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On the stability of the equation with a partial boundary value condition

Boundary Value Problems20182018:30

https://doi.org/10.1186/s13661-018-0953-7

Received: 11 September 2017

Accepted: 7 March 2018

Published: 12 March 2018

Abstract

The degenerate parabolic equation with a convection term is considered. Let Ω be a bounded domain with \(C^{2}\) smooth boundary and \(d(x)=\operatorname{dist}(x, \partial\Omega)\) be the distance function from the boundary. If \(\Delta d\leq0\) when x is near to the boundary, then the stability of the entropy solutions is proved independent of the boundary value conditions. The degeneracy of the convection term on the boundary can take place of the usual boundary value condition.

Keywords

Degenerate parabolic equationEntropy solutionKruzkov’s bi-variables methodStability

MSC

35L6535L8535R35

1 Introduction

The degenerate parabolic equation
$$ \frac{\partial u}{\partial t} =\Delta A(u) +\operatorname{div}\bigl(b(u,x,t)\bigr), \quad(x,t)\in\Omega\times (0,T), $$
(1.1)
comes from many reaction–diffusion problems [1]. It has been widely researched for a long time, the first well-known paper goes back to the work [2] by Vol’pert and Hudjaev in 1967. Let
$$ A(u)= \int_{0}^{u}a(s)\,ds,\quad a(s)\geq0, a(0)=0. $$
(1.2)
Then the degeneracy of \(a(s)\) may lead to the equation with the hyperbolic characteristic, and the uniqueness of the usual weak solution is not true. In other words, the usual weak solution (for example, the measured value solution) is so weak that it lacks the regularity to ensure the stability or the uniqueness. The entropy condition is considered in this context. Till now, this has been one of the most well-known conditions in the degenerate parabolic equation theory. In the sense of the entropy solution, there are a lot of important papers devoted to equation (1.1), one can refer to [315] and the references therein. Based on the papers, we can conclude that the initial value condition
$$ u(x,0)=u_{0}(x),\quad x\in\Omega, $$
(1.3)
is always indispensable, while the usual Dirichlet boundary value condition
$$ u(x,t)=0,\quad (x,t)\in\partial\Omega\times(0,T), $$
(1.4)
might be overdetermined. Instead of (1.4), a partial boundary value condition
$$ u(x,t)=0,\quad (x,t)\in\Sigma_{1}\times(0,T), $$
(1.5)
should be imposed, where \(\Sigma_{1}\) is a relative open subset of Ω. If
$$ b_{i}(u,x,t)\equiv b_{i}(u), $$
(1.6)
the explicit formula of \(\Sigma_{1}\) was studied in our previous works [1618].
For small \(\eta>0\), let
$$S_{\eta}(s)= \int_{0}^{s}h_{\eta}(\tau)\,d\tau,\quad h_{\eta}(s)=\frac{2}{\eta}\biggl(1-\frac{ \vert s \vert }{\eta } \biggr)_{+}. $$
Then \(h_{\eta}(s)\in C(\mathbb{R})\), and
$$ \begin{gathered} h_{\eta}(s)\geq0,\qquad \bigl\vert sh_{\eta }(s) \bigr\vert \leq1, \qquad \bigl\vert S_{\eta}(s) \bigr\vert \leq 1;\\ \lim _{\eta\rightarrow0}S_{\eta}(s)=\operatorname{sgn}s,\qquad \lim _{\eta\rightarrow0}sS_{\eta}^{\prime}(s)=0. \end{gathered} $$
(1.7)
The definition of the entropy solution of equation (1.1) is given as follows.

Definition 1.1

A function u is said to be the entropy solution of equation (1.1) with the initial value condition (1.3) if
  1. 1.
    u satisfies
    $$ u\in \operatorname{BV}(Q_{T})\cap L^{\infty}(Q_{T}),\qquad \frac{\partial}{\partial x_{i}} \int_{0}^{u}\sqrt{a(s)}\,ds\in L^{2}(Q_{T}). $$
     
  2. 2.
    For any \(\varphi\in C_{0}^{2}(Q_{T})\), \(\varphi\geq0\), for any \(k\in \mathbb{R}\), for any small \(\eta >0\), u satisfies
    $$ \begin{gathered}[b] \iint_{Q_{T}} \Biggl[I_{\eta}(u-k)\varphi_{t}- \sum_{i=1}^{N}B_{\eta }^{i}(u,x,t,k) \varphi_{x_{i}}+A_{\eta}(u,k)\Delta\varphi\\ \qquad {} -S_{\eta}^{\prime}(u-k) \biggl\vert \nabla \int_{0}^{u}\sqrt {a(s)}\,ds \biggr\vert ^{2}\varphi \Biggr]\,dx \,dt \\ \qquad{}-\sum_{i=1}^{N} \iint_{Q_{T}} \int_{k}^{u}b_{ix_{i}}S'_{\eta }(s-k) \,ds\varphi \,dx \,dt \\ \quad \geq0. \end{gathered} $$
    (1.8)
     
  3. 3.
    Condition (1.3) is true in the sense that
    $$ \lim_{t\rightarrow0} \int_{\Omega} \bigl\vert u(x,t)-u_{0}(x) \bigr\vert \,dx=0,\quad\text{a.e. } x\in\Omega. $$
    (1.9)
     
Here,
$$\begin{aligned}& B_{\eta}^{i}(u,x,t,k) = \int_{k}^{u}\frac{\partial b_{i}(s,x,t)}{\partial s}S_{\eta} (s-k) \,ds, \\& A_{\eta}(u,k) = \int_{k}^{u}a(s)S_{\eta}(s-k)\,ds, \end{aligned}$$
and
$$I_{\eta}(u-k)= \int_{0}^{u-k}S_{\eta}(s)\,ds. $$

We would like to suggest that these kinds of entropy solutions were introduced by the author in [14, 15]. For any small \(\eta>0\), any \(k\in\mathbb{R}\), by multiplying with \(\varphi S_{\eta} (u-k)\) in equation (1.1), we can obtain the entropy inequality (1.8).

