Abstract
Consider the equation
where Ω is a bounded domain, \(d(x)\) is the distance function from the boundary ∂Ω. Since the nonlinearity, the boundary value condition cannot be portrayed by the Fichera function. If \(\alpha< p-1\), a partial boundary value condition is portrayed by a new way, the stability of the weak solutions is proved by this partial boundary value condition. If \(\alpha>p-1\), the stability of the weak solutions may be proved independent of the boundary value condition.
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1 Introduction and the main results
Benedikt et al. [1] considered the equation
and showed that the uniqueness of the solution is not true [1]. Here, \(0<\gamma<1\), Ω is a bounded domain in \(R^{N}\) with appropriately smooth boundary, \(q(x)\geq0\) and at least there is a \(x_{0}\in\Omega\) such that \(q(x_{0})>0\). Zhan [2] had shown that the stability of the solutions to the equation
is true, where \(d(x) = \operatorname{dist} (x,\partial\Omega)\) is distance function, \(\alpha>0\) is a constant. The result of [2] is in complete antithesis to that of [1]. So, when the well-posedness of the solutions is considered, the degeneracy of the diffusion coefficient \(d^{\alpha}\) plays an important role.
Yin and Wang [3, 4] studied the equation
and showed that there is a constant \(\gamma>1\) such that, if \(\alpha< p-1\), then
Recently, Zhan [5] had generalized the Yin and Wang result to the equation
In this paper, we continue to consider a more general equation,
and study the well-posedness of the weak solutions. As usual, the initial value
is necessary. But, since the coefficient \(d^{\alpha}\) is degenerate on the boundary, when \(\alpha< p-1\), though (1.4) is true, and the boundary value condition
can be imposed in the sense of the trace, it may be overdetermined. While \(\alpha\geq p-1\), it is almost impossible to prove (1.4). How to impose a suitable boundary value condition to match up with Eq. (1.6) becomes very troublesome [4]. Stated succinctly, instead of the Dirichlet boundary value condition (1.8), only a partial boundary value condition,
is needed, where \(\Sigma_{p}\subseteq\partial\Omega\) is a relatively open subset. The main difficulty comes from the fact that, since Eq. (1.6) is a nonlinear parabolic equation, \(\Sigma_{p}\) cannot be expressed by the Fichera function (one can refer to Sect. 6 of this paper). In this paper, we will try to depict the geometric characteristic of \(\Sigma_{1}\), and establish the stability of the weak solutions based on the partial boundary value condition (1.9).
We denote
Definition 1.1
Let
and
Here \(\varphi_{1} \in{ C_{0}^{1} ({Q_{T}})}\), \(\varphi_{2}(x,t) \in W^{1,p}_{\alpha}\) for any given t, and \(\vert \varphi_{2}(x,t) \vert \leq c\) for any given x. If the initial value (1.7) is satisfied in the sense of
then we say \(u(x,t)\) is a solution of Eq. (1.6) with the initial condition (1.7).
Theorem 1.2
If \(p>2\) and \(\alpha<\frac{p-2}{2}\), for any \(i \in\{ 1,2,\ldots, N\} \), \(b_{i}(s,x,t)\) is a \(C^{1}\) function, and there are constants β, c such that
then there is a solution of Eq. (1.6) with the initial value (1.7).
Certainly, we suggest that the conditions in Theorem 1.2 are not the optimal, we only provide a basic result of the existence here. The main aim of this paper is to research the stability of the weak solutions.
Theorem 1.3
Let \(\alpha>p-1>0\), \(b_{i}\) satisfy
If u and v are two solutions of Eq. (1.6) with the initial values \(u_{0}(x)\) and \(v_{0}(x)\), respectively, then
Remark 1.4
If \(\alpha< p-1\), we can prove the stability of the weak solutions for the initial-boundary value problem (1.6), (1.7), and (1.8) in a standard way [6]. We ask whether the spatial variable x in the nonlinear convection term \(b_{i}(u,x,t)\) can bring about the essential change. In particular, when \(b_{i}(s,x,t)\equiv0\), then only if \(\alpha\geq p-1\), Yin and Wang [3] had shown that
Without the condition (1.15), we can prove a result of the local stability of the weak solutions. This is the following theorem.
