On stability with respect to boundary conditions for anisotropic parabolic equations with variable exponents

Abstract

The anisotropic parabolic equations with variable exponents are considered. If some of diffusion coefficients $\{b_{i}(x)\}$ are degenerate on the boundary, the others are always positive, then how to impose a suitable boundary value condition is researched. The existence of weak solutions is proved by the parabolically regularized method. The stability of weak solutions, based on the partial boundary value condition, is established by choosing a suitable test function.

1 Introduction and the main results

Recently, the anisotropic parabolic equations with the variable exponents

$$\begin{aligned} v_{t}=\sum_{i=1}^{N} \frac{\partial}{\partial x_{i}} \bigl( \vert v_{x_{i}} \vert ^{p_{i}(x)-2}v_{x_{i}} \bigr), \quad (x,t)\in Q_{T}=\Omega\times (0,T), \end{aligned}$$

(1.1)

were studied by Antontsev and Shmarev [1], Tersenov [2, 3], and some essential characteristics different from the evolutionary p-Laplacian equations were revealed. Zhan [4, 5] studied the equations

$$\begin{aligned} v_{t}=\sum_{i=1}^{N} \frac{\partial}{\partial x_{i}} \bigl(b_{i}(x) \vert v_{x_{i}} \vert ^{p_{i}(x)-2}v_{x_{i}} \bigr) \end{aligned}$$

(1.2)

and showed some essential characteristics different from equation (1.1). Here,

$$\begin{aligned} b_{i}(x)>0, \quad x\in\Omega;\qquad b_{i}(x)=0, \quad x\in\partial \Omega. \end{aligned}$$

(1.3)

In this paper, we study the equation

$$\begin{aligned} v_{t}=\sum_{i=1}^{N} \frac{\partial}{\partial x_{i}} \bigl(b_{i}(x) \vert v_{x_{i}} \vert ^{p_{i}(x)-2}v_{x_{i}} \bigr)+\sum_{i=1}^{N}g^{i}(x) \frac {\partial a(v)}{\partial x_{i}} \end{aligned}$$

(1.4)

with the initial value condition

$$\begin{aligned} v(x,0) = v_{0}(x),\quad x\in\Omega, \end{aligned}$$

(1.5)

and with a partial boundary value condition

$$\begin{aligned} v(x,t)=0,\quad (x,t)\in\Sigma_{1}\times(0,T), \end{aligned}$$

(1.6)

where $\Sigma_{1}\subseteq\partial\Omega$ is a relatively open subset. A similar partial boundary value condition was imposed on the equation

$$\begin{aligned} \frac{\partial v}{\partial t}-\operatorname{div} \bigl(a(x) \vert \nabla v \vert ^{p-2} \nabla v \bigr)-\sum_{i=1}^{N}b^{i}(x)D_{i}v+c(x,t)v=f(x,t),\quad (x,t) \in Q_{T}, \end{aligned}$$

(1.7)

and a new approach to prescribe the boundary value condition rather than define the Fichera function was formulated by Yin and Wang [6]. However, since equation (1.4) is anisotropic and with the variable exponents, the method of [6] seems difficult to be applied to equation (1.4). In what follows, we will try to depict $\Sigma_{1}$ in another way. Moreover, instead of depicting the explicit formula of $\Sigma _{1}$, we will try to find the other conditions to substitute the boundary value condition.

Instead of condition (1.3), we assume that $x\in\Omega$, $b_{i}(x)>0$, and

$$\begin{aligned} &b_{i_{1}}(x)>0, b_{i_{2}}(x)>0, \ldots, b_{i_{k}}(x)>0,\quad x \in\overline {\Omega}, \end{aligned}$$

(1.8)

$$\begin{aligned} &b_{j_{1}}(x)=0, b_{j_{2}}(x)=0, \ldots, b_{j_{l}}(x)=0,\quad x \in \partial \Omega. \end{aligned}$$

(1.9)

Here, $\{i_{1}, i_{2}, \ldots, i_{k}\}\cup\{j_{1}, j_{2}, \ldots, j_{l}\}=\{ 1, 2, \ldots, N\}$, $k+l=N$. For the sake of simplicity, we denote that

$$\begin{aligned} &p_{0}=\min_{x\in\overline{\Omega}} \bigl\{ p_{1}(x), p_{2}(x), \ldots, p_{N-1}(x), p_{N}(x) \bigr\} , \\ &p^{0}=\max_{x\in\overline{\Omega}} \bigl\{ p_{1}(x), p_{2}(x), \ldots, p_{N-1}(x), p_{N}(x) \bigr\} , \end{aligned}$$

and assume that $p_{0}>1$.

Let us introduce the basic definition and the main results. First of all, for any small constant $\eta>0$, we define

$$\begin{aligned} \Omega_{\eta}= \Biggl\{ x\in\Omega:\sum_{s=1}^{l}b_{j_{s}}(x)> \eta \Biggr\} . \end{aligned}$$

Conditions (1.8)–(1.9) assure that this set is an open subset of Ω.

Definition 1.1

If a function $v(x,t)$ satisfies

$$\begin{aligned} v \in{L^{\infty}}({Q_{T}}),\quad v_{t}\in L^{{p^{0}}'} \bigl(0,T;W^{-1,{p^{0}}'}(\Omega) \bigr),\qquad b_{i}(x) \vert v_{x_{i}} \vert ^{p_{i}(x)} \in L^{2} \bigl(0,T; L^{1}(\Omega) \bigr), \end{aligned}$$

(1.10)

and for $\varphi\in L^{2}(0,T; W^{1,p^{0}}(\Omega)), \varphi |_{x\in\partial\Omega}=0$,

$$\begin{aligned} \iint_{Q_{T}} \Biggl\{ \frac{\partial v}{\partial t}\varphi+ \sum _{i=1}^{N} \bigl[b_{i}(x) \vert v_{x_{i}} \vert ^{p_{i}(x)- 2} v_{x_{i}} \varphi_{x_{i}}+ a(v)g^{i}(x)\varphi_{x_{i}}+ a(v)g^{i}_{x_{i}} \varphi \bigr] \Biggr\} \,dx\,dt=0, \end{aligned}$$

(1.11)

then we say that $v(x,t)$ is a weak solution of equation (1.1) with initial value condition (1.5), provided that

$$\begin{aligned} \lim_{t\rightarrow0} \int_{\Omega}\bigl\vert v(x,t) - v_{0}(x) \bigr\vert \,dx = 0. \end{aligned}$$

(1.12)

Besides, if the partial boundary value condition (1.6) is satisfied in the sense of the trace, then we say that $v(x,t)$ is a weak solution of the initial-boundary value problem (1.4)–(1.6).

Here and in what follows, $p'=\frac{p}{p-1}$ as usual.

Theorem 1.2

If $p_{0}>1$, $b_{i}(x)$ satisfies conditions (1.8), (1.9), $g^{i}(x)\in C^{1}(\overline{\Omega})$, $a(s)$ is a continuous function,

$$\begin{aligned} {v_{0}} \in{L^{\infty}}(\Omega), \bigl(b_{i}(x) \bigr)^{\frac{1}{p_{i}(x)}}v_{0x_{i}} \in L^{p_{i}(x)}(\Omega), \end{aligned}$$

(1.13)

then equation (1.4) with initial value (1.5) has a solution. If

$$\begin{aligned} \int_{\Omega}b_{j_{r}}^{-\frac{1}{p_{j_{r}}-1}}(x)\,dx< \infty,\quad 1\leq r \leq l, \end{aligned}$$

then there exists a solution of the initial-boundary value problem (1.4)–(1.6).

Theorem 1.3

Let $p_{0}>1$, $b_{i}(x)$ satisfy conditions (1.8), (1.9), $g^{i}(x)\in C^{1}(\overline{\Omega})$, $a(s)$ be a Lipschitz function and for every $1\leq r\leq l$, $\int_{\Omega}b_{j_{r}}^{-\frac {1}{p_{j_{r}}-1}}(x)\,dx<\infty$. If $v(x,t)$ and $u(x,t)$ are two solutions of equation (1.4),

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

(1.14)

then

$$\begin{aligned} \int_{\Omega} \bigl\vert v(x,t)-u(x,t) \bigr\vert \,dx\leq \int_{\Omega} \bigl\vert v(x,0)-u(x,0) \bigr\vert \,dx. \end{aligned}$$

(1.15)

Theorem 1.4

If $p_{0}>1$, $b_{i}(x)$ satisfies conditions (1.8), (1.9), $g^{i}(x)\in C^{1}(\overline{\Omega})$, $a(s)$ is a Lipschitz function. Let $v(x,t)$ and $u(x,t)$ be two solutions of equation (1.4). If

$$\begin{aligned} v(x,t)=u(x,t)=0, \quad (x,t)\in\Sigma_{1}\times(0,T), \end{aligned}$$

(1.16)

and for every $1\leq r\leq l$,

$$\begin{aligned} \frac{1}{\eta} \Biggl( \int_{\Omega\setminus\Omega_{\eta}} b_{j_{r}}(x) \Biggl\vert \Biggl(\sum _{s=1}^{l} b_{j_{s}}(x) \Biggr)_{x_{j_{r}}} \Biggr\vert ^{p_{j_{r}}}\,dx \Biggr)^{\frac{1}{p^{+}_{j_{r}}}}\leq c, \end{aligned}$$

