On anisotropic parabolic equations with a nonlinear convection term depending on the spatial variable

Abstract

Consider an anisotropic parabolic equation with a nonlinear convection term depending on the spatial variable. If the diffusion coefficients are degenerate, in general, the boundary trace cannot be defined for the weak solution. The existence and the uniqueness of weak solution are researched without the boundary value condition. Moreover, a general method to prove stability of weak solutions independent of the boundary value condition is introduced for the first time.

1 Introduction

In this paper, the anisotropic parabolic equation

$$ u_{t}=\sum_{i=1}^{N} \frac{\partial}{\partial x_{i}} \bigl(a_{i}(x)|u_{x_{i}}|^{p_{i}-2}u_{x_{i}} \bigr)+\sum_{i=1}^{N}\frac{\partial b_{i}(u,x,t)}{\partial x_{i}}, \quad (x,t) \in{Q_{T}}, $$

(1.1)

is considered, where Ω is a bounded domain in $\mathbb{R}^{N}$ with a $C^{2}$ smooth boundary ∂Ω, $p_{i}>1$, ${Q_{T}} = \varOmega \times(0,T)$, $a_{i}(x) \in C^{1}(\overline{\varOmega})$, $b_{i}(\cdot, x, t)\in C(\overline{Q_{T}})$.

Equation (1.1) arises in the mathematical modeling of various physical processes such as flows of incompressible turbulent fluids or gases in pipes, and processes of filtration in glaciology [1,2,3]. A particular case of Eq. (1.1) is the usual non-Newtonian fluid equation,

$$ u_{t} = \operatorname{div} \bigl( \vert \nabla u \vert ^{p - 2}\nabla u\bigr), $$

(1.2)

which has been researched far and widely, one can refer to [4,5,6] and the references therein. In recent years, there are more and more mathematicians interested in the anisotropic parabolic equations

$$u_{t}=\sum_{i=1}^{N} \frac{\partial}{\partial x_{i}} \bigl(|u_{x_{i}}|^{p_{i}-2}u_{x_{i}} \bigr)+f(x,t, \nabla u),\quad (x,t) \in{Q_{T}}, $$

one can refer to [7,8,9,10,11,12,13,14].

In this paper, we suppose that

$$ a_{i}(x)|_{x\in\varOmega}> 0, \qquad a_{i}(x)|_{x\in\partial\varOmega }=0, \quad i=1, 2, \ldots, N, $$

(1.3)

then Eq. (1.1) is always degenerate on the boundary. To study the well-posedness of the solutions of Eq. (1.1), the initial value

$$ u(x,0)=u_{0}(x), \quad x\in\varOmega, $$

(1.4)

is always indispensable. Moreover, the usual boundary value condition

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

(1.5)

may be invalid. This is due to the fact that the weak solution of Eq. (1.1) may lack the enough regularity to be defined the trace on the boundary [15]. Accordingly, one has tried to study the uniqueness of weak solution only depending on the initial value condition (1.4) [16, 17]. In fact, for a degenerate parabolic equation, that the boundary value (1.5) may be overdetermined is well known, one can refer to [18,19,20,21,22,23,24,25,26,27]. But how to impose a suitable boundary value condition instead of (1.5) has been a difficult and interesting unsolved problem for a long time.

Inspired by [15,16,17,18,19,20,21,22,23,24,25,26,27], we may conjecture that the degeneracy of $a_{i}(x)$ on the boundary may take the place of the usual boundary value condition (1.5). In other words, the stability of weak solutions can be proved without the condition (1.5). Comparing with our previous work [16, 17], not only the anisotropic case is more complicated than the isotropic case, but also the nonlinear convection term $\sum_{i=1}^{N}\frac{\partial b_{i}(u,x,t)}{\partial x_{i}}$ adds difficulties. We employ some special techniques to overcome these difficulties. Moreover, we will introduce a general method to study the stability of weak solutions for a parabolic equation without the boundary value condition.

2 Definitions and main results

We denote

$$\begin{aligned}& p_{-}=\min\{p_{1}, p_{2}, \ldots, p_{N-1}, p_{N}\},\quad p_{-}>1, \\& p_{+}=\max\{p_{1}, p_{2}, \ldots, p_{N-1}, p_{N}\}. \end{aligned}$$

In the first place, we introduce definition of weak solutions.

Definition 2.1

A function $u(x,t)$ is said to be a weak solution of Eq. (1.1) with the initial value (1.4), if

$$ u \in L^{\infty}({Q_{T}}), \qquad a_{i}(x) \vert u_{x_{i}} \vert ^{p_{i}} \in L^{1}(Q_{T}), \qquad u_{t}\in L^{2}(Q_{T}), $$

(2.1)

and for any function $\varphi \in C_{0}^{1}(Q_{T})$,

$$ \iint_{Q_{T}}u_{t}\varphi \,dx\,dt+ \sum _{i=1}^{N} \iint_{Q_{T}} \bigl[a_{i}(x) \vert u_{x_{i}} \vert ^{p_{i} - 2}u_{x_{i}} \cdot \varphi_{x_{i}} + b_{i}(u,x,t) \cdot\varphi_{x_{i}} \bigr]\,dx\,dt= 0. $$

(2.2)

The initial value is satisfied in the following sense:

$$ \lim_{t\rightarrow0} \int_{\varOmega} \bigl\vert u(x,t)-u_{0}(x) \bigr\vert \,dx=0. $$

(2.3)

Definition 2.2

The function $u(x,t)$ is said to be the weak solution of Eq. (1.1) with the initial boundary values (1.4)–(1.5) if u satisfies Definition 2.1, and the boundary value condition (1.5) is satisfied in the sense of trace.

Theorem 2.3

If $p_{-}>2$, $a_{i}(x)\in C^{1}(\overline{\varOmega })$ satisfies (1.3), $b_{i}(s,x,t)$ is a $C^{1}$ function on $\mathbb {R}\times\overline{ \varOmega}\times[0,T]$,

$$ u_{0} \in L^{\infty}(\varOmega), \qquad \vert u_{0x_{i}} \vert \in L^{p_{i}}(\varOmega),\quad i=1, 2,\ldots, N, $$

(2.4)

either

$$ \int_{\varOmega}a_{i}^{-\frac{2}{p_{i}-2}}(x)\,dx< \infty, $$

(2.5)

or

$$ \bigl\vert b_{is}(s,x,t) \bigr\vert \leq ca^{\frac{1}{p_{i}}}(x), $$

(2.6)

then Eq. (1.1) with initial value (1.4) has a weak solution.

Theorem 2.4

Let $p_{-}>2$, for every $1\leq i\leq N$, either condition (2.5) be true, or $\int_{\varOmega}a_{i}^{-\frac{1}{p_{i}-1}}(x)\,dx<\infty$ and condition (2.6) be true, $a_{i}(x)\in C^{1}(\overline{\varOmega})$ satisfy (1.3), $b_{i}(s, x, t)$ be a $C^{1}$ function on $\mathbb{R}\times\overline{ \varOmega}\times[0,T]$. Then the initial boundary value problem (1.1)–(1.4)–(1.5) has a solution.

If $b_{i}\equiv0$, then only if $p_{-}>1$ and $\int_{\varOmega }a_{i}^{-\frac{1}{p_{i}-1}}(x)\,dx<\infty$, Theorem 2.3 and Theorem 2.4 are still true. However, if $b_{i}\equiv0$ is not valid, when $p_{-}>1$, then it is difficult to prove that $u_{t}\in L^{2}(Q_{T})$. If we do not require $u_{t}\in L^{2}(Q_{T})$, in other words, if we admit $u_{t}$ belonging to another kind of Banach space, then the conditions (2.5) and (2.6) may not be necessary, one can refer to our previous work [28]. Moreover, the condition (2.6) (also the condition (2.9)) reflects that there are some relationships between the diffusion coefficient and the convection term. At least, one of our motivations on condition (2.6) (also the condition (2.9)) initially comes from the study of a model of strong degenerate parabolic equation arising in mathematical finance [29], which has the form

$$ \frac{\partial u}{\partial t} =\sum_{i=1}^{2} \frac{\partial }{\partial x_{i}} \biggl(a^{ij}(u)\frac{\partial u}{\partial x_{j}} \biggr)+\sum _{i=1}^{2}\frac{\partial b_{i}(u)}{\partial x_{i}}, \quad \text{in } Q_{T}=\varOmega\times(0,T), $$

and satisfies

$$\bigl\vert b_{i}'(s) \bigr\vert \leq a^{ii}(s),\quad i=1,2,\ldots, N, $$

where $\varOmega\subset\mathbb{R}^{2}$ is a bounded domain with the smooth boundary ∂Ω. From this, one can see that there are some relationships between the diffusion coefficient and the convection term.

Since we mainly are concerned about how the degeneracy of the coefficient $a_{i}(x)$ affects the uniqueness or the stability of weak solutions, we have no intention to make a deep research on the existence. The main results of this paper are the following stability theorems.

Theorem 2.5

Let $p_{-}>1$, for $1\leq i\leq N$, $a_{i}(x)\in C^{1}(\overline{\varOmega})$ satisfy (1.3), $\int_{\varOmega}a_{i}^{-\frac {1}{p_{i}-1}}(x)\,dx<\infty$ and $b_{i}(s,x,t)$ be a Lipchitz function $\mathbb{R}\times\overline{ \varOmega}\times[0,T]$. If u and v are two solutions of Eq. (1.1) with the same homogeneous boundary value condition

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

(2.7)

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

$$ \int_{\varOmega} \bigl\vert u(x,t)-v(x,t) \bigr\vert \,dx\leq \int_{\varOmega} \bigl\vert u_{0}(x)-v_{0}(x) \bigr\vert \,dx, \quad t\in[0,T). $$

(2.8)

Roughly speaking, the condition $\int_{\varOmega}a_{i}^{-\frac {1}{p_{i}-1}}(x)\,dx<\infty$ can guarantee that the boundary value condition (1.5) is true in the sense of trace. If this condition is invalid, for example,

$$\int_{\varOmega}a_{1}^{-\frac{1}{p_{1}-1}}(x)\,dx< \infty $$

and

$$\int_{\varOmega}a_{2}^{-\frac{1}{p_{2}-1}}(x)\,dx=\infty, $$

that whether Theorems 2.4–2.5 are true or not is an open problem. Fortunately, by adding some restrictions on $a_{i}(x)$ and $b_{i}(s,x,t)$, we are able to prove the following stability of weak solutions without any boundary value condition, no matter whether $\int_{\varOmega }a_{i}^{-\frac{1}{p_{i}-1}}(x)\,dx<\infty$ or not.

Theorem 2.6

Let $p_{-}>1$, $a_{i}(x)\in C^{1}(\overline{\varOmega })$ satisfy (1.3), $b_{i}(s,x, t)$ be a Lipschitz function on $\mathbb {R}\times\overline{ \varOmega}\times[0,T]$. Let u and v be two solutions of (1.1) with the initial values $u_{0}(x)$ and $v_{0}(x)$, respectively. If $b_{i}(s,x,t)$ satisfies

$$ \bigl\vert b_{i}(u,x,t)-b_{i}(v,x,t) \bigr\vert \leq c a_{i}^{\frac{1}{p_{i}}}|u-v|,\quad i=1, 2,\ldots, N, $$

(2.9)

and, for η small enough,

$$ \frac{1}{\eta} \Biggl( \int_{\varOmega\setminus\varOmega_{\eta}} a_{i}(x) \Biggl\vert \Biggl(\prod _{j=1}^{N} a_{j}(x) \Biggr)_{x_{i}} \Biggr\vert ^{p_{i}}\,dx \Biggr)^{\frac{1}{p_{i}}} \leq c, i=1, 2,\ldots, N, $$

(2.10)

then the stability (2.8) is true.

Here, $\varOmega_{\eta}=\{x\in\varOmega: (\prod_{j=1}^{N} a_{j}(x))>\eta\}$.

Comparing Theorem 2.6 with Theorem 2.5, we find that, in some cases, the degeneracy of $a_{i}(x)$ on the boundary can take the place of the usual boundary value condition (1.5). Even, for some given kind of the weak solutions, the condition (2.10) may not be necessary. For example, we have the following result.

Theorem 2.7

Let $p_{-}>1$, $a_{i}(x)\in C^{1}(\overline{\varOmega })$ satisfy (1.3), $b_{i}(s,x, t)$ be a Lipschitz function on $\mathbb {R}\times\overline{ \varOmega}\times[0,T]$. Let u and v be two solutions of (1.1) with the initial values $u_{0}(x)$ and $v_{0}(x)$, respectively, and for η small enough,

$$ \begin{aligned} &\frac{1}{\eta} \biggl( \int_{\varOmega\setminus\varOmega_{\eta}}a_{i}(x)| u_{x_{i}}|^{p_{i}} \,dx \biggr)^{\frac{p_{i}-1}{p_{i}}}\leq c, \\ &\frac{1}{\eta} \biggl( \int_{\varOmega\setminus\varOmega_{\eta}}a_{i}(x)| v_{x_{i}}|^{p_{i}} \,dx \biggr)^{\frac{p_{i}-1}{p_{i}}}\leq c,\quad i=1, 2,\ldots, N. \end{aligned} $$

(2.11)

If $b_{i}(s,x,t)$ satisfies (2.9), then the stability (2.8) is true.