Theorem 1.5
Let \(p>1\), \(b_{i}(s,x,t)\) be a Lipschitz function. If u and v are two solutions of Eq. (1.6), then there exists a constant β large enough such that
Theorem 1.5 implies that the uniqueness of the weak solutions is true only if \(\alpha>0\). When \(b_{i}(u,x,t)=b_{i}(x)D_{i}u\), i.e., the convection term is just linear, Theorem 1.5 had been proved in paper [7]. When \(b_{i}(u,x,t)=b_{i}(u)\), Theorem 1.5 had been proved in [8] very recently. For the sake of simplicity, we will not give the details of the proof of Theorem 1.5 in this paper.
Once more, by introducing a new kind of the weak solutions, choosing a suitable test function, we can prove the following theorems.
Theorem 1.6
Let \(\alpha>p-1\), \(p>2\), \(b_{i}\) satisfying
If u and v are two solutions of Eq. (1.6) with the initial values \(u_{0}(x)\) and \(v_{0}(x)\), respectively, then
Theorem 1.6 seems just a minor version of Theorem 1.3. However, on the right hand side of (1.19), there is no constant c as in (1.16).
Last but no the least, we will prove the stability of the solutions based on a partial boundary value condition.
Theorem 1.7
Let \(b(s,x,t)\) be a Lipschitz function, u and v be two weak solutions of Eq. (1.6) with the same partial homogeneous boundary value
If
and there is nonnegative function \(a_{i}(x)\) such that
then
Here,
The paper is arranged as follows. In Sect. 1, we have given the basic definition and introduced the main results. In Sect. 2, we prove the existence of the solution to Eq. (1.6) with initial value (1.7). In Sect. 3, we prove Theorem 1.3. In Sect. 4, we give another kind of the weak solutions. By this new definition, we can prove Theorem 1.6. In Sect. 5, we will prove Theorem 1.7. In Sect. 7, we will give an explanation of the reasonableness of the partial boundary value condition.
2 The proof of existence
Consider the regularized equation
with the initial boundary conditions
Here, \(u_{0\varepsilon} \in C^{\infty}_{0}(\Omega)\) and \(u_{0 \varepsilon}\) converges to \(u_{0}\) in \(W_{0}^{1,p}(\Omega)\).
Proof of Theorem 1.2
Similar to [9], we can easily prove that there exists a weak solution \({u_{\varepsilon}}\in L^{\infty}(0,T;W ^{1,p}_{0}(\Omega))\) of the initial-boundary value problem (2.1)–(2.3),
Multiplying (2.1) by \(u_{\varepsilon}\) and integrating it over \(Q_{T}\), by the fact
we are able to deduce that
Then
for any \(\Omega_{\lambda}=\{x\in\Omega, d(x,\partial\Omega)> \lambda\}\subseteq\Omega\), λ being a small constant.
Multiplying (2.5) by \(u_{\varepsilon t}\), integrating it over \(Q_{T}\), then it yields
Notice that
Thus,
By condition (1.13),
By Hölder’s inequality and \(\alpha\leq\frac{p-2}{2}\),
Combining (2.7)–(2.10), we have
by the inequality, we have
Hence, by (2.4), (2.6), (2.11), there exist a function u and a n-dimensional vector \(\overrightarrow{\zeta}= ({\zeta_{1}},\ldots,{\zeta_{n}})\) satisfying
and \(u_{\varepsilon}\rightarrow u\) a.e. \(\in Q_{T}\),
Here, if \(p\geq2\), \(r=2\), while \(1< p<2\), \(1< r<\frac{Np}{N-p}\).
In order to prove that u is the solution of Eq. (1.6), for any function \(\varphi\in C_{0}^{1} ({Q_{T}})\), we have
we let \(\varepsilon\rightarrow0\).