(1.17)

then the stability (1.15) is true. Here,

$$\begin{aligned} \Sigma_{1}= \biggl\{ x\in\partial\Omega:\frac{ \vert (\sum_{s=1}^{l} b_{j_{s}}(x))_{x_{i_{r}}} \vert ^{p_{i_{r}}(x)}}{[\sum_{j=1}^{l} b_{j}(x)]^{p_{i_{r}(x)}-1}}\neq0 \biggr\} ,\quad r=1, 2, \ldots, k. \end{aligned}$$

(1.18)

If $a(s)\equiv0$ in equation (1.4), the similar conclusion as Theorem 1.4 was obtained in [5], where the partial boundary $\Sigma_{1}$ was depicted as follows:

$$\begin{aligned} \Sigma_{1}= \biggl\{ x\in\partial\Omega:\frac{ \vert (\prod_{s=1}^{l} b_{j_{s}}(x))_{x_{i_{r}}} \vert ^{p_{i_{r}}(x)}}{[\prod_{j=1}^{l} b_{j}(x)]^{p_{i_{r}(x)}-1}}\neq0 \biggr\} ,\quad r=1, 2, \ldots, k. \end{aligned}$$

(1.19)

In fact, letting φ be a nonnegative $C^{1}$ function, satisfying

$$\begin{aligned} \varphi(x)>0,\quad x\in\Omega, \qquad \varphi(x)=0, \quad x\in\partial\Omega , \end{aligned}$$

(1.20)

the partial boundary $\Sigma_{1}$ can be depicted by φ as

$$\begin{aligned} \Sigma_{\varphi}= \biggl\{ x\in\partial\Omega:\frac{ \vert \varphi _{x_{i_{r}}}(x) \vert ^{p_{i_{r}}(x)}}{[\varphi(x)]^{p_{i_{r}(x)}-1}}\neq0 \biggr\} ,\quad r=1, 2, \ldots, k. \end{aligned}$$

(1.21)

By this token, the exact partial boundary $\Sigma_{1}$, such that the partial boundary value condition (1.6) matches up the nonlinear degenerate parabolic equation, should satisfy that

$$\begin{aligned} \Sigma_{1}\subseteq\Sigma_{\varphi}, \end{aligned}$$

and we can depict it as

$$\begin{aligned} \Sigma_{1}=\bigcap_{\varphi} \Sigma_{\varphi} \end{aligned}$$

(1.22)

for any φ satisfying (1.20). However, if we really choose $\Sigma_{1}$ as (1.22), it lacks the technical support to obtain the stability of the weak solutions for the time being. Anyway, by adopting some ideas and techniques in [4, 5], in some special cases, we can prove the stability of the weak solutions independent of the boundary value condition.

Theorem 1.5

If $p_{0}>1$, $b_{i}(x)$ satisfies conditions (1.8), (1.9), $g^{i}(x)\in C^{1}(\overline{\Omega})$, $a(s)$ is a Lipschitz function. Let $v(x,t)$ and $u(x,t)$ be two solutions of equation (1.4) only with the initial values $v_{0}(x)$ and $u_{0}(x)$, respectively, but without any boundary value condition. If condition (1.17) is true, and

$$\begin{aligned} \frac{1}{\eta} \Biggl( \int_{\Omega\setminus\Omega_{\eta}} \Biggl\vert \Biggl(\sum _{s=1}^{l} b_{j_{s}}(x) \Biggr)_{x_{i_{r}}} \Biggr\vert ^{p_{i_{r}}}\,dx \Biggr)^{\frac {1}{p^{+}_{i_{r}}}}\leq c \end{aligned}$$

(1.23)

for every $1\leq r\leq k$, then the stability (1.15) is true.

One can see that no boundary value condition is required in Theorem 1.5. From my own perspective, condition (1.23) is an alternative of the partial boundary value condition (1.6). By the way, for the following reaction–diffusion equation

$$\begin{aligned} \frac{\partial v}{\partial t}=\operatorname{div} \bigl(a(v,x,t)\nabla v \bigr) +\operatorname{div} \bigl(b(v) \bigr),\quad (x,t)\in \Omega\times(0,T), \end{aligned}$$

(1.24)

with

$$\begin{aligned} a(v,x,t)|_{x\in\partial\Omega}=0, \end{aligned}$$

(1.25)

we had conjectured that a partial boundary value condition should be imposed. This conjecture was partially proved in [7, 8].

2 The proof of existence

By a similar method as in [4], we can prove the following.

Lemma 2.1

If $\int_{\Omega}b_{i}^{-\frac {1}{p_{i}-1}}(x)\,dx<\infty$, $v(x,t)$ is a weak solution of equation (1.4) with initial condition (1.5). Then, for any given $t\in[0,T)$,

$$\begin{aligned} \int_{\Omega} \vert v_{x_{i}} \vert \,dx\leq c,\quad i=1,2, \ldots, N, \end{aligned}$$

(2.1)

and the trace of v on the boundary ∂Ω can be defined in the traditional way.

We omit the details of the proof here. By this lemma, we know that if $b_{i}(x)$ satisfies (1.8), (1.9) and if for every $1\leq r\leq l$, $\int _{\Omega}b_{j_{r}}^{-\frac{1}{p_{j_{r}}-1}}(x)\,dx<\infty$, then (2.1) is satisfied. Thus, we can define the trace of v on the boundary ∂Ω.

Consider the regularized equation

$$\begin{aligned} {v_{t}}={}& \sum_{r=1}^{k} \bigl(b_{i_{r}}(x) \vert v_{x_{i_{r}}} \vert ^{p_{i_{r}}(x)-2}v_{x_{i_{r}}} \bigr)_{x_{i_{r}}} +\sum_{r=1}^{l} \bigl( \bigl(b_{j_{r}}(x)+\varepsilon \bigr) \vert v_{x_{j_{r}}} \vert ^{p_{j_{r}}(x)-2}v_{x_{j_{r}}} \bigr)_{x_{j_{r}}} \\ &{}+\sum _{i=1}^{N}g^{i}(x)\frac{\partial a(v)}{\partial x_{i}},\quad (x,t) \in Q_{T} \end{aligned}$$

(2.2)

with the initial-boundary condition

$$\begin{aligned} &v(x,0) = v_{0\varepsilon}(x),\quad x\in\Omega, \end{aligned}$$

(2.3)

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

(2.4)

Here, $v_{0\varepsilon}(x)\in C_{0}^{\infty}(\Omega)$ and is strongly convergent to $v_{0}(x)$ in $W_{0}^{1,p^{0}}(\Omega)$.

Then, by Wu [9], we know that problem (2.2)–(2.4) has a unique solution $v_{\varepsilon}\in L^{\infty}(Q_{T})$, $v_{\varepsilon}\in W^{1,p_{0}}_{0}(\Omega)$.

Proof of Theorem 1.2

Multiplying (2.2) by $v_{\varepsilon}$ and integrating it over $Q_{T}$ yield

$$\begin{aligned} &\frac{1}{2} \int_{\Omega}v_{\varepsilon}^{2}\,dx + \sum _{i=1}^{N} \iint _{{Q_{T}}} b_{i}(x) \vert v_{\varepsilon x_{i}} \vert ^{p_{i}(x)}\,dx\,dt +\varepsilon\sum_{r=1}^{l} \iint_{{Q_{T}}} \vert v_{\varepsilon x_{j_{r}}} \vert ^{p_{j_{r}}(x)}\,dx\,dt \\ &\quad= \frac{1}{2} \int_{\Omega} v_{0}^{2}(x)\,dx-\sum _{i=1}^{N} \iint _{{Q_{T}}} \bigl[a(v_{\varepsilon})g^{i}(x)v_{\varepsilon x_{i}}+ a(v_{\varepsilon})g^{i}_{x_{i}}v_{\varepsilon} \bigr]\,dx\,dt, \end{aligned}$$

(2.5)

then

$$\begin{aligned} \varepsilon\sum_{r=1}^{l} \iint_{{Q_{T}}} \vert v_{\varepsilon x_{j_{r}}} \vert ^{p_{j_{r}}(x)}\,dx\,dt \leqslant c \end{aligned}$$

(2.6)

and

$$\begin{aligned} \sum_{i=1}^{N} \iint_{{Q_{T}}}b_{i}(x) \vert v_{\varepsilon x_{i}} \vert ^{p_{i}(x)}\,dx\,dt\leq c. \end{aligned}$$

(2.7)

Hence, by (2.5), (2.6), and (2.7), there exists a function v and an n-dimensional vector $\overrightarrow{\xi}= ({\xi_{1}}, \ldots,{\xi _{n}})$ such that

$$\begin{aligned} v \in{L^{\infty}}({Q_{T}}), \qquad \frac{{\partial v}}{{\partial t}} \in {L^{2}}({Q_{T}}), \qquad \xi_{i}\in{L^{1} \bigl(0,T; L^{\frac{p_{i}(x)}{{p_{i}(x) - 1}}}}(\Omega) \bigr), \end{aligned}$$

and $v_{\varepsilon}\rightarrow v$ $a.e.\in Q_{T}$,

$$\begin{aligned} &{v_{\varepsilon}}\rightharpoonup v,\quad \mbox{weakly star in }L^{\infty}(Q_{T}), \\ &v_{\varepsilon} \to v, \quad\mbox{in }L^{2} \bigl(0,T;L_{\mathrm{loc}}^{r}( \Omega) \bigr), \\ &\frac{{\partial{v_{\varepsilon}}}}{{\partial t}} \rightharpoonup\frac {{\partial v}}{{\partial t}}\quad \mbox{in }{L^{2}}({Q_{T}}), \\ &\varepsilon v_{\varepsilon x_{j_{r}}}\rightharpoonup0,\quad \mbox{in }L^{{p_{j_{r}}(x)}}(Q_{T}), \\ &b_{i}(x) \vert v_{\varepsilon x_{i}} \vert ^{p_{i}(x) - 2} v_{\varepsilon x_{i} } \rightharpoonup\xi_{i}\quad \mbox{in }L^{1} \bigl(0,T; L^{\frac{p_{i}(x)}{p_{i}(x) - 1}}(\Omega) \bigr). \end{aligned}$$

Here, $r<\frac{Np_{0}}{N-p^{0}}$.

