Abstract
A parabolic equation related to the p-Laplacian is considered. If the equation is degenerate on the boundary, then demonstrating the regularity on the boundary is difficult, the trace on the boundary cannot be defined, in general. The existence and uniqueness of weak solutions are researched. Based on uniqueness, the stability of solutions can be proved without any boundary condition.
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1 Introduction and main results
Consider a parabolic equation related to the p-Laplacian
with the initial value
where Ω is a bounded domain in \(\mathbb{R}^{N}\) with appropriately smooth boundary, \(p>1\), \(u_{0}(x)\) is a \(C_{0}^{1}( \varOmega )\) function, \(a(u, x,t)\geq 0\). If \(a(u, x,t)=1\), Eq. (1.1) is the evolutionary p-Laplacian equation with a convective term
and the usual boundary condition
can be imposed. The initial-boundary boundary value problem of Eq. (1.3) has been studied in many monographs or textbooks, one can refer to [1,2,3] and the references therein. Benedikt et al. [4, 5] had studied the equation
with \(0<\alpha <1\), and such that there exists an \(x_{0}\in \varOmega \) satisfying \(q(x_{0})>0\). They showed that the uniqueness of a solution does not hold. Meanwhile, the author of [6] had studied the equation
with \(\alpha >0\), and has shown that the stability of solutions can be proved without any boundary condition, where \(d=d(x)=\operatorname{dist}(x,\partial \varOmega )\) is the distance function from the boundary and \(f(s,x,t)\) is a Lipschitz function. Certainly, \(|u|^{\alpha -1}u\) is not a Lipschitz function with respect to u, the result of [6] is compatible with those of [4, 5]. But then, the result of [6] shows that the degeneracy of the coefficient \(d^{\alpha }\) can eliminate the action from the source term \(f(u,x,t)\). Moreover, we have shown that a weak solution to the equation
is unique independent of the boundary value condition [7], and the stability of the weak solutions can be proved in some cases [8].
For a degenerate parabolic equation, the phenomenon that the solution is free from the limitation of the boundary condition has been studied for a long time, one can refer to [9,10,11,12,13,14,15]. Roughly speaking, instead of the whole boundary condition (1.4), we may conjecture that only a partial boundary condition
should be imposed, where \(\varSigma _{1}\) is a relatively open subset of ∂Ω. In this paper, we will show that a weak solution to Eq. (1.1) is unique independent of the boundary value condition. In other words, the degeneracy of the diffusion \(a(\cdot , x,t)\) on the boundary can take place regardless of the boundary value condition.
To simplify exposition, in what follows, we assume that
where \(r>0\) is a constant, \(\rho (x)\) is a \(C^{1}(\overline{\varOmega })\) nonnegative function and
Let
Then Eq. (1.1) becomes
where
The initial value matching up to Eq. (1.7) is
Definition 1.1
Function \(v(x,t)\) is said to be a weak solution of Eq. (1.7) with the initial value (1.8), if v satisfies
and, for any function \(\phi (x,t) \in {C_{0}^{1}}(Q_{T})\), there holds
If v is a weak solution of Eq. (1.7) with the initial value (1.8), then we say that \(u=|v|^{\beta -1}v\) is a weak solution of Eq. (1.1) with the initial value (1.2).
We will give a basic result of the existence of a weak solution.
Theorem 1.2
If \(p\geq 2\), \(u_{0}(x)\geq 0\), \(\rho (x) \mid _{x\in \partial \varOmega }=0\) and \(\int _{\varOmega }\rho (x)^{- \frac{2}{p-2}}\,dx<\infty \), for any given \(i\in \{1, 2, \dots , N\}\), \(a_{i}(s)\) is a \(C^{1}\) function and there exist constants α and c such that
then there is a nonnegative weak solution of Eq. (1.7) with the initial value (1.8) in the sense of Definition 1.1.
This theorem may not be optimal, the conditions \(p\geq 2\), \(\int _{ \varOmega }\rho (x)^{-\frac{2}{p-2}}\,dx<\infty \) and (1.11) may all be weakened. However, the main aim of this paper is to probe the uniqueness and stability of weak solutions, the main results of our paper are the following theorems.