However, for some weak solutions, condition (2.9) may not be necessary. In fact, if the convection term is independent of the diffusion coefficient, we have the following result.

Theorem 2.8

Let $p_{-}>1$, $a_{i}(x)\in C^{1}(\overline{\varOmega })$ satisfy (1.3), $b_{i}(s,x, t)$ be a Lipschitz function on $\mathbb {R}\times\overline{ \varOmega}\times[0,T]$. If u and v are two solutions of Eq. (1.1) with the initial values $u_{0}(x)$ and $v_{0}(x)$, respectively, then, for any $\varOmega_{1}\subset\subset\varOmega$,

$$ \int_{\varOmega_{1}} \bigl\vert u(x,t)-v(x,t) \bigr\vert ^{2}\,dx\leq c(\varOmega_{1}) \int_{\varOmega } \bigl\vert u_{0}(x)-v_{0}(x) \bigr\vert ^{2}\,dx, $$

(2.12)

which implies that the uniqueness of weak solution is true.

Actually, by the general method introduced in the last section of this paper, many kinds of stability theorems of weak solutions can be found.

3 The weak solutions dependent on the initial value

We consider the following regularized problem:

$$\begin{aligned}& u_{\varepsilon t} - \sum_{i=1}^{N} \frac{\partial}{\partial x_{i}} \bigl( \bigl(a_{i}(x)+\varepsilon\bigr) \vert {u_{\varepsilon x_{i}}} \vert ^{p_{i} - 2}{u_{\varepsilon x_{i}}} \bigr) - \sum _{i = 1}^{N} \frac{\partial b_{i}(u_{\varepsilon},x,t)}{\partial x_{i}} = 0,\quad (x,t)\in{Q_{T}}, \end{aligned}$$

(3.1)

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

(3.2)

$$\begin{aligned}& {u_{\varepsilon}}(x,0) = {u_{\varepsilon0}}(x), \quad x\in\varOmega . \end{aligned}$$

(3.3)

Here, $u_{\varepsilon0} \in C^{\infty}_{0}(\varOmega)$, $|u_{\varepsilon 0}|_{L^{\infty}(\varOmega)}\leq|u_{0}|_{L^{\infty}(\varOmega)}$, $\vert \nabla u_{\varepsilon0} \vert$ converges to $|\nabla u_{0}(x)|$ in $L^{p_{+}}(\varOmega)$. It is well known that the above problem has an unique weak solution $u_{\varepsilon}\in L^{\infty}(0,T; W_{0}^{1,\vec {p}}(a_{i}(x), \varOmega))$ [5, 30].

By the maximum principle [5], there is a constant c only dependent on ${ \Vert {{u_{0}}} \Vert _{{L^{\infty}}(\varOmega)}}$ but independent on ε, such that

$$\Vert u_{\varepsilon} \Vert _{L^{\infty}(Q_{T})} \leqslant c. $$

Multiplying (3.1) by $u_{\varepsilon}$ and integrating it over $Q_{T}$, then

$$\begin{aligned}& \frac{1}{2} \int_{\varOmega}u_{\varepsilon}^{2}\,dx+\sum _{i=1}^{N} \iint _{{Q_{T}}} \bigl(a_{i}(x)+\varepsilon \bigr)|u_{\varepsilon x_{i}}|^{p_{i}}\,dx\,dt + \iint_{Q_{T}}\frac{\partial b_{i}(u_{\varepsilon},x,t)}{\partial x_{i}} u_{\varepsilon}\,dx\,dt \\& \quad = \frac{1}{2} \int_{\varOmega}u_{0}^{2}\,dx. \end{aligned}$$

(3.4)

If $\int_{\varOmega}a_{i}^{-\frac{2}{p_{i}-2}}(x)\,dx<\infty$, we know that $\int_{\varOmega}a_{i}^{-\frac{1}{p_{i}-1}}(x)\,dx<\infty$, then

$$\begin{aligned} \biggl\vert \int_{\varOmega}\frac{\partial b_{i}(u_{\varepsilon},x,t)}{\partial x_{i}} u_{\varepsilon}\,dx \biggr\vert &\leq \int_{\varOmega} \biggl\vert \frac {\partial b_{i}(s,x,t)}{\partial s} \bigg|_{s= u_{\varepsilon }}u_{\varepsilon x_{i}} \biggr\vert |u_{\varepsilon}|\,dx \\ &\leq c \int_{\varOmega} \biggl\vert \frac{\partial b_{i}(s,x,t)}{\partial s}\bigg| _{s= u_{\varepsilon}}u_{\varepsilon x_{i}} \biggr\vert \,dx \\ &\leq\frac{1}{2} \int_{\varOmega}a_{i}(x)|u_{\varepsilon x_{i}}|^{p_{i}} \,dx+\frac{c}{2} \int_{\varOmega}a_{i}^{-\frac{1}{p_{i}-1}}(x)\,dx \\ &\leq\frac{1}{2} \int_{\varOmega}a_{i}(x)|u_{\varepsilon x_{i}}|^{p_{i}} \,dx+c. \end{aligned}$$

If the condition (2.6) is true, then

$$\begin{aligned} \biggl\vert \int_{\varOmega}\frac{\partial b_{i}(u_{\varepsilon},x,t)}{\partial x_{i}} u_{\varepsilon}\,dx \biggr\vert &\leq c \int_{\varOmega} \biggl\vert \frac{\partial b_{i}(s,x,t)}{\partial s}\bigg| _{s= u_{\varepsilon}}u_{\varepsilon x_{i}} \biggr\vert \,dx \\ &\leq\frac{1}{2} \int_{\varOmega}a_{i}(x)|u_{\varepsilon x_{i}}|^{p_{i}} \,dx+c, \end{aligned}$$

clearly. Accordingly, by (3.4), we have

$$ \int_{\varOmega}u_{\varepsilon}^{2}\,dx+\sum _{i=1}^{N} \iint_{{Q_{T}}} \bigl(a_{i}(x)+\varepsilon \bigr)|u_{\varepsilon x_{i}}|^{p_{i}}\,dx\,dt \leqslant c. $$

(3.5)

For any $\varOmega_{1}\subset\subset\varOmega$, since $p_{-}=\min\{p_{i}\} >2$, $a_{i}(x)$ satisfies (1.3),

$$a_{i}(x)\geq c(\varOmega_{1})>0, \quad i=1,2,\ldots, N, $$

by (3.5),

$$\begin{aligned} \int_{0}^{T} \int_{\varOmega_{1}}|\nabla u_{\varepsilon}|^{2}\,dx\,dt&\leq c \biggl( \int_{0}^{T} \int_{\varOmega_{1}}|\nabla u_{\varepsilon }|^{p_{-}}\,dx\,dt \biggr)^{\frac{2}{p_{-}}} \\ &\leq c(\varOmega_{1})\sum_{i=1}^{N} \biggl( \int_{0}^{T} \int_{\varOmega _{1}}a_{i}(x)| u_{\varepsilon x_{i}}|^{p_{i}} \,dx\,dt \biggr)^{\frac{2}{p_{i}}} \\ &\leq c(\varOmega_{1})\sum_{i=1}^{N} \biggl( \int_{0}^{T} \int_{\varOmega }a_{i}(x)|u_{\varepsilon x_{i}} |^{p_{i}}\,dx\,dt \biggr)^{\frac{2}{p_{i}}} \\ &\leq c, \end{aligned}$$

(3.6)

where $c(\varOmega_{1})$ represents the constant depending upon the compact subset $\varOmega_{1}$, but it may be different from one to another.

Multiplying (2.5) by $u_{\varepsilon t}$, integrating it over $Q_{T}$, it yields

$$\begin{aligned}& \iint_{{Q_{T}}} |u_{\varepsilon t}|^{2}\,dx\,dt \\& \quad = \sum_{i=1}^{N} \iint_{{Q_{T}}}\frac{\partial}{\partial x_{i}} \bigl(\bigl(a_{i}(x)+ \varepsilon\bigr) \vert u_{\varepsilon x_{i}} \vert ^{p_{i} - 2}u_{\varepsilon x_{i}} \bigr) u_{\varepsilon t}\,dx\,dt \\& \qquad {}+\sum_{i=1}^{N} \iint_{{Q_{T}}}u_{\varepsilon t}\frac{\partial b_{i}(u_{\varepsilon},x,t)}{\partial x_{i}}\,dx\,dt . \end{aligned}$$

(3.7)

Noticing that

$$|u_{\varepsilon x_{i}}|^{p_{i}-2}u_{\varepsilon x_{i}} u_{\varepsilon x_{i} t} = \frac{1}{2}\frac{d}{dt} \int_{0}^{ \vert u_{\varepsilon x_{i}} \vert ^{2}} s^{\frac{p_{i} - 2}{2}}\,ds , $$

then

$$\begin{aligned}& \iint_{Q_{T}} \frac{\partial}{\partial x_{i}} \bigl(\bigl(a_{i}(x)+ \varepsilon \bigr)|u_{\varepsilon x_{i}}|^{p_{i}-2}u_{\varepsilon x_{i}} \bigr) u_{\varepsilon t}\,dx\,dt \\& \quad = - \iint_{Q_{T}} \bigl(a_{i}(x)+\varepsilon \bigr)|u_{\varepsilon x_{i}}|^{p_{i}-2}u_{\varepsilon x_{i}} u_{\varepsilon x_{i} t}\,dx\,dt \\& \quad = -\frac{1}{2} \iint_{Q_{T}} \bigl(a_{i}(x)+\varepsilon\bigr) \frac{d}{dt} \int _{0}^{ \vert u_{\varepsilon x_{i}} \vert ^{2}} s^{\frac{p_{i} - 2}{2}}\,ds \,dx \,dt. \end{aligned}$$

(3.8)

If $\int_{\varOmega}a_{i}^{-\frac{2}{p_{i}-2}}(x)\,dx<\infty$,

$$\begin{aligned} \iint_{Q_{T}} u_{\varepsilon t} \frac{\partial b_{i}(u_{\varepsilon},x,t)}{\partial x_{i}} \,dx\,dt & \leqslant \iint_{Q_{T}} \bigl\vert b_{iu}(u_{\varepsilon},x,t) \bigr\vert \vert u_{\varepsilon x_{i}} \vert \vert u_{\varepsilon t} \vert \,dx\,dt \\ &\quad {}+ \iint_{Q_{T}} \bigl\vert b_{ix_{i}}(u_{\varepsilon}, x,t) \bigr\vert \vert u_{\varepsilon t} \vert \,dx\,dt \\ &\leqslant\frac{1}{2} \iint_{Q_{T}} |u_{\varepsilon t}|^{2}\,dx\,dt + c \iint _{Q_{T}} \vert u_{\varepsilon x_{i}} \vert ^{2} \,dx\,dt +c, \end{aligned}$$

(3.9)

by the Hölder inequality

$$\begin{aligned} \iint_{Q_{T}} \vert u_{\varepsilon x_{i}} \vert ^{2} \,dx\,dt &= c \iint_{Q_{T}} a^{-\frac{2}{p_{i}}} a^{\frac{2}{p_{i}}} \vert u_{\varepsilon x_{i}} \vert ^{2}\,dx\,dt \\ &\leq c \biggl( \iint_{Q_{T}} a^{-\frac{2}{p_{i}- 2}}\,dx\,dt \biggr)^{\frac {p_{i}- 2}{p_{i}}} \biggl( \iint_{Q_{T}} a_{i}(x) \vert u_{\varepsilon x_{i}} \vert ^{p_{i}}\,dx\,dt \biggr)^{\frac{2}{p_{i}}} \\ &\leq c. \end{aligned}$$

(3.10)