Now, similar to [1], we can show that

$$\begin{aligned} v_{t}\in L^{{p^{0}}'} \bigl(0,T;W^{-1,{p^{0}}'}(\Omega) \bigr), \end{aligned}$$

and by Wu [9], by a process of the limit, we are able to prove that

$$\begin{aligned} \sum_{i=1}^{N} \iint_{{Q_{T}}} b_{i}(x) \vert v_{x_{i}} \vert ^{p_{i}(x) - 2}v_{x_{i}} \varphi_{x_{i}} \,dx\,dt =\sum _{i=1}^{N} \iint_{{Q_{T}}} \xi_{i}(x) \varphi_{x_{i}} \,dx\,dt \end{aligned}$$

(2.8)

for any function $\varphi\in L^{2}(0,T; W^{1,p^{0}}(\Omega))$, $\varphi |_{x\in\partial\Omega}=0$. Thus, $v(x,t)$ satisfies (1.10) and (1.11). Moreover, according to Lemma 2.1, the partial boundary value condition (1.6) is satisfied in the sense of trace.

Now, we can prove the initial value (1.5) in a similar way as that in [10]. In detail, for small given $r>0$, denote $D_{r}=\{x\in\Omega: \operatorname{dist}(x,\partial\Omega)\leq r\}$. For large enough $m, n$, denoting that $v_{m}(x,t)=v_{\varepsilon=\frac{1}{m}}(x,t)$, we declare that

$$\begin{aligned} \int_{D_{2r}} \bigl\vert v_{m}(x,t)-v_{n}(x,t) \bigr\vert \,dx\leq \int_{D_{r}} \bigl\vert v_{0m}(x)-v_{0n}(x) \bigr\vert \,dx+c_{r}(t), \end{aligned}$$

(2.9)

where $c_{r}(t)$ is independent of $m,n$, and $\lim_{t\rightarrow 0}c_{r}(t)=0$. In fact, by (2.2), for any $t\in[0, T)$, we have

$$\begin{aligned} &\int_{0}^{t} \int_{D_{r}}\varphi(v_{mt}-v_{nt})\,dx\,d\tau \\ &\qquad{}+\sum_{s=1}^{k} \int_{0}^{t} \int_{D_{r}}b_{i_{s}}(x) \bigl( \vert v_{mx_{i_{s}}} \vert ^{p_{i_{s}}(x)-2} v_{mx_{i_{s}}}- \vert v_{nx_{i_{s}}} \vert ^{p_{i_{s}}(x)-2} v_{nx_{i_{s}}} \bigr)\varphi_{x_{i_{s}}}\,dx\,d\tau \\ &\qquad{}+\sum_{s=1}^{l} \int_{0}^{t} \int_{D_{r}} \biggl[ \biggl(b_{j_{s}}(x)+ \frac{1}{m} \biggr) \vert v_{mx_{j_{s}}} \vert ^{p_{j_{s}}(x)-2} v_{mx_{j_{s}}} \\ &\qquad{}- \biggl(b_{j_{s}}(x)+\frac{1}{n} \biggr) \vert v_{nx_{j_{s}}} \vert ^{p_{j_{s}}(x)-2} v_{nx_{j_{s}}} \biggr] \varphi_{x_{j_{s}}}\,dx\,d\tau \\ &\qquad{}+\sum_{i=1}^{N} \int_{0}^{t} \int_{D_{r}} \bigl\{ \bigl[a(v_{m})-a(v_{n}) \bigr]g^{i}(x)\varphi_{x_{i}}+ \bigl[a(v_{m})-a(v_{n}) \bigr]g^{i}_{x_{i}}\varphi \bigr\} \,dx\,d\tau \\ &\quad=0, \quad\forall\varphi\in L^{2} \bigl(0,T; W_{0}^{1,p_{0}}( \Omega) \bigr). \end{aligned}$$

(2.10)

For small $\eta>0$, let

$$\begin{aligned} L_{\eta}(s)= \int_{0}^{s}l_{\eta}(\tau)\,d\tau,\qquad l_{\eta}(s)=\frac{2}{\eta} \biggl(1-\frac{|s|}{\eta } \biggr)_{+}. \end{aligned}$$

(2.11)

Obviously, $l_{\eta}(s)\in C(\mathbb{R})$ and

$$ l_{\eta}(s)\geq0, \qquad \bigl|sl_{\eta}(s)\bigr|\leq1, \qquad \bigl|L_{\eta }(s)\bigr|\leq1. $$

(2.12)

Clearly, if we denote $A_{\eta}(s)=\int_{0}^{s}L_{\eta}(s)\,ds$,

$$\begin{aligned} \lim_{\eta\rightarrow0}L_{\eta}(s)=\operatorname{sgn}(s),\qquad \lim _{\eta \rightarrow0}A_{\eta}(s)= \vert s \vert , \quad s\in(-\infty, + \infty), \end{aligned}$$

(2.13)

and

$$\begin{aligned} \lim_{\eta\rightarrow0}l_{\eta}(s)s= 0. \end{aligned}$$

(2.14)

Suppose that $\xi(x)\in C_{0}^{1}(D_{r})$ such that

$$\begin{aligned} 0\leq\xi\leq1;\qquad \xi |_{D_{2r}}=1, \end{aligned}$$

and choose $\varphi=\xi L_{\eta}(v_{m}-v_{n})$ in (2.10), then

$$\begin{aligned} &\int_{0}^{t} \int_{D_{r}}\xi L_{\eta }(v_{m}-v_{n}) (v_{mt}-v_{nt})\,dx\,d\tau \\ &\qquad{}+\sum_{s=1}^{k} \int_{0}^{t} \int_{D_{r}}b_{i_{s}}(x) \bigl( \vert v_{mx_{i_{s}}} \vert ^{p_{i_{s}}(x)-2} v_{nx_{i_{s}}}- \vert v_{nx_{i_{s}}} \vert ^{p_{i_{s}}(x)-2} v_{nx_{i_{s}}} \bigr) \\ &\qquad{}\times \xi l_{\eta}(v_{m}-v_{n}) (v_{m}-v_{n})_{x_{i_{s}}}\,dx\,d\tau \\ &\qquad{}+\sum_{s=1}^{k} \int_{0}^{t} \int_{D_{r}}b_{i_{s}}(x) \bigl( \vert v_{mx_{i_{s}}} \vert ^{p_{i_{s}}(x)-2} v_{mx_{i_{s}}}- \vert v_{nx_{i_{s}}} \vert ^{p_{i_{s}}(x)-2} v_{nx_{i_{s}}} \bigr) L_{\eta}(v_{m}-v_{n}) \xi_{x_{i_{s}}}\,dx\,d\tau \\ &\qquad{}+\sum_{s=1}^{l} \int_{0}^{t} \int_{D_{r}} \biggl[ \biggl(b_{j_{s}}(x)+ \frac{1}{m} \biggr) \vert v_{mx_{j_{s}}} \vert ^{p_{j_{s}}(x)-2} v_{mx_{j_{s}}}- \biggl(b_{j_{s}}(x)+\frac{1}{n} \biggr) \vert v_{nx_{j_{s}}} \vert ^{p_{j_{s}}(x)-2} v_{nx_{j_{s}}} \biggr] \\ &\qquad{}\times \xi l_{\eta}(v_{m}-v_{n}) (v_{m}-v_{n})_{x_{j_{s}}}\,dx\,d\tau \\ &\qquad{}+\sum_{s=1}^{l} \int_{0}^{t} \int_{D_{r}} \biggl[ \biggl(b_{j_{s}}(x)+ \frac{1}{m} \biggr) \vert v_{mx_{j_{s}}} \vert ^{p_{j_{s}}(x)-2} v_{mx_{j_{s}}}- \biggl(b_{j_{s}}(x)+\frac{1}{n} \biggr) \vert v_{nx_{j_{s}}} \vert ^{p_{j_{s}}(x)-2} v_{nx_{j_{s}}} \biggr] \\ &\qquad{}\times L_{\eta}(v_{m}-v_{n}) \xi_{x_{j_{s}}}\,dx\,d\tau \\ &\qquad{}+\sum_{i=1}^{N} \int_{0}^{t} \int_{D_{r}} \bigl[a(v_{m})-a(v_{n}) \bigr]g^{i}(x)\xi l_{\eta}(v_{m}-v_{n}) (v_{m}-v_{n})_{x_{i}}\,dx\,d\tau \\ &\qquad{}+\sum_{i=1}^{N} \int_{0}^{t} \int_{D_{r}} \bigl[a(v_{m})-a(v_{n}) \bigr]g^{i}(x) L_{\eta }(v_{m}-v_{n}) \xi_{x_{i}}\,dx\,d\tau \\ &\qquad{}+\sum_{i=1}^{N} \int_{0}^{t} \int_{D_{r}} \bigl[a(v_{m})-a(v_{n}) \bigr]g^{i}_{x_{i}}\xi L_{\eta}(v_{m}-v_{n}) \,dx\,d\tau \\ &\quad =0. \end{aligned}$$