Theorem 1.3
Let \(p>1\), \(a_{i}(s)\) be a Lipschitz function, and let \(\rho (x)\) satisfy (1.6). Then a weak solution of Eq. (1.7) with the initial value (1.8) is unique.
Theorem 1.4
Let \(u(x,t)\) and \(v(x,t)\) be two weak solutions of Eq. (1.7) with different initial values \(u_{0}(x)\) and \(v_{0}(x)\), respectively. If \(p\geq 2\),
and \(a_{i}(s)\) is a Lipschitz function, then
It is well-known that the usual evolutionary p-Laplacian equation needs to be subjected to the whole boundary condition (1.4) [2, 3]. Clearly, condition \(a(u, x,t)|_{x\in \partial \varOmega }=0\) excludes the usual evolutionary p-Laplacian equation, while condition (1.12) excludes the conservation law equation. The uniqueness of solutions for a conservation law equation only holds in the sense of the entropy solution [2]. The equations considered in [6,7,8, 16,17,18,19], as well as Eq. (1.1), have apparently different characteristics from both the usual evolutionary p-Laplacian equation and the conservation term. Roughly speaking, in the interior of Ω, Eq. (1.1) has the characteristic of the usual evolutionary p-Laplacian equation, while on the boundary ∂Ω, Eq. (1.1) has the characteristic of the conservation law equation. Comparing with our previous works [7, 17,18,19] and [6, 8], the main difficulty comes from the nonlinearity of the diffusion coefficient \(a(u,x,t)\). Moreover, unlike our previous works, the stability of the weak solutions is based on the uniqueness of the weak solution.
Theorem 1.3 shows that the uniqueness of the weak solution holds independently of the boundary value condition. Once we have the uniqueness of the weak solution, Theorem 1.4 shows that the stability of the weak solutions is also true without the boundary value condition. Accordingly, Theorems 1.3 and 1.4 show that not only the degeneracy of the coefficient \(a(u,x,t)\) can eliminate the action from the source term \(f(u,x,t)\) [6], but it may also eliminate the action of the convection term \(\sum_{i=1}^{N}\frac{\partial a_{i}(v)}{\partial x _{i}}\).
2 Existence of a solution
Consider an approximate problem of Eq. (1.7), namely
with the initial boundary value conditions
Definition 2.1
A function \(v(x,t)\) is said to be a weak solution of problem (2.1)–(2.3), if v satisfies
and for any \(\phi (x) \in {C_{0}^{1}}( Q_{T})\), there holds
For any \(k>0\), we define \(\varphi ^{+}_{k}(s)=\beta s^{\beta -1}\) when \(s\geq k^{-1}\), \(\varphi ^{+}_{k}(s)=\beta (a_{k}s^{2}+b_{k} s)\) when \(0\leq s< k^{-1}\), where
Extending \(\varphi ^{+}(s)\) to be an even function on the whole \(\mathbb{R}^{1}\), and denoting it as \(\varphi _{k}(s)\), we have \(\varphi _{k}(s)\in C^{1}\), \(\varphi _{k}(s)\rightarrow \beta s^{\beta -1}\), \(s\neq 0\) as \(k\rightarrow \infty \). By considering the following approximate problem:
where \(\|v_{0k}(x)-v_{0}(x)\|_{p}\rightarrow 0\) as \(k\rightarrow 0\) and \(|v|^{\beta -1}v_{0}(x)=u_{0}(x)\), we obtain that there is a unique classical solution \(v_{k\varepsilon }\) of problem (2.6)–(2.8). Let \(k\rightarrow \infty \). Similarly as in [20], we can prove that
and \(v_{\varepsilon }\) is a solution of problem (2.1)–(2.3) in the sense of Definition 2.1; we omit the details here. In particular, if \(u_{0}(x)\in L^{\infty }(\varOmega )\), then we have
where c is a constant independent of k and ε, but depending on \(\|u_{0}\|_{L^{\infty }(\varOmega )}\). In what follows, we call \(v_{\varepsilon }\) an asymptotic solution.