If $|b_{is}(s,x,t)|\leq ca_{i}^{\frac{2}{p_{i}}}(x)$, $p_{i}\geq2$, then by the Young inequality

$$\begin{aligned} \iint_{Q_{T}} u_{\varepsilon t} \frac{\partial b_{i}(u_{\varepsilon},x,t)}{\partial x_{i}} \,dx\,dt \leqslant& \iint_{Q_{T}} \bigl\vert b_{iu}(u_{\varepsilon},x,t) \bigr\vert \vert u_{\varepsilon x_{i}} \vert \vert u_{\varepsilon t} \vert \,dx\,dt \\ &{}+ \iint_{Q_{T}} \bigl\vert b_{ix_{i}}(u_{\varepsilon}, x,t) \bigr\vert \vert u_{\varepsilon t} \vert \,dx\,dt \\ \leq& \iint_{Q_{T}}a_{i}^{\frac{2}{p_{i}}}(x) \vert u_{\varepsilon x_{i}} \vert ^{2}\,dx\,dt +\frac{1}{2} \iint_{Q_{T}} \vert u_{\varepsilon t} \vert ^{2} \,dx\,dt+c \\ \leq& c \iint_{Q_{T}}a_{i}(x) \vert u_{\varepsilon x_{i}} \vert ^{p_{i}}\,dx\,dt +\frac{1}{2} \iint_{Q_{T}} \vert u_{\varepsilon t} \vert ^{2} \,dx\,dt+c. \end{aligned}$$

Combining (3.7)–(3.10), we have

$$\iint_{{Q_{T}}} |u_{\varepsilon t}|^{2}\,dx\,dt + \sum _{i=1}^{N} \iint_{{Q_{T}}} \bigl(a_{i}(x)+\varepsilon\bigr) \frac{d}{dt} \int_{0}^{ \vert u_{\varepsilon x_{i}} \vert ^{2}} s^{\frac{p_{i} - 2}{2}}\,ds \,dx\,dt \leqslant c, $$

by the above inequality, we have

$$ \iint_{{Q_{T}}} |u_{\varepsilon t}|^{2}\,dx\,dt \leqslant c + c\sum_{i=1}^{N} \int_{\varOmega}\bigl(a_{i}(x)+\varepsilon\bigr) \vert u_{\varepsilon0 x_{i}} \vert ^{p_{i}}\,dx \leqslant c. $$

(3.11)

Now, by (3.4), (3.5), (3.6) and (3.11), there exist a function u and an n-dimensional vector function $\overrightarrow{\zeta}= ({\zeta _{1}}, \ldots,{\zeta_{n}})$ satisfying $u_{\varepsilon}\rightarrow u$ a.e. in $Q_{T}$, and

$$\begin{aligned}& u \in L^{\infty}(Q_{T}),\qquad \vert \zeta_{i} \vert \in L^{\frac {p_{i}}{p_{i} - 1}}(Q_{T}), \\& {u_{\varepsilon}} \rightharpoonup* u,\quad \text{in } {L^{\infty}(Q_{T})}, \\& b_{i}(u_{\varepsilon},x,t)\rightarrow b_{i}(u,x,t),\quad \text{a.e. in } Q_{T}, \\& {u_{\varepsilon x_{i}}} \rightharpoonup u_{x_{i}}, \quad \text{in } L_{\mathrm{loc}}^{p_{i}}(Q_{T}), \\& a_{i}(x) \vert u_{\varepsilon x_{i}} \vert ^{p_{i} - 2} u_{\varepsilon x_{i}} \rightharpoonup\zeta_{i},\quad \text{in } {L^{\frac{p_{i}}{p_{i}-1}}}({Q_{T}}). \end{aligned}$$

(3.12)

It is easy to show that

$$\begin{aligned}& \lim_{\varepsilon\rightarrow0}\sum_{i=1}^{N} \iint _{Q_{T}}\bigl(a_{i}(x)+\varepsilon\bigr) \vert u_{\varepsilon x_{i}} \vert ^{p_{i} - 2}u_{\varepsilon x_{i}}\varphi_{x_{i}} \,dx\,dt \\& \quad =\lim_{\varepsilon\rightarrow0}\sum_{i=1}^{N} \iint_{Q_{T}}a_{i}(x) \vert u_{\varepsilon x_{i}} \vert ^{p_{i} - 2}u_{\varepsilon x_{i}}\varphi_{x_{i}}\,dx\,dt \\& \quad = \iint_{Q_{T}} \overrightarrow{\zeta} \cdot\nabla\varphi \,dx\,dt, \end{aligned}$$

(3.13)

for any $\varphi\in C_{0}^{1}(Q_{T})$.

Now, we will prove that

$$ \sum_{i=1}^{N} \iint_{Q_{T}} a_{i}(x) \vert u_{x_{i}} \vert ^{p_{i} - 2} u_{x_{i}}\varphi_{1x_{i}} \,dx\,dt = \iint_{Q_{T}} \overrightarrow{\zeta} \cdot\nabla \varphi_{1} \,dx\,dt, $$

(3.14)

for any given function $\varphi_{1} \in C_{0}^{1} ({Q_{T}})$. In detail, we notice that, for any function $\varphi \in C_{0}^{1} ({Q_{T}})$,

$$ \iint_{Q_{T}} \Biggl[\frac{\partial u_{\varepsilon}}{\partial t}\varphi + \sum _{i=1}^{N} \bigl(a_{i}(x)+\varepsilon \bigr)| u_{\varepsilon x_{i}}|^{p_{i}-2}u_{\varepsilon x_{i}} \varphi_{x_{i}} + \sum_{i = 1}^{N} b_{i}(u_{\varepsilon},x,t ) \varphi_{x_{i}} \Biggr]\,dx\,dt= 0. $$

(3.15)

Let $\varepsilon\rightarrow0$. Then

$$ \iint_{Q_{T}} \Biggl[\frac{\partial u}{\partial t}\varphi +\sum _{i=1}^{N}\zeta_{i}\varphi_{x_{i}} + \sum_{i = 1}^{N} b_{i}(u,x,t) \varphi_{x_{i}} \Biggr]\,dx\,dt = 0. $$

(3.16)

Let $0 \leqslant\psi \in C_{0}^{\infty}({Q_{T}})$ and $\psi=1$ on $\operatorname{supp}\varphi_{1}$. Let $v \in {L^{\infty}}({Q_{T}})$, $a_{i}(x) \vert v_{x_{i}} \vert ^{p_{i}} \in {L^{1}}({Q_{T}})$. One has

$$ \iint_{Q_{T}} \psi a_{i}(x) \bigl( \vert u_{\varepsilon x_{i}} \vert ^{p_{i} - 2}u_{\varepsilon x_{i}} - \vert v_{x_{i}} \vert ^{p_{i} - 2} v_{x_{i}} \bigr) (u_{\varepsilon x_{i}} - v_{x_{i}})\,dx\,dt \geqslant0. $$

(3.17)

By choosing $\varphi = \psi u_{\varepsilon}$ in (3.15),

$$\begin{aligned}& \iint_{Q_{T}} \Biggl[\frac{\partial u_{\varepsilon}}{\partial t}\psi u_{\varepsilon}+ \sum_{i=1}^{N} \bigl(a_{i}(x)+ \varepsilon\bigr)|u_{\varepsilon x_{i}}|^{p_{i}-2}u_{\varepsilon x_{i}} (\psi u_{\varepsilon})_{x_{i}} + \sum_{i = 1}^{N} b_{i}(u_{\varepsilon},x,t ) (\psi u_{\varepsilon })_{x_{i}} \Biggr]\,dx\,dt \\& \quad = 0. \end{aligned}$$

(3.18)

By (3.17)–(3.18), we have

$$\begin{aligned}& \frac{1}{2} \iint_{Q_{T}} \psi_{t}u_{\varepsilon}^{2} \,dx\,dt-\sum_{i=1}^{N} \iint _{Q_{T}}\bigl(a_{i}(x)+\varepsilon\bigr)| u_{\varepsilon x_{i}}|^{p_{i}-2}u_{\varepsilon x_{i}}\psi_{x_{i}}u_{\varepsilon} \,dx\,dt \\& \qquad {}-\sum_{i=1}^{N} \iint_{Q_{T}}\bigl(a_{i}(x)+\varepsilon\bigr)| v_{ x_{i}}|^{p_{i}-2}v_{ x_{i}}( u_{\varepsilon x_{i}}-v_{x_{i}}) \psi \,dx\,dt \\& \qquad {}-\sum_{i=1}^{N} \iint_{Q_{T}}\bigl(a_{i}(x)+\varepsilon\bigr)| u_{\varepsilon x_{i}}|^{p_{i}-2}u_{\varepsilon x_{i}} v_{x_{i}}\psi \,dx\,dt \\& \qquad {}- \sum_{i = 1}^{N} \iint_{Q_{T}} b_{i}(u_{\varepsilon},x,t ) (u_{\varepsilon x_{i}}\psi + u_{\varepsilon}\psi_{x_{i}})\,dx\,dt \\& \quad \geq0. \end{aligned}$$

(3.19)

Let $\varepsilon\rightarrow0$. Then

$$\begin{aligned}& \frac{1}{2} \iint_{Q_{T}} \psi_{t}u^{2}\,dx\,dt - \sum _{i = 1}^{N} \iint_{Q_{T}} u\zeta_{i} \psi_{x_{i}} \,dx \,dt \\& \qquad {}- \sum_{i = 1}^{N} \iint _{Q_{T}}a_{i}(x)|v_{x_{i}}|^{p_{i}-2}v_{x_{i}}(u_{x_{i}}-v_{x_{i}}) \,dx\,dt- \sum_{i = 1}^{N} \iint_{{Q_{T}}} a_{i}(x) \zeta_{x_{i}} v_{x_{i}}\psi \,dx\,dt \\& \qquad {}-\sum_{i = 1}^{N} \iint_{Q_{T}} b_{i}(u,x,t) (u_{x_{i}}\psi + u \psi _{x_{i}})\,dx\,dt \\& \quad \geqslant0. \end{aligned}$$

(3.20)

Let $\varphi=\psi u$ in (3.16). We get

$$\begin{aligned}& \frac{1}{2} \iint_{Q_{T}} u^{2}\psi_{t}\,dx\,dt-\sum _{i = 1}^{N} \iint _{Q_{T}} \psi\zeta_{i} u_{x_{i}}\,dx \,dt- \sum_{i = 1}^{N} \iint_{Q_{T}} u \zeta_{i} \psi_{x_{i}} \,dx \,dt \\& \qquad {} - \sum_{i = 1}^{N} \iint_{Q_{T}} b_{i}(u,x,t) (u_{x_{i}}\psi + u \psi_{x_{i}})\,dx\,dt \\& \quad = 0. \end{aligned}$$

(3.21)

Thus

$$ \sum_{i = 1}^{N} \iint_{{Q_{T}}} \psi \bigl(\zeta_{i} - a_{i}(x) \vert v_{x_{i}} \vert ^{p_{i} - 2}v_{x_{i}} \bigr) ( u_{x_{i}} - v_{x_{i}})\,dx\,dt \geqslant0. $$

(3.22)

Let $v=u- \lambda\varphi_{1}$, $\lambda>0$. Then

$$\sum_{i = 1}^{N} \iint_{{Q_{T}}} \psi \bigl(\zeta_{i} - a_{i}(x) \bigl\vert ( u- \lambda\varphi_{1})_{x_{i}} \bigr\vert ^{p_{i} - 2} \bigr) \varphi _{1x_{i}}\,dx\,dt \geqslant0. $$

If $\lambda\rightarrow0$, then

$$\sum_{i = 1}^{N} \iint_{{Q_{T}}} \psi \bigl(\zeta_{i} - a_{i}(x) \vert u_{x_{i}} \vert ^{p_{i} - 2}u_{x_{i}} \bigr) \varphi_{1x_{i}}\,dx\,dt \geqslant0. $$

Moreover, if $\lambda<0$, similarly we can get

$$\sum_{i = 1}^{N} \iint_{{Q_{T}}} \psi \bigl(\zeta_{i} - a_{i}(x) \vert u_{x_{i}} \vert ^{p_{i} - 2}u_{x_{i}} \bigr) \varphi_{1x_{i}}\,dx\,dt \leq0. $$

Thus

$$\sum_{i = 1}^{N} \iint_{{Q_{T}}} \psi \bigl(\zeta_{i} - a_{i}(x) \vert u_{x_{i}} \vert ^{p_{i} - 2}u_{x_{i}} \bigr) \varphi_{1x_{i}}\,dx\,dt = 0. $$

Noticing that $\psi= 1$ on ${\operatorname{supp}}\varphi_{1}$, then (3.14) holds.

At last, we are able to prove (2.3) as in [31], then u is a solution of Eq. (1.1) with the initial value (1.4) in the sense of Definition 2.1. Thus we have Theorem 2.3.

Now, by a similar method as in [32], we can prove the following.

Lemma 3.1

If $\int_{\varOmega}a_{i}^{-\frac {1}{p_{i}-1}}(x)\,dx<\infty$, u is a weak solution of Eq. (1.1) with the initial condition (1.4). Then, for any given $t\in[0,T)$,

$$ \int_{\varOmega}|u_{x_{i}}|\,dx\leq c, \quad i=1,2, \ldots, N. $$

(3.23)

For simplicity, we omit the details of the proof of Lemma 3.1 here. By (3.23) and the fact $\iint_{Q_{T}}|u_{t}|\,dx\,dt\leq c$, we know that $u\in BV(Q_{T})$, $C_{0}^{\infty}(Q_{T})$ is dense in $BV(Q_{T})$ and the trace of u on the boundary ∂Ω can be defined in the traditional way. By Theorem 2.3 and Lemma 3.1, we clearly have Theorem 2.4.