Since as \(\varepsilon\rightarrow0\), by \(d(x)>0\), \(x\in\Omega\), then \(c>\sup_{\operatorname{supp} \varphi}\frac{ \vert \nabla\varphi \vert }{d^{\alpha}}>0\) due to \(\varphi\in C_{0}^{1} ({Q_{T}})\), we have
By this note, we have
Now, similar to the general evolutionary p-Laplician equation [6], we are able to prove that (the details are omitted here)
and
for any function \(\varphi\in C_{0}^{1} ({Q_{T}})\). Then
If for any given \(t\in[0, T)\), we denote \(\Omega_{\varphi}=\operatorname{supp} \varphi\), then
Now, for any \(\varphi_{1}\in C_{0}^{1} ({Q_{T}})\), \(\varphi_{2}(x,t) \in W^{1,p}_{\alpha}\) for any given t, and \(\vert \varphi_{2}(x,t) \vert \leq c\) for any given x, it is clear that \(\varphi_{2}\in W^{1, p}( \Omega_{\varphi_{1}})\). By the fact that \(C_{0}^{\infty}( \Omega_{\varphi_{1}})\) is dense in \(W^{1, p}(\Omega_{\varphi_{1}})\), by a process of limits, we have
which implies that
Then u satisfies Eq. (1.6) in the sense of Definition 1.1. □
3 Proof of Theorem 1.3
Proof
Let u and v be two weak solutions of Eq. (1.6) with the initial values \(u_{0}(x)\) and \(v_{0}(x)\), respectively. For large enough \(n>0\), let
Obviously \(h_{n}(s)\in C(\mathbb{R})\), and
We define
Since for any given t, \(\varphi_{1}=g_{n}(u-v)\in W^{1,p}_{\alpha}\), by a process of limit, we can choose \(d_{n}{g_{n}}(u - v)\) as the test function, then
Thus
Denoting \(D_{n}=\{x\in\Omega:d(x)>\frac{1}{n}\}\), \(q=\frac{p}{p-1}\), clearly
which goes to zero since that \(\alpha>p-1\).
By this fact, \(\vert \nabla d_{n} \vert =n\), \(x\in{\Omega\setminus D_{n}}\), we have
which goes to 0 as \(n\rightarrow0\).
Once more, since
by the Lebesgue dominated convergence theorem, we have
Once again,
Now, let \(n\rightarrow\infty\) in (3.3). Then
It implies that
Theorem 1.3 is proved. □
4 Another kind of weak solution
In this section, we introduce another kind of weak solution and prove another stability theorem.
Definition 4.1
If a function \(u(x,t)\) satisfies (1.10), and
for \(\varphi\in{ C_{0}^{1} ({Q_{T}})}\), \(g(s)\) is a \(C^{1}\) function with \(g(0)=0\), the initial value (1.7) is satisfied in the sense of (1.12), then we say \(u(x,t)\) is a weak solution of Eq. (1.6) with the initial value (1.7).
Only if we choose \(\varphi_{1}=g(\varphi)\), \(\varphi_{2}=1\) in Definition 1.1, one can obtain the existence of the weak solutions in the sense of Definition 4.1.
Theorem 4.2
If \(b_{i}\) is a Lipchitz function,
and one of the following conditions is true:
-
(i)
\(\alpha\geq p\);
-
(ii)
\(p>\alpha>p-1\), \(p>2\);
then the stability
is true for the solutions u and v with the initial values \(u_{0}(x)\) and \(v_{0}(x)\), respectively.
Proof
By a process of limit, we may choose \(\varphi= \chi_{[\tau,s]}g_{n}(d^{\beta}(u-v))\) as a test function, where β is a constant to be chosen later. Then
Now, let us calculate every term in (4.5). For the first term on the right hand side of (4.5),
Clearly,
By the fact that \(\vert \nabla d \vert =1\) is true almost everywhere, \(\alpha>p-1\), we have
accordingly, using the Lebesgue dominated convergent theorem and the limit \(\lim_{n \rightarrow\infty}s h_{n}(s)=0\), we have
which goes to zero as \(n\rightarrow\infty\).
As for the second term on the right hand side of (5.5),
Since for any given \((x,t)\), \(b_{i}(s,x,t)\) is a Lipschitz function, \(u,v\in L^{\infty}(Q_{T})\), we have
which goes to zero when \(n\rightarrow0\). This is due to \([b_{i}(u,x,t)-b _{i}(v,x,t)]d^{-1}(x)\in L^{1}(Q_{T})\) by (4.2)–(4.3), using the Lebesgue dominated convergent theorem in (4.10) and using \(\lim_{n \rightarrow\infty}sh_{n}(s)=0\) again.
Meanwhile, also using the dominated convergent theorem, we have
which goes to zero provided that one of the conditions (i) and (ii) is true. Here \(q=\frac{p}{p-1}\) as usual.
