The partial boundary value conditions of nonlinear degenerate parabolic equation

Abstract

The stability of the solutions to a parabolic equation

$$ \frac{\partial u}{\partial t} = \Delta A(u) +\sum_{i=1}^{N}b_{i}(x,t)D_{i}u-c(x,t)u-f(x,t) $$

with homogeneous boundary condition is considered. Since the set $\{s: A'(s)=a(s)=0\}$ may have an interior point, the equation is with strong degeneracy and the Dirichlet boundary value condition is overdetermined generally. How to find a partial boundary value condition to match up with the equation is studied in this paper. By choosing a suitable test function, the stability of entropy solutions is obtained by Kruzkov bi-variables method.

1 The boundary condition

We consider the equation

$$ \frac{\partial u}{\partial t} = \Delta A(u) +\sum _{i=1}^{N}b_{i}(x,t)D_{i}u-c(x,t)u-f(x,t), \quad (x,t)\in Q_{T}=\Omega \times (0,T), $$

(1)

and assume that

$$ A(u)= \int _{0}^{u}a(s)\,ds,\quad a(s)\geq 0,\, a(0)=0, $$

(2)

where $\Omega \subset \mathbb{R}^{N}$ is an appropriately smooth bounded domain, $D_{i}=\frac{\partial }{\partial x_{i}}$, $b_{i}(x,t)\in C^{1}(\overline{Q}_{T})$, $c(x,t), f(x,t)\in C(\overline{Q}_{T})$. Equation (1) has a widely applied background, for example, the reaction diffusion problem [10] and the spread of an epidemic disease in heterogeneous environments.

For the Cauchy problem of equation (1), the paper [2] by Vol’pert and Hudjaev was the first one to study its solvability. Since then there have been many papers to study its well-posedness ceaselessly, one can refer to book [18] and references [1–6, 8, 9, 11, 13–20], and [25, 26].

If we want to consider the initial boundary value problem of equation (1), the initial value condition

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

(3)

is always necessary. But the Dirichlet homogeneous boundary value condition

$$ u(x,t)=0,\quad (x,t)\in \Sigma _{T}=\partial \Omega \times (0,T), $$

(4)

may be overdetermined generally. In [21, 22, 24], a version of equation (1)

$$ \frac{\partial u}{\partial t} = \Delta A(u) +\sum _{i=1}^{N} \frac{\partial b_{i}(u)}{\partial x_{i}},\quad (x,t)\in Q_{T}, $$

(5)

was studied. Instead of it, a partial boundary value

$$ u(x,t)=0,\quad (x,t)\in \Sigma _{1}\times (0,T), $$

(6)

is enough, where $\Sigma _{1}\subset \partial \Omega$ is a relative open subset. One can refer to [21, 22, 24] for details, in which the equation $\Sigma _{1}\subset \partial \Omega$ was depicted out in some special ways. Such a fact was found firstly in [19], in which the non-Newtonian fluid equation

$$ \begin{gathered}[b] \frac{\partial u}{\partial t}- \operatorname{div} \bigl(a(x) \vert \nabla u \vert ^{p-2}\nabla u \bigr)-\sum_{i=1}^{N}b_{i}(x)D_{i}u+c(x,t)u=f(x,t), \\ \quad (x,t)\in Q_{T}=\Omega \times (0,T), \end{gathered} $$

(7)

was considered. Here $p> 1$, $D_{i} =\frac{\partial }{\partial x_{i}}$, $0\leq a(x) \in C(\overline{\Omega })$, $b_{i}(x) \in C^{1}(\overline{\Omega })$, $c(x,t)$ and $f(x,t)$ are continuous functions on $\overline{Q}_{T}$.

However, in [21] and [22], because there are two parameters including in $\Sigma _{1}$, the expression $\Sigma _{1}$ seems very complicated and hard to be verified, and the stability of entropy solutions is proved under the assumptions

$$ \vert \triangle d \vert \leq c, \quad \frac{1}{\lambda } \int _{\Omega _{\lambda }} dx\,dt \leq c. $$

(8)

Here $d(x)=\text{dist}(x,\partial \Omega )$ and $\Omega _{\lambda }=\{x\in \Omega , d(x)<\lambda \}$, λ is small enough, while [24] considered the case of Ω being unbounded and satisfying some harsh terms. Moreover, all partial boundary value conditions appearing in [19, 21, 22] and [24] are with the form as (6).

The dedications of this paper lie in that, for any given bounded domain Ω, due to the fact that the coefficient $b_{i}(x,t)$ depends on the time variable t, we find that, unlike (6), the partial boundary value condition matching up with equation (1) must be of the following form:

$$ u(x,t)=0, \quad (x,t)\in \Sigma _{p}\subset \partial \Omega \times (0,T), $$

(9)

where $\Sigma _{p}$ is just a submanifold of $\partial \Omega \times (0,T)$ and it cannot be a cylinder as $\Sigma _{1}\times (0,T)$. By choosing some technical test functions, the stability of entropy solutions is established by Kruzkov’s bi-variables method.

2 The definition and the main results

For small $\eta >0$, let

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

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

$$ \underset{\eta \rightarrow 0}{\lim }S_{\eta }(s)=\operatorname{sgn}s, \qquad \underset{\eta \rightarrow 0}{\lim }sS_{\eta }^{\prime }(s)=0. $$

(10)

Let

$$\begin{aligned} A_{\eta }(u,k) =& \int _{k}^{u}a(s)S_{\eta }(s-k)\,ds, \qquad I_{\eta }(u-k)= \int _{0}^{u-k}S_{\eta }(s)\,ds. \end{aligned}$$

Define that $u\in \operatorname{BV}(Q_{T})$ if and only if $u\in L_{\mathrm{loc}}^{1}(Q_{T})$ and

$$ \int _{0}^{T} \int _{\Omega } \bigl\vert u(x_{1}+h_{1}, \ldots ,x_{N}+h_{N},t+h_{N+1})-u(x,t) \bigr\vert \,dx\,dt\leq C \vert h \vert , $$

where $h=(h_{1},h_{2},\ldots, h_{N},h_{N+1})$. This is equivalent to that the generalized derivatives of every function in $\operatorname{BV}(\Omega )$ are regular measures on Ω. Under the norm

$$ \Vert f \Vert _{\operatorname{BV}}= \Vert f \Vert _{L^{1}}+ \int _{\Omega } \vert Df \vert , $$

$\operatorname{BV}(\Omega )$ is a Banach space.