4 The stability of the initial boundary value problem

In order to prove the stability of the weak solutions, for small $\eta >0$, let

$$S_{\eta}(s)= \int_{0}^{s}h_{\eta}(\tau)\,d\tau,\qquad h_{\eta}(s)=\frac{2}{\eta} \biggl(1-\frac{| s|}{\eta } \biggr)_{+}. $$

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

$$ \begin{aligned} &h_{\eta}(s)\geq0, \qquad \bigl\vert sh_{\eta}(s) \bigr\vert \leq1,\qquad \bigl\vert S_{\eta }(s) \bigr\vert \leq1; \\ &\lim_{\eta\rightarrow0}S_{\eta }(s)=\operatorname{sgn}s,\qquad \lim _{\eta\rightarrow 0}sS_{\eta}^{\prime}(s)=0. \end{aligned} $$

(4.1)

Clearly, if we denote $H_{\eta}(s)=\int_{0}^{s}S_{\eta}(\tau)\,d\tau$, then we have

$$ \lim_{\eta\rightarrow0}H_{\eta}(s)=|s|, \quad s\in(-\infty, + \infty ). $$

(4.2)

Lemma 4.1

Let $p_{-}>1$, for $1\leq i\leq N$, $\int_{\varOmega }a_{i}^{-\frac{1}{p_{i}-1}}(x)\,dx<\infty$ and

$$ \bigl\vert b_{i}(u,x,t)-b_{i}(v,x,t) \bigr\vert \leq c a_{i}^{\frac{1}{p_{i}}}|u-v|. $$

(4.3)

If u and v are two solutions of Eq. (1.1) with the same homogeneous value condition

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

(4.4)

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

$$\int_{\varOmega} \bigl\vert u(x,t)-v(x,t) \bigr\vert \,dx\leq \int_{\varOmega} \bigl\vert u_{0}(x)-v_{0}(x) \bigr\vert \,dx,\quad t\in[0,T). $$

Proof

Let u and v be two weak solutions of Eq. (1.1). Since $\int_{\varOmega}a_{i}^{-\frac{1}{p_{i}-1}}(x)\,dx<\infty$, by Lemma 3.1, $u,v\in BV(Q_{T})$ we can choose $\varphi=\chi_{[\tau,s]}S_{\eta }(u - v)$ as the test function. Here $\chi_{[\tau,s]}$ is the characteristic function of $[\tau, s]\subset(0, T)$. Then

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

(4.5)

As usual, one has

$$ \int_{\tau}^{s} \int_{\varOmega} a_{i}(x) \bigl( \vert u_{x_{i}} \vert ^{p_{i}- 2} u_{x_{i}} - \vert v _{x_{i}} \vert ^{p_{i} - 2}v_{x_{i}} \bigr) (u - v)_{x_{i}}h_{\eta}(u-v) \,dx \,dt\geq0. $$

(4.6)

Since $\iint_{Q_{T}}|u_{t}|\,dx\,dt\leq c$, $\iint_{Q_{T}}|v_{t}|\,dx\,dt\leq c$, using the dominated convergence theorem, one has

$$\begin{aligned}& \lim_{\eta\rightarrow0} \int_{\tau}^{s} \int_{\varOmega}S_{\eta}(u - v)\frac{\partial(u - v)}{\partial t}\,dx\,dt \\& \quad =\lim_{\eta\rightarrow0} \int_{\varOmega}\bigl[H_{\eta}(u-v) (x,s)-H_{\eta }(u-v) (x,\tau)\bigr]\,dx \\& \quad = \int_{\varOmega}|u-v|(x,s)\,dx- \int_{\varOmega}|u-v|(x,\tau) \,dx, \end{aligned}$$

(4.7)

where $H_{\eta}(u-v)(x,s)=H_{\eta}(u(x,s)-v(x,s))$.

Moreover, since $b_{i}(s,x,t)$ satisfies the condition (4.3), one has

$$\begin{aligned}& \lim_{\eta\rightarrow0}\sum_{i=1}^{N} \biggl\vert \int_{\tau}^{s} \int _{{\varOmega}}\bigl[b_{i}(u,x,t)-b_{i}(v,x,t) \bigr](u-v)_{x_{i}}\phi_{\eta}h_{\eta }(u - v)\,dx\,dt \biggr\vert \\& \quad \leq c\lim_{\eta\rightarrow0}\sum_{i=1}^{N} \int_{\tau}^{s} \int _{{\varOmega}} \bigl\vert h_{\eta}(u - v) (u-v)a_{i}^{\frac {1}{p_{i}}}(u-v)_{x_{i}}\phi_{\eta} \bigr\vert \,dx\,dt \\& \quad \leq c\lim_{\eta\rightarrow0}\sum_{i=1}^{N} \biggl( \int_{\tau }^{s} \int_{\varOmega}a_{i}\bigl(|u_{x_{i}}|^{p_{i}}+| v_{x_{i}}|^{p_{i}}\bigr)\,dx\,dt \biggr)^{\frac{1}{p_{i}}} \\& \qquad {}\cdot\biggl( \int_{\tau}^{s} \int_{\varOmega} \bigl\vert (u-v)h_{\eta }(u - v) \bigr\vert ^{\frac{p_{i}}{p_{i}-1}}\,dx\,dt \biggr)^{\frac{p_{i}-1}{p_{i}}} \\& \quad =0. \end{aligned}$$

(4.8)

Now, let $\eta\rightarrow0$ in (4.5). By (4.6)–(4.8), one has

$$\int_{\varOmega}\bigl\vert u(x,s) - v(x,s) \bigr\vert \,dx \leqslant \int _{\varOmega}\bigl\vert u(x,\tau)- v(x,\tau) \bigr\vert \,dx. $$

Let $\tau\rightarrow0$. Then

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

Lemma 4.1 is proved. □

In fact, the condition (4.3) in Lemma 4.1 is not the optimal. Without the condition (4.3), we have Theorem 2.5.

Proof of Theorem 2.5

From the above proof of Lemma 4.1, we only need to prove that

$$ \lim_{\eta\rightarrow0} \biggl\vert \int_{\varOmega }\bigl[b_{i}(u,x,t)-b_{i}(v,x,t) \bigr](u - v)_{x_{i}}h_{\eta}(u-v)\,dx \biggr\vert =0, $$

(4.9)

without the condition (4.3). In detail, we have

$$\begin{aligned}& \int_{\varOmega}\bigl[b_{i}(u,x,t)-b_{i}(v,x,t) \bigr](u - v)_{x_{i}}h_{\eta}(u-v)\,dx \\& \quad = \int_{\{\varOmega: |u-v|< \eta\}}\bigl[b_{i}(u,x,t)-b_{i}(v,x,t) \bigr](u - v)_{x_{i}}h_{\eta}(u-v)\,dx. \end{aligned}$$

If the set $\{\varOmega: |u-v|=0\}$ has zero a measure, then

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

If the set $\{\varOmega: |u-v|=0\}$ only has a positive measure, then by, $a_{i}^{-\frac{1}{p_{i}-1}}\in L^{1}(\varOmega)$,

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

Thus, we have the conclusion. □

5 The global stability without the boundary value condition

Proof of Theorem 2.6

Let u and v be two weak solutions of Eq. (1.1) with the initial values $u_{0}(x)$, $v_{0}(x)$, respectively.

Let $\varOmega_{\eta}= \{x\in\varOmega:\prod_{i=1}^{N}a_{i}(x)>\eta \}$, and

$$ \phi_{\eta}(x)= \textstyle\begin{cases} 1, & \text{if } x\in\varOmega_{\eta}, \\ \frac{1}{\eta}\prod_{i=1}^{N}a_{i}(x), & \text{if } x\in\varOmega \setminus\varOmega_{\eta}. \end{cases} $$

(5.1)

Let us recall

$$J(x) = \textstyle\begin{cases} k\exp [\frac{-1}{1-|x|^{2}} ],&|x| < 1, \\ 0, &|x|\geq1, \end{cases} $$

where k is a constant such that $\int_{\mathbb{R}^{N}}J(x)\,dx=1$. The usual mollifier is defined as

$$J_{\varepsilon}(x)=\frac{1}{\varepsilon^{N}}J \biggl(\frac {x}{\varepsilon} \biggr) $$

for small $\varepsilon>0$. Let

$$f_{\varepsilon}(x)=J_{\varepsilon}f(x)=J_{\varepsilon}\ast f(x)= \int _{\mathbb{R}^{N}}J_{\varepsilon}(x-y)f(y)\,dy, $$

for any $f(x)\in L^{1}_{\mathrm{loc}}(\overline{\varOmega})$.

Let $\phi_{\eta\varepsilon}(x)$ be the mollified function of $\phi _{\eta}(x)$. We can choose $\chi_{[\tau,s]}\phi_{\eta\varepsilon }(x)S_{\eta}(u - v)$ as the test function. By the process of taking the limit, $\varepsilon\rightarrow0$, we can choose $\chi_{[\tau ,s]}\phi_{\eta}(x)S_{\eta}(u - v)$ as the test function finally. Then

$$\begin{aligned}& \int_{\tau}^{s} \int_{{\varOmega}} \phi_{\eta}S_{\eta}(u - v) \frac {\partial(u - v)}{\partial t}\,dx\,dt \\& \qquad {}+ \sum_{i=1}^{N} \int_{\tau}^{s} \int_{{\varOmega}} a_{i}(x) \bigl( \vert u_{x_{i}} \vert ^{p_{i} - 2} u_{x_{i}} - \vert v_{x_{i}} \vert ^{p_{i}- 2}\nabla v\bigr) (u_{x_{i}} - v_{x_{i}} )h_{\eta}(u-v)\phi_{\eta }(x)\,dx\,dt \\& \qquad {} + \sum_{i=1}^{N} \int_{\tau}^{s} \int_{{\varOmega}} a_{i}(x) \bigl( \vert u_{x_{i}} \vert ^{p_{i} - 2}u_{x_{i}} - \vert v_{x_{i}} \vert ^{p_{i}- 2} v_{x_{i}} \bigr) (u - v)S_{\eta}(u-v) \phi_{\eta x_{i}}\,dx\,dt \\& \qquad {} + \sum_{i=1}^{N} \int_{\tau}^{s} \int_{{\varOmega }}\bigl[b_{i}(u,x,t)-b_{i}(v,x,t) \bigr]\phi_{\eta x_{i}}{S_{\eta}}(u - v)\,dx\,dt \\& \qquad {} + \sum_{i=1}^{N} \int_{\tau}^{s} \int_{{\varOmega }}\bigl[b_{i}(u,x,t)-b_{i}(v,x,t) \bigr](u-v)_{x_{i}}\phi_{\eta}{h_{\eta}}(u - v)\,dx\,dt \\& \quad =0. \end{aligned}$$

(5.2)

Let us observe every term on the left-hand side of (5.2).