(2.15)

Clearly,

$$\begin{aligned} &\int_{0}^{t} \int_{D_{r}}b_{i_{s}}(x) \bigl( \vert v_{mx_{i_{s}}} \vert ^{p_{i_{s}}(x)-2} v_{mx_{i_{s}}}- \vert v_{nx_{i_{s}}} \vert ^{p_{i_{s}}(x)-2} v_{nx_{i_{s}}} \bigr)\xi l_{\eta}(v_{m}-v_{n}) (v_{m}-v_{n})_{x_{i_{s}}}\,dx\,d\tau \\ &\quad\geq0 \end{aligned}$$

(2.16)

and

$$\begin{aligned} &\int_{0}^{t} \int_{D_{r}} \biggl[ \biggl(b_{j_{s}}(x)+ \frac{1}{m} \biggr) \vert v_{mx_{j_{s}}} \vert ^{p_{j_{s}}(x)-2} v_{mx_{j_{s}}}- \biggl(b_{j_{s}}(x)+\frac{1}{n} \biggr) \vert v_{nx_{j_{s}}} \vert ^{p_{j_{s}}(x)-2} v_{nx_{j_{s}}} \biggr] \\ &\qquad{}\times \xi l_{\eta}(v_{m}-v_{n}) (v_{m}-v_{n})_{x_{j_{s}}}\,dx\,d\tau \\ &\quad \geq0. \end{aligned}$$

(2.17)

Noticing that $\xi\in C_{0}^{1}(\Omega)$, $a(s)$ is a Lipschitz function, using Hölder’s inequality of the variable exponent Sobolev space, by (2.14), we easily deduce that

$$\begin{aligned} \lim_{\eta\rightarrow0} \int_{D_{r}} \bigl[a(v_{m})-a(v_{n}) \bigr]g^{i}(x)\xi l_{\eta }(v_{m}-v_{n}) (v_{m}-v_{n})_{x_{i}}\,dx=0. \end{aligned}$$

(2.18)

At the same time,

$$\begin{aligned} &\lim_{\eta\rightarrow0} \int_{0}^{t} \int_{D_{r}} \xi L_{\eta}(v_{m}-v_{n}) (v_{mt}-v_{nt})\,dx\,d\tau \\ &\quad =\lim_{\eta\rightarrow 0} \int_{0}^{t} \int_{D_{r}} \xi \biggl( \int_{0}^{v_{m}-v_{n}}L_{\eta}(s)\,ds \biggr)_{\tau}\,dx\,d\tau \\ &\quad =\lim_{\eta\rightarrow0} \int_{0}^{t} \int_{D_{r}} \xi \int_{0}^{v_{m}-v_{n}}\operatorname{sgn}_{\eta}(s)\,ds \Big|_{0}^{t}\,dx \\ &\quad= \int_{\Omega_{r}}\xi \vert v_{m}-v_{n} \vert \,dx- \int_{D_{r}}\xi \vert v_{0m}-v_{0n} \vert \,dx. \end{aligned}$$

(2.19)

Let $\eta\rightarrow0$. By (2.11)–(2.19), we can obtain

$$\begin{aligned} &\int_{D_{2r}}\xi \vert v_{m}-v_{n} \vert \,dx\\ &\quad\leq \int_{D_{r}} \vert v_{0m}-v_{0n} \vert \,dx \\ &\qquad{}+\frac{c}{r}\sum_{s=1}^{k} \int_{0}^{t} \int _{D_{r}}b_{i_{s}}(x) \bigl( \vert v_{mx_{i_{s}}} \vert ^{p_{i_{s}}(x)-1}+ \vert v_{nx_{i_{s}}} \vert ^{p_{i_{s}}(x)-1} \bigr)\,dx\,d\tau \\ &\qquad{}+\frac{c}{r}\sum_{s=1}^{l} \int_{0}^{t} \int_{D_{r}} \biggl[ \biggl(b_{j_{s}}(x)+ \frac {1}{m} \biggr) \vert v_{mx_{j_{s}}} \vert ^{p_{j_{s}}(x)-1}+ \biggl(b_{j_{s}}(x)+ \frac{1}{n} \biggr) \vert v_{nx_{j_{s}}} \vert ^{p_{j_{s}}(x)-1} \biggr]\,dx\,d\tau \\ &\qquad{}+\frac{c}{r}\sum_{i=1}^{N} \int_{0}^{t} \int _{D_{r}} \vert v_{m}-v_{n} \vert \,dx\,d\tau \\ &\quad\leq \int_{D_{r}} \vert v_{0m}-v_{0n} \vert \,dx+c_{r}(t). \end{aligned}$$

By $v_{\varepsilon}\in L^{\infty}(Q_{T})$ and the unform estimates (2.6)–(2.7), we know that $c_{r}(t)$ is independent of $m,n$.

Now, for any given small r, if $m,n$ are large enough, by (2.9), we have

$$\begin{aligned} &\int_{D_{2r}} \bigl\vert v(x,t)-v_{0}(x) \bigr\vert \,dx \\ &\quad\leq \int_{D_{2r}} \bigl\vert v(x,t)-v_{m}(x,t) \bigr\vert \,dx+ \int_{D_{2r}} \bigl\vert v_{m}(x,t)-v_{n}(x,t) \bigr\vert \,dx \\ &\qquad{}+ \int_{D_{2r}} \bigl\vert v_{n}(x,t)-v_{0n}(x) \bigr\vert \,dx+ \int_{D_{2r}} \bigl\vert v_{0n}(x)-v_{0}(x) \bigr\vert \,dx \\ &\quad\leq \int_{D_{2r}} \bigl\vert v(x,t)-v_{m}(x,t) \bigr\vert \,dx+ \int _{D_{r}} \bigl\vert v_{m}(x,0)-v_{n}(x,0) \bigr\vert \,dx+c_{r}(t) \\ &\qquad{}+ \int_{D_{2r}} \bigl\vert v_{n}(x,t)-v_{0n}(x) \bigr\vert \,dx+ \int_{D_{2r}} \bigl\vert v_{0n}(x)-v_{0}(x) \bigr\vert \,dx. \end{aligned}$$

(2.20)

By (2.20), similar to the usual evolutionary p-Laplacian equation (Chap. 2, [9]), we have (1.12). Then Theorem 1.2 is proved.

Certainly, the initial value condition (1.5) can be right in the other sense; for example, in [11], it has the form

$$\begin{aligned} \lim_{t\rightarrow0} \int_{\Omega}v(x,t)\phi(x)\,dx= \int_{\Omega }v_{0}(x)\phi(x)\,dx,\quad\forall\phi(x)\in C_{0}^{\infty}(\Omega). \end{aligned}$$

Also, the existence of weak solutions can be proved in other ways. Here, we would like to mention some recent related papers [12–14]. □

3 The stability of the initial-boundary value problem

Theorem 3.1

If $p_{0}>1$, $b_{i}(x)$ satisfies conditions (1.8), (1.9), $g^{i}(x)\in C^{1}(\overline{\Omega})$, $a(s)$ is a Lipschitz function and for every $1\leq r\leq l$, $\int_{\Omega}b_{j_{r}}^{-\frac {1}{p_{j_{r}}-1}}(x)\,dx<\infty$, $g^{i}(x)$ satisfies

$$\begin{aligned} g^{i}(x)b_{i}^{-\frac{1}{p_{0}}}(x)\leq c. \end{aligned}$$

(3.1)

If $v(x,t)$ and $u(x.t)$ are two solutions of equation (1.4) with the same homogeneous value

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

and with different initial values $u_{0}(x)$ and $v_{0}(x)$, then

$$\begin{aligned} \int_{\Omega} \bigl\vert v(x,t)-u(x,t) \bigr\vert \,dx\leq \int_{\Omega} \bigl\vert v_{0}(x)-u_{0}(x) \bigr\vert \,dx. \end{aligned}$$

Proof

Since $\int_{\Omega}b_{j_{r}}^{-\frac {1}{p_{j_{r}}-1}}(x)\,dx<\infty$, by Lemma 2.1, we can choose $\varphi =\chi_{[\tau,s]}L_{\eta}(v - u)$ in (1.11), where $\chi_{[\tau ,s]}$ is the characteristic function of $[\tau, s]\subset(0, T)$. Then

$$\begin{aligned} &\int_{\tau}^{s} \int_{\Omega} L_{\eta}(v - u)\frac{\partial(v - u)}{\partial t}\,dx\,dt \\ &\qquad{} +\sum _{i=1}^{N} \int_{\tau}^{s} \int_{\Omega} b_{i}(x) \bigl( \vert v_{x_{i}} \vert ^{p_{i}(x)- 2}v_{x_{i}} - \vert u_{x_{i}} \vert ^{p_{i}(x)- 2}u_{x}{x_{i}} \bigr) (v- u)_{x_{i}}l_{\eta}(v-u) \,dx\,dt \\ &\qquad{}+\sum_{i=1}^{N} \int_{\tau}^{s} \int_{\Omega } \bigl[ \bigl(a(v)-a(u) \bigr)g^{i}(x)l_{\eta}(v-u) (v-u)_{x_{i}} \\ &\qquad{}+ \bigl(a(v)-a(u) \bigr)g^{i}_{x_{i}}L_{\eta}(v-u) \bigr] \,dx\,dt \\ &\quad=0. \end{aligned}$$