Definition 1.2

Let \(u(x,t)\) be the entropy solution of equation (1.1) with the initial value condition (1.3). If, moreover, the partial boundary value condition (1.5) is satisfied in the sense of the trace, then we say that \(u(x,t)\) is the entropy solution of the initial-boundary value problem of equation (1.1).

If \(b_{i}(u,x,t)\equiv b_{i}(u)\) is independent of the variables \((x,t)\), the existence of the entropy solution of initial-boundary value problem was obtained in [18]. By the aid of Fichera–Oleinik theory, we conjectured that the partial boundary value condition (1.5) should be
$$ u(x,t)=0,\qquad (x,t)\in\Sigma_{1}\times(0,T),\qquad \Sigma_{1}=\bigl\{ x\in\partial \Omega: b_{i}(0)n_{i}< 0 \bigr\} , $$
(1.10)
where \(\vec{n}=\{n_{i}\}\) is the inner normal vector of Ω, and proved the following theorem.

Theorem 1.3

. Suppose that \(A(s)\) is \(C^{2}\) and \(b_{i}(s,x,t)\equiv b_{i}(s)\) is \(C^{1}\). Let \(u(x,t)\) and \(v(x,t)\) be two entropy solutions of equation (1.1) with the different initial values \(u_{0}(x)\) and \(v_{0}(x)\), respectively, and with the same partial homogeneous boundary value condition
$$ \gamma u=\gamma v=0,\quad x\in\Sigma_{1}. $$
(1.11)
If Ω is with the property
$$\vert \triangle d \vert \leq c, \qquad \frac{1}{\lambda} \int _{\Omega_{\lambda}}\,dx \,dt\leq c, $$
then
$$ \int_{\Omega} \bigl\vert u(x,t)-v(x,t) \bigr\vert \,dx\leq \int _{\Omega} \vert u_{0}-v_{0} \vert \,dx+\operatorname {ess}\sup_{(x,t)\in\Sigma_{2}\times(0,T)} \bigl\vert u(x,t)-v(x,t) \bigr\vert , $$
(1.12)
where \((x,t)\in\mathbb{R}^{N+1}\), \(\Sigma_{2}=\partial\Omega \setminus\Sigma_{1}\), \(\operatorname{ess}\sup_{(x,t)\in\Sigma_{2}\times (0,T)}|u(x,t)-v(x,t)|\) is in the sense of N-dimensional Hausdorff measure.

If \(\Omega\subset\mathbb{R}^{N}\) is a bounded domain, Theorem 1.3 can be found in [16]. If \(\Omega= \mathbb{R}_{+}^{N}\) is the half space, Theorem 1.3 can be found in [17]. Also, if Ω is some special given domains, Theorem 1.3 was obtained in [19].

However, since there is an unknown term \(\operatorname{ess}\sup_{(x,t)\in\Sigma_{2}\times(0,T)}|u(x,t)-v(x,t)|\) in (1.12), Theorem 1.3 is far from perfection. The root of the problem lies in that equation (1.1) is with strong nonlinearity, it is almost impossible to verify whether conjecture (1.10) is true or not.

In this paper, we leave conjecture (1.10) out of consideration. We suppose that the convection term satisfies \(b_{i}(\cdot, x,t)=0\) when \(x\in\partial\Omega\). By ingeniously choosing the test function φ in the entropy inequality (1.8), we can deduce an explicit formula of \(\Sigma_{1}\) and establish the stability of the entropy solutions based on a partial boundary value condition (1.5). This is the following theorem.

Theorem 1.4

Let the domain Ω be with a \(C^{2}\) smooth boundary Ω, \(A(s)\) be \(C^{2}\) and \(b_{i}(s,x,t)\) be \(C^{1}\). Let \(u(x,t)\) and \(v(x,t)\) be two solutions of equation (1.1) with the different initial values \(u_{0}(x)\) and \(v_{0}(x)\), respectively, with the same partial boundary value condition (1.5), and
$$ \Sigma_{1}=\{x\in\partial\Omega: \Delta d >0\}. $$
(1.13)
If \(u_{0}(x),v_{0}(x)\in L^{\infty}(\Omega)\), and \(b_{i}(s,x,t)\) satisfies
$$ \bigl\vert b_{i}(u,x,t)-b_{i}(v,x,t) \bigr\vert \leq c d(x), $$
(1.14)
then
$$ \int_{\Omega} \bigl\vert u(x,t)-v(x,t) \bigr\vert \,dx\leq c \int _{\Omega} \vert u_{0}-v_{0} \vert \,dx. $$
(1.15)

Theorem 1.4 is proved by Kruzkov’s bi-variables method. Comparing (1.15) with (1.12), we find that the unperfect term \(\operatorname {ess}\sup_{(x,t)\in\Sigma_{2}\times(0,T)}|u(x,t)-v(x,t)|\) in (1.12) has disappeared. Considering different typical techniques used in [1419], the novelty of this paper lies in that we endow condition (1.14), and by this assumption, we can choose a suitable test function φ in (1.8) to obtain the perfect stability (1.15).

A direct corollary of Theorem 1.4 is the following theorem.