For the first term, using the dominated convergence theorem, we have

$$\begin{aligned}& \lim_{\eta\rightarrow0} \int_{\tau}^{s} \int_{\varOmega}\phi_{\eta }(x){S_{\eta}}(u - v) \frac{\partial(u - v)}{\partial t}\,dx\,dt \\& \quad = \lim_{\eta\rightarrow0} \int_{\tau}^{s} \int_{\varOmega}\frac {\partial[\phi_{\eta}(x)H_{\eta}(u-v)]}{\partial t}\,dx\,dt \\& \quad = \lim_{\eta\rightarrow0} \int_{\varOmega}\phi_{\eta}(x)\bigl[H_{\eta }(u-v) (x,s)-H_{\eta}(u-v) (x,\tau)\bigr]\,dx \\& \quad = \int_{\varOmega}|u-v|(x,s)\,dx- \int_{\varOmega}|u-v|(x,\tau)\,dx. \end{aligned}$$

(5.3)

For the second term, we have

$$ \int_{\varOmega} a_{i}(x) \bigl( \vert u_{x_{i}} \vert ^{p_{i} - 2}u_{x_{i}} - \vert v_{x_{i}} \vert ^{p_{i} - 2} v_{x_{i}} \bigr) (u_{x_{i}} - v_{x_{i}})h{_{\eta}}(u-v)\phi_{\eta}(x)\,dx\geq0. $$

(5.4)

For the third term, obviously, $\phi_{\eta x_{i}}=\frac{1}{\eta} (\prod_{j=1}^{N} a_{j}(x) )_{x_{i}}$ when $x\in\varOmega\setminus \varOmega_{\eta}$, in the other places, it is identical to zero. By the condition (2.10), we have

$$\begin{aligned}& \biggl\vert \int_{\varOmega} a_{i}(x) \bigl( \vert u_{x_{i}} \vert ^{p_{i}- 2}u_{x_{i}} - \vert v_{x_{i}} \vert ^{p_{i}- 2}v_{x_{i}} \bigr) \phi _{\eta x_{i}}S_{\eta}(u-v) \,dx \biggr\vert \\& \quad = \biggl\vert \int_{\varOmega\setminus\varOmega_{\eta}}a_{i}(x) \bigl( \vert u_{x_{i}} \vert ^{p_{i}- 2}u_{x_{i}} - \vert v_{x_{i}} \vert ^{p_{i} - 2}v_{x_{i}} \bigr) \phi_{\eta x_{i}} S_{\eta}(u-v) \,dx \biggr\vert \\& \quad \leq\frac{1}{\eta} \int_{\varOmega\setminus\varOmega_{\eta}} a_{i}(x) \bigl( \vert u_{x_{i}} \vert ^{p_{i} - 1}+ \vert v_{x_{i}} \vert ^{p_{i}- 1} \bigr) \Biggl\vert \Biggl(\prod_{j=1}^{N} a_{j}(x) \Biggr)_{x_{i}} S_{\eta}(u-v)\Biggr\vert \,dx \\& \quad \leq c \frac{1}{\eta} \biggl( \int_{\varOmega\setminus\varOmega_{\eta}} \bigl(a_{i}(x) \vert u_{x_{i}} \vert ^{p_{i}}+ \vert v_{x_{i}} \vert ^{p_{i}} \bigr) \,dx \biggr)^{\frac{p_{i}-1}{p_{i}}} \Biggl( \int_{\varOmega\setminus\varOmega_{\eta}} a_{i}(x) \Biggl\vert \Biggl(\prod _{j=1}^{N} a_{j}(x) \Biggr)_{x_{i}} \Biggr\vert ^{p_{i}}\,dx \Biggr)^{\frac{1}{p_{i}}} \\& \quad \leq c \biggl[ \biggl( \int_{\varOmega\setminus\varOmega_{\eta}}a_{i}(x)| u_{x_{i}}|^{p_{i}} \,dx \biggr)^{\frac{p_{i}-1}{p_{i}}}+ \biggl( \int_{\varOmega \setminus\varOmega_{\eta}}a_{i}(x)|v_{x_{i}}|^{p_{i}} \,dx \biggr)^{\frac {p_{i}-1}{p_{i}}} \biggr] \\& \qquad {}\cdot \Biggl[\frac{1}{\eta} \Biggl( \int_{\varOmega\setminus\varOmega _{\eta}} a_{i}(x) \Biggl\vert \Biggl(\prod _{j=1}^{N} a_{j}(x) \Biggr)_{x_{i}} \Biggr\vert ^{p_{i}}\,dx \Biggr)^{\frac{1}{p_{i}}} \Biggr] \\& \quad \leq c \biggl[ \biggl( \int_{\varOmega\setminus\varOmega_{\eta}}a_{i}(x)| u_{x_{i}}|^{p_{i}} \,dx \biggr)^{\frac{p_{i}-1}{p_{i}}}+ \biggl( \int_{\varOmega \setminus\varOmega_{\eta}}a_{i}(x)|v_{x_{i}}|^{p_{i}} \,dx \biggr)^{\frac {p_{i}-1}{p_{i}}} \biggr]. \end{aligned}$$

(5.5)

Then

$$\begin{aligned}& \lim_{\eta\rightarrow0} \biggl\vert \int_{\tau}^{s} \int_{\varOmega} a_{i}(x) \bigl( \vert u_{x_{i}} \vert ^{p- 2} u_{x_{i}} - \vert v_{x_{i}} \vert ^{p_{i}- 2} v_{x_{i}} \bigr) \phi_{\eta x_{i}} S_{\eta }(u-v)\,dx\,dt \biggr\vert \\& \quad \leq c\lim_{\eta\rightarrow0} \int_{\tau}^{s} \biggl[ \biggl( \int _{\varOmega\setminus\varOmega_{\eta}}a_{i}(x)| u_{x_{i}}|^{p_{i}} \,dx \biggr)^{\frac{p_{i}-1}{p_{i}}}+ \biggl( \int_{\varOmega\setminus\varOmega_{\eta }}a_{i}(x)|v_{x_{i}}|^{p_{i}} \,dx \biggr)^{\frac{p_{i}-1}{p_{i}}} \biggr]\,dt \\& \quad =0. \end{aligned}$$

(5.6)

For the fourth term, since $b_{i}(s,x,t)$ satisfies the condition (2.9), we have

$$ \lim_{\eta\rightarrow0}\sum_{i=1}^{N} \int_{\tau}^{s} \int _{{\varOmega}}\bigl[b_{i}(u,x,t)-b_{i}(v,x,t) \bigr](u-v)_{x_{i}}\phi_{\eta}h_{\eta }(u - v)\,dx \,dt=0, $$

(5.7)

as before.

Finally, for the fifth term, by the condition (2.10), we have

$$\begin{aligned}& \lim_{\eta\rightarrow0} \biggl\vert \int_{{\varOmega }}\bigl[b_{i}(u,x,t)-b_{i}(v,x,t) \bigr]\phi_{\eta x_{i}}S_{\eta}(u - v)\,dx \biggr\vert \\& \quad =\lim_{\eta\rightarrow0} \biggl\vert \int_{\varOmega\setminus\varOmega_{\eta }}\bigl[b_{i}(u,x,t)-b_{i}(v,x,t) \bigr]\phi_{\eta x_{i}}S_{\eta}(u - v)\,dx \biggr\vert \\& \quad \leq\lim_{\eta\rightarrow0}\frac{1}{\eta} \int_{\varOmega\setminus \varOmega_{\eta}}a_{i}^{\frac{1}{p_{i}}} \Biggl\vert \Biggl(\prod_{j=1}^{N} a_{j}(x) \Biggr)_{x_{i}} \Biggr\vert \bigl\vert S_{\eta}(u - v) (u-v) \bigr\vert \,dx \\& \quad \leq\lim_{\eta\rightarrow0}\frac{1}{\eta} \Biggl( \int_{\varOmega \setminus\varOmega_{\eta}} a_{i}(x) \Biggl\vert \Biggl(\prod _{j=1}^{N} a_{j}(x) \Biggr)_{x_{i}} \Biggr\vert ^{p_{i}}\,dx \Biggr)^{\frac{1}{p_{i}}} \biggl( \int_{\varOmega} \bigl\vert S_{\eta}(u - v) (u-v) \bigr\vert ^{\frac {p_{i}}{p_{i}-1}}\,dx \biggr)^{\frac{p_{i}-1}{p_{i}}} \\& \quad \leq c\lim_{\eta\rightarrow0} \biggl( \int_{\varOmega} \bigl\vert S_{\eta}(u - v) (u-v) \bigr\vert ^{\frac{p_{i}}{p_{i}-1}}\,dx \biggr)^{\frac{p_{i}-1}{p_{i}}} \\& \quad \leq c \biggl( \int_{\varOmega}|u-v|\,dx \biggr)^{\frac{p_{i}-1}{p_{i}}}, \end{aligned}$$

(5.8)

since $u,v\in L^{\infty}(Q_{T})$.

Now, let $\eta\rightarrow0$ in (5.2). Then

$$ \int_{\varOmega}\bigl\vert u(x,s) - v(x,s) \bigr\vert \,dx \leqslant \int _{\varOmega}\bigl\vert u(x,\tau) - {v(x,\tau)} \bigr\vert \,dx +c \biggl( \int_{0}^{t} \int_{\varOmega}|u-v|\,dx\,dt \biggr)^{l}, $$

(5.9)

where $l<1$.

Let $\kappa(s)=\int_{\varOmega}|u(x,s)-v(x,s)|\,dx$. Without loss of the generality, we may assume that there exists $\tau \in[0, T)$, $\kappa(\tau)>0$. Then, for any $s> \tau$, $\int_{\tau }^{s}k(t)\,dt>0$. If we denote

$$\tau_{0}=\max\bigl\{ t\in[\tau,s], \kappa(t)>0\bigr\} ,\qquad \int_{\tau}^{\tau _{0}}k(t)\,dt=c_{1}, $$

then $\tau<\tau_{0}\leq s$, and

$$\int_{\tau}^{s}k(t)\,dt\geq \int_{\tau}^{\tau_{0}}k(t)\,dt=c_{1}. $$

By $u,v\in L^{\infty}(Q_{T})$, there exists a constant $C>0$ such that

$$ \frac{c(\int_{\tau}^{s}k(t)\,dt)^{l}}{\int_{\tau}^{s}k(t)\,dt}\leq \frac{c(\int_{\tau}^{s}k(t)\,dt)^{l}}{c_{1}}\leq C=C(c,c_{1},T,q). $$

(5.10)

By (5.9) and (5.10), we have

$$\kappa(s)-\kappa(\tau)\leq(C+c) \int_{\tau}^{s}k(t)\,dt, $$

using the Gronwall inequality, we easily get

$$\int_{\varOmega}\bigl\vert u(x,s) - v(x,s) \bigr\vert \,dx \leqslant c \int _{\varOmega}\bigl\vert {u(x,\tau)} - {v(x,\tau)} \bigr\vert \,dx. $$

then, by the arbitrariness of τ,

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

 □

Proof of Theorem 2.7

Similar to the proof of Theorem 2.6, we have (5.1)–(5.4). Now, by the condition (2.11), we have

$$\begin{aligned}& \lim_{\eta\rightarrow0} \biggl\vert \int_{\varOmega} a_{i}(x) \bigl( \vert u_{x_{i}} \vert ^{p_{i}- 2}u_{x_{i}} - \vert v_{x_{i}} \vert ^{p_{i}- 2}v_{x_{i}} \bigr) \phi_{\eta x_{i}}S_{\eta}(u-v) \,dx \biggr\vert \\& \quad =\lim_{\eta\rightarrow0} \biggl\vert \int_{\varOmega\setminus\varOmega_{\eta }}a_{i}(x) \bigl( \vert u_{x_{i}} \vert ^{p_{i}- 2}u_{x_{i}} - \vert v_{x_{i}} \vert ^{p_{i} - 2}v_{x_{i}} \bigr) \phi_{\eta x_{i}} S_{\eta }(u-v) \,dx \biggr\vert \\& \quad \leq\lim_{\eta\rightarrow0}\frac{1}{\eta} \int_{\varOmega\setminus \varOmega_{\eta}} a_{i}(x) \bigl( \vert u_{x_{i}} \vert ^{p_{i} - 1}+ \vert v_{x_{i}} \vert ^{p_{i}- 1} \bigr) \Biggl\vert \Biggl(\prod_{j=1}^{N} a_{j}(x) \Biggr)_{x_{i}} S_{\eta}(u-v)\Biggr\vert \,dx \\& \quad \leq c \lim_{\eta\rightarrow0}\frac{1}{\eta} \biggl( \int_{\varOmega \setminus\varOmega_{\eta}} a_{i}(x) \bigl( \vert u_{x_{i}} \vert ^{p_{i}}+ \vert v_{x_{i}} \vert ^{p_{i}} \bigr) \,dx \biggr)^{\frac{p_{i}-1}{p_{i}}} \Biggl( \int_{\varOmega\setminus\varOmega_{\eta}} a_{i}(x) \Biggl\vert \Biggl(\prod _{j=1}^{N} a_{j}(x) \Biggr)_{x_{i}} \Biggr\vert ^{p_{i}}\,dx \Biggr)^{\frac{1}{p_{i}}} \\& \quad \leq c \lim_{\eta\rightarrow0} \Biggl( \int_{\varOmega\setminus\varOmega _{\eta}} a_{i}(x) \Biggl\vert \Biggl(\prod _{j=1}^{N} a_{j}(x) \Biggr)_{x_{i}} \Biggr\vert ^{p_{i}}\,dx \Biggr)^{\frac{1}{p_{i}}} \\& \quad =0. \end{aligned}$$

(5.11)

Last but not least, since $a_{i}(x)\in C^{1}(\overline{\varOmega})$, $a_{i}(x)=0$ when $x\in\partial\varOmega$, we have

$$ a_{i}^{\frac{1}{p_{i}}}(x) \Biggl(\prod_{j=1}^{N}a_{j}(x) \Biggr)_{x_{i}}=0, \quad x\in\partial\varOmega. $$

(5.12)