At last,
Then
The proof is complete. □
Proof of Theorem 1.6
Since \(\alpha>p-1\), \(p>2\) and the condition (1.18) in Theorem 1.6, one can see that (4.2)–(4.3) are all right. Thus, Theorem 1.6 is true. □
5 Proof of Theorem 1.7
Proof
For a small positive constant \(\lambda>0\), define
where
Then
u and v are two weak solutions of Eq. (1.6) with the same partial homogeneous boundary value (1.20) and with the different initial values \(u_{0}(x)\) and \(v_{0}(x)\), respectively. According to Definition 4.1, we choose \({g_{n}}(\phi(u - v))\) as the test function. Thus
For the terms on the left hand side of (5.2),
By the fact that
using the Young inequality, we have
which goes to 0 as \(\lambda\rightarrow0\), by \(p-1>\alpha\geq \frac{p-1}{p-2}\), implying
Meanwhile,
We use \(\vert b_{i}(u,x,t)-b_{i}(v,x,t) \vert \leq a_{i}(x) \vert u-v \vert \). According to the definition of the trace, by the partial boundary value condition (1.6),
and
we have
Moreover, as in [10], we can prove that
In detail,
Since \(\alpha< p-1\), \(\vert b_{i}(u,x,t)-b_{i}(v,x,t) \vert \leq c \vert u-v \vert \),
If \(\{ x \in\Omega: \vert u - v \vert = 0\}\) is a set with 0 measure, then
If the set \(\{ x \in\Omega: \vert u - v \vert = 0\}\) has a positive measure, then
Therefore, in both cases, (5.10) goes to 0 as \(\eta\rightarrow0\).
Now, after letting \(\lambda\rightarrow0\), let \(n\rightarrow\infty \) in (5.2). Then, by (5.3), (5.4), (5.6), (5.8), and (5.9), we have
by the Gronwall inequality, we have
Theorem 1.7 is proved. □
6 The partial boundary condition
Let us simply review Fichera–Oleǐnik theory. For a linear degenerate elliptic equation,
the symmetric matrix \(( a^{rs}(x) ) \) has nonnegative characteristic value, to study its well-posedness problem, one only needs to give a partial boundary condition. In detail, let \(\{n_{s}\}\) be the unit inner normal vector of ∂Ω̃ and denote
Then, to ensure the well-posedness of Eq. (1.7), Fichera–Oleǐnik theory tells us that the suitable boundary condition is
In particular, if the matrix \((a^{rs})\) is positive definite, (6.2) is just the usual Dirichlet boundary condition. Considering the classical parabolic equation
with the matrix \((a^{ij})\) is positive definite, besides the initial condition
only a parabolic boundary value condition
is imposed. However, for Eq. (1.6) considered in this paper, since the equations are strongly nonlinear and degenerate, including the extremely case of \(a\equiv0\), Fichera–Oleǐnik theory is invalid, the corresponding problem becomes more complicated. To show that the partial boundary value condition imposed on the main equation (1.6) is reasonable, we can come back to the linear case. In other words, let us suppose that \(p=2\) and
Then Eq. (1.6) has the form
where \(\overrightarrow{a}=\{a_{i}\}\). According to Fichera–Oleǐnik theory, the optional boundary value condition is
with
where \(\vec{n}=\{n_{i}\}\) is the inner normal vector of Ω.
Now, by reviewing the partial boundary value condition (1.24)
we have found
Though the condition (1.24) may be not the optimal, it is reasonable.
7 Conclusion
Besides the diffusion coefficient \(d^{\alpha}\) being degenerate on the boundary, Eq. (1.6) has a convection term \(\sum_{i=1}^{N}\frac{\partial b_{i}(u,x,t)}{\partial x_{i}}\), which depends on the spatial variable x. Such a characteristic can bring about essential changes on the boundary value condition. A reasonable partial boundary value condition is proposed for the first time, the stability of the weak solutions based on this partial boundary value condition is established. One can see that, if the convection term is independent of the spatial variable x, putting up a reasonable partial boundary condition becomes more difficult. We hope we can solve this problem in our follow-up work.
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The paper is supported by Natural Science Foundation of Fujian province, supported by Science Foundation of Xiamen University of Technology, China.
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Zhan, H. On the evolutionary p-Laplacian equation with a partial boundary value condition. J Inequal Appl 2018, 227 (2018). https://doi.org/10.1186/s13660-018-1820-x
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DOI: https://doi.org/10.1186/s13660-018-1820-x