Theorem 1.5

Let the domain Ω be with a \(C^{2}\) smooth boundary Ω, \(A(s)\) be \(C^{2}\), and \(b_{i}(s,x,t)\) be \(C^{1}\). Let \(u(x,t)\) and \(v(x,t)\) be solutions of equation (1.1) with the different initial values \(u_{0}(x)\) and \(v_{0}(x)\), respectively, but without any boundary value condition. Suppose \(u_{0}(x),v_{0}(x)\in L^{\infty}(\Omega)\), when x is near to the boundary,
$$ \Delta d\leq0, $$
(1.16)
and \(b_{i}(s,x,t)\) satisfies (1.14), then the stability (1.15) is true.
One can see that, since \(u_{0}(x)\), \(v_{0}(x)\in L^{\infty}(\Omega)\), \(b_{i}(s,x,t)\) is \(C^{1}\), then
$$\bigl\vert b_{i}(u,x,t)-b_{i}(v,x,t) \bigr\vert \leq c \vert u-v \vert $$
is always true. If we assume that
$$\bigl\vert u(x,t) \bigr\vert \leq c d(x),\qquad \bigl\vert v(x,t \bigr\vert \leq cd(x), $$
which is stronger than the homogeneous boundary value condition (1.4), then condition (1.14) is true naturally. However, the condition has its independent significance. In fact, only if
$$ \bigl\vert b_{i}(s,x,t) \bigr\vert \leq c d(x), $$
(1.17)
is true, when \(|s|\leq c\) and x is near to the boundary Ω, then condition (1.14) can be guaranteed. Certainly, if condition (1.14) is not true, how to clarify the partial boundary \(\Sigma_{1}\) remains an open problem. If possible, we will present a follow-up work in the future.
Let us give two examples of the domains which satisfy condition (1.16). For example, if \(\Omega=D_{1}=\{x\in\mathbb{R}^{N}: x_{1}^{2}+x_{2}^{2}+\cdots +x_{N}^{2}<1\}\) is the unit disc, then
$$ d(x)=1-\sqrt{\bigl(x_{1}^{2}+x_{2}^{2}+ \cdots+x_{N}^{2}\bigr)}. $$
(1.18)
For another example, if \(\Omega=\{x\in\mathbb{R}^{N}: 0< x_{i}<1, i=1,2,\ldots, N\}\) is the N-dimensional unit cube, then
$$ d(x)=x_{i} \quad \text{or}\quad (1-x_{i}), $$
(1.19)
when \(x=(x_{1},x_{2}, \ldots, x_{N})\) is near to the hyperplane \(\{x_{i}=0\}\) or \(\{x_{i}=1\}\) respectively. The distance functions (1.18)–(1.19) all satisfy that
$$\Delta d\leq0. $$
The following example shows what \(\Sigma_{1}\) is. Let \(\Omega=\{x\in \mathbb{R}^{N}: r_{0}^{2}< x_{1}^{2}+x_{2}^{2}+\cdots+x_{N}^{2}< R_{0}^{2}\}\). When x is near to \(\{x\in\mathbb{R}^{N}: x_{1}^{2}+x_{2}^{2}+\cdots +x_{N}^{2}=R_{0}^{2}\}\),
$$\Delta d\leq0. $$
But when x is near to \(\{x\in\mathbb{R}^{N}: x_{1}^{2}+x_{2}^{2}+\cdots +x_{N}^{2}=r_{0}^{2}\}\),
$$0< \Delta d=\frac{N-1}{r}, $$
then
$$\Sigma_{1}=\bigl\{ x\in\mathbb{R}^{N}: x_{1}^{2}+x_{2}^{2}+ \cdots+x_{N}^{2}=r_{0}^{2}\bigr\} . $$

In Sect. 2, we will introduce Kruzkov’s bi-variables method. Theorem 1.4 and Theorem 1.5 are proved in Sect. 3.

2 Kruzkov’s bi-variables method

The context in this section is just a minor version of Kruzkov’s bi-variables method used in our previous works [1419].

Let \(\Gamma_{u}\) be the set of all jump points of \(u\in \operatorname{BV}(Q_{T}),v\) be the normal of \(\Gamma_{u}\) at \(X=(x,t)\), \(u^{+}(X)\) and \(u^{-}(X)\) be the approximate limits of u at \(X\in\Gamma_{u}\) with respect to \((v,Y-X)>0\) and \((v,Y-X)<0\), respectively. For a continuous function \(p(u,x,t)\) and \(u\in \operatorname{BV}(Q_{T})\), define
$$ \widehat{p}(u,x,t)= \int_{0}^{1}p\bigl(\tau u^{+}+(1- \tau)u^{-},x,t\bigr)\,d\tau , $$
(2.1)
which is called the composite mean value of p. For given t, we denote \(\Gamma_{u}^{t}\), \(H^{t}\), \((v_{1}^{t},\ldots,v_{N}^{t})\) and \(u_{\pm}^{t}\) as all jump points of \(u(\cdot,t)\), Hausdorff measure of \(\Gamma_{u}^{t}\), the unit normal vector of \(\Gamma_{u}^{t}\), and the asymptotic limit of \(u(\cdot,t)\), respectively. Moreover, if \(f(s)\in C^{1}( \mathbb{R})\), \(u\in \operatorname{BV}(Q_{T})\), then \(f(u)\in \operatorname{BV}(Q_{T})\) and
$$ \frac{\partial f(u)}{\partial x_{i}}=\widehat{f^{\prime}}(u)\frac {\partial u}{\partial x_{i}}, \quad i=1, 2, \ldots, N, N+1, $$
(2.2)
where \(x_{N+1}=t\) as usual.