According to the definition of $\varOmega_{\eta}$, we have

$$\begin{aligned}& \lim_{\eta\rightarrow0} \biggl\vert \int_{{\varOmega }}\bigl[b_{i}(u,x,t)-b_{i}(v,x,t) \bigr]\phi_{\eta x_{i}}S_{\eta}(u - v)\,dx \biggr\vert \\& \quad =\lim_{\eta\rightarrow0} \biggl\vert \int_{\varOmega\setminus\varOmega_{\eta }}\bigl[b_{i}(u,x,t)-b_{i}(v,x,t) \bigr]\phi_{\eta x_{i}}S_{\eta}(u - v)\,dx \biggr\vert \\& \quad \leq\lim_{\eta\rightarrow0}\frac{1}{\eta} \int_{\varOmega\setminus \varOmega_{\eta}}a_{i}^{\frac{1}{p_{i}}} \Biggl\vert \Biggl(\prod_{j=1}^{N} a_{j}(x) \Biggr)_{x_{i}} \Biggr\vert \bigl\vert S_{\eta}(u - v) (u-v) \bigr\vert \,dx \\& \quad \leq c\lim_{\eta\rightarrow0}\frac{1}{\eta} \int_{\varOmega\setminus \varOmega_{\eta}}a_{i}^{\frac{1}{p_{i}}} \Biggl\vert \Biggl(\prod_{j=1}^{N} a_{j}(x) \Biggr)_{x_{i}} \Biggr\vert \,dx \\& \quad \leq c\max_{x\in\partial\varOmega}a_{i}^{\frac{1}{p_{i}}}(x) \Biggl(\prod_{j=1}^{N}a_{j}(x) \Biggr)_{x_{i}} \\& \quad =0. \end{aligned}$$

(5.13)

Now, let $\eta\rightarrow0$ in (5.2). Then

$$\int_{\varOmega}\bigl\vert u(x,s) - v(x,s) \bigr\vert \,dx \leqslant \int _{\varOmega}\bigl\vert u(x,\tau) - {v(x,\tau)} \bigr\vert \,dx. $$

By the arbitrariness of τ,

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

 □

6 The uniqueness of the solution

Theorem 2.6 and Theorem 2.7 both imply that the uniqueness of the weak solution is true, their proofs are based on the condition (2.9). Actually, without the condition (2.9), we still can prove the uniqueness of the solution without any boundary value condition.

Theorem 6.1

Let $p_{-}>1$, $a_{i}(x)\in C^{1}(\overline{\varOmega })$ satisfy (1.3), $b_{i}(s,x, t)$ be a Lipschitz function on $\mathbb {R}\times\overline{ \varOmega}\times[0,T]$. If u and v are two solutions of Eq. (1.1) with the initial values $u_{0}(x)$ and $v_{0}(x)$, respectively, then there exists a positive constant $\beta_{j}\geq2$ such that

$$ \int_{\varOmega} \Biggl(\prod_{j=1}^{N} a_{j}^{\beta_{j}}(x) \Biggr) \bigl\vert u(x,t)-v(x,t) \bigr\vert ^{2}\,dx\leq c \int_{\varOmega} \Biggl(\prod_{j=1}^{N} a_{j}^{\beta_{j}}(x) \Biggr) \bigl\vert u_{0}(x)-v_{0}(x) \bigr\vert ^{2}\,dx. $$

(6.1)

In particular, for any small enough constant $\delta>0$,

$$ \int_{\varOmega_{\delta}} \bigl\vert u(x,t)-v(x,t) \bigr\vert ^{2}\,dx\leq c(\delta,\beta _{j}) \int_{\varOmega} \bigl\vert u_{0}(x)-v_{0}(x) \bigr\vert ^{2}\,dx, $$

(6.2)

where $\varOmega_{\delta}= \{x\in\varOmega: \prod_{j=1}^{N} a_{j}^{\beta_{j}}(x)>\delta \}$.

Proof

Let u and v be two solutions of Eq. (1.1) with the initial values $u_{0}(x)$ and $v_{0}(x)$, respectively. By the process of taking the limit, we may choose $\varphi=\chi_{[\tau,s]}\prod_{j=1}^{N}a_{j}^{\beta_{j}}(u-v)$ as a test function. Denoting that $Q_{\tau s}=\varOmega\times[\tau, s]$, then

$$\begin{aligned}& \iint_{Q_{\tau s}}(u-v)\prod_{j=1}^{N}a_{j}^{\beta_{j}} \frac{\partial (u-v)}{\partial t}\,dx\,dt \\& \quad =-\sum_{i=1}^{N} \iint_{Q_{\tau s}}a_{i}(x) \bigl(|u_{x_{i}}|^{p_{i}-2}u_{x_{i}}-|v_{x_{i}}|^{p_{i}-2} v_{x_{i}} \bigr) \Biggl[(u-v)\prod_{j=1}^{N}a_{j}^{\beta_{j}} \Biggr]_{x_{i}} \,dx\,dt \\& \qquad {}-\sum_{i=1}^{N} \iint_{Q_{\tau s}}\bigl[b_{i}(u,x,t)-b_{i}(v,x,t) \bigr] \Biggl[(u-v)\prod_{j=1}^{N}a_{j}^{\beta_{j}} \Biggr]_{x_{i}} \,dx\,dt. \end{aligned}$$

(6.3)

In the first place, we have

$$ \iint_{Q_{\tau s}}a_{i}(x) \bigl(|u_{x_{i}}|^{p_{i}-2}u_{x_{i}}-|v_{x_{i}}|^{p_{i}-2} v_{x_{i}} \bigr) (u-v)_{x_{i}}\prod_{j=1}^{N}a_{j}^{\beta_{j}} \,dx\,dt\geq0 $$

(6.4)

and

$$\begin{aligned}& \Biggl\vert \iint_{Q_{\tau s}}(u-v)a_{i}(x) \bigl(|u_{x_{i}}|^{p_{i}-2}u_{x_{i}}-|v_{x_{i}}|^{p_{i}-2} v_{x_{i}} \bigr) \Biggl(\prod_{j=1}^{N}a_{j}^{\beta_{j}} \Biggr)_{x_{i}}\,dx\,dt \Biggr\vert \\& \quad \leq \iint_{Q_{\tau s}}|u-v|a_{i}(x) \bigl(|u_{x_{i}}|^{p_{i}-1}+| v_{x_{i}}|^{p_{i}-1} \bigr) \Biggl\vert \Biggl(\prod _{j=1}^{N}a_{j}^{\beta _{j}} \Biggr)_{x_{i}} \Biggr\vert \,dx\,dt \\& \quad \leq c \biggl( \int_{\tau}^{s} \int_{\varOmega}a_{i}(x) \bigl(|u_{x_{i}}|^{p_{i}}+| v_{x_{i}}|^{p_{i}} \bigr)\,dx\,dt \biggr)^{\frac{p_{i}-1}{p_{i}}} \\& \qquad {}\cdot \Biggl( \int_{\tau}^{s} \int_{\varOmega}a_{i}(x) \Biggl\vert \Biggl(\prod _{j=1}^{N}a_{j}^{\beta_{j}} \Biggr)_{x_{i}} \Biggr\vert ^{p_{i}}|u-v|^{p_{i}}\,dx\,dt \Biggr)^{\frac{1}{p_{i}}} \\& \quad \leq c \biggl( \int_{\tau}^{s} \int_{\varOmega}a_{i}(x) \bigl(|u_{x_{i}}|^{p_{i}}+| v_{x_{i}}|^{p_{i}} \bigr)\,dx\,dt \biggr)^{\frac{p_{i}-1}{p_{i}}} \\& \qquad {}\cdot \Biggl( \int_{\tau}^{s} \int_{\varOmega}a_{i}(x)\prod _{j=1}^{N} \bigl\vert a_{j}^{\beta_{j}-1}a_{jx_{i}} \bigr\vert \Biggr)^{p_{i}}|u-v|^{p_{i}}\,dx\,dt )^{\frac{1}{p_{i}}} \\& \quad \leq c \Biggl( \int_{\tau}^{s} \int_{\varOmega}a_{i}(x)\prod _{j=1}^{N} \bigl\vert a_{j}^{\beta_{j}-1}a_{jx_{i}} \bigr\vert ^{p_{i}}|u-v|^{p_{i}}\,dx\,dt \Biggr)^{\frac {1}{p_{i}}}. \end{aligned}$$

(6.5)

Here, we have used the fact that $|a_{x_{i}}|\leq c$. Now, we choose $\beta_{i}\geq2$. If $p_{i}\geq2$,

$$\begin{aligned}& \Biggl( \int_{\tau}^{s} \int_{\varOmega}a_{i}(x)\prod _{j=1}^{N} \bigl\vert a_{j}^{\beta_{j}-1}a_{jx_{i}} \bigr\vert ^{p_{i}}|u-v|^{p_{i}}\,dx\,dt \Biggr)^{\frac {1}{p_{i}}} \\& \quad \leq c \Biggl( \int_{\tau}^{s} \int_{\varOmega}\prod_{j=1}^{N}a_{j}^{\beta_{j}}|u-v|^{2} \,dx\,dt \Biggr)^{\frac{1}{p_{i}}}. \end{aligned}$$

(6.6)

If $1< p_{i}<2$, by the Hölder inequality

$$\begin{aligned}& \Biggl( \int_{\tau}^{s} \int_{\varOmega}a_{i}(x)\prod _{j=1}^{N} \bigl\vert a_{j}^{\beta_{j}-1}a_{jx_{i}} \bigr\vert ^{p_{i}}|u-v|^{p_{i}}\,dx\,dt \Biggr)^{\frac {1}{p_{i}}} \\& \quad \leq c \Biggl( \int_{\tau}^{s} \int_{\varOmega}\prod_{j=1}^{N}a_{j}^{\beta_{j}}|u-v|^{2} \,dx\,dt \Biggr)^{\frac{1}{2}}. \end{aligned}$$

(6.7)

Combining (6.5)–(6.7), we obtain

$$\begin{aligned}& \Biggl\vert \iint_{Q_{\tau s}}(u-v)a_{i}(x) \bigl(|u_{x_{i}}|^{p_{i}-2}u_{x_{i}}-|v_{x_{i}}|^{p_{i}-2} v_{x_{i}} \bigr) \Biggl(\prod_{j=1}^{N}a_{j}^{\beta_{j}} \Biggr)_{x_{i}}\,dx\,dt \Biggr\vert \\& \quad \leq c \Biggl( \int_{\tau}^{s} \int_{\varOmega}\prod_{j=1}^{N}a_{j}^{\beta _{j}}|u-v|^{2} \,dx\,dt \Biggr)^{l}, \end{aligned}$$

(6.8)

where $l<1$.

In the second place,

$$\begin{aligned}& \iint_{Q_{\tau s}}\bigl[b_{i}(u,x,t)-b_{i}(v,x,t) \bigr] \Biggl[(u-v)\prod_{j=1}^{N}a_{j}^{\beta_{j}} \Biggr]_{x_{i}} \,dx\,dt \\& \quad = \iint_{Q_{\tau s}}\bigl[b_{i}(u,x,t)-b_{i}(v,x,t) \bigr](u-v) \Biggl(\prod_{j=1}^{N}a_{j}^{\beta_{j}} \Biggr)_{ x_{i}} \,dx\,dt \\& \qquad {}+ \iint_{Q_{s}}\bigl[b_{i}(u,x,t)-b_{i}(v,x,t) \bigr](u-v)_{x_{i}}\prod_{j=1}^{N}a_{j}^{\beta_{j}} \,dx\,dt. \end{aligned}$$

(6.9)

For the first term on the right-hand side of (6.7), since $\beta_{j}\geq 2$, $| a_{jx_{i}}|\leq c$, by the Hölder inequality,

$$\begin{aligned}& \iint_{Q_{\tau s}}\bigl[b_{i}(u,x,t)-b_{i}(v,x,t) \bigr](u-v) \Biggl(\prod_{j=1}^{N}a_{j}^{\beta_{j}} \Biggr)_{x_{i}} \,dx\,dt \\& \quad = \int_{\tau}^{s} \int_{\varOmega}\bigl[b_{i}(u,x,t)-b_{i}(v,x,t) \bigr](u-v)\sum_{k=1}^{N} \Biggl( \beta_{k}a_{k}^{\beta_{k}-1} a_{kx_{i}}\prod _{j=1, j\neq k}^{N}a_{j}^{\beta_{j}} \Biggr)\,dx\,dt \\& \quad \leq c \int_{\tau}^{s} \int_{\varOmega}|u-v|\sum_{k=1}^{N} \Biggl(\beta _{k}a_{k}^{\beta_{k}-1} a_{kx_{i}} \prod_{j=1, j\neq k}^{N}a_{j}^{\beta _{j}} \Biggr)\,dx\,dt \\& \quad \leq c \Biggl( \int_{\tau}^{s} \int_{\varOmega}\prod_{j=1}^{N}a_{j}^{\beta _{j}}|u-v|^{2} \,dx\,dt \Biggr)^{\frac{1}{2}}. \end{aligned}$$

(6.10)