A basic property of BV function is that (see [17, 18]): if $f\in \operatorname{BV}(Q_{T})$, then there exists a sequence $\{f_{n}\}\subset C^{\infty }(Q_{T})$ such that

$$\begin{aligned}& \lim_{n\rightarrow \infty } \iint _{Q_{T}} \vert f_{n}-f \vert \,dx\,dt=0, \\& \lim_{n\rightarrow \infty } \iint _{Q_{T}} \vert \nabla f_{n} \vert \,dx \,dt= \iint _{Q_{T}} \vert \nabla f \vert . \end{aligned}$$

So, we can define the trace of the functions in a BV space as in a Sobolev space i.e. the trace of $f(x)\in \operatorname{BV}(Q_{T})$ on the boundary ∂Ω is defined as the limit of a sequence $f_{n}(x)$ as follows:

$$ f(x)|_{x\in \partial \Omega }=\lim_{n\rightarrow \infty } f_{n}(x)|_{x \in \partial \Omega }. $$

(11)

Then it is well known that the BV function space is the weakest space such that the trace of $u\in \operatorname{BV}(Q_{T})$ can be defined as (11) and the integration by parts can be used. Also, one can refer to [7] for the definition of the trace of $u\in \operatorname{BV}(Q_{T})$ on the boundary value in another way.

Definition 1

A function u is said to be the entropy solution of equation (1) with the initial value condition (3) and with the boundary value condition (9) if

1. u satisfies

$$ u\in \operatorname{BV}(Q_{T})\cap L^{\infty }(Q_{T}), \; \frac{\partial }{\partial x_{i}} \int _{0}^{u}\sqrt{a(s)}\,ds\in L^{2}(Q_{T}). $$

2. For any $\varphi \in C_{0}^{2}(Q_{T})$, $\varphi \geq 0$, for any $k\in \mathbb{R}$, for any small $\eta >0$, u satisfies

$$ \begin{aligned}[b] & \iint _{Q_{T}} \Biggl[I_{\eta }(u-k)\varphi _{t}- \sum_{i=1}^{N} b_{i}(x,t)I_{\eta }(u-k)\varphi _{x_{i}}+A_{\eta }(u,k) \Delta \varphi \Biggr]\,dx\,dt \\ &\qquad {}- \iint _{Q_{T}}S_{\eta }^{\prime }(u-k) \biggl\vert \nabla \int _{0}^{u} \sqrt{a(s)}\,ds \biggr\vert ^{2}\varphi \,dx\,dt \\ &\qquad {}- \iint _{Q_{T}} \Biggl[ \Biggl(c(x,t)+\sum _{i=1}^{N}b_{ix_{i}}(x,t) \Biggr)u+f(x,t) \Biggr]\varphi S_{\eta }(u-k)\,dx\,dt \\ &\quad \geq 0. \end{aligned} $$

(12)

3. The partial homogeneous boundary value condition (9) is true in the sense of trace.

4. The initial value condition (3) is true in the sense that

$$ \underset{t\rightarrow 0}{\lim } \int _{\Omega } \bigl\vert u(x,t)-u_{0}(x) \bigr\vert \,dx=0. $$

(13)

If $a(0)=0$, $b_{i}(x,t)\in C^{1}(\overline{Q}_{T})$, $c(x,t)$ and $f(x,t)$ are bounded functions, the existence of the entropy solution in the sense of Definition 1 can be proved by a similar way as that in [21, 26], we omit the details here.

In this paper, we study the stability of the entropy solutions of equation (1) without condition (8). In order to display the method used in our paper, the unite n-dimensional cube

$$ D_{1}= \bigl\{ (x_{1}, x_{2}, \ldots , x_{N}): 0< x_{i}< 1, i=1, 2, \ldots , N \bigr\} $$

is considered firstly. By choosing special test functions, we can prove the following theorem.

Theorem 2

Let $u(x,t)$, $v(x,t)$ be solutions of equation (1) with the initial values $u_{0}(x)$, $v_{0}(x)\in L^{\infty }(D_{1})$, respectively, and with the same partial boundary value condition

$$ u(x,t)=v(x,t)=0,\quad (x,t)\in \Sigma _{p}. $$

(14)

Then

$$ \int _{D_{1}} \bigl\vert u(x,t)-v(x,t) \bigr\vert \,dx\leq \int _{D_{1}} \vert u_{0}-v_{0} \vert \,dx. $$

(15)

Here, if

$$ b_{i}(x,t)\geq 0, \quad i=1,2,\ldots , N, $$

(16)

then $\Sigma _{p}=\emptyset$. While if

$$ b_{i}(x,t)\leq 0, \quad i=1,2,\ldots , N, $$

(17)

is true, then

$$ \Sigma _{p}= \Biggl\{ (x,t)\in \partial D_{1}\times (0,T): \sum_{i=1}^{N}b_{i}(x,t)< 0 \Biggr\} . $$

(18)

Secondly, we generalize Theorem 2 to a general bounded domain Ω.

The main result of this paper is the following stability theorem.

Theorem 3

Suppose that Ω is a bounded smooth domain in $\mathbb{R}^{N}$, and when x is near to the boundary ∂Ω, the distance function $\rho (x)$ is a $C^{2}$ function, $A(s)$ is a Lipschitz function, and $a(0)=0$. Let $u(x,t)$ and $v(x,t)$ be the solutions of equation (1) with the initial values $u_{0}(x)\in L^{\infty }(\Omega )$ and $v_{0}(x)\in L^{\infty }(\Omega )$, respectively, and with the same partial boundary value condition

$$ \gamma u=\gamma v=0,\quad (x,t)\in \Sigma _{p}\subset \partial \Omega \times (0,T). $$

Then we have

$$ \int _{\Omega } \bigl\vert u(x,t)-v(x,t) \bigr\vert \,dx\leq \int _{\Omega } \bigl\vert u_{0}(x)-v_{0}(x) \bigr\vert \,dx. $$

(19)

Here,

$$ \Sigma _{p}= \Biggl\{ (x,t)\in \partial \Omega \times (0,T): \sum_{i=1}^{N}b_{i}(x,t)n_{i}(x)< 0 \Biggr\} , $$

(20)

where $\vec{n}=\{n_{i}\}$ $(i=1,2,\ldots,N)$ is the outer normal vector of Ω.