Lemma 2.1

Let u be a solution of equation (1.1). Then
$$ a(s)=0, \quad s\in I\bigl(u^{+}(x,t),u^{-}(x,t)\bigr) \textit{ a.e. on }\Gamma_{u}, $$
(2.3)
where \(I(\alpha,\beta)\) denotes the closed interval with endpoints α and β, and (2.3) is in the sense of Hausdorff measure \(H_{N}(\Gamma_{u})\).
Now, let \(u(x,t)\) and \(v(x,t)\) be two entropy solutions of equation (1.1) with initial values
$$ u(x,0)=u_{0}(x), \qquad v(x,0)=v_{0}(x), $$
respectively.
By Definition 1.1, for \(\varphi\in C_{0}^{2}(Q_{T})\), we have
$$\begin{aligned}& \begin{aligned}[b] & \iint_{Q_{T}} \Biggl[I_{\eta}(u-k)\varphi_{t}- \sum_{i=1}^{N}B_{\eta }^{i}(u,x,t,k) \varphi_{x_{i}}+A_{\eta}(u,k)\Delta\varphi\\ &\qquad {}-S_{\eta }^{\prime}(u-k) \biggl\vert \nabla \int_{0}^{u}\sqrt{a(s)}\,ds \biggr\vert ^{2}\varphi \Biggr]\,dx \,dt \\ &\qquad {}-\sum_{i=1}^{N} \iint_{Q_{T}} \int _{k}^{u}b_{ix_{i}}(s,x,t)S'_{\eta}(s-k) \,ds\varphi \,dx \,dt \\ &\quad\geq0, \end{aligned} \end{aligned}$$
(2.4)
$$\begin{aligned}& \begin{aligned}[b] & \iint_{Q_{T}} \Biggl[I_{\eta}(v-l)\varphi_{\tau}- \sum_{i=1}^{N}B_{\eta}^{i}(v,y, \tau, l)\varphi_{y_{i}}+A_{\eta }(v,l)\Delta\varphi\\ &\qquad {}-S_{\eta}^{\prime}(v-l) \biggl\vert \nabla \int _{0}^{v}\sqrt{a(s)}\,ds \biggr\vert ^{2}\varphi \Biggr] \,dy \,d\tau \\ &\qquad {}-\sum_{i=1}^{N} \iint_{Q_{T}} \int_{l}^{v}b_{iy_{i}}(s,y,\tau )S'_{\eta}(s-l)\,ds\varphi \,dx \,dt \\ &\quad \geq0 . \end{aligned} \end{aligned}$$
(2.5)
Let \(\psi(x,t,y,\tau)=\phi(x,t)j_{h}(x-y,t-\tau)\). Here, \(\phi (x,t)\geq0\), \(\phi(x,t)\in C_{0}^{\infty}(Q_{T})\), and
$$\begin{aligned}& j_{h}(x-y,t-\tau)=\omega_{h}(t-\tau)\prod_{i=1}^{N} \omega _{h}(x_{i}-y_{i}), \end{aligned}$$
(2.6)
$$\begin{aligned}& \begin{gathered} \omega_{h}(s)=\frac{1}{h}\omega\biggl(\frac{s}{h} \biggr),\qquad \omega(s)\in C_{0}^{\infty}(R), \qquad \omega(s)\geq0, \\ \omega(s)=0\quad \mbox{if } \vert s \vert >1, \qquad \int_{-\infty}^{\infty}\omega(s)\,ds=1. \end{gathered} \end{aligned}$$
(2.7)
We choose \(k=v(y,\tau)\), \(l=u(x,t)\), \(\varphi=\psi(x,t,y,\tau)\) in (2.4), (2.5), integrate over \(Q_{T}\) respectively, add them together. Then
$$ \begin{gathered}[b] \iint_{Q_{T}} \iint_{Q_{T}} \bigl[I_{\eta}(u-v) (\psi_{t}+ \psi_{\tau })+A_{\eta}(u,v)\Delta_{x} \psi+A_{\eta}(v,u)\Delta_{y}\psi \bigr] \\ \qquad{}-\sum_{i=1}^{N} \bigl[B_{\eta}^{i}(u,x,t,v)\psi_{x_{i}}+B_{\eta }^{i}(v,y, \tau, u)\psi_{y_{i}}\bigr] \\ \qquad{}-\sum_{i=1}^{N} \biggl[ \int_{k}^{u}b_{ix_{i}}(s,x,t)S'_{\eta }(s-k) \,ds+ \int_{l}^{v}b_{iy_{i}}(s,y, \tau)S'_{\eta}(s-l)\,ds \biggr] \\ \qquad{}-S_{\eta}^{\prime}(u-v) \biggl( \biggl\vert \nabla \int _{0}^{u}\sqrt{a(s)}\,ds \biggr\vert ^{2}+ \biggl\vert \nabla \int _{0}^{v}\sqrt{a(s)}\,ds \biggr\vert ^{2} \biggr)\psi \,dx \,dt \,dy \,d\tau \\ \quad\geq0. \end{gathered} $$
(2.8)
Clearly,
$$\begin{aligned}& \frac{\partial j_{h}}{\partial t}+\frac{\partial j_{h}}{\partial\tau }=0, \qquad \frac{\partial j_{h}}{\partial x_{i}}+ \frac{\partial j_{h}}{\partial y_{i}}=0, \quad i=1,\ldots,N; \\& \frac{\partial\psi}{\partial t}+\frac{\partial\psi}{\partial\tau }=\frac{\partial\phi}{\partial t}j_{h}, \qquad \frac{\partial\psi }{\partial x_{i}}+\frac{\partial\psi}{\partial y_{i}}=\frac {\partial\phi}{\partial x_{i}}j_{h}. \end{aligned}$$
Noticing that
$$\begin{aligned} \lim_{\eta\rightarrow0}B_{\eta}^{i}(u,x,t,v) =& \lim _{\eta \rightarrow0}B_{\eta}^{i}(v,y,\tau, u) \\ =& \operatorname{sgn}(u-v) \bigl(b_{i}(u,x,t)-b_{i}(v,y, \tau)\bigr), \end{aligned}$$
then
$$\begin{gathered} \lim_{\eta\rightarrow0} \iint_{Q_{T}} \iint_{Q_{T}}\bigl[B_{\eta }^{i}(u,x,t,v) \psi_{x_{i}}+ B_{\eta}^{i}(v,y,\tau,u) \psi_{y_{i}}\bigr]\,dx \,dt \,dy \,d\tau \\ \quad = \iint_{Q_{T}} \iint_{Q_{T}}\operatorname {sgn}(u-v)\bigl[b_{i}(u,x,t)-b_{i}(v,y, \tau)\bigr]\phi_{x_{i}}j_{h}\,dx \,dt \,dy \,d\tau \end{gathered} $$
and
$$ \begin{gathered}[b] \lim_{h \rightarrow0} \iint_{Q_{T}} \iint_{Q_{T}}\operatorname {sgn}(u-v)\bigl[b_{i}(u, x,t)-b_{i}(v,y,\tau)\bigr]\phi_{x_{i}}j_{h}\,dx \,dt \,dy \,d\tau \\ \quad= \iint_{Q_{T}}\operatorname{sgn} (u-v)\bigl[b_{i}(u,x,t)-b_{i}(v,x,t) \bigr]\phi_{x_{i}}\,dx \,dt. \end{gathered} $$