(3.2)

At first, we have

$$\begin{aligned} \int_{\tau}^{s} \int_{\Omega} b_{i}(x) \bigl( \vert v_{x_{i}} \vert ^{p_{i}(x)- 2} v_{x_{i}} - \vert u _{x_{i}} \vert ^{p_{i}(x) - 2}u_{x_{i}} \bigr) (v- u)_{x_{i}}l_{\eta}(v-u)\,dx\, dt\geq0. \end{aligned}$$

(3.3)

By Lemma 3.1 from [11], we have

$$\begin{aligned} &\lim_{\eta\rightarrow0} \int_{\tau}^{s} \int_{\Omega}L_{\eta}(v - u) \frac{\partial(v - u)}{\partial t}\,dx\,dt \\ &\quad=\lim_{\eta\rightarrow0} \int_{\Omega} \bigl[A_{\eta}(v-u) (x,s)-A_{\eta }(v-u) (x,\tau) \bigr]\,dx \\ &\quad= \int_{\Omega} \vert v-u \vert (x,s)\,dx- \int_{\Omega} \vert v-u \vert (x,\tau) \,dx. \end{aligned}$$

(3.4)

Moreover, since $g^{i}(x)$ satisfies condition (3.1)

$$\begin{aligned} g^{i}(x)b_{i}^{-\frac{1}{p_{0}}}\leq c, \end{aligned}$$

by (3.4), $a(s)$ is a Lipschitz function, we have

$$\begin{aligned} &\lim_{\eta\rightarrow0}\sum_{i=1}^{N} \biggl\vert \int_{\tau}^{s} \int _{{\Omega}} \bigl[a(v)-a(u) \bigr]g^{i}(x)l_{\eta}(v-u) (v-u)_{x_{i}}\,dx\,dt \biggr\vert \\ &\quad=\lim_{\eta\rightarrow0}\sum_{i=1}^{N} \biggl\vert \int_{\tau}^{s} \int _{{\Omega}} \bigl[a(v)-a(u) \bigr]g^{i}(x)b_{i}^{-\frac{1}{p_{0}}}l_{\eta }(v-u)b_{i}^{-\frac{1}{p_{0}}}(v-u)_{x_{i}}\,dx\,dt \biggr\vert \\ &\quad\leq\sum_{i=1}^{N} \lim _{\eta\rightarrow0} \biggl( \int_{\tau }^{s} \int_{{\Omega}} \bigl\vert \bigl[a(v)-a(u) \bigr]g^{i}(x)b_{i}^{-\frac {1}{p_{0}}}l_{\eta}(v-u) \bigr\vert ^{\frac{p_{0}}{p_{0}-1}}\,dx\,dt \biggr)^{\frac{p_{0}-1}{p_{0}}} \\ &\qquad{}\times \biggl( \int_{\tau}^{s} \int_{{\Omega }}b_{i}(x) \bigl( \vert v_{x_{i}} \vert ^{p_{0}}+ \vert u_{x_{i}} \vert ^{p_{0}} \bigr)\,dx\,dt \biggr)^{\frac{1}{p_{0}}} \\ &\quad\leq c\sum_{i=1}^{N} \lim _{\eta\rightarrow0} \biggl( \int_{\tau }^{s} \int_{{\Omega}} \bigl\vert (v-u)l_{\eta}(v-u) \bigr\vert ^{\frac {p_{0}}{p_{0}-1}}\,dx\,dt \biggr)^{\frac{p_{0}-1}{p_{0}}} \\ &\qquad{}\times \biggl( \int_{\tau}^{s} \int_{{\Omega }}b_{i}(x) \bigl( \vert v_{x_{i}} \vert ^{p_{0}}+ \vert u_{x_{i}} \vert ^{p_{0}} \bigr)\,dx\,dt \biggr)^{\frac{1}{p_{0}}} \\ &\quad =0 . \end{aligned}$$

(3.5)

$$\begin{aligned} &\lim_{\eta\rightarrow0}\sum_{i=1}^{N} \biggl\vert \int_{\tau}^{s} \int _{{\Omega}} \bigl[a(v)-a(u) \bigr]g^{i}_{x_{i}}L_{\eta}(v-u)\,dx\,dt \biggr\vert \\ &\quad \leq c \int_{\tau}^{s} \int_{{\Omega}} \vert v-u \vert \,dx\,dt. \end{aligned}$$

(3.6)

Now, let $\eta\rightarrow0$ in (3.2). Then

$$\begin{aligned} &\int_{\Omega}\bigl\vert v(x,s) - u(x,s) \bigr\vert \,dx \\ &\quad\leqslant \int _{\Omega}\bigl\vert v(x,\tau)- u(x,\tau) \bigr\vert \,dx+c \int_{\tau }^{s} \int_{{\Omega}} \vert v-u \vert \,dx\,dt. \end{aligned}$$

(3.7)

By Gronwall’s inequality, letting $\tau\rightarrow0$, we have

$$\begin{aligned} \int_{\Omega}\bigl\vert v(x,s) - u(x,s) \bigr\vert \,dx \leqslant \int _{\Omega}\bigl\vert v_{0}(x) - u_{0}(x) \bigr\vert \,dx. \end{aligned}$$

Theorem 3.1 is proved.

One can see that condition (3.1) is used to prove (3.4). In fact, without this condition, the conclusion of Theorem 3.1 is still true. This is Theorem 1.3. □

Proof of Theorem 1.3

From the above proof of Theorem 3.1, we only need to prove that

$$\begin{aligned} &\lim_{\eta\rightarrow0} \biggl\vert \int_{\Omega} \bigl[a(v)-a(u) \bigr]g^{i}(x) (v - u)_{x_{i}}L'_{\eta}(v-u)\,dx \biggr\vert \\ &\quad\leq c\lim_{\eta\rightarrow0} \int_{\Omega} \bigl\vert \bigl[a(v)-a(u) \bigr](v - u)_{x_{i}}L'_{\eta}(u-v) \bigr\vert \,dx =0, \end{aligned}$$

(3.8)

without condition (3.1). Let us give an explanation. Noticing

$$\begin{aligned} &\int_{\Omega} \bigl\vert \bigl[a(v)-a(u) \bigr](v - u)_{x_{i}}L'_{\eta}(v-u) \bigr\vert \,dx \\ &\quad= \int_{\Omega} \bigl\vert \bigl[a(v)-a(u) \bigr](v - u)_{x_{i}}l_{\eta}(v-u) \bigr\vert \,dx \\ &\quad= \int_{\{\Omega: \vert v-u \vert < \eta\}} \bigl\vert \bigl[a(v)-a(u) \bigr](v - u)_{x_{i}}l_{\eta}(v-u) \bigr\vert \,dx, \end{aligned}$$

(3.9)

if $\{\Omega: \vert v-u \vert =0\}$ is a subset of Ω with a positive measure, then

$$ \begin{aligned} &\lim_{\eta\rightarrow0} \int_{\{\Omega: \vert v-u \vert < \eta\} } \bigl| \bigl[a(v)-a(u) \bigr](v - u)_{x_{i}}l_{\eta}(v-u)\bigr| \,dx \\ &\quad\leq c \biggl( \int_{\{\Omega: \vert v-u \vert =0\}} \bigl(b_{i}^{\frac {1}{p_{i}}} \vert v_{x_{i}}-u_{x_{i}} \vert \bigr)^{p_{i}}\,dx \biggr)^{\frac{1}{p_{i}}} \biggl( \int _{\Omega}b_{i}^{-\frac{1}{p_{i}-1}}\,dx \biggr)^{\frac{p_{i}-1}{p_{i}}} \\ &\quad =0. \end{aligned} $$

At the same time, if $\{\Omega: \vert v-u \vert =0\}$ is a subset of Ω with zero measure, since $b_{i}(x)$ satisfies (1.8)–(1.9), and for every $1\leq r\leq l$, $\int_{\Omega}b_{j_{r}}^{-\frac{1}{p_{j_{r}}-1}}(x)\,dx<\infty$, we have

$$\begin{aligned} b_{i}^{-\frac{1}{p_{i}-1}}\in L^{1}(\Omega),\quad i=1, 2, \ldots, N. \end{aligned}$$

Then

$$ \begin{aligned} &\lim_{\eta\rightarrow0} \int_{\{\Omega: \vert v-u \vert < \eta\} } \bigl| \bigl[a(v)-a(u) \bigr](v- u)_{x_{i}}l_{\eta}(v- u)\bigr| \,dx \\ &\quad\leq c \biggl( \int_{\Omega} \bigl(b_{i}^{\frac {1}{p_{i}}} \vert v_{x_{i}}-u_{x_{i}} \vert \bigr)^{p_{i}}\,dx \biggr)^{\frac{1}{p_{i}}} \biggl( \int _{\{\Omega: \vert v- u \vert =0\}}b_{i}^{-\frac{1}{p_{i}-1}}\,dx \biggr)^{\frac{p_{i}-1}{p_{i}}} \\ &\quad\leq c \biggl( \int_{\Omega }b_{i}(x) \bigl( \vert v_{x_{i}} \vert ^{p_{i}}+ \vert u_{x_{i}} \vert ^{p_{i}} \bigr)\,dx \biggr)^{\frac {1}{p_{i}}} \biggl( \int_{\{\Omega: \vert v- u \vert =0\}}b_{i}^{-\frac {1}{p_{i}-1}}\,dx \biggr)^{\frac{p_{i}-1}{p_{i}}} \\ &\quad=0. \end{aligned} $$