We give a simple comment. For a linear degenerate parabolic equation

$$ \frac{\partial u}{\partial t}-\operatorname{div} \bigl(a(x)\nabla u \bigr)- \sum_{i=1}^{N}b_{i}(x)D_{i}u+c(x,t)u=f(x,t), $$

(21)

where $a(x)$, $b(x)$, $c(x,t)$, and $f(x,t)$ are continuous functions, if

$$ a(x)=0,\quad x\in \partial \Omega ;\qquad a(x)>0, \quad x\in \Omega , $$

which implies equation (21) is only degenerate on the boundary ∂Ω, according to the Fichera–Oleinik theory [12], the optimal boundary value condition matching up with equation (21) is

$$ u(x,t)=0,\quad (x,t)\in \Sigma _{1}\times [0,T) $$

with

$$ \Sigma _{1}= \Biggl\{ x\in \partial \Omega : \sum _{i=1}^{N}b_{i}(x)n_{i}(x)< 0 \Biggr\} . $$

(22)

For the nonlinear degenerate parabolic equation (1), the most important characteristic lies in that the set $\{s\in \mathbb{R}: a(s)=0\}$ may have interior points, and so it is a strongly degenerate parabolic equation. In addition, when the Dirichlet boundary value condition (4) is imposed, $a(0)=0$ exactly implies that equation (1) is degenerate on the boundary $\Sigma _{T}=\partial \Omega \times (0,T)$. On the other hand, when a partial boundary value condition (9) is imposed, we only know that (1) is degenerate on the boundary $\Sigma _{p}$, while on $\Sigma _{T}\setminus \Sigma _{p}$, whether equation (1) is degenerate or not is uncertain. To the best knowledge of the author, this is the first paper to study the stability of entropy solutions to equation (1) when the partial boundary value condition is imposed on a submanifold $\Sigma _{p}\subset \Sigma _{T}$.

3 An important inequality

In this section, we use the Kruzkov bi-variables method to deduce an important inequality. Such a method was used in [18, 21, 26] and many other references. We begin with some basic denotations. For $u\in \operatorname{BV}(Q_{T})$, we denote by that $\Gamma _{u}$ is the set of all jump points, ν is the normal of $\Gamma _{u}$ at $X=(x,t)$, $u^{+}(X)$ and $u^{-}(X)$ are the approximate limits of u at $X\in \Gamma _{u}$ with respect to $(\nu ,Y-X)>0$ and $(\nu ,Y-X)<0$, respectively. For a continuous function $f(u)$, the composite mean value of f is defined as

$$ \widehat{f}(u)= \int _{0}^{1}f \bigl(\tau u^{+}+(1- \tau )u^{-} \bigr)\,d\tau . $$

When $f(s)\in C^{1}( \mathbb{R})$, $u\in BV(Q_{T})$, by [17, 18], we know $f(u)\in \operatorname{BV}(Q_{T})$ and

$$ \frac{\partial f(u)}{\partial x_{i}}=\widehat{f^{\prime }}(u) \frac{\partial u}{\partial x_{i}}, \quad i=1, 2, \ldots , N, N+1, $$

where $x_{N+1}=t$.

Just as that in [23, 26], we have the following lemma.

Lemma 4

Let u be an entropy solution of (1). Then

$$ a(s)=0,\quad s\in I \bigl(u^{+}(x,t),u^{-}(x,t) \bigr)\quad \textit{a.e. on } \Gamma ^{u}, $$

(23)

where $I(\alpha ,\beta )$ denotes the closed interval with endpoints α and β, and (23) is in the sense of Hausdorff measure $H_{N}(\Gamma ^{u})$.

Let $u(x,t)$, $v(x,t)$ be two entropy solutions of equation (1.1) with the initial value conditions

$$ u(x,0)=u_{0}(x) $$

and

$$ v(x,0)=v_{0}(x), $$

respectively.

From Definition 1, for any $\varphi \in C_{0}^{2}(Q_{T})$, we have

$$\begin{aligned}& \begin{aligned}[b] & \iint _{Q_{T}} \Biggl[I_{\eta }(u-v)\varphi _{t}- \sum_{i=1}^{N} b_{i}(x,t)I_{\eta }(u-v)\varphi _{x_{i}}+A_{\eta }(u,v) \Delta \varphi \Biggr]\,dx\,dt \\ &\qquad {}- \iint _{Q_{T}}S_{\eta }^{\prime }(u-v) \biggl\vert \nabla \int _{0}^{u} \sqrt{a(s)}\,ds \biggr\vert ^{2}\varphi \,dx\,dt \\ &\qquad {}- \iint _{Q_{T}} \Biggl[ \Biggl(c(x,t)+\sum _{i=1}^{N}b_{ix_{i}}(x,t) \Biggr)u+f(x,t) \Biggr]\varphi S_{\eta }(u-v)\,dx\,dt \\ &\quad \geq 0. \end{aligned} \end{aligned}$$

(24)

$$\begin{aligned}& \begin{aligned}[b] & \iint _{Q_{T}} \Biggl[I_{\eta }(v-u)\varphi _{ \tau }-\sum_{i=1}^{N} b_{i}(y,\tau )I_{\eta }(v-u)\varphi _{y_{i}}+A_{ \eta }(v,u) \Delta \varphi \Biggr]\,dy\,d\tau \\ &\qquad {}- \iint _{Q_{T}}S_{\eta }^{\prime }(v-u) \biggl\vert \nabla \int _{0}^{v} \sqrt{a(s)}\,ds \biggr\vert ^{2}\varphi \,dy\,d\tau \\ &\qquad {}- \iint _{Q_{T}} \Biggl[ \Biggl(c(y,\tau )+\sum _{i=1}^{N}b_{iy_{i}}(y, \tau ) \Biggr)v+f(y,\tau ) \Biggr]\varphi S_{\eta }(v-u)\,dy\,d\tau \\ &\quad \geq 0. \end{aligned} \end{aligned}$$

(25)