(2.9)
Once more, we have
$$ \begin{gathered}[b] \iint_{Q_{T}}\bigl[A_{\eta}(u,v)\Delta_{x} \psi+A_{\eta}(v,u)\Delta _{y}\psi\bigr]\,dx \,dt \,dy \,d\tau \\ \quad= \iint_{Q_{T}} \iint_{Q_{T}} \Biggl\{ A_{\eta}(u,v) \Biggl(\Delta _{x}\phi j_{h}+2\sum_{i=1}^{N} \phi_{x_{i}}j_{hx_{i}}+\phi\Delta j_{h} \Biggr)+A_{\eta}(v,u)\phi\Delta_{y}j_{h} \Biggr\} \,dx \,dt \,dy \,d\tau\hspace{-30pt} \\ \quad= \iint_{Q_{T}} \iint_{Q_{T}} \Biggl\{ A_{\eta}(u,v)\Delta _{x} \phi j_{h}+\sum_{i=1}^{N} \bigl[A_{\eta}(u,v)\phi _{x_{i}}j_{hx_{i}}+A_{\eta}(v,u) \phi_{x_{i}}j_{hy_{i}}\bigr] \Biggr\} \,dx \,dt \,dy \,d\tau\hspace{-30pt} \\ \qquad{} -\sum_{i=1}^{N} \iint_{Q_{T}} \iint_{Q_{T}} \biggl\{ \biggl[\widehat{a(u)S_{\eta}(u-v)} \frac{\partial u}{\partial x_{i}}\\ \qquad {}-\widehat{ \int_{u}^{v}a(s)S_{\eta}^{\prime}(s-v) \,ds}\frac{ \partial u}{\partial x_{i}} \biggr]\phi j_{hx_{i}} \biggr\} \,dx \,dt \,dy \,d \tau, \end{gathered} $$
(2.10)
where
$$\begin{aligned}& \widehat{a(u)S_{\eta}(u-v)}= \int_{0}^{1}a\bigl(su^{+}+(1-s)u^{-} \bigr)S_{\eta }\bigl(su^{+}+(1-s)u^{-}-v\bigr)\,ds, \\& \int_{u}^{v}\widehat{a(s)S_{\eta}^{\prime}(s-v)} \,ds= \int_{0}^{1} \int_{su^{+}+(1-s)u^{-}}^{v}a(\sigma)S_{\eta}\bigl(\sigma -su^{+}-(1-s)u^{-}\bigr)\,d\sigma \,ds. \end{aligned}$$
Notice that
$$ \begin{gathered}[b] \iint_{Q_{T}} \iint_{Q_{T}}S_{\eta}^{\prime}(u-v) \biggl( \biggl\vert \nabla_{x} \int_{0}^{u}\sqrt{a(s)}\,ds \biggr\vert ^{2}+ \biggl\vert \nabla_{y} \int_{0}^{v}\sqrt{a(s)}\,ds \biggr\vert ^{2} \biggr)\psi \,dx \,dt \,dy \,d\tau\hspace{-20pt} \\ \quad= \iint_{Q_{T}} \iint_{Q_{T}}S_{\eta}^{\prime}(u-v) \biggl( \biggl\vert \nabla_{x} \int_{0}^{u}\sqrt{a(s)}\,ds \biggr\vert - \biggl\vert \nabla_{y} \int_{0}^{v}\sqrt{a(s)}\,ds \biggr\vert \biggr)^{2}\psi \,dx \,dt \,dy \,d\tau\hspace{-20pt} \\ \qquad{} +2 \iint_{Q_{T}} \iint_{Q_{T}}S_{\eta}^{\prime}(u-v)\nabla _{x} \int_{0}^{u} \sqrt{a(s)}\,ds\cdot \nabla_{y} \int_{0}^{v}\sqrt {a(s)}\,ds\psi \,dx \,dt \,dy \,d \tau,\hspace{-20pt} \end{gathered} $$
(2.11)
and, by Lemma 2.1, we can show that
$$ \begin{gathered}[b] \sum_{i=1}^{N} \iint_{Q_{T}} \iint_{Q_{T}}\biggl(\widehat{a(u)S_{\eta }(u-v)} \frac{\partial u}{\partial x_{i}}-\widehat{ \int _{u}^{v}a(s)S_{\eta}^{\prime}(s-u) \,ds}\frac{\partial u}{\partial x_{i}}\biggr)j_{hx_{i}}\phi \,dx \,dt \,dy \,d\tau \\ \qquad{} +2 \iint_{Q_{T}} \iint_{Q_{T}}S_{\eta}^{\prime}(u-v)\nabla _{x} \int_{0}^{u} \sqrt{a(s)}\,ds\cdot \nabla_{y} \int_{0}^{v}\sqrt {a(s)}\,ds\psi \,dx \,dt \,dy \,d \tau \\ \quad=-\sum_{i=1}^{N} \iint_{Q_{T}} \iint_{Q_{T}} \int_{0}^{1} \int _{su^{+}+(1-s)u^{-}}^{v}\bigl[ \sqrt{a(\sigma)}-\sqrt{a \bigl(su^{+}+(1-s)u^{-}\bigr)}\bigr] \\ \qquad{} \times S^{\prime}_{\eta} \bigl(\sigma-su^{+}-(1-s)u^{-} \bigr)\,d\sigma \,ds\frac{\partial u}{ \partial x_{i}}j_{hx_{i}}\phi \,dx \,dt \,dy \,d\tau \\ \quad \rightarrow0 \end{gathered} $$
(2.12)
as \(\eta\rightarrow0\).
Moreover, since
$$\lim_{\eta\rightarrow0}A_{\eta}(u,v)=\lim_{\eta\rightarrow 0}A_{\eta}(v,u)= \operatorname{sgn}(u-v)\bigl[A(u)-A(v)\bigr], $$
we have
$$ \lim_{\eta\rightarrow0}\bigl[A_{\eta}(u,v)\phi _{x_{i}}j_{hx_{i}}+A_{\eta}(u,v) \phi_{y_{i}}j_{hy_{i}}\bigr]=0. $$
(2.13)
Combining (2.8)–(2.13) and letting \(\eta\rightarrow0\), \(h\rightarrow 0\) in (2.8), we get
$$ \begin{gathered}[b] \iint_{Q_{T}} \bigl[ \bigl\vert u(x,t)-v(x,t) \bigr\vert \phi_{t}+ \bigl\vert A(u)-A(v) \bigr\vert \Delta\phi \bigr]\,dx \,dt \\ \qquad{} -\sum_{i=1}^{N} \iint_{Q_{T}}\operatorname {sgn}(u-v)\bigl[b_{i}(u,x,t)-b_{i}(v,x,t) \bigr]\phi_{x_{i}}\,dx \,dt \\ \qquad{} -\sum_{i=1}^{N} \iint_{Q_{T}}\bigl[b_{ix_{i}}(v,x,t)\operatorname {sgn}(u-v) \phi+b_{ix_{i}}(u,x,t)\operatorname{sgn}(v-u)\phi\bigr]\,dx \,dt \\ \quad\geq0. \end{gathered} $$
(2.14)
By (2.14), by choosing a suitable test function φ, one may obtain the stability of the entropy solution.