Thus, Theorem 1.3 is true. □

4 The stability based on the partial boundary value condition

Theorem 4.1

If $p_{0}>1$, $b_{i}(x)$ satisfies conditions (1.8), (1.9), $g^{i}(x)\in C^{1}(\overline{\Omega})$ satisfies (3.5), $a(s)$ is a Lipschitz function. Let $v(x,t)$ and $u(x,t)$ be two solutions of equation (1.4). If the initial values $u_{0}(x)$ and $v_{0}(x)$ are different, while the partial boundary values satisfy

$$\begin{aligned} v(x,t)=u(x,t)=0,\quad (x,t)\in\Sigma_{1}\times[0,T), \end{aligned}$$

then the stability (1.15) is true, provided that for every $1\leq r\leq l$, condition (1.17) is true, i.e.,

$$\begin{aligned} \frac{1}{\eta} \Biggl( \int_{\Omega\setminus\Omega_{\eta}} b_{j_{r}}(x) \Biggl\vert \Biggl(\sum _{s=1}^{l} b_{j_{s}}(x) \Biggr)_{x_{j_{r}}} \Biggr\vert ^{p_{j_{r}}}\,dx \Biggr)^{\frac{1}{p^{+}_{j_{r}}}}\leq c. \end{aligned}$$

Proof

Let $\Omega_{\eta}=\{x\in\Omega:\sum_{r=1}^{l}b_{j_{r}}(x)>\eta\}$, and

$$\begin{aligned} \phi_{\eta}(x)= \textstyle\begin{cases} 1 & \text{if }x\in\Omega_{\eta},\\ \frac{1}{\eta}\sum_{r=1}^{l}b_{j_{r}}(x)& \text{if }x\in\Omega \setminus\Omega_{\eta}. \end{cases}\displaystyle \end{aligned}$$

(4.1)

Then, if $x\in\Omega\setminus\Omega_{\eta}$, $\phi_{\eta x_{i}}=\frac{1}{\eta}(\sum_{r=1}^{l} b_{j_{r}}(x))_{x_{i}}$, while $x\in \Omega_{\eta}$, $\phi_{x_{i}}=0$.

Let $\varphi=\chi_{[\tau,s]}\phi_{\eta}L_{\eta}(v-u)$ be the test function in (1.11). Then

$$\begin{aligned} &\int_{\tau}^{s} \int_{{\Omega}} \phi_{\eta}L_{\eta}(v-u) \frac {\partial(v-u)}{\partial t}\,dx\,dt \\ &\qquad{}+ \sum_{i=1}^{N} \int_{\tau}^{s} \int_{\Omega} b_{i}(x) \bigl( \vert v_{x_{i}} \vert ^{p_{i}(x) - 2} v_{x_{i}} - \vert u_{x_{i}} \vert ^{p_{i}(x)- 2} u_{x_{i}} \bigr) (u_{x_{i}} - v_{x_{i}})l_{\eta}(u-v) \phi_{\eta }(x)\,dx\,dt \\ &\qquad{}+ \sum_{r=1}k \int_{\tau}^{s} \int_{\Omega} b_{i_{r}}(x) \bigl( \vert v_{x_{i_{r}}} \vert ^{p_{i_{r}}(x) - 2}v_{x_{i_{r}}} - \vert u_{x_{i_{r}}} \vert ^{p_{i_{r}}(x)- 2} u_{x_{i_{r}}} \bigr)L_{\eta}(v-u) \phi_{\eta x_{i_{r}}}\,dx\,dt \\ &\qquad{}+ \sum_{r=l}^{l} \int_{\tau}^{s} \int_{\Omega} b_{j_{r}}(x) \bigl( \vert v_{x_{j_{r}}} \vert ^{p_{j_{r}}(x) - 2}v_{x_{j_{r}}} - \vert u_{x_{j_{r}}} \vert ^{p_{j_{r}}(x)- 2} u_{x_{j_{r}}} \bigr)L_{\eta}(v-u) \phi_{\eta x_{j_{r}}}\,dx\,dt \\ &\qquad{}+\sum_{i=1}^{N} \int_{\tau}^{s} \int_{\Omega } \bigl[ \bigl(a(v)-a(u) \bigr)g^{i}(x)l_{\eta}(v-u) (v-u)_{x_{i}}\phi_{\eta} \\ &\qquad{}+ \bigl(a(v)-a(u) \bigr)g^{i}_{x_{i}}L_{\eta}(v-u) \phi_{\eta x_{i}} \bigr] \,dx\,dt \\ &\quad=0. \end{aligned}$$

(4.2)

At first,

$$\begin{aligned} \int_{\Omega} b_{i}(x) \bigl( \vert v_{x_{i}} \vert ^{p_{i}(x) - 2}v_{x_{i}} - \vert u_{x_{i}} \vert ^{p_{i}(x) - 2} v_{x_{i}} \bigr) (v_{x_{i}} - u_{x_{i}})l_{\eta}(v-u) \phi_{\eta}(x)\,dx\geq0, \end{aligned}$$

(4.3)

and

$$\begin{aligned} &\lim_{\eta\rightarrow0} \int_{\tau}^{s} \int_{\Omega}\phi_{\eta }(x)L_{\eta}(v-u) \frac{\partial(v-u)}{\partial t}\,dx\,dt \\ &\quad= \int_{\Omega} \vert v-u \vert (x,s)\,dx- \int_{\Omega} \vert v-u \vert (x,\tau)\,dx. \end{aligned}$$

(4.4)

Secondly, by Hölder’s inequality of the variable exponent Sobolev space, we have

$$\begin{aligned} & \biggl\vert \int_{\Omega} b_{i_{r}}(x) \bigl( \vert v_{x_{i_{r}}} \vert ^{p_{i_{r}}(x) - 2}v_{x_{i_{r}}} - \vert u_{x_{i_{r}}} \vert ^{p_{i_{r}}(x)- 2} u_{x_{i_{r}}} \bigr)L_{\eta}(v-u) \phi_{\eta x_{i_{r}}}\,dx \biggr\vert \\ &\quad= \biggl\vert \int_{\Omega\setminus\Omega_{\eta}}b_{i_{r}}(x) \bigl( \vert v_{x_{i_{r}}} \vert ^{p_{i_{r}}(x) - 2}v_{x_{i_{r}}} - \vert u_{x_{i_{r}}} \vert ^{p_{i_{r}}(x)- 2} u_{x_{i_{r}}} \bigr)L_{\eta}(v-u) \phi_{\eta x_{i_{r}}}\,dx \biggr\vert \\ &\quad\leq \int_{\Omega\setminus\Omega_{\eta}} b_{i_{r}}(x) \bigl( \vert v_{x_{i_{r}}} \vert ^{p_{i_{r}}(x) - 1}+ \vert u_{x_{i_{r}}} \vert ^{p_{i_{r}}(x)- 1} \bigr) \bigl\vert L_{\eta}(v-u)\phi_{\eta x_{i_{r}}} \bigr\vert \,dx \\ &\quad\leq \int_{\Omega\setminus\Omega_{\eta}} \Biggl[\frac{1}{\eta }b_{i_{r}}(x)\sum _{s=1}^{l} b_{j_{s}}(x) \Biggr]^{\frac {p_{i_{r}(x)}-1}{p_{i_{r}(x)}}} \bigl( \vert v_{x_{i_{r}}} \vert ^{p_{i_{r}}(x) - 1}+ \vert u_{x_{i_{r}}} \vert ^{p_{i_{r}}(x)- 1} \bigr) \\ &\qquad{}\times \bigl(b_{i_{r}}(x) \bigr)^{\frac{1}{p_{i_{r}(x)}}} \biggl\vert L_{\eta}(v-u)\frac{(\frac {1}{\eta})^{\frac{1}{p_{i_{r}}(x)}}[\sum_{s=1}^{l} b_{j_{s}}(x)]_{x_{i_{r}}}}{[\sum_{s=1}^{l} b_{j_{s}} (x)]^{\frac {p_{i_{r}(x)}-1}{p_{i_{r}(x)}}}} \biggr\vert \,dx \\ &\quad\leq \Biggl( \int_{\Omega\setminus\Omega_{\eta}}\frac{1}{\eta }\sum_{s=1}^{l} b_{j_{s}}(x) b_{i_{r}}(x) \bigl( \vert v_{x_{i_{r}}} \vert ^{p_{i_{r}}(x)}+ \vert u_{x_{i_{r}}} \vert ^{p_{i_{r}}(x)} \bigr)\,dx \Biggr)^{\frac{1}{q^{1}_{i_{r}}}} \\ &\qquad{}\times \biggl( \int_{\Omega\setminus\Omega_{\eta}} \frac{1}{\eta }b_{i_{r}}(x) \bigl\vert L_{\eta}(v-u) \bigr\vert ^{p_{i_{r}}(x)} \frac{ \vert (\sum_{s=1}^{l} b_{j_{s}}(x))_{x_{i_{r}}} \vert ^{p_{i_{r}}(x)}}{[\sum_{s=1}^{l} b_{j_{s}} (x)]^{p_{i_{r}(x)}-1}}\,dx \biggr)^{\frac{1}{p^{1}_{i_{r}}}} \\ &\quad\leq c \biggl( \int_{\Omega\setminus\Omega_{\eta}}b_{i_{r}}(x) \bigl( \vert v_{x_{i_{r}}} \vert ^{p_{i_{r}}(x)}+ \vert u_{x_{i_{r}}} \vert ^{p_{i_{r}} (x)} \bigr)\,dx \biggr)^{\frac{1}{q^{1}_{i_{r}}}} \\ &\qquad{}\times \biggl(\frac{1}{\eta} \int_{\Omega\setminus\Omega_{\eta }} \bigl\vert L_{\eta}(v-u) \bigr\vert ^{p_{i_{r}}(x)} \frac{ \vert (\sum_{s=1}^{l} b_{j_{s}}(x))_{x_{i_{r}}} \vert ^{p_{i_{r}}(x)}}{[\sum_{s=1}^{l} b_{j_{s}} (x)]^{p_{i_{r}(x)}-1}}\,dx \biggr)^{\frac{1}{p^{1}_{i_{r}}}}. \end{aligned}$$