Let $\omega _{h}$ be the mollifier which is defined as

$$ \omega _{h}(s)=\frac{1}{h}\omega \biggl( \frac{s}{h} \biggr),\omega (s) \in C_{0}^{\infty }( \mathbb{R}), \omega (s)\geq 0, \omega (s)=0 \quad \text{if } \vert s \vert >1, \int _{-\infty }^{\infty }\omega (s)\,ds=1. $$

Define $\psi (x,t,y,\tau )=\phi (x,t)j_{h}(x-y,t-\tau )$, where $\phi (x,t)\geq 0$, $\phi (x,t)\in C_{0}^{\infty }(Q_{T})$, and

$$ j_{h}(x-y,t-\tau )=\omega _{h}(t-\tau )\prod _{i=1}^{N}\omega _{h}(x_{i}-y_{i}). $$

Now, we choose $\varphi =\psi$ in (24) and (25), then we have

$$\begin{aligned} & \iint _{Q_{T}} \iint _{Q_{T}} \bigl[I_{ \eta }(u-v) (\psi _{t}+\psi _{\tau })+A_{\eta }(u,v)\Delta _{x}\psi +A_{ \eta }(v,u)\Delta _{y}\psi \bigr]\,dx\,dt\,dy\,d\tau \\ &\qquad {}-\sum_{i=1}^{N} \iint _{Q_{T}} \iint _{Q_{T}} \bigl(b_{i}(x,t) \psi _{x_{i}}+b_{i}(y,\tau )\psi _{y_{i}} \bigr)I_{\eta }(u-v)\,dx\,dt\,dy\,d\tau \\ &\qquad {}- \iint _{Q_{T}} \iint _{Q_{T}}S_{\eta }^{\prime }(u-v) \biggl( \biggl\vert \nabla \int _{0}^{u}\sqrt{a(s)}\,ds \biggr\vert ^{2}+ \biggl\vert \nabla \int _{0}^{v}\sqrt{a(s)}\,ds \biggr\vert ^{2} \biggr)\psi \,dx\,dt\,dy\,d \tau \\ &\qquad {}- \iint _{Q_{T}} \Biggl[ \Biggl(c(x,t)+\sum _{i=1}^{N}b_{ix_{i}}(x,t) \Biggr)u+f(x,t) \Biggr]\psi S_{\eta }(u-v)\,dx\,dt \\ &\qquad {}- \iint _{Q_{T}} \iint _{Q_{T}} \Biggl[ \Biggl(c(y,\tau )+ \sum _{i=1}^{N}b_{iy_{i}}(y,\tau ) \Biggr)v+f(y, \tau ) \Biggr]\psi S_{\eta }(v-u)\,dx\,dt \\ &\quad \geq 0. \end{aligned}$$

(26)

Since

$$\begin{aligned}& \frac{\partial j_{h}}{\partial t}+ \frac{\partial j_{h}}{\partial \tau }=0,\qquad \frac{\partial j_{h}}{\partial x_{i}}+ \frac{\partial j_{h}}{\partial y_{i}}=0,\quad i=1,\ldots ,N; \\& \frac{\partial \psi }{\partial t}+\frac{\partial \psi }{\partial \tau }= \frac{\partial \phi }{\partial t}j_{h}, \qquad \frac{\partial \psi }{\partial x_{i}}+ \frac{\partial \psi }{\partial y_{i}}= \frac{\partial \phi }{\partial x_{i}}j_{h}, \end{aligned}$$

we have

$$ \begin{aligned}[b] &\lim_{h\rightarrow 0}\lim _{\eta \rightarrow 0} \iint _{Q_{T}} \iint _{Q_{T}} \bigl(b_{i}(x,t)\psi _{x_{i}}+b_{i}(y,\tau )\psi _{y_{i}} \bigr)I_{ \eta }(u-v)\,dx\,dt\,dy\,d\tau \\ &\quad =\lim_{h\rightarrow 0} \iint _{Q_{T}} \iint _{Q_{T}} \bigl(b_{i}(x,t)+b_{i}(y, \tau ) \bigr) \vert u-v \vert \phi _{x_{i}}j_{h}\,dx\,dt\,dy\,d \tau \\ &\quad =2 \iint _{Q_{T}}b_{i}(x,t) \vert u-v \vert \phi _{x_{i}}\,dx\,dt. \end{aligned} $$

(27)

Meanwhile, we have

$$ \begin{aligned}[b] & \iint _{Q_{T}} \iint _{Q_{T}} \bigl[A_{\eta }(u,v)\Delta _{x} \psi +A_{\eta }(v,u)\Delta _{y}\psi \bigr]\,dx\,dt\,dy\,d\tau \\ &\quad = \iint _{Q_{T}} \iint _{Q_{T}} \bigl\{ A_{\eta }(u,v) (\Delta _{x}\phi j_{h}+2 \phi _{x_{i}}j_{hx_{i}}+ \phi \Delta j_{h})+A_{\eta }(v,u)\phi \Delta _{y}j_{h} \bigr\} \,dx\,dt\,dy\,d\tau \\ &\quad = \iint _{Q_{T}} \iint _{Q_{T}} \bigl\{ A_{\eta }(u,v)\Delta _{x}\phi j_{h}+A_{ \eta }(u,v)\phi _{x_{i}}j_{hx_{i}}+A_{\eta }(v,u)\phi _{x_{i}}j_{hy_{i}} \bigr\} \,dx\,dt\,dy\,d\tau \\ &\qquad {}- \iint _{Q_{T}} \iint _{Q_{T}} \biggl\{ \widehat{a(u)S_{\eta }(u-v)} \frac{\partial u}{ \partial x_{i}}\\ &\qquad {}-\widehat{ \int _{u}^{v}a(s)S_{\eta }^{\prime }(s-v)\,ds} \frac{ \partial u}{\partial x_{i}})\phi j_{hx_{i}} \biggr\} \,dx\,dt\,dy\,d\tau , \end{aligned} $$

(28)

where

$$\begin{aligned}& \widehat{a(u)S_{\eta }(u-v)}= \int _{0}^{1}a \bigl(su^{+}+(1-s)u^{-} \bigr)S_{ \eta } \bigl(su^{+}+(1-s)u^{-}-v \bigr)\,ds, \\& \int _{u}^{v}\widehat{a(s)S_{\eta }^{\prime }(s-v)}\,ds= \int _{0}^{1} \int _{su^{+}+(1-s)u^{-}}^{v}a(\sigma )S_{\eta } \bigl( \sigma -su^{+}-(1-s)u^{-} \bigr)d \sigma \,ds. \end{aligned}$$

