{n}(u_{\varepsilon} - v_{\varepsilon}) \,dx\,dt \\ &\quad=-\sum_{i = 1}^{N} \iint_{Q_{\tau s}} \bigl[\gamma_{i}(u) - \gamma _{i}(v)\bigr]h_{n}(u_{\varepsilon} - v_{\varepsilon}) \frac{\partial g^{i}(x,t)}{\partial x_{i}} \,dx\,dt. \end{aligned} $$

(4.5)

In the first place, (4.4) yields

$$ \begin{aligned}[b] &\lim_{\varepsilon\rightarrow0} \iint_{Q_{\tau s}} \bigl[b(x,t)^{\frac{p(x,t)-1}{p(x,t)}} \bigl( \vert \nabla u \vert ^{p(x,t)-2}\nabla u- \vert \nabla v \vert ^{p(x,t)-2}\nabla v \bigr) \bigr] \\ &\qquad\cdot b(x,t)^{\frac{1}{p(x,t)}}\bigl[\nabla(u_{\varepsilon }-v_{\varepsilon})- \nabla(u-v)\bigr] \,dx\,dt \\ &\quad=0. \end{aligned} $$

(4.6)

In the second place, by (4.6) and the second mean value theorem, we have

$$ \begin{aligned}[b] &\lim_{\varepsilon\rightarrow0} \iint_{Q_{\tau s}}b(x,t)^{\frac {p(x,t)-1}{p(x,t)}} \bigl( \vert \nabla u \vert ^{p(x,t)-2}\nabla u- \vert \nabla v \vert ^{p(x,t)-2}\nabla v \bigr) \\ &\qquad\cdot b(x,t)^{\frac{1}{p(x,t)}}\bigl[\nabla(u_{\varepsilon }-v_{\varepsilon})- \nabla(u-v)\bigr]h'_{n}(u-v) \,dx\,dt \\ &\quad=h'_{n}(\zeta)\lim_{\varepsilon\rightarrow0} \iint_{Q_{\tau s}}b(x,t)^{\frac{p(x,t)-1}{p(x,t)}} \bigl( \vert \nabla u \vert ^{p(x,t)-2}\nabla u- \vert \nabla v \vert ^{p(x,t)-2}\nabla v \bigr) \\ &\qquad\cdot b(x,t)^{\frac{1}{p(x,t)}}\bigl[\nabla(u_{\varepsilon }-v_{\varepsilon})- \nabla(u-v)\bigr]\,dx\,dt \\ &\quad=0, \end{aligned} $$

(4.7)

and since $(u_{\varepsilon}-v_{\varepsilon})\rightarrow(u-v)$, a.e. in Ω,

$$\bigl\vert \bigl[h'_{n}(u_{\varepsilon}-v_{\varepsilon})-h'_{n}(u-v) \bigr] \bigr\vert \leq c(n), $$

by (4.6),

$$ \begin{aligned}[b] &\lim_{\varepsilon\rightarrow0} \int_{\varOmega}b(x,t)^{\frac {p(x,t)-1}{p(x,t)}} \bigl( \vert \nabla u \vert ^{p(x,t)-2}\nabla u- \vert \nabla v \vert ^{p(x,t)-2}\nabla v \bigr) \\ &\qquad\cdot b(x,t)^{\frac{1}{p(x,t)}}\bigl[\nabla(u_{\varepsilon }-v_{\varepsilon})- \nabla(u-v)\bigr] \bigl[h'_{n}(u_{\varepsilon}-v_{\varepsilon })-h'_{n}(u-v) \bigr] \,dx\,dt \\ &\quad=0. \end{aligned} $$

(4.8)