(4.5)

Here, $p^{1}_{i_{r}}=p^{+}_{i_{r}}$ or $p^{-}_{i_{r}}$ according to

$$\begin{aligned} \int_{\Omega\setminus\Omega_{\eta}} \frac{1}{\eta }b_{i_{r}}(x) \bigl\vert L_{\eta}(v-u) \bigr\vert ^{p_{i_{r}}(x)} \frac{ \vert (\sum_{s=1}^{l} b_{j_{s}}(x))_{x_{i_{r}}} \vert ^{p_{i_{r}}(x)}}{[\sum_{s=1}^{l} b_{j_{s}} (x)]^{p_{i_{r}(x)}-1}}\,dx\geq1, \end{aligned}$$

or

$$\begin{aligned} \int_{\Omega\setminus\Omega_{\eta}} \frac{1}{\eta }b_{i_{r}}(x) \bigl\vert L_{\eta}(v-u) \bigr\vert ^{p_{i_{r}}(x)} \frac{ \vert (\sum_{s=1}^{l} b_{j_{s}}(x))_{x_{i_{r}}} \vert ^{p_{i_{r}}(x)}}{[\sum_{s=1}^{l} b_{j_{s}} (x)]^{p_{i_{r}(x)}-1}}\,dx< 1, \end{aligned}$$

one can refer to Lemma 2.1 of [4]. $q_{j_{r}}(x)=\frac {p_{j_{r}}(x)}{p_{j_{r}}(x)-1}$, $q^{1}_{i_{r}}$ has a similar sense.

Let $\Sigma_{2}=\partial\Omega\setminus\Sigma_{1}$, and

$$\begin{aligned} &\Omega_{\eta1}= \bigl\{ x\in\Omega\setminus\Omega_{\eta}:\operatorname {dist}(x, \Sigma_{2})>\operatorname{dist}(x, \Sigma_{1}) \bigr\} , \\ &\Omega_{\eta2}= \bigl\{ x\in\Omega\setminus\Omega_{\eta}:\operatorname {dist}(x, \Sigma_{2})\leq\operatorname{dist}(x, \Sigma_{1}) \bigr\} . \end{aligned}$$

Then

$$\begin{aligned} &\frac{1}{\eta} \int_{\Omega\setminus\Omega_{\eta}} \bigl\vert L_{\eta }(v-u) \bigr\vert ^{p_{i_{r}}(x)} \frac{ \vert (\sum_{s=1}^{l} b_{j_{s}}(x))_{x_{i_{r}}} \vert ^{p_{i_{r}}(x)}}{[\sum_{s=1}^{l} b_{j_{s}}(x)]^{p_{i_{r}(x)}-1}}\,dx \\ &\quad\leq\frac{1}{\eta} \int_{\Omega_{\eta1}} \bigl\vert L_{\eta }(v-u) \bigr\vert ^{p_{i_{r}}(x)} \frac{ \vert (\sum_{s=1}^{l} b_{j_{s}}(x))_{x_{i_{r}}} \vert ^{p_{i_{r}}(x)}}{[\sum_{s=1}^{l} b_{j_{s}}(x)]^{p_{i_{r}(x)}-1}}\,dx \\ &\qquad{}+n \int_{\Omega_{\eta2}} \bigl\vert L_{\eta}(v-u) \bigr\vert ^{p_{i_{r}}(x)} \frac{ \vert (\sum_{s=1}^{l} b_{j_{s}}(x))_{x_{i_{r}}} \vert ^{p_{i_{r}}(x)}}{[\sum_{s=1}^{l} b_{j_{s}}(x)]^{p_{i_{r}(x)}-1}}\,dx. \end{aligned}$$

(4.6)

Since

$$\begin{aligned} v=u=0,\qquad x\in\Sigma_{1}, \end{aligned}$$

by the definition of the trace, we have

$$\begin{aligned} &\lim_{\eta\rightarrow0}\frac{1}{\eta} \int_{\Omega_{\eta1}} \bigl\vert L_{\eta}(v-u) \bigr\vert ^{p_{i_{r}}(x)} \frac{ \vert (\sum_{s=1}^{l} b_{j_{s}}(x))_{x_{i_{r}}} \vert ^{p_{i_{r}}(x)}}{[\sum_{s=1}^{l} b_{j_{s}} (x)]^{p_{i_{r}(x)}-1}}\,dx \\ &\quad = \int_{\Sigma_{1}}\operatorname{sign}(v-u)\frac{ \vert (\sum_{s=1}^{l} b_{j_{s}}(x))_{x_{i_{r}}} \vert ^{p_{i_{r}}(x)}}{[\sum_{s=1}^{l} b_{j_{s}}(x)]^{p_{i_{r}(x)}-1}}\,d \Sigma=0. \end{aligned}$$

(4.7)

Moreover, since

$$\begin{aligned} \frac{ \vert (\sum_{s=1}^{l} b_{j_{s}}(x))_{x_{i_{r}}} \vert ^{p_{i_{r}}(x)}}{[\sum_{s=1}^{l} b_{j_{s}}(x)]^{p_{i_{r}(x)}-1}}=0,\quad x\in\Sigma_{2}, \end{aligned}$$

we have

$$\begin{aligned} &\lim_{\eta\rightarrow\infty}\frac{1}{\eta} \int_{\Omega_{\eta 2}} \bigl\vert L_{\eta}(v-u) \bigr\vert ^{p_{i_{r}}(x)} \frac{ \vert (\sum_{s=1}^{l} b_{j_{s}}(x))_{x_{i_{r}}} \vert ^{p_{i_{r}}(x)}}{[\sum_{s=1}^{l} b_{j_{s}} (x)]^{p_{i_{r}(x)}-1}}\,dx \\ &\quad\leq\lim_{\eta\rightarrow\infty}\frac{1}{\eta} \int_{\Omega _{\eta2}} \frac{ \vert (\sum_{s=1}^{l} b_{j_{s}}(x))_{x_{i_{r}}} \vert ^{p_{i_{r}}(x)}}{[\sum_{s=1}^{l} b_{j_{s}}(x)]^{p_{i_{r}(x)}-1}}\,dx \\ &\quad= \int_{\Sigma_{2}}\frac{ \vert (\sum_{s=1}^{l} b_{j_{s}}(x))_{x_{i_{r}}} \vert ^{p_{i_{r}}(x)}}{[\sum_{s=1}^{l} b_{j_{s}}(x)]^{p_{i_{r}(x)}-1}}\,d\Sigma=0. \end{aligned}$$

(4.8)

According to (4.5)–(4.8), we have

$$\begin{aligned} \lim_{\eta\rightarrow0} \biggl\vert \int_{\Omega} b_{i_{r}}(x) \bigl( \vert u_{x_{i_{r}}} \vert ^{p_{i_{r}}(x) - 2}u_{x_{i_{r}}} - \vert v_{x_{i_{r}}} \vert ^{p_{i_{r}}(x)- 2} v_{x_{i_{r}}} \bigr)L_{\eta}(u-v) \phi_{\eta x_{i_{r}}}\,dx \biggr\vert =0. \end{aligned}$$

(4.9)

Thirdly, for the last term of the left-hand side of (4.2), we have

$$\begin{aligned} & \biggl\vert \int_{\Omega} b_{j_{r}}(x) \bigl( \vert v_{x_{j_{r}}} \vert ^{p_{j_{r}}(x) - 2}u_{x_{j_{r}}} - \vert u_{x_{j_{r}}} \vert ^{p_{j_{r}}(x)- 2} v_{x_{j_{r}}} \bigr)L_{\eta}(v-u) \phi_{\eta x_{j_{r}}}\,dx \biggr\vert \\ &\quad= \biggl\vert \int_{\Omega\setminus\Omega_{\eta}}b_{j_{r}}(x) \bigl( \vert v_{x_{j_{r}}} \vert ^{p_{j_{r}}(x) - 2}v_{x_{j_{r}}} - \vert u_{x_{j_{r}}} \vert ^{p_{j_{r}}(x)- 2} u_{x_{j_{r}}} \bigr)L_{\eta}(v-u) \phi_{\eta x_{j_{r}}}\,dx \biggr\vert \\ &\quad\leq \int_{\Omega\setminus\Omega_{\eta}} b_{j_{r}}(x) \bigl(\vert v_{x_{j_{r}}} \vert ^{p_{j_{r}}(x) - 1}+ \vert u_{x_{j_{r}}} \vert ^{p_{j_{r}}(x)- 1} \bigr) \Biggl\vert \Biggl(\sum_{s=1}^{l} b_{j_{s}} (x) \Biggr)_{x_{j_{r}}} L_{\eta}(v-u)\Biggr\vert \,dx \\ &\quad\leq c \biggl( \int_{\Omega\setminus\Omega_{\eta}} b_{j_{r}}(x)\bigl(\vert v_{x_{j_{r}}} \vert ^{p_{j_{r}}(x)}+ \vert u_{x_{j_{r}}} \vert ^{p_{j_{r}}(x)}\bigr) \,dx \biggr)^{\frac{1}{q^{+}_{j_{r}}}} \\ &\qquad{}\times \frac{1}{\eta} \Biggl( \int_{\Omega\setminus\Omega_{\eta}} b_{j_{r}}(x) \Biggl\vert \Biggl(\sum _{s=1}^{l} b_{j_{s}}(x) \Biggr)_{x_{i}} \Biggr\vert ^{p_{j_{r}}(x)}\,dx \Biggr)^{\frac{1}{p^{+}_{j_{r}}}} . \end{aligned}$$