By that

$$ \begin{aligned}[b] & \iint _{Q_{T}} \iint _{Q_{T}}S_{\eta }^{\prime }(u-v) \biggl( \biggl\vert \nabla _{x} \int _{0}^{u}\sqrt{a(s)}\,ds \biggr\vert ^{2}+ \biggl\vert \nabla _{y} \int _{0}^{v}\sqrt{a(s)}\,ds \biggr\vert ^{2} \biggr)\psi \,dx\,dt\,dy\,d \tau \\ &\quad = \iint _{Q_{T}} \iint _{Q_{T}}S_{\eta }^{\prime }(u-v) \biggl( \biggl\vert \nabla _{x} \int _{0}^{u}\sqrt{a(s)}\,ds \biggr\vert - \biggl\vert \nabla _{y} \int _{0}^{v} \sqrt{a(s)}\,ds \biggr\vert \biggr)^{2}\psi \,dx\,dt\,dy\,d\tau \\ &\qquad {}+2 \iint _{Q_{T}} \iint _{Q_{T}}S_{\eta }^{\prime }(u-v)\nabla _{x} \int _{0}^{u} \sqrt{a(s)}\,ds\cdot \nabla _{y} \int _{0}^{v}\sqrt{a(s)}\,ds\psi \,dx\,dt\,dy\,d \tau , \end{aligned} $$

(29)

and using Lemma 4, we obtain the facts

$$ \begin{aligned}[b] & \iint _{Q_{T}} \iint _{Q_{T}}\nabla _{x}\nabla _{y} \int _{v}^{u} \sqrt{a(\delta )} \int _{\delta }^{v}\sqrt{a(\sigma )}S_{\eta }^{\prime }( \sigma -\delta )\,d\sigma \,d\delta \psi \,dx\,dt\,dy\,d\tau \\ &\quad = \iint _{Q_{T}} \iint _{Q_{T}} \int _{0}^{1} \int _{0}^{1}\sqrt{ a \bigl(su^{+}+(1-s)u^{-} \bigr)} \sqrt{a(\sigma v^{+}+(1-\sigma )v^{-}} \\ &\qquad {}\times S_{\eta }^{\prime } \bigl[\sigma v^{+}+(1- \sigma )v^{-}-su^{+}-(1-s)u^{-} \bigr]\,ds\,d \sigma \nabla _{x}u\nabla _{y}v\,dx\,dt\,dy\,d\tau \\ &\quad = \iint _{Q_{T}} \iint _{Q_{T}} \int _{0}^{1} \int _{0}^{1}S_{\eta }^{ \prime } \bigl[\sigma v^{+}+(1-\sigma )v^{-}-su^{+}-(1-s)u^{-} \bigr]\,ds\,d\sigma \\ &\qquad {}\times \widehat{\sqrt{a(u)}}\nabla _{x}u\widehat{\sqrt{a(v)}} \nabla _{y}v\,dx\,dt\,dy\,d \tau \\ &\quad = \iint _{Q_{T}} \iint _{Q_{T}} \int _{0}^{1} \int _{0}^{1}S_{\eta }^{ \prime }(v-u) \nabla _{x} \int _{0}^{u}\sqrt{a(s)}\,ds\nabla _{y} \int _{0}^{v} \sqrt{a(s)}\,ds\,dx\,dt\,dy\,d\tau \end{aligned} $$

(30)

and

$$ \begin{aligned}[b] & \iint _{Q_{T}} \iint _{Q_{T}}\nabla _{x}\nabla _{y} \int _{v}^{u} \sqrt{a(\delta )} \int _{\delta }^{v}\sqrt{a(\sigma )}S_{\eta }^{\prime }( \sigma -\delta )\,d\sigma \,d\delta \psi \,dx\,dt\,dy\,d\tau \\ &\quad = \iint _{Q_{T}} \iint _{Q_{T}} \int _{0}^{1}\sqrt{a \bigl(su^{+}+(1-s)u^{-} \bigr)} \\ &\qquad {}\times \int _{su^{+}+(1-s)u^{-}}^{v}\sqrt{a(\sigma )}S_{\eta }^{ \prime } \bigl(\sigma -su^{+}-(1-s)u^{-} \bigr)\,d\sigma \,ds \frac{\partial u}{\partial x_{i}}j_{hx_{i}}\phi \,dx\,dt\,dy\,d\tau . \end{aligned} $$

(31)

Then we have

$$\begin{aligned} & \iint _{Q_{T}} \iint _{Q_{T}} \biggl( \widehat{a(u)S_{\eta }(u-v)} \frac{\partial u}{\partial x_{i}}- \widehat{ \int _{u}^{v}a(s)S_{\eta }^{\prime }(s-u)\,ds} \frac{\partial u}{\partial x_{i}} \biggr)j_{hx_{i}}\phi \,dx\,dt\,dy\,d\tau \\ &\qquad {}+2 \iint _{Q_{T}} \iint _{Q_{T}}S_{\eta }^{\prime }(u-v)\nabla _{x} \int _{0}^{u} \sqrt{a(s)}\,ds\cdot \nabla _{y} \int _{0}^{v}\sqrt{a(s)}\,ds\psi \,dx\,dt\,dy\,d \tau \\ &\quad = \iint _{Q_{T}} \iint _{Q_{T}} \biggl[ \int _{0}^{1}a \bigl(su^{+}+(1-s)u^{-} \bigr)S_{ \eta } \bigl(su^{+}+(1-s)u^{-}-v \bigr)\,ds \\ &\qquad {}- \int _{0}^{1} \int _{su^{+}+(1-s)u^{-}}^{v}a(\sigma )S_{\eta }^{ \prime } \bigl(\sigma -su^{+}-(1-s)u^{-} \bigr)\,d\sigma \,ds \\ &\qquad {}+2 \int _{0}^{1}\sqrt{a \bigl(su^{+}+(1-s)u^{-} \bigr)} \int _{su^{+}+(1-s)u^{-}}^{v} \sqrt{a(\sigma )}S_{\eta }^{\prime } \bigl(\sigma -su^{+}-(1-s)u^{-} \bigr)\,d \sigma \,ds \biggr] \\ &\qquad {}\times \frac{\partial u}{\partial x_{i}}j_{hx_{i}}\phi \,dx\,dt\,dy\,d\tau \\ &\quad =- \iint _{Q_{T}} \iint _{Q_{T}} \int _{0}^{1} \int _{su^{+}+(1-s)u^{-}}^{v} \bigl[ \sqrt{a(\sigma )}-\sqrt{a \bigl(su^{+}+(1-s)u^{-} \bigr)} \bigr]^{2} \\ &\qquad {}\times S^{\prime }_{\eta } \bigl(\sigma -su^{+}-(1-s)u^{-} \bigr)\,d\sigma \,ds \frac{\partial u}{ \partial x_{i}}j_{hx_{i}}\phi \,dx\,dt\,dy\,d\tau \\ &\quad \rightarrow 0 \end{aligned}$$

(32)

as $\eta \rightarrow 0$.