By (4.7)–(4.8), we have

$$ \begin{aligned}[b] &\lim_{\varepsilon\rightarrow0} \iint_{Q_{\tau s}}b(x,t)^{\frac {p(x,t)-1}{p(x,t)}} \bigl( \vert \nabla u \vert ^{p(x,t)-2}\nabla u- \vert \nabla v \vert ^{p(x,t)-2}\nabla v \bigr) \\ &\qquad\cdot b(x,t)^{\frac{1}{p(x,t)}}\bigl[\nabla(u_{\varepsilon }-v_{\varepsilon})- \nabla(u-v)\bigr]h'_{n}(u_{\varepsilon}-v_{\varepsilon }) \,dx\,dt \\ &\quad=\lim_{\varepsilon\rightarrow0} \iint_{Q_{\tau s}}b(x,t)^{\frac {p(x,t)-1}{p(x,t)}} \bigl( \vert \nabla u \vert ^{p(x,t)-2}\nabla u- \vert \nabla v \vert ^{p(x,t)-2}\nabla v \bigr) \\ &\qquad\cdot b(x,t)^{\frac{1}{p(x,t)}}\bigl[\nabla(u_{\varepsilon}-v_{\varepsilon })- \nabla(u-v)\bigr]h'_{n}(u-v) \,dx\,dt \\ &\qquad+\lim_{\varepsilon\rightarrow0} \iint_{Q_{\tau s}}b(x,t)^{\frac {p(x,t)-1}{p(x,t)}}(x) \bigl( \vert \nabla u \vert ^{p(x,t)-2}\nabla u- \vert \nabla v \vert ^{p(x,t)-2}\nabla v \bigr) \\ &\qquad\cdot b(x,t)^{\frac{1}{p(x,t)}}\bigl[\nabla(u_{\varepsilon }-v_{\varepsilon})- \nabla(u-v)\bigr] \bigl[h'_{n}(u_{\varepsilon}-v_{\varepsilon })-h'_{n}(u-v) \bigr] \,dx\,dt \\ &\quad=\lim_{\varepsilon\rightarrow0} \iint_{Q_{\tau s}}b(x,t)^{\frac {p(x,t)-1}{p(x,t)}} \bigl( \vert \nabla u \vert ^{p(x,t)-2}\nabla u- \vert \nabla v \vert ^{p(x,t)-2}\nabla v \bigr) \\ &\qquad\cdot b(x,t)^{\frac{1}{p(x,t)}}\bigl[\nabla(u_{\varepsilon}-v_{\varepsilon })- \nabla(u-v)\bigr]h'_{n}(u-v) \,dx\,dt \\ &\quad=0. \end{aligned} $$

(4.9)

In the third place,

$$ \begin{aligned}[b] &\lim_{\varepsilon\rightarrow0} \iint_{Q_{\tau s}}b(x,t) \bigl( \vert \nabla u \vert ^{p(x,t)-2}\nabla u- \vert \nabla v \vert ^{p(x,t)-2}\nabla v \bigr) \nabla(u_{\varepsilon}-v_{\varepsilon}) h'_{n}(u_{\varepsilon }-v_{\varepsilon})\,dx\,dt \\ &\quad= \iint_{Q_{\tau s}}b(x,t) \bigl( \vert \nabla u \vert ^{p(x,t)-2}\nabla u- \vert \nabla v \vert ^{p(x,t)-2}\nabla v \bigr) \nabla(u-v) h'_{n}(u-v)\,dx\,dt \\ &\quad\geq0. \end{aligned} $$

(4.10)

In the fourth place, since

$$\begin{gathered} \int_{\varOmega} \bigl\vert b(x,t)^{\frac{1}{p(x,t)}} (u_{\varepsilon} - v_{\varepsilon})_{x_{i}}h'_{n}(u_{\varepsilon} - v_{\varepsilon}) \bigr\vert ^{p(x,t)}\,dx \\ \quad\leq c(n) \int_{\varOmega} \bigl\vert b(x,t)^{\frac{1}{p(x,t)}} (u_{\varepsilon} - v_{\varepsilon})_{x_{i}} \bigr\vert ^{p(x,t)}\,dx\leq c(n),\end{gathered} $$

as $\varepsilon\rightarrow0$, we have

$$b(x,t)^{\frac{1}{p(x,t)}}h'_{n}(u_{\varepsilon} - v_{\varepsilon}) (u_{\varepsilon}-v_{\varepsilon})_{x_{i}} \rightharpoonup b(x,t)^{\frac {1}{p(x,t)}}(u-v)_{x_{i}}h_{n}'(u-v), \quad\text{in } L^{1}\bigl(0,T; L^{p(x,t)}(\varOmega)\bigr). $$