3 Proofs of Theorem 1.4 and Theorem 1.5

Proof of Theorem 1.4

For small enough λ, we set
$$ \varphi_{\lambda}(x)= \textstyle\begin{cases} \sin\frac{d(x)}{\lambda}, &\mbox{if } 0\leq d(x)< \frac{\pi \lambda}{2},\\ 1, &\mbox{if } d(x)\geq\frac{\pi\lambda}{2}. \end{cases} $$
(3.1)
Let \(0\leq\eta(t)\in C_{0}^{2}(t)\) and
$$\phi(x,t)=\eta(t)\varphi_{\lambda}(x). $$
By (3.1), when \(0\leq d(x)<\frac{\pi\lambda}{2}\), we clearly have
$$ \partial_{x_{i}}\phi(x,t)=\eta(t)\partial_{x_{i}} \varphi_{\lambda }(x) =\eta(t)\frac{1}{\lambda}\cos\frac{d(x)}{\lambda }d_{x_{i}}(x) $$
(3.2)
and
$$ \begin{aligned}[b] \triangle\phi(x,t)&=\frac{1}{\lambda}\eta(t) \Biggl[-\frac{1}{\lambda }\sin\frac{d(x)}{\lambda}\sum_{i=1}^{N}d_{x_{i}}^{2}+ \cos\frac {d(x)}{\lambda}\Delta d(x)\Biggr] \\ &=-\frac{1}{\lambda^{2}}\eta(t) \sin\frac{d(x)}{\lambda}\sum _{i=1}^{N}d_{x_{i}}^{2}+ \frac{1}{\lambda}\eta(t)\cos\frac {d(x)}{\lambda}\Delta d(x). \end{aligned} $$
(3.3)
In another place, i.e., when \(d(x)\geq\frac{\pi\lambda}{2}\),
$$ \partial_{x_{i}}\phi(x,t)=0=\triangle\phi(x,t),\quad i=1,2, \ldots, N. $$
(3.4)
If we denote \(\Omega_{1\lambda}=\{x\in\Omega: d(x)<\frac{\lambda \pi}{2}\}\), by that \(|\nabla d|=1\), according to (3.3)–(3.4), we have
$$ \begin{gathered}[b] \iint_{Q_{T}} \bigl\vert A(u)-A(v) \bigr\vert \Delta\phi \,dx \,dt \\ \quad=-\frac{1}{\lambda^{2}} \int_{0}^{T} \int_{\Omega_{1\lambda}} \bigl\vert A(u)-A(v) \bigr\vert \eta(t) \sin \frac{d(x)}{\lambda }\,dx \,dt \\ \qquad{}+\frac{1}{\lambda} \int_{0}^{T} \int_{\Omega_{1\lambda}}\eta (t) \bigl\vert A(u)-A(v) \bigr\vert \cos \frac{d(x)}{\lambda }\Delta d(x)\,dx \,dt. \end{gathered} $$
(3.5)
Substituting into (2.14), we have
$$ \begin{gathered}[b] \iint_{Q_{T}} \bigl\vert u(x,t)-v(x,t) \bigr\vert \eta_{t}\varphi _{\lambda}(x)\,dx \,dt \\ \qquad{} -\sum_{i=1}^{N} \int_{0}^{T} \int_{\Omega_{1\lambda}}\frac{\eta (t)}{\lambda}\operatorname{sgn} (u-v) \bigl(b_{i}(u,x,t)-b_{i}(v,x,t)\bigr)\cos \frac{d(x)}{\lambda}d_{x_{i}}(x)\,dx \,dt \\ \qquad{} + \int_{0}^{T} \int_{\Omega_{1\lambda}}\frac{\eta(t)}{\lambda} \bigl\vert A(u)-A(v) \bigr\vert \cos\frac{d(x)}{\lambda}\Delta d(x)\,dx \,dt \\ \qquad{} -\sum_{i=1}^{N} \iint_{Q_{T}}\bigl[b_{ix_{i}}(v,x,t)\operatorname {sgn}(u-v)+b_{ix_{i}}(u,x,t)\operatorname{sgn}(v-u)\bigr]\eta(t)\varphi _{\lambda}(x)\,dx \,dt \\ \quad \geq0. \end{gathered} $$
(3.6)
Let
$$\Omega_{+}=\Omega_{1\lambda}\cap\{x\in\Omega: \Delta d>0\},\qquad \Omega_{-}=\Omega_{1\lambda}\cap\{x\in \Omega: \Delta d< 0\}. $$
Then
$$ \begin{gathered}[b] \iint_{Q_{T}} \bigl\vert u(x,t)-v(x,t) \bigr\vert \eta_{t}\varphi _{\lambda}(x)\,dx \,dt \\ \qquad{} -\sum_{i=1}^{N} \int_{0}^{T} \int_{\Omega_{1\lambda}}\frac{\eta (t)}{\lambda}\operatorname{sgn} (u-v) \bigl(b_{i}(u,x,t)-b_{i}(v,x,t)\bigr)\cos \frac{d(x)}{\lambda}d_{x_{i}}(x)\,dx \,dt \\ \qquad{} + \int_{0}^{T} \int_{\Omega_{\lambda_{+}}}\frac{\eta(t)}{\lambda} \bigl\vert A(u)-A(v) \bigr\vert \Delta d(x)\,dx \,dt \\ \qquad{} -\sum_{i=1}^{N} \iint_{Q_{T}}\bigl[b_{ix_{i}}(v,x,t)\operatorname {sgn}(u-v)+b_{ix_{i}}(u,x,t)\operatorname{sgn}(v-u)\bigr]\eta(t)\varphi _{\lambda}(x)\,dx \,dt \\ \quad \geq0. \end{gathered} $$