Once more,

$$ \lim_{\eta \rightarrow 0}A_{\eta }(u,v)= \underset{\eta \rightarrow 0}{\lim }A_{\eta }(v,u)=\operatorname{sgn}(u-v) \bigl[A(u)-A(v) \bigr], $$

we have

$$ \lim_{\eta \rightarrow 0} \bigl[A_{\eta }(u,v) \phi _{x_{i}}j_{hx_{i}}+A_{ \eta }(u,v)\phi _{y_{i}}j_{hy_{i}} \bigr]=0. $$

(33)

Also, we clearly have

$$\begin{aligned} &-\lim_{h\rightarrow 0}\lim _{\eta \rightarrow 0} \iint _{Q_{T}} \iint _{Q_{T}} \Biggl[ \Biggl(c(x,t)+\sum _{i=1}^{N}b_{ix_{i}}(x,t) \Biggr)u+f(x,t) \Biggr]\psi S_{\eta }(u-v)\,dx\,dt\,dy\,d\tau \\ &\qquad {}-\lim_{h\rightarrow 0}\lim_{\eta \rightarrow 0} \iint _{Q_{T}} \iint _{Q_{T}} \Biggl[ \Biggl(c(y,\tau )+\sum _{i=1}^{N}b_{iy_{i}}(y, \tau ) \Biggr)v+f(y,\tau ) \Biggr]\psi S_{\eta }(v-u)\,dx\,dt\,dy\,d\tau \\ &\quad =-\lim_{h\rightarrow 0} \iint _{Q_{T}} \iint _{Q_{T}} \Biggl[ \Biggl(c(x,t)+\sum _{i=1}^{N}b_{ix_{i}}(x,t) \Biggr)u+f(x,t) \Biggr]\psi \operatorname{sgn}(u-v)\,dx\,dt\,dy\,d\tau \\ &\qquad {}-\lim_{h\rightarrow 0} \iint _{Q_{T}} \iint _{Q_{T}} \Biggl[ \Biggl(c(y,\tau )+\sum _{i=1}^{N}b_{iy_{i}}(y,\tau ) \Biggr)v+f(y, \tau ) \Biggr]\psi \operatorname{sgn}(v-u)\,dx\,dt\,dy\,d\tau \\ &\quad =- \iint _{Q_{T}} \Biggl(c(x,t)+\sum _{i=1}^{N}b_{ix_{i}}(x,t) \Biggr) \phi \vert u-v \vert \,dx\,dt. \end{aligned}$$

(34)

Letting $\eta \rightarrow 0$, $h\rightarrow 0$ in (26) and combining (27)–(34), we get

$$\begin{aligned} & \iint _{Q_{T}} \Biggl[ \bigl\vert u(x,t)-v(x,t) \bigr\vert \phi _{t}-2 \vert u-v \vert \sum_{i=1}^{N}b_{i}(x,t) \phi _{x_{i}} \\ &\qquad {}- \Biggl(c(x,t)+\sum_{i=1}^{N}b_{ix_{i}}(x,t) \Biggr)\phi \vert u-v \vert + \bigl\vert A(u)-A(v) \bigr\vert \Delta \phi \Biggr]\,dx\,dt \\ &\quad \geq 0. \end{aligned}$$

(35)

This is the most important inequality to prove Theorem 2 and Theorem 3.

4 Proof of Theorem 2

The proof of Theorem 2

Let

$$ d_{i}(x_{i})=\textstyle\begin{cases} x_{i}, & \text{if } 0\leq x_{i}\leq \frac{1}{2}, \\ 1-x_{i}, & \text{if } \frac{1}{2}< x_{i}\leq 1. \end{cases} $$

For small enough λ, we set

$$ \varphi _{i\lambda }(x_{i})=\textstyle\begin{cases} \sin \frac{1}{\lambda }(d_{i}(x_{i})), & \text{if } 0\leq d_{i}(x_{i}) \leq \frac{\pi \lambda }{2}, \\ 1, &\text{if } d_{i}(x)\geq \frac{\pi \lambda }{2}. \end{cases} $$

Let $0\leq \eta (t)\in C_{0}^{2}(t)$ and

$$ \phi (x,t)=\eta (t)\prod_{j=1}^{N} \varphi _{j\lambda }(x_{j}). $$

Then

$$\begin{aligned}& \begin{aligned}[b]\partial _{x_{i}}\phi (x,t)&= \eta (t)\partial _{x_{i}} \varphi _{i\lambda }(x_{i}) \prod_{j=1,j\neq i}^{N}\varphi _{j\lambda }(x_{j}) \\ &=\eta (t)\frac{1}{\lambda }\cos \frac{1}{\lambda } \bigl(d_{i}(x_{i}) \bigr)d_{ix_{i}}(x_{i}) \prod_{j=1,j\neq i}^{N} \varphi _{j\lambda }(x_{j}), \quad 0\leq d_{i}(x_{i}) \leq \frac{\pi \lambda }{2}. \end{aligned} \end{aligned}$$