By (1.12), $|\sum_{i=1}^{N}g^{i}(x,t)|\leq cb(x,t)^{\frac {1}{p(x,t)}}$, we extrapolate that

$$\begin{gathered} \lim_{\varepsilon\rightarrow0} \iint_{Q_{\tau s}}\sum_{i=1}^{N} g^{i}(x,t)\bigl[\gamma_{i}(u) - \gamma_{i}(v) \bigr] (u_{\varepsilon} - v_{\varepsilon})_{x_{i}}h'_{n}(u_{\varepsilon} - v_{\varepsilon}) \,dx\,dt \\ \quad= \iint_{Q_{\tau s}} \sum_{i=1}^{N}g^{i}(x,t) \bigl[\gamma_{i}(u) - \gamma _{i}(v)\bigr] \cdot(u - v)_{x_{i}}h'_{n}(u-v)\,dx\,dt. \end{gathered} $$

Moreover, since

$$\bigl\vert \bigl[\gamma_{i}(u) -\gamma_{i}(v) \bigr]h_{n}'(u - v) \bigr\vert \leq c \bigl\vert (u-v)h_{n}'(u - v) \bigr\vert \leq c, $$

if we denote $\varOmega_{1n}=\{x\in\varOmega: |u - v| < \frac{1}{n}\}$, we have

$$ \begin{aligned}[b] & \Biggl\vert \iint_{Q_{\tau s}} \sum_{i=1}^{N}g^{i}(x,t) \bigl[\gamma_{i}(u) -\gamma_{i}(v)\bigr]h'_{n}(u - v) (u - v)_{x_{i}}\,dx\,dt \Biggr\vert \\ &\quad= \Biggl\vert \int_{\tau}^{s} \int_{\varOmega_{1n}} \sum_{i=1}^{N}g^{i}(x,t) \bigl[\gamma_{i}(u) -\gamma_{i}(v)\bigr]h'_{n}(u - v) (u - v)_{x_{i}}\,dx\,dt \Biggr\vert \\ &\quad\leqslant c \int_{\tau}^{s} \int_{\varOmega_{1n}} \Biggl\vert \sum_{i=1}^{N}g^{i}(x,t) (u - v)_{x_{i}} \Biggr\vert \,dx\,dt \\ &\quad\leqslant c { \biggl[ \int_{\tau}^{s} \int_{\varOmega_{1n}} \bigl\vert b^{\frac{1}{p(x,t)}}\nabla(u - v) \bigr\vert ^{p(x,t)}\,dx\,dt \biggr]^{\frac{1}{p_{1}}}} \\ &\quad\leq c. \end{aligned} $$

(4.11)

If $\varOmega_{1n}$ has 0 measure, from (4.11), letting $n\rightarrow\infty$, we have

$$\lim_{n \to\infty} \int_{\varOmega_{1n}} \bigl\vert b^{\frac {1}{p(x,t)}}\nabla(u - v) \bigr\vert ^{p(x,t)}\,dx = 0. $$

While $\varOmega_{1n}$ is with a positive measure, from (4.11), using the dominated convergence theorem, we directly have

$$\lim_{n \to\infty} \iint_{Q_{\tau s}} \sum_{i=1}^{N}g^{i}(x,t) \bigl[\gamma_{i}(u) -\gamma_{i}(v)\bigr]h_{n}(u - v) (u - v)_{x_{i}}\,dx\,dt = 0. $$

Therefore, we have

$$ \lim_{n \to\infty} \iint_{Q_{\tau s}} \sum_{i=1}^{N}g^{i}(x,t) \bigl[\gamma _{i}(u) - \gamma_{i}(v)\bigr] \cdot(u - v)_{x_{i}}h'_{n}(u-v)\,dx\,dt = 0. $$

(4.12)

Once more,

$$ \begin{aligned} &{-}\lim_{\varepsilon\rightarrow0}\sum _{i = 1}^{N} \iint_{Q_{\tau s}} \bigl[\gamma_{i}(u) - \gamma_{i}(v)\bigr]h_{n}(u_{\varepsilon} - v_{\varepsilon})\frac{\partial g^{i}(x,t)}{\partial x_{i}} \,dx\,dt \\ &\quad=-\sum_{i = 1}^{N} \iint_{Q_{\tau s}} \bigl[\gamma_{i}(u) - \gamma _{i}(v)\bigr]h_{n}(u - v)\frac{\partial g^{i}(x,t)}{\partial x_{i}} \,dx\,dt, \end{aligned} $$