(4.10)

By condition (1.17),

$$\begin{aligned} \frac{1}{\eta} \Biggl( \int_{\Omega\setminus\Omega_{\eta}} b_{j_{r}}(x) \Biggl\vert \Biggl(\sum _{s=1}^{l} b_{j_{s}}(x) \Biggr)_{x_{j_{r}}} \Biggr\vert ^{p_{j_{r}}(x)}\,dx \Biggr)^{\frac{1}{p^{+}_{j_{r}}}}\leq c. \end{aligned}$$

Then

$$\begin{aligned} \lim_{\eta\rightarrow0} \biggl\vert \int_{\Omega} b_{j_{r}}(x) \bigl( \vert v_{x_{j_{r}}} \vert ^{p_{j_{r}}(x) - 2}v_{x_{j_{r}}} - \vert u_{x_{j_{r}}} \vert ^{p_{j_{r}}(x)- 2} u_{x_{j_{r}}} \bigr)L_{\eta}(v-u) \phi_{\eta x_{j_{r}}}\,dx \biggr\vert =0. \end{aligned}$$

(4.11)

Fourthly, since $g^{i}(x)$ satisfies condition (3.5), we have

$$\begin{aligned} &\lim_{\eta\rightarrow0}\sum_{i=1}^{N} \biggl\vert \int_{\tau}^{s} \int _{{\Omega}} \bigl[a(v)-a(u) \bigr]g^{i}(x)l_{\eta}(v-u) (v-u)_{x_{i}}\phi_{\eta}(x)\,dx\,dt \biggr\vert \\ &\quad=\lim_{\eta\rightarrow0}\sum_{i=1}^{N} \biggl\vert \int_{\tau}^{s} \int _{{\Omega}} \bigl[a(v)-a(u) \bigr]g^{i}(x)a^{-\frac{1}{p_{0}}}l_{\eta }(v-u)b_{i}^{-\frac{1}{p_{0}}}(v-u)_{x_{i}} \phi_{\eta}(x)\,dx\,dt \biggr\vert \\ &\quad\leq\sum_{i=1}^{N} \lim _{\eta\rightarrow0} \biggl( \int_{\tau }^{s} \int_{{\Omega}} \bigl\vert \bigl[a(v)-a(u) \bigr]g^{i}(x)b_{i}^{-\frac{1}{p_{0}}}l_{\eta }(v-u) \bigr\vert ^{\frac{p_{0}}{p_{0}-1}}\,dx\,dt \biggr)^{\frac{p_{0}-1}{p_{0}}} \\ &\qquad{}\times \biggl( \int_{\tau}^{s} \int_{{\Omega }}b_{i}(x) \bigl( \vert v_{x_{i}} \vert ^{p_{0}}+ \vert u_{x_{i}} \vert ^{p_{0}} \bigr)\,dx\,dt \biggr)^{\frac{1}{p_{0}}} \\ &\quad=0. \end{aligned}$$

(4.12)

At last, we have

$$\begin{aligned} \lim_{\eta\rightarrow0}\sum_{i=1}^{N} \biggl\vert \int_{\tau}^{s} \int _{{\Omega}} \bigl[a(v)-a(u) \bigr]g^{i}_{x_{i}}L_{\eta}(u-v) \phi_{\eta}(x)\,dx\,dt \biggr\vert \leq c \int_{\tau}^{s} \int_{{\Omega}} \vert v-u \vert \,dx\,dt. \end{aligned}$$

(4.13)

Now, let $\eta\rightarrow0$ in (4.2). By (4.3), (4.4), (4.9), (4.11), (4.12), and (4.13), we have

$$\begin{aligned} \int_{\Omega}\bigl\vert v(x,s) - u(x,s) \bigr\vert \,dx \leqslant \int_{\Omega}\bigl\vert v(x,\tau) - u(x,\tau) \bigr\vert \,dx+c \int_{\tau }^{s} \bigl\vert v(x,t)-u(x,t) \bigr\vert \,dx\,dt. \end{aligned}$$

(4.14)

By Gronwall’s inequality, we have

$$\begin{aligned} \int_{\Omega}\bigl\vert v(x,s) - u(x,s) \bigr\vert \,dx \leqslant \int_{\Omega}\bigl\vert v(x,\tau) - u(x,\tau) \bigr\vert \,dx. \end{aligned}$$

(4.15)

Let $\tau\rightarrow0$. Then

$$\begin{aligned} \int_{\Omega}\bigl\vert v(x,s) - u(x,s) \bigr\vert \,dx \leqslant \int_{\Omega}\bigl\vert v_{0}(x)- u_{0}(x) \bigr\vert \,dx. \end{aligned}$$

 □

Proof of Theorem 1.4

If $\vert g^{i}(x) \vert \leq c$ and $a(s)$ is a Lipschitz function, for every r satisfies (1.17), similar to the proof of Theorem 1.3 in Sect. 3, combining with Theorem 4.1, we know Theorem 1.4 is true. □

5 The stability without boundary value condition

Theorem 5.1

Let $v(x,t)$ and $u(x,t)$ be two solutions of equation (1.4) with different initial values $v_{0}(x)$ and $u_{0}(x)$, respectively. If $p_{0}>1$, $b_{i}(x)$ satisfies conditions (1.8), (1.9), $g^{i}(x)\in C^{1}(\overline{\Omega})$ satisfies (3.5), $a(s)$ is a Lipschitz function, conditions (1.17) and (1.23) are true, then the stability (1.15) is true.

Proof

As in the proof of Theorem 4.1, let $\varphi=\chi _{[\tau,s]}\phi_{\eta}L_{\eta}(v-u)$ be the test function and obtain (4.2)–(4.4).

Since

$$\begin{aligned} & \biggl\vert \int_{\Omega} b_{i_{r}}(x) \bigl( \vert v_{x_{i_{r}}} \vert ^{p_{i_{r}}(x) - 2}v_{x_{i_{r}}} - \vert u_{x_{i_{r}}} \vert ^{p_{i_{r}}(x)- 2} u_{x_{i_{r}}} \bigr)L_{\eta}(v-u) \phi_{\eta x_{i_{r}}}\,dx \biggr\vert \\ &\quad= \biggl\vert \int_{\Omega\setminus\Omega_{\eta}}b_{i_{r}}(x) \bigl( \vert v_{x_{i_{r}}} \vert ^{p_{i_{r}}(x) - 2}v_{x_{i_{r}}} - \vert u_{x_{i_{r}}} \vert ^{p_{i_{r}}(x)- 2} u_{x_{i_{r}}} \bigr)L_{\eta}(v-u) \phi_{\eta x_{i_{r}}}\,dx \biggr\vert \\ &\quad\leq \int_{\Omega\setminus\Omega_{\eta}} b_{i_{r}}(x) \bigl( \vert V_{x_{i_{r}}} \vert ^{p_{i_{r}}(x) - 1}+ \vert U_{x_{i_{r}}} \vert ^{p_{i_{r}}(x)- 1} \bigr) \bigl\vert L_{\eta}(v-u)\phi_{\eta x_{i_{r}}} \bigr\vert \,dx \\ &\quad\leq \Biggl( \int_{\Omega\setminus\Omega_{\eta}}\sum_{s=1}^{l} b_{j_{s}}(x) b_{i_{r}}(x) \bigl( \vert v_{x_{i_{r}}} \vert ^{p_{i_{r}}(x)}+ \vert u_{x_{i_{r}}} \vert ^{p_{i_{r}}(x)} \bigr)\,dx \Biggr)^{\frac{1}{q^{1}_{i_{r}}}} \\ &\qquad{}\times \frac{c}{\eta} \Biggl( \int_{\Omega\setminus\Omega_{\eta}} \Biggl|\sum_{s=1}^{l} b_{j_{s}}(x)_{x_{i_{r}}}\Biggr| ^{p_{i_{r}}(x)}\,dx \Biggr)^{\frac {1}{p^{1}_{i_{r}}}}, \end{aligned}$$

(5.1)

by condition (1.23), we have (4.9). At the same time, we have (4.10)–(4.15). The proof is complete. □

Proof of Theorem 1.5

If $\vert g^{i}(x) \vert \leq c$ and $a(s)$ is a Lipschitz function, condition (1.23) is true. Similar to the proof of Theorem 1.3 in Sect. 3, combining with Theorem 5.1, we know Theorem 1.5 is true. □