(36)

$$\begin{aligned}& \begin{aligned}[b]\triangle \phi (x,t)&= \frac{1}{\lambda }\eta (t)\prod_{j=1,j \neq i}^{N} \varphi _{j\lambda }(x_{j}) \biggl[-\frac{1}{\lambda }\sin \frac{1}{\lambda } \bigl(d_{i}(x_{i}) \bigr)d_{ix_{i}}^{2}+\frac{1}{\lambda }\cos \frac{1}{\lambda } \bigl(d_{i}(x_{i}) \bigr)\triangle d_{i}(x_{i}) \biggr] \\ &=-\frac{1}{\lambda ^{2}}\eta (t)\prod_{j=1,j\neq i}^{N} \varphi _{j \lambda }(x_{j}) \sin \frac{1}{\lambda } \bigl(d_{i}(x_{i}) \bigr)d_{ix_{i}}^{2}, \quad 0\leq d_{i}(x_{i})\leq \frac{\pi \lambda }{2}. \end{aligned} \end{aligned}$$

(37)

When λ is small enough, $0\leq d_{i}(x_{i})\leq \frac{\pi \lambda }{2}<\frac{1}{2}$,

$$ d_{ix_{i}}(x_{i})=1. $$

Then (36) yields

$$ \begin{aligned}[b] &-2 \vert u-v \vert \sum _{i=1}^{N}b_{i}(x,t)\phi _{x_{i}}(x,t) \\ &\quad =-2 \vert u-v \vert \eta (t)\frac{1}{\lambda }\cos \frac{1}{\lambda } \sum_{i=1}^{N}d_{i}(x_{i})b_{i}(x,t) \prod_{j=1,j\neq i}^{N}\varphi _{j\lambda }(x_{j}). \end{aligned} $$

(38)

We now substitute these formulas into (35), if $b_{i}(x,t)\geq 0$, by (38), we have

$$ \begin{aligned}[b] & \iint _{Q_{T}} \Biggl[ \bigl\vert u(x,t)-v(x,t) \bigr\vert \phi _{t}- \Biggl(c(x,t)+ \sum_{i=1}^{N}b_{ix_{i}}(x,t) \Biggr)\phi \vert u-v \vert \Biggr]\,dx\,dt \\ &\quad \geq 0. \end{aligned} $$

(39)

Accordingly, we have

$$ \iint _{Q_{T}} \bigl[ \bigl\vert u(x,t)-v(x,t) \bigr\vert \eta _{t}+c \vert u-v \vert \bigr]\,dx\,dt \geq 0. $$

(40)

If $b_{i}(x,t)\leq 0$, we have

$$ \begin{aligned}[b] & \iint _{Q_{T}} \Biggl[ \bigl\vert u(x,t)-v(x,t) \bigr\vert \phi _{t}- \Biggl(c(x,t)+ \sum_{i=1}^{N}b_{ix_{i}}(x,t) \Biggr)\phi \vert u-v \vert \Biggr]\,dx\,dt \\ &\qquad {}-2 \int _{0}^{T} \int _{\Omega _{\lambda }}\eta (t)\frac{1}{\lambda } \sum _{i=1}^{N}b_{i}(x,t)| \vert u-v \vert \,dx\,dt \\ &\quad \geq 0, \end{aligned} $$

(41)

where $\Omega _{\lambda }=\{(x\in \Omega : d_{i}(x_{i})< \frac{\lambda \pi }{2}\}$. According to the definition of the trace of BV functions (see [7]), when $x\in \Sigma _{1t}= \{ \partial \Omega : \sum_{i=1}^{N}b_{i}(x,t)< 0 \}$, $u(x,t)=v(x,t)=0$, let $\lambda \rightarrow 0$ in (41). We also have (40).

Let $0< s<\tau <T$, and

$$ \eta (t)= \int _{\tau -t}^{s-t}\alpha _{\varepsilon }(\sigma )\,d \sigma , \quad \varepsilon < \min \{\tau , T-s\}, $$

where $\alpha _{\varepsilon }(t)$ is the kernel of mollifier with $\alpha _{\varepsilon }(t)=0$ for $t\notin (-\varepsilon ,\varepsilon )$. Then

$$ \int _{0}^{T} \bigl[\alpha _{\varepsilon }(t-s)- \alpha _{\varepsilon }(t- \tau ) \bigr] \vert u-v \vert _{L^{1}(\Omega )}\,dt+c \int _{\tau }^{s} \int _{\Omega } \vert u-v \vert \,dx\,dt \geq 0. $$

Let $\varepsilon \rightarrow 0$. Then

$$ \bigl\vert u(x,\tau )-v(x,\tau ) \bigr\vert _{L^{1}(\Omega )}\leq \bigl\vert u(x,s)-v(x,s) \bigr\vert _{L^{1}( \Omega )}+c \int _{\tau }^{s} \int _{\Omega } \vert u-v \vert \,dx\,dt. $$

Let $s \rightarrow 0$, then the desired result follows by Gronwall’s inequality. □

Thus, by the Kruzkov bi-variables method, we have proved Theorem 2.

5 Proof of Theorem 3

Proof of Theorem 3

Let $u(x,t)$ and $v(x,t)$ be two entropy solutions of equation (1) with the initial values

$$ u(x,0)=u_{0}(x) \quad \text{and}\quad v(x,0)=v_{0}(x). $$

Recalling (35), for any $\phi (x,t)\in C^{2}_{0}(Q_{T})$, we have

$$ \begin{aligned}[b] & \iint _{Q_{T}} \Biggl[ \bigl\vert u(x,t)-v(x,t) \bigr\vert \phi _{t}-2 \vert u-v \vert \sum_{i=1}^{N}b_{i}(x,t) \phi _{x_{i}} \\ &\qquad {}- \Biggl(c(x,t)+\sum_{i=1}^{N}b_{ix_{i}}(x,t) \Biggr)\phi \vert u-v \vert + \bigl\vert A(u)-A(v) \bigr\vert \Delta \phi \Biggr]\,dx\,dt \\ &\quad \geq 0. \end{aligned} $$

(42)

Since when x is near to the boundary ∂Ω, the distance function $\rho (x)$ is a $C^{2}$ function, we can define

$$ \phi (x,t)=\eta (t)\varphi _{\lambda }(x), $$

(43)

where $0\leq \eta (t)\in C_{0}^{2}(t)$ and

$$ \varphi _{\lambda }(x)=\textstyle\begin{cases} \sin \frac{\rho (x)}{\lambda }, &\text{if } 0\leq \rho (x)< \frac{\pi \lambda }{2}, \\ 1, &\text{if } \rho (x)\geq \frac{\pi \lambda }{2}. \end{cases} $$