(3.7)
In the first place, by that \(|d_{x_{i}}(x)|\leq|\nabla d|=1\), by condition (1.14),
$$ \begin{gathered}[b] \biggl\vert \int_{0}^{T} \int_{\Omega_{1\lambda}}\frac{\eta (t)}{\lambda}\operatorname{sgn}(u-v) \bigl(b_{i}(u,x,t)-b_{i}(v,x,t)\bigr)\cos \frac{d(x)}{\lambda}d_{x_{i}}(x) \biggr\vert \\ \quad \leq c \int_{0}^{T} \int_{\Omega_{1\lambda}}\frac{\eta(t)}{\lambda }d(x)\,dx \,dt \\ \quad \leq c \int_{0}^{T} \int_{\Omega_{1\lambda}}{\eta(t)}\,dx \,dt \end{gathered} $$
(3.8)
goes to zero when \(\lambda\rightarrow0\).
In the second place, by the partial boundary value condition (1.5), we have
$$ \begin{gathered}[b] \lim_{\lambda\rightarrow0}\frac{1}{\lambda} \int_{0}^{T} \int_{\Omega _{\lambda-{+}}}\frac{\eta(t)}{\lambda} \bigl\vert A(u)-A(v) \bigr\vert \cos\frac{d(x)}{\lambda} \Delta d \,dx \,dt \\ \quad \leq c\lim_{\lambda\rightarrow0}\frac{1}{\lambda} \int_{0}^{T} \int _{\Omega_{\lambda_{+}}} \vert u-v \vert \,dx \,dt \\ \quad =c \int_{0}^{T} \int_{\Sigma_{1}} \vert u-v \vert \,d\sigma \,dt =0. \end{gathered} $$
(3.9)
Moreover,
$$ \begin{gathered}[b] \lim_{\lambda\rightarrow0} \biggl\vert \iint _{Q_{t}}\bigl[b_{ix_{i}}(v,x,t)\operatorname {sgn}(u-v)+b_{ix_{i}}(u,x,t)\operatorname{sgn}(v-u)\bigr]\eta(t)\varphi _{\lambda}(x)\,dx \,dt \biggr\vert \\ \quad \leq c \iint_{Q_{T}}\eta(t) \vert u-v \vert \,dx \,dt. \end{gathered} $$
(3.10)
By (3.6)–(3.10), letting \(\lambda\rightarrow0\), we can deduce that
$$ \iint_{Q_{T}} \bigl\vert u(x,t)-v(x,t) \bigr\vert \eta'_{t}\,dx \,dt+c \int_{0}^{T} \int_{\Omega} \vert u-v \vert \eta (t)\,dx \,dt. $$
(3.11)
Let \(0< s<\tau<T\), and
$$\eta(t)= \int_{\tau-t}^{s-t}\alpha_{\varepsilon}(\sigma)\,d\sigma, \quad \varepsilon< \min\{\tau, T-s\}. $$
Here, \(\alpha_{\varepsilon}(t)\) is the kernel of mollifier with \(\alpha_{\varepsilon}(t)=0\) for \(t\notin(-\varepsilon,\varepsilon )\). Then
$$c \int_{0}^{T}{\eta(t)} \vert u-v \vert \,dx \,dt + \int _{0}^{T} \bigl[\alpha_{\varepsilon}(t-s)- \alpha_{\varepsilon }(t-\tau) \bigr] \vert u-v \vert _{L^{1}(\Omega)}\,dt\geq0. $$
Let \(\varepsilon\rightarrow0\). Then
$$\bigl\vert u(x,\tau)-v(x,\tau) \bigr\vert _{L^{1}(\Omega)}\leq \bigl\vert u(x,s)-v(x,s) \bigr\vert _{L^{1}(\Omega)}+c \int _{0}^{t} \int_{\Omega} \vert u-v \vert \,dx \,dt. $$
By Gronwall’s inequality, we have
$$\int_{\Omega} \bigl\vert u(x,\tau)-v(x,\tau) \bigr\vert \,dx \leq c \int_{\Omega} \bigl\vert u(x,s)-v(x,s) \bigr\vert \,dx. $$
Let \(s \rightarrow0\). Then
$$\int_{\Omega} \bigl\vert u(x,\tau)-v(x,\tau) \bigr\vert \,dx \leq c \int_{\Omega} \bigl\vert u_{0}(x)-v_{0}(x) \bigr\vert \,dx. $$
The proof is complete. □

