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.
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1 Introduction
The degenerate parabolic equation
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
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 [3–15] and the references therein. Based on the papers, we can conclude that the initial value condition
is always indispensable, while the usual Dirichlet boundary value condition
might be overdetermined. Instead of (1.4), a partial boundary value condition
should be imposed, where \(\Sigma_{1}\) is a relative open subset of ∂Ω. If
the explicit formula of \(\Sigma_{1}\) was studied in our previous works [16–18].
For small \(\eta>0\), let
Then \(h_{\eta}(s)\in C(\mathbb{R})\), and
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.
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.
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.
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,
and
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
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
If Ω is with the property
then
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
If \(u_{0}(x),v_{0}(x)\in L^{\infty}(\Omega)\), and \(b_{i}(s,x,t)\) satisfies
then
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 [14–19], 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,
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
is always true. If we assume that
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
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
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
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
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 \(\{x\in\mathbb{R}^{N}: x_{1}^{2}+x_{2}^{2}+\cdots +x_{N}^{2}=R_{0}^{2}\}\),
But when x is near to \(\{x\in\mathbb{R}^{N}: x_{1}^{2}+x_{2}^{2}+\cdots +x_{N}^{2}=r_{0}^{2}\}\),
then
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 [14–19].
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
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
where \(x_{N+1}=t\) as usual.
Lemma 2.1
Let u be a solution of equation (1.1). Then
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
respectively.
By Definition 1.1, for \(\varphi\in C_{0}^{2}(Q_{T})\), we have
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
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
Clearly,
Noticing that
then
and
Once more, we have
where
Notice that
and, by Lemma 2.1, we can show that
as \(\eta\rightarrow0\).
Moreover, since
we have
Combining (2.8)–(2.13) and letting \(\eta\rightarrow0\), \(h\rightarrow 0\) in (2.8), we get
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
Let \(0\leq\eta(t)\in C_{0}^{2}(t)\) and
By (3.1), when \(0\leq d(x)<\frac{\pi\lambda}{2}\), we clearly have
and
In another place, i.e., when \(d(x)\geq\frac{\pi\lambda}{2}\),
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
Substituting into (2.14), we have
Let
Then
In the first place, by that \(|d_{x_{i}}(x)|\leq|\nabla d|=1\), by condition (1.14),
goes to zero when \(\lambda\rightarrow0\).
In the second place, by the partial boundary value condition (1.5), we have
Moreover,
By (3.6)–(3.10), letting \(\lambda\rightarrow0\), we can deduce that
Let \(0< s<\tau<T\), and
Here, \(\alpha_{\varepsilon}(t)\) is the kernel of mollifier with \(\alpha_{\varepsilon}(t)=0\) for \(t\notin(-\varepsilon,\varepsilon )\). Then
Let \(\varepsilon\rightarrow0\). Then
By Gronwall’s inequality, we have
Let \(s \rightarrow0\). Then
The proof is complete. □
Proof of Theorem 1.5
If, when x is near to the boundary, condition (1.16)
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.
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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.
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Zhan, H. On the stability of the equation with a partial boundary value condition. Bound Value Probl 2018, 30 (2018). https://doi.org/10.1186/s13661-018-0953-7
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DOI: https://doi.org/10.1186/s13661-018-0953-7