and by assumption (i) $\beta\leq1$, or (ii)

$$\bigl\vert \gamma_{i}(t_{1})-\gamma_{i}(t_{2}) \bigr\vert \leq c \bigl\vert \vert t_{1} \vert ^{\beta-1}t_{1}- \vert t_{2} \vert ^{\beta-1}t_{2} \bigr\vert , $$

we easily deduce that

$$ \begin{aligned}[b] &\lim_{n\rightarrow\infty} \Biggl\vert -\sum_{i = 1}^{N} \iint_{Q_{\tau s}} \bigl[\gamma_{i}(u) - \gamma_{i}(v)\bigr]h_{n}(u - v)\frac{\partial g^{i}(x,t)}{\partial x_{i}} \,dx\,dt \Biggr\vert \\ &\quad\leq c \iint_{Q_{\tau s}} \bigl\vert \vert u \vert ^{\beta-1}u- \vert v \vert ^{\beta-1}v \bigr\vert \,dx\,dt. \end{aligned} $$

(4.13)

Last but not least,

$$ \begin{aligned}[b] &\lim_{n\rightarrow\infty}\lim _{\varepsilon\rightarrow0} \iint _{Q_{\tau s}} h_{n}(u_{\varepsilon}- v_{\varepsilon})\frac{\partial ( \vert u \vert ^{\beta-1}u - \vert v \vert ^{\beta-1}v )}{\partial t}\,dx\,dt \\ &\quad=\lim_{n\rightarrow\infty} \iint_{Q_{\tau s}} h_{n}(u- v)\frac {\partial ( \vert u \vert ^{\beta-1}u - \vert v \vert ^{\beta-1}v )}{\partial t}\,dx\,dt \\ &\quad= \iint_{Q_{\tau s}}\operatorname{sign}(u-v)\frac{\partial ( \vert u \vert ^{\beta -1}u - \vert v \vert ^{\beta-1}v )}{\partial t}\,dx\,dt \\ &\quad= \iint_{Q_{\tau s}}\operatorname{sign} \bigl( \vert u \vert ^{\beta-1}u - \vert v \vert ^{\beta -1}v \bigr)\frac{\partial ( \vert u \vert ^{\beta-1}u - \vert v \vert ^{\beta -1}v )}{\partial t}\,dx\,dt \\ &\quad= \int_{\tau}^{s}\frac{d}{dt} \bigl\Vert \vert u \vert ^{\beta-1}u - \vert v \vert ^{\beta -1}v \bigr\Vert _{L^{1}(\varOmega)}\,dt. \end{aligned} $$

(4.14)

Then, by (4.5), (4.6), (4.7), (4.9), (4.10), (4.12), (4.14), we have

$$\int_{\tau}^{s}\frac{d}{dt} \bigl\Vert \vert u \vert ^{\beta-1}u - \vert v \vert ^{\beta -1}v \bigr\Vert _{L^{1}(\varOmega)}\,dt \leq c \iint_{Q_{\tau s}}\bigl| \vert u \vert ^{\beta-1}u- \vert v \vert ^{\beta-1}v\bigr|\,dx\,dt. $$

By the Gronwall inequality,

$$\begin{gathered} \int_{\varOmega}\bigl| \vert u \vert ^{\beta-1}u(x,s) - \vert v \vert ^{\beta-1}v(x,s) \bigr|\,dx \\\quad\leqslant \int_{\varOmega}\bigl| \bigl\vert u(x,\tau) \bigr\vert ^{\beta -1}u(x,\tau) - \bigl\vert v(x,\tau) \bigr\vert ^{\beta-1}v(x,\tau) \bigr|\,dx,\quad \forall t \in[0,T).\end{gathered} $$

By the arbitrariness of τ, we extrapolate that

$$\int_{\varOmega}\bigl| \vert u \vert ^{\beta-1}u(x,s) - \vert v \vert ^{\beta-1}v(x,s) \bigr|\,dx \leqslant \int_{\varOmega}\bigl| \vert u_{0} \vert ^{\beta-1}u_{0} - \vert v_{0} \vert ^{\beta-1}v_{0} \bigr|\,dx,\quad\forall s \in[0,T). $$

The proof is complete. □