For the second term on the right-hand side of (6.9), since $\beta _{i}\geq1$, denoting $p_{i}'=\frac{p_{i}}{p_{i}-1}$ as usual, we have

$$\biggl(\beta_{i}-\frac{1}{p_{i}} \biggr)p_{i}' \geq\beta_{i}. $$

By this inequality, we have

$$\begin{aligned}& \Biggl\vert \iint_{Q_{\tau s}}\bigl[b_{i}(u,x,t)-b_{i}(v,x,t) \bigr](u-v)_{x_{i}}\prod_{j=1}^{N}a_{j}^{\beta_{j}} \,dx\,dt \Biggr\vert \\& \quad \leq c \Biggl( \int_{\tau}^{s} \int_{\varOmega}a_{i}^{(\beta_{i}-\frac {1}{p_{i}})p_{i}'} \Biggl(\prod _{j=1, j\neq i}^{N}a_{j}^{\beta _{j}} \bigl\vert b_{i}(u,x,t)-b_{i}(v,x,t) \bigr\vert \Biggr)^{p_{i}'}\,dx\,dt \Biggr)^{\frac{1}{p_{i}'}} \\& \qquad {}\cdot \biggl( \int_{\tau}^{s} \int_{\varOmega}a_{i} \bigl(|u_{x_{i}}|^{p_{i}}+|v_{x_{i}}|^{p_{i}} \bigr) \,dx\,dt \biggr)^{\frac{1}{p_{i}}} \\& \quad \leq c \Biggl( \int_{\tau}^{s} \int_{\varOmega}a_{i}^{(\beta_{i}-\frac {1}{p_{i}})p_{i}'} \Biggl(\prod _{j=1, j\neq i}^{N}a_{j}^{\beta _{j}} \bigl\vert b_{i}(u,x,t)-b_{i}(v,x,t) \bigr\vert \Biggr)^{p_{i}'}\,dx\,dt \Biggr)^{\frac{1}{p_{i}'}} \\& \quad \leq c \Biggl( \int_{\tau}^{s} \int_{\varOmega}\prod_{j=1}^{N}a_{j}^{\beta _{j}}|u-v|^{p_{i}'} \,dx\,dt \Biggr)^{\frac{1}{p_{i}'}}. \end{aligned}$$

(6.11)

If $p_{i}>2$, then $1< p_{i}'<2$. By the Hölder inequality,

$$ \Biggl( \int_{\tau}^{s} \int_{\varOmega}\prod_{j=1}^{N}a_{j}^{\beta _{j}}|u-v|^{p_{i}'} \,dx\,dt \Biggr)^{\frac{1}{p_{i}'}} \leq c \Biggl( \int_{\tau}^{s} \int_{\varOmega}\prod_{j=1}^{N}a_{j}^{\beta _{j}}|u-v|^{2} \,dx\,dt \Biggr)^{\frac{1}{2}}. $$

(6.12)

If $1< p_{i}\leq2$, then $p_{i}'\geq2$,

$$ \Biggl( \int_{\tau}^{s} \int_{\varOmega}\prod_{j=1}^{N}a_{j}^{\beta _{j}}|u-v|^{p_{i}'} \,dx\,dt \Biggr)^{\frac{1}{p_{i}'}}\leq c \Biggl( \int_{\tau }^{s} \int_{\varOmega}\prod_{j=1}^{N}a_{j}^{\beta_{j}}|u-v|^{2} \,dx\,dt \Biggr)^{\frac{1}{p_{i}'}}. $$

(6.13)

Combining (6.11)–(6.13), we have

$$\begin{aligned}& \Biggl\vert \iint_{Q_{\tau s}}\bigl[b_{i}(u,x,t)-b_{i}(v,x,t) \bigr](u-v)_{x_{i}}\prod_{j=1}^{N}a_{j}^{\beta_{j}} \,dx\,dt \Biggr\vert \\& \quad \leq c \Biggl( \int_{\tau}^{s} \int_{\varOmega}\prod_{j=1}^{N}a_{j}^{\beta _{j}}|u-v|^{2} \,dx\,dt \Biggr)^{l}, \end{aligned}$$

(6.14)

where $l<1$.

Moreover,

$$\begin{aligned}& \iint_{Q_{\tau s}}(u-v)\prod_{j=1}^{N}a_{j}^{\beta_{j}} \frac{\partial (u-v)}{\partial t}\,dx\,dt \\& \quad = \iint_{Q_{\tau s}}(u-v)\sqrt{\prod _{j=1}^{N}a_{j}^{\beta_{j}}} \frac{\partial [\sqrt{\prod_{j=1}^{N}a_{j}^{\beta_{j}}} (u-v) ]}{\partial t}\,dx\,dt \\& \quad = \int_{\varOmega}\prod_{j=1}^{N}a_{j}^{\beta_{j}} \bigl[u(x,s)-v(x,s)\bigr]^{2}\,dx- \int _{\varOmega}\prod_{j=1}^{N}a_{j}^{\beta_{j}} \bigl[u(x,\tau)-v(x,\tau )\bigr]^{2}\,dx. \end{aligned}$$

(6.15)

According to (6.3), (6.4), (6.8), (6.10), (6.14) and (6.15), we have

$$\begin{aligned}& \int_{\varOmega}\prod_{j=1}^{N}a_{j}^{\beta_{j}} \bigl[u(x,s)-v(x,s)\bigr]^{2}\,dx- \int _{\varOmega}\prod_{j=1}^{N}a_{j}^{\beta_{j}} \bigl[u(x,\tau)-v(x,\tau)\bigr]^{2}\,dx \\& \quad \leq c \Biggl( \int_{0}^{s} \int_{\varOmega}\prod_{j=1}^{N}a_{j}^{\beta _{j}} \bigl\vert u(x,t)-v(x,t) \bigr\vert ^{2}\,dx\,dt \Biggr)^{l}, \end{aligned}$$

(6.16)

where $l<1$. By (6.16), we easily show that

$$ \int_{\varOmega}\prod_{j=1}^{N}a_{j}^{\beta_{j}} \bigl\vert u(x,\tau)-v(x,\tau ) \bigr\vert ^{2} \,dx\leq \int_{\varOmega}\prod_{j=1}^{N}a_{j}^{\beta_{j}} \bigl\vert u(x,\tau)-v(x,\tau) \bigr\vert ^{2} \,dx. $$

(6.17)

Thus, by the arbitrariness of τ, we have

$$ \int_{\varOmega}\prod_{j=1}^{N}a_{j}^{\beta_{j}} \bigl\vert u(x,s)-v(x,s) \bigr\vert ^{2} \,dx\leq \int_{\varOmega}\prod_{j=1}^{N}a_{j}^{\beta_{j}} \bigl\vert u_{0}(x)-v_{0}(x) \bigr\vert ^{2} \,dx. $$

(6.18)

By (6.18), we clearly have (6.1) and (6.2). The proof is complete. □

By this theorem, Theorem 2.8 is true.

7 The general method to prove the stability of weak solutions

We can generalize the method used in Sect. 6 to prove various kinds of stability of weak solutions.

Let $\chi(x)$ be a $C^{1}(\overline{\varOmega})$ function satisfying

$$ \chi(x)=0, \quad \text{if } x\in\partial\varOmega;\qquad \chi(x)>0, \quad \text{if } x\in\varOmega. $$

(7.1)

Theorem 7.1

Let $p_{-}\geq2$, $a_{i}(x)\in C^{1}(\overline {\varOmega})$ satisfy (1.3), $b_{i}(s,x, t)$ is bounded when s is bounded and $(x,t)\in\varOmega\times[0,T)$. If there exist constants $0<\sigma_{i}<1$, $0<\delta_{i}<1$, and there exists $\chi(x)$ satisfying (7.1) and

$$\begin{aligned}& \int_{\varOmega} \biggl\vert \frac{(\chi(x))_{x_{i}}}{\chi^{\frac{\delta _{i}}{2}}(x)} \biggr\vert ^{\frac{2}{2-\delta_{i}}}\,dx\leq c, \end{aligned}$$

(7.2)

$$\begin{aligned}& \int_{\varOmega} \bigl\vert \bigl(\chi(x)\bigr)^{p_{i}'-\frac{\sigma_{i}}{2}}a_{i}^{-\frac {1}{p_{i}-1}}(x) \bigr\vert ^{\frac{2}{2-\sigma_{i}}}\,dx\leq c. \end{aligned}$$

(7.3)

Let u and v are be solutions of Eq. (1.1) with the initial values $u_{0}(x)$ and $v_{0}(x)$, respectively. Then, for any $\varOmega_{1}\subset \subset\varOmega$,

$$\int_{\varOmega_{1}} \bigl\vert u(x,t)-v(x,t) \bigr\vert ^{2}\,dx\leq c(\varOmega_{1}) \int_{\varOmega } \bigl\vert u_{0}(x)-v_{0}(x) \bigr\vert ^{2}\,dx. $$

Here, $p_{i}'=\frac{p_{i}}{p_{i}-1}$ as usual.

Proof

Let u and v be two solutions of Eq. (1.1) with the initial values $u_{0}(x)$ and $v_{0}(x)$, respectively. By the process of taking the limit, we may choose $\varphi=\chi_{[\tau,s]}\chi (x)(u-v)$ as a test function. Denoting $Q_{\tau s}=\varOmega\times[\tau , s]$, then

$$\begin{aligned}& \iint_{Q_{\tau s}}(u-v)\chi(x)\frac{\partial(u-v)}{\partial t}\,dx\,dt \\& \quad =-\sum_{i=1}^{N} \iint_{Q_{\tau s}}a_{i}(x) \bigl(|u_{x_{i}}|^{p_{i}-2}u_{x_{i}}-|v_{x_{i}}|^{p_{i}-2} v_{x_{i}} \bigr)\bigl[(u-v)\chi (x)\bigr]_{x_{i}} \,dx\,dt \\& \qquad {}-\sum_{i=1}^{N} \iint_{Q_{\tau s}}\bigl[b_{i}(u,x,t)-b_{i}(v,x,t) \bigr] \bigl[(u-v)\chi(x)\bigr]_{x_{i}} \,dx\,dt. \end{aligned}$$

(7.4)

In the first place, we have

$$ \iint_{Q_{\tau s}}a_{i}(x) \bigl(|u_{x_{i}}|^{p_{i}-2}u_{x_{i}}-|v_{x_{i}}|^{p_{i}-2} v_{x_{i}} \bigr) (u-v)_{x_{i}}\chi(x) \,dx\,dt \geq0, $$

(7.5)

and using (7.2) we deduce that

$$\begin{aligned}& \biggl\vert \iint_{Q_{\tau s}}(u-v)a_{i}(x) \bigl(|u_{x_{i}}|^{p_{i}-2}u_{x_{i}}-|v_{x_{i}}|^{p_{i}-2} v_{x_{i}} \bigr) \bigl(\chi (x)\bigr)_{x_{i}}\,dx\,dt \biggr\vert \\& \quad \leq \iint_{Q_{\tau s}}|u-v|a_{i}(x) \bigl(|u_{x_{i}}|^{p_{i}-1}+| v_{x_{i}}|^{p_{i}-1} \bigr) \bigl\vert \bigl(\chi(x) \bigr)_{x_{i}} \bigr\vert \,dx\,dt \\& \quad \leq c \biggl( \int_{\tau}^{s} \int_{\varOmega}a_{i}(x) \bigl(|u_{x_{i}}|^{p_{i}}+| v_{x_{i}}|^{p_{i}} \bigr)\,dx\,dt \biggr)^{\frac{p_{i}-1}{p_{i}}} \\& \qquad {}\cdot \biggl( \int_{\tau}^{s} \int_{\varOmega}a_{i}(x) \bigl\vert \bigl(\chi (x) \bigr)_{x_{i}} \bigr\vert ^{p_{i}}|u-v|^{p_{i}}\,dx\,dt \biggr)^{\frac{1}{p_{i}}} \\& \quad \leq c \biggl( \int_{\tau}^{s} \int_{\varOmega}a_{i}(x) \bigl\vert \bigl(\chi (x) \bigr)_{x_{i}} \bigr\vert ^{p_{i}}|u-v|^{p_{i}}\,dx\,dt \biggr)^{\frac{1}{p_{i}}} \\& \quad = c \biggl( \int_{\tau}^{s} \int_{\varOmega}a_{i}(x)\frac{|(\chi (x))_{x_{i}}|^{p_{i}}}{\chi^{\frac{\delta_{i}}{2}}(x)} \chi^{\frac{\delta _{i}}{2}}|u-v|^{\delta_{i}} |u-v|^{p_{i}-\delta_{i}}\,dx\,dt \biggr)^{\frac{1}{p_{i}}} \\& \quad \leq c \biggl( \int_{\tau}^{s} \int_{\varOmega} \biggl(\frac{|(\chi(x))_{x_{i}}|^{p_{i}}}{\chi^{\frac{\delta _{i}}{2}}(x)} \biggr)^{\frac{2}{2-\delta_{i}}}\,dx \,dt \biggr)^{\frac {1}{p_{i}}\frac{2-\delta_{i}}{2}} \\& \qquad {}\cdot \biggl( \int_{\tau}^{s} \int_{\varOmega}\chi(x)|u-v|^{2}\,dx \biggr)^{\frac{\delta_{i}}{2}\frac{1}{p_{i}}} \\& \quad \leq c \biggl( \int_{\tau}^{s} \int_{\varOmega}\chi(x)|u-v|^{2}\,dx \biggr)^{\frac{\delta_{i}}{2}\frac{1}{p_{i}}}. \end{aligned}$$