Proof of Theorem 1.5

If, when x is near to the boundary, condition (1.16)
$$\Delta d\leq0 $$
is true, then \(\Sigma_{1}=\emptyset\). By Theorem 1.4, we have the conclusion of Theorem 1.5, i.e., the stability of the entropy solutions is true independent of the boundary value conditions. □

4 Conclusion

The equation considered in this paper comes from many applied fields. It is of a hyperbolic–parabolic mixed type, and only has a discontinuous solution generally. The most characteristic feature of the equation lies in that the usual Dirichlet boundary value condition may be overdetermined. Using Kruzkov’s bi-variables method, by choosing a suitable test function, the stability of the entropy solutions is proved based on the partial boundary value condition, provided that the convection term has the degeneracy on the boundary.

Declarations

Acknowledgements

The author would like to thank SpringerOpen Accounts Team for kindly agreeing to give me a discount of the paper charge if my paper can be accepted.

Availability of data and materials

Not applicable.

Funding

The paper is supported by the Natural Science Foundation of Fujian province, supported by the Science Foundation of Xiamen University of Technology (Grant No. XYK201448), China.

Authors’ contributions

The author read and approved the final manuscript.

Ethics approval and consent to participate

Not applicable.

Competing interests

The author declares that he has no competing interests.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
School of Applied Mathematics, Xiamen University of Technology, Xiamen, P.R. China

References

  1. Wu, Z., Zhao, J., Yin, J., Li, H.: Nonlinear Diffusion Equations. World Scientific, Singapore (2001) View ArticleMATHGoogle Scholar
  2. Vol’pert, A.I., Hudjaev, S.I.: On the problem for quasilinear degenerate parabolic equations of second order. Mat. Sb. 3, 374–396 (1967) (Russian) Google Scholar
  3. Brezis, H., Crandall, M.G.: Uniqueness of solutions of the initial value problem for \(u_{t}-\Delta\varphi(u)=0\). J. Math. Pures Appl. 58, 153–163 (1979) MathSciNetMATHGoogle Scholar
  4. Kružkov, S.N.: First order quasilinear equations in several independent variables. Math. USSR Sb. 10, 217–243 (1970) View ArticleGoogle Scholar
  5. Cockburn, B., Gripenberg, G.: Continuous dependence on the nonlinearities of solutions of degenerate parabolic equations. J. Differ. Equ. 151, 231–251 (1999) MathSciNetView ArticleMATHGoogle Scholar
  6. Chen, G.Q., Perthame, B.: Well-posedness for non-isotropic degenerate parabolic-hyperbolic equations. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 20(4), 645–668 (2003) MathSciNetView ArticleMATHGoogle Scholar
  7. Bendahamane, M., Karlsen, K.H.: Reharmonized entropy solutions for quasilinear anisotropic degenerate parabolic equations. SIAM J. Math. Anal. 36(2), 405–422 (2004) MathSciNetView ArticleGoogle Scholar
  8. Carrillo, J.: Entropy solutions for nonlinear degenerate problems. Arch. Ration. Mech. Anal. 147, 269–361 (1999) MathSciNetView ArticleMATHGoogle Scholar
  9. Li, Y., Wang, Q.: Homogeneous Dirichlet problems for quasilinear anisotropic degenerate parabolic-hyperbolic equations. J. Differ. Equ. 252, 4719–4741 (2012) MathSciNetView ArticleMATHGoogle Scholar
  10. Lions, P.L., Perthame, B., Tadmor, E.: A kinetic formation of multidimensional conservation laws and related equations. J. Am. Math. Soc. 7, 169–191 (1994) View ArticleMATHGoogle Scholar
  11. Kobayasi, K., Ohwa, H.: Uniqueness and existence for anisotropic degenerate parabolic equations with boundary conditions on a bounded rectangle. J. Differ. Equ. 252, 137–167 (2012) MathSciNetView ArticleMATHGoogle Scholar
  12. Mascia, C., Porretta, A., Terracina, A.: Qualitative behavior for one-dimensional strongly degenerate parabolic problems. Interfaces Free Bound. 8(3), 263–280 (2006) MathSciNetView ArticleMATHGoogle Scholar
  13. Michel, A., Vovelle, J.: Entropy formulation for parabolic degenerate equations with general Dirichlet boundary conditions and application to the convergence of FV methods. SIAM J. Numer. Anal. 41(6), 2262–2293 (2003) MathSciNetView ArticleMATHGoogle Scholar
  14. Zhan, H.: The study of the Cauchy problem of a second order quasilinear degenerate parabolic equation and the parallelism of a Riemannian manifold. Doctor Thesis, Xiamen University, Xiamen, China (2004) Google Scholar
  15. Zhao, J., Zhan, H.: Uniqueness and stability of solution for Cauchy problem of degenerate quasilinear parabolic equations. Sci. China Ser. A 48, 583–593 (2005) MathSciNetView ArticleMATHGoogle Scholar
  16. Zhan, H.: On a hyperbolic-parabolic mixed type equation. Discrete Contin. Dyn. Syst., Ser. S 10(3), 605–624 (2017) MathSciNetView ArticleGoogle Scholar
  17. Zhan, H.: The solutions of a hyperbolic-parabolic mixed type equation on half-space domain. J. Differ. Equ. 259, 1449–1481 (2015) MathSciNetView ArticleMATHGoogle Scholar
  18. Zhan, H.: Reaction diffusion equations with boundary degeneracy. Electron. J. Differ. Equ. 2016, 81 (2016) MathSciNetView ArticleMATHGoogle Scholar
  19. Zhan, H.: The entropy solution of a hyperbolic-parabolic mixed type equation. SpringerPlus 5(1), 1–13 (2016) View ArticleGoogle Scholar

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