Then we have

$$\begin{aligned} \partial _{x_{i}}\phi (x,t) =&\eta (t)\partial _{x_{i}}\varphi _{ \lambda }(x) \\ =&\eta (t)\frac{1}{\lambda }\cos \frac{\rho (x)}{\lambda }\rho _{x_{i}}(x) \end{aligned}$$

(44)

and

$$\begin{aligned} \triangle \phi (x,t) =& \frac{1}{\lambda }\eta (t) \Biggl[- \frac{1}{\lambda }\sin \frac{\rho (x)}{\lambda }\sum_{i=1}^{N} \rho _{x_{i}}^{2}+ \cos \frac{\rho (x)}{\lambda }\triangle \rho (x) \Biggr] \\ =&-\frac{1}{\lambda ^{2}}\eta (t) \sin \frac{\rho (x)}{\lambda }\sum _{i=1}^{N} \rho _{x_{i}}^{2}+ \frac{1}{\lambda }\eta (t)\cos \frac{\rho (x)}{\lambda }\triangle \rho (x), \end{aligned}$$

(45)

where $0\leq \rho (x)< \frac{\pi \lambda }{2}$ for small λ.

We define $\Omega _{\lambda }= \{ x\in \Omega : \rho (x)< \frac{\pi \lambda }{2} \}$. Since

$$\begin{aligned}& \lim_{\lambda \rightarrow 0} \int _{0}^{\frac{\lambda \pi }{2}} \frac{1}{\lambda ^{2}} \sin \frac{\rho (x)}{\lambda }\,d\rho =\infty , \\& \lim_{\lambda \rightarrow 0} \int _{0}^{\frac{\lambda \pi }{2}} \frac{1}{\lambda }\cos \frac{\rho (x)}{\lambda } \vert \triangle \rho \vert \,d\rho < \infty , \end{aligned}$$

we have

$$\begin{aligned}& \int _{\Omega } \bigl\vert A(u)-A(v) \bigr\vert \Delta \phi \,dx \\& \quad = \int _{\Omega _{\lambda }} \biggl[\frac{1}{\lambda }\eta (t)\cos \frac{\rho (x)}{\lambda }\triangle \rho -\frac{1}{\lambda ^{2}}\eta (t) \sin \frac{\rho (x)}{\lambda } \biggr] \bigl\vert A(u)-A(v) \bigr\vert \,dx \\& \quad \leq \int _{\Omega _{\lambda }} \biggl[\frac{1}{\lambda }\eta (t) \cos \frac{\rho (x)}{\lambda } \vert \triangle \rho \vert -\frac{1}{\lambda ^{2}} \eta (t) \sin \frac{\rho (x)}{\lambda } \bigl\vert A(u)-A(v) \bigr\vert \biggr]\,dx \\& \quad < 0. \end{aligned}$$

(46)

Notice that

$$ 2 \vert u-v \vert \sum_{i=1}^{N}b_{i}(x,t) \leq c_{i} \vert u-v \vert . $$

We denote

$$ \Omega _{\lambda 1}= \Biggl\{ x\in \Omega _{\lambda }:-\sum _{i=1}^{N}b_{i}(x,t) \rho _{x_{i}}>0 \Biggr\} $$

and derive that

$$\begin{aligned}& - 2\sum_{i=1}^{N} \lim_{\lambda \rightarrow 0} \int _{\Omega _{ \lambda }} \vert u-v \vert \sum _{i=1}^{N}b_{i}(x,t)\phi _{x_{i}}\,dx \\& \quad =-2\sum_{i=1}^{N}\lim _{\lambda \rightarrow 0} \int _{\Omega _{ \lambda }}\eta (t) \vert u-v \vert \sum _{i=1}^{N}b_{i}(x,t) \frac{1}{\lambda }\cos \frac{\rho (x)}{\lambda }\rho _{x_{i}}\,dx \\& \quad \leq 2\eta (t)\lim_{\lambda \rightarrow 0}\frac{1}{\lambda } \int _{ \Omega _{\lambda }} \vert u-v \vert \Biggl(-\sum _{i=1}^{N}b_{i}(x,t)\rho _{x_{i}} \Biggr)\,dx \\& \quad \leq 2\eta (t)\lim_{\lambda \rightarrow 0}\frac{1}{\lambda } \int _{ \Omega _{\lambda 1}} \vert u-v \vert \Biggl(-\sum _{i=1}^{N}b_{i}(x,t)\rho _{x_{i}} \Biggr)\,dx \\& \quad =-2 \int _{\Sigma _{p}} \vert u-v \vert \sum _{i=1}^{N}b_{i}(x,t)n_{i}(x) \,d\Sigma \\& \quad =0, \end{aligned}$$

(47)

where $\vec{n}=\{n_{i}\}$ ($i=1,2,\ldots,N$) is the outer normal vector of Ω.

At the same time, we have

$$ \begin{aligned}[b] &- \iint _{Q_{T}} \Biggl(c(x,t)+\sum_{i=1}^{N}b_{ix_{i}}(x,t) \Biggr) \phi \vert u-v \vert + \bigl\vert A(u)-A(v) \bigr\vert \Delta \phi ]\,dx\,dt \\ &\quad \leq- \iint _{Q_{T}} \Biggl(c(x,t)+\sum_{i=1}^{N}b_{ix_{i}}(x,t) \Biggr)\phi \vert u-v \vert \,dx\,dt \\ &\quad \leq c \iint _{Q_{T}}\phi \vert u-v \vert \,dx\,dt. \end{aligned} $$

(48)

Processing in an analogous manner as we did in the discussion of (40), letting $\lambda \rightarrow 0$ in (42), we arrive at the desired result. □

Thus, we have proved Theorem 3 by the Kruzkov bi-variables method. One can see that the partial boundary value condition is imposed on a submanifold $\Sigma _{p}\subset \partial \Omega \times (0,T)$. Such a conclusion reflects how the time variable t affects the well-posedness problem of a degenerate parabolic equation. By the way, the assumption that, when x is near to the boundary ∂Ω, the distance function $\rho (x)$ is a $C^{2}$ function can be weakened as follows: there is a subdomain $\Omega _{\lambda }=\{x\in \Omega : \rho (x)<\lambda \}$, $\rho (x)$ is an almost everywhere $C^{2}$ function on $\Omega _{\lambda }$, and $\int _{\Omega _{\lambda }}|\Delta \rho |\,dx\leq c$.