(7.6)

In the second place,

$$\begin{aligned}& \iint_{Q_{\tau s}}\bigl[b_{i}(u,x,t)-b_{i}(v,x,t) \bigr] \bigl[(u-v)\chi(x)\bigr]_{x_{i}} \,dx\,dt \\& \quad = \iint_{Q_{\tau s}}\bigl[b_{i}(u,x,t)-b_{i}(v,x,t) \bigr](u-v) \bigl(\chi(x)\bigr)_{ x_{i}} \,dx\,dt \\& \qquad {}+ \iint_{Q_{s}}\bigl[b_{i}(u,x,t)-b_{i}(v,x,t) \bigr](u-v)_{x_{i}}\chi(x) \,dx\,dt. \end{aligned}$$

(7.7)

For the first term on the right-hand side of (7.7), since $b_{i}(u,x,t)$ and $b_{i}(v,x,t)$ are bounded when $u\in L^{\infty }(Q_{T})$, $v\in L^{\infty}(Q_{T})$, and by that $p_{-}\geq2$ implies $p_{i}'-\frac{2\delta_{i}}{(2-\delta _{i})(p_{i}-1)}\geq0$, by the Hölder inequality, using (7.2), we have

$$\begin{aligned}& \biggl\vert \iint_{Q_{\tau s}}\bigl[b_{i}(u,x,t)-b_{i}(v,x,t) \bigr](u-v) \bigl(\chi (x)\bigr)_{x_{i}} \,dx\,dt \biggr\vert \\& \quad \leq c \biggl[ \iint_{Q_{\tau s}} \biggl(\frac{|(\chi(x))_{x_{i}}|^{\frac {2}{2-\delta_{i}}}}{(\chi(x))^{\frac{\delta_{i}}{(2-\delta _{i})p_{i}}}} \biggr)^{p_{i}} \,dx \biggr]^{\frac{1}{p_{i}}} \biggl( \iint _{Q_{\tau s}}\bigl[\chi^{\frac{\delta_{i}}{(2-\delta _{i})p_{i}}}|u-v|\bigr]^{p_{i}'} \,dx \biggr)^{\frac{1}{p_{i}'}} \\& \quad \leq c \biggl( \iint_{Q_{\tau s}}\bigl[\chi^{\frac{\delta_{i}}{(2-\delta _{i})p_{i}}}|u-v|\bigr]^{p_{i}'} \,dx \biggr)^{\frac{1}{p_{i}'}} \\& \quad =c \biggl( \iint_{Q_{\tau s}} \bigl[\chi^{\frac{\delta_{i}}{(2-\delta _{i})(p_{i}-1)}}|u-v| \bigr]^{\frac{2\delta_{i}}{(2-\delta _{i})(p_{i}-1)}}|u-v|^{p_{i}'-\frac{2\delta_{i}}{(2-\delta _{i})(p_{i}-1)}}\,dx \biggr)^{\frac{1}{p_{i}'}} \\& \quad \leq c \biggl( \iint_{Q_{\tau s}}\chi(x)|u-v|^{2}\,dx \biggr)^{\frac {\delta_{i}}{(2-\delta_{i})p_{i}}}. \end{aligned}$$

(7.8)

For the second term on the right-hand side of (7.7), by this inequality and the condition (7.3), we have

$$\begin{aligned}& \biggl\vert \iint_{Q_{\tau s}}\bigl[b_{i}(u,x,t)-b_{i}(v,x,t) \bigr](u-v)_{x_{i}}\chi (x)\,dx\,dt \biggr\vert \\& \quad = \biggl\vert \iint_{Q_{\tau s}}\bigl(a_{i}(x)\bigr)^{\frac{1}{p_{i}}}(u-v)_{x_{i}} \chi (x) \bigl(a_{i}(x)\bigr)^{-\frac{1}{p_{i}}}(u-v)\,dx\,dt \biggr\vert \\& \quad \leq c \biggl( \int_{\tau}^{s} \int_{\varOmega} \bigl\vert \chi(x)a_{i}^{-\frac {1}{p_{i}}}(u-v) \bigr\vert ^{p_{i}'}\,dx\,dt \biggr)^{\frac{1}{p_{i}'}} \biggl( \int_{\tau}^{s} \int_{\varOmega}a_{i}(x) \bigl(|u_{x_{i}}|^{p_{i}}+|v_{x_{i}}|^{p_{i}} \bigr) \,dx\,dt \biggr)^{\frac{1}{p_{i}}} \\& \quad \leq c \biggl( \int_{\tau}^{s} \int_{\varOmega} \bigl\vert \chi(x)a_{i}^{-\frac {1}{p_{i}}}(u-v) \bigr\vert ^{p_{i}'}\,dx\,dt \biggr)^{\frac{1}{p_{i}'}} \\& \quad =c \biggl( \int_{\tau}^{s} \int_{\varOmega}|\chi(x)|^{p_{i}'-\frac{\sigma _{i}}{2}}a_{i}^{-\frac{1}{p_{i}-1}}| \chi(x)|^{\frac{\sigma _{i}}{2}}|u-v|^{\sigma_{i}} (u-v)|^{p_{i}'-\sigma_{i}}\,dx\,dt \biggr)^{\frac{1}{p_{i}'}} \\& \quad \leq c \biggl( \int_{\tau}^{s} \int_{\varOmega} \bigl(\chi^{p_{i}'-\frac {\sigma_{i}}{2}}a_{i}^{-\frac{1}{p_{i}-1}} \bigr)^{\frac{2}{2-\sigma _{i}}}\,dx\,dt \biggr)^{\frac{2-\sigma_{i}}{2p_{i}'}} \biggl( \int_{\tau}^{s} \int_{\varOmega}\chi(x)|u-v|^{2}\,dx\,dt \biggr)^{\frac{\sigma_{i}}{2p_{i}'}} \\& \quad \leq c \biggl( \int_{\tau}^{s} \int_{\varOmega}\chi(x)|u-v|^{2}\,dx\,dt \biggr)^{\frac{\sigma_{i}}{2p_{i}'}}. \end{aligned}$$

(7.9)

Moreover,

$$\begin{aligned}& \iint_{Q_{\tau s}}(u-v)\chi(x)\frac{\partial(u-v)}{\partial t}\,dx\,dt \\& \quad = \iint_{Q_{\tau s}}(u-v)\sqrt{\chi(x)}\frac{\partial [\sqrt{\chi(x)} (u-v) ]}{\partial t}\,dx\,dt \\& \quad = \int_{\varOmega}\chi(x)\bigl[u(x,s)-v(x,s)\bigr]^{2}\,dx- \int_{\varOmega}\chi (x)\bigl[u(x,\tau)-v(x,\tau)\bigr]^{2} \,dx. \end{aligned}$$

(7.10)

According to (7.4)–(7.10), we have

$$\begin{aligned}& \int_{\varOmega}\chi(x)\bigl[u(x,s)-v(x,s)\bigr]^{2}\,dx- \int_{\varOmega}\chi (x)\bigl[u(x,\tau)-v(x,\tau)\bigr]^{2} \,dx \\& \quad \leq c \biggl( \int_{\tau}^{s} \int_{\varOmega}\chi(x) \bigl\vert u(x,t)-v(x,t) \bigr\vert ^{2}\,dx\,dt \biggr)^{l}, \end{aligned}$$

(7.11)

where $l<1$. By (7.11), we easily can show that

$$ \int_{\varOmega}\chi(x) \bigl\vert u(x,\tau)-v(x,\tau) \bigr\vert ^{2} \,dx\leq \int _{\varOmega}\chi(x) \bigl\vert u(x,\tau)-v(x,\tau) \bigr\vert ^{2} \,dx. $$

(7.12)

By the arbitrariness of τ, then

$$ \int_{\varOmega}\chi(x) \bigl\vert u(x,s)-v(x,s) \bigr\vert ^{2} \,dx\leq \int_{\varOmega }\chi(x) \bigl\vert u_{0}(x)-v_{0}(x) \bigr\vert ^{2} \,dx. $$

(7.13)

Since (7.1), by (7.13), the inequality (7.3) is true clearly.

One can see that the condition $p_{-}\geq2$ is only used to estimate (7.8). We are sure that it can weakened to $p_{-}>1$. For example, if there exists constant $\gamma_{i}>0$ such that

$$ \bigl\vert b_{i}(u,x,t) \bigr\vert \leq c \bigl\vert \chi(x) \bigr\vert ^{\gamma_{i}}, $$

(7.14)

then we obtain

$$\begin{aligned}& \biggl\vert \iint_{Q_{\tau s}}\bigl[b_{i}(u,x,t)-b_{i}(v,x,t) \bigr](u-v)_{x_{i}}\chi (x)\,dx\,dt \biggr\vert \\& \quad \leq c \biggl( \iint_{Q_{\tau s}}\chi (x)|u-v|^{2}\,dx\,dt \biggr)^{l}, \end{aligned}$$

(7.15)

where $l<1$. Thus, we still have the conclusion of Theorem 7.1. □

However, we are not ready to discuss how to weaken the condition $p_{-}\geq2$ again in what follows. We prefer to explain the importance of Theorem 7.1. That is, if we choose various kinds of functions $\chi(x)$, we can obtain the corresponding stability theorems. Let us give some examples.

If we choose $\chi(x)=\prod_{i=1}^{N}a^{\beta}_{i}(x)$, we have a similar conclusion to Theorem 2.8. By the process of taking the limit, we can choose $\chi(x)=d^{\alpha}(x)$, where $\alpha>0$ is a constant, $d(x)=\operatorname{dist}(x, \partial\varOmega)$ is the distance function from the boundary. Then we have the following theorem.

Theorem 7.2

Let $p_{-}\geq2$, $a_{i}(x)\in C^{1}(\overline {\varOmega})$ satisfy (1.3), $b_{i}(s,x, t)$ is bounded when s is bounded and $(x,t)\in\varOmega\times[0,T)$. Assume $\alpha>1$, and assume there exist constants $0<\sigma_{i}<1$,

$$ \int_{\varOmega} \bigl\vert \bigl(d^{\alpha}(x) \bigr)^{p_{i}'-\frac{\sigma _{i}}{2}}a_{i}^{-\frac{1}{p_{i}-1}}(x) \bigr\vert ^{\frac{2}{2-\sigma _{i}}}\,dx\leq c. $$

(7.16)

Let u and v are be solutions of Eq. (1.1) with the initial values $u_{0}(x)$ and $v_{0}(x)$, respectively. Then, for any $\varOmega_{1}\subset \subset\varOmega$,

$$\int_{\varOmega_{1}} \bigl\vert u(x,t)-v(x,t) \bigr\vert ^{2}\,dx\leq c(\varOmega_{1}) \int_{\varOmega } \bigl\vert u_{0}(x)-v_{0}(x) \bigr\vert ^{2}\,dx. $$

Proof

Since $\chi(x)=d^{\alpha}(x)$, $\alpha>1$, for any $0<\sigma_{i}<1$, it is not difficult to show the inequality (7.1) is true. Then we have the conclusion. □

As long as one wants, one can choose other types of the functions $\chi (x)$, e.g. $\chi(x)=\sum_{i=1}^{N} a_{i}(x)$, $\chi(x)=e^{a_{i}(x)}-1$ for any given $i\in\{1,2,\ldots, N\}$, or $\chi(x)=\max\{a_{i}(x)\}$, to obtain the corresponding stability theorems.

8 Conclusion

The anisotropic parabolic equations considered in this paper arise from many applied fields such as non-Newtonian fluid theory, reaction–diffusion problems. If the convection term depends on the diffusion coefficient which is degenerate on the boundary, then the stability of weak solutions may be proved without any boundary value condition. If the convection term is independent of the diffusion coefficient, the uniqueness of the weak solution is still true only if the convection function $b_{i}(u,x,t)$ is bounded when $|u|\leq c$. Moreover, a general method to prove the stability of the weak solutions without the boundary value condition is introduced for the first time in this paper. We believe such a method can be used in many kinds of parabolic equations, especially those lacking the regularity for the trace on the boundary to be defined.