Proof of Theorem 1.2
Multiplying (2.1) by \(v_{\varepsilon }\) and integrating over \(Q_{T}\), we have
where \(\rho _{\varepsilon }=\rho +\varepsilon \). Using the fact
we have
and in particular,
For small enough \(\lambda >0\), let \(\varOmega _{\lambda }=\{x\in \varOmega : \rho (x)>\lambda \}\). Since \(p\geq 2\), by (1.6) and (2.10),
Multiplying (2.1) with \(v_{\varepsilon t}\), and integrating over Ω,
By the assumption of (1.11),
Here, we have used the assumption \(\int _{\varOmega }\rho (x)^{- \frac{2}{p-2}}\,dx<\infty \), which implies \(\int _{\varOmega }(\rho +\varepsilon )^{-\frac{2}{p-2}}\,dx<\infty \). Then
and
From (2.10), (2.11), and (2.16), one knows that
where \(Q_{\lambda T}=\varOmega _{\lambda }\times (0,T)\). Then \(v_{\varepsilon }\rightarrow v\) in \(L^{2}(Q_{\lambda T})\). By the arbitrariness of λ, \(v_{\varepsilon }\rightarrow v\) a.e. in \(Q_{T}\). Thus \(a_{i}(v_{\varepsilon })\rightarrow a_{i}(v)\) a.e. in \(Q_{T}\). □
We use (2.10), (2.15), and let \(\varepsilon \rightarrow 0\). Similarly as in [3, 20], we can prove that
so that \(a_{i}(v_{\varepsilon })\rightarrow a_{i}(v)\) a.e. in \(Q_{T}\), and then there is a solution of Eq. (1.7) with the initial value (1.8) in the sense of Definition 1.1.
3 The uniqueness
Theorem 3.1
Let \(u(x,t)\) and \(v(x,t)\) be two weak solutions of Eq. (1.7) with different initial values \(u_{0}(x)\) and \(v_{0}(x)\), respectively, \(0< m\leq \|u\|_{L^{\infty }(Q_{T})}\leq M\), \(0< m\leq \|v \|_{L^{\infty }(Q_{T})}\leq M\). Let \(p>1\), \(a_{i}(s)\) be a Lipschitz function, and let \(\rho (x)\) satisfy (1.6). Then there exists a constant \(\alpha _{1}\geq \max \{p, 2, 2(p-1)\}\) such that
Proof
Denote \(\varOmega _{\lambda }=\{x\in \varOmega : \rho (x)> \lambda \}\) as before. Let
For any fixed \(\tau ,s\in [0,T]\), we may choose \(\chi _{[\tau ,s]}(u _{\varepsilon }-v_{\varepsilon })\xi _{\lambda }\) as a test function in (3.1), where \(\chi _{[\tau ,s]}\) is the characteristic function on \([\tau ,s]\), where \(u_{\varepsilon }\) and \(v_{\varepsilon }\) are the mollified functions of the solutions u and v, respectively. Then, denoting \(Q_{\tau s}=\varOmega \times [\tau , s]\), we have
For any given small \(\lambda >0\), denoting \(Q_{T\lambda }= \varOmega _{\lambda }\times (0,T)\), since \(\rho (x)\in C^{1}(\overline{ \varOmega })\) and \(\rho (x)>0\) when \(x\in \varOmega \), then \(\nabla u\in L ^{p}(Q_{T\lambda })\), \(\nabla v\in L^{p}(Q_{T\lambda })\). According to the definition of the mollified functions \(u_{\varepsilon }\) and \(v_{\varepsilon }\), we have
Since on \(\varOmega _{\lambda }\), by Young inequality,
by (3.2), (3.4) and (3.5), using the Lebesgue dominated convergence theorem, we have
The first term on the right-hand side of (3.6) satisfies
The last term on the right-hand side of (3.6) can be bounded as follows:
Here, we have used the fact that \(|\nabla \rho (x)|\leq c\). Then,
If \(p\geq 2\), clearly, since \(u,v\in L^{\infty }\), \(|u-v|\leq c\), and we have
if \(1< p<2\), for \(\alpha _{1}\geq 2(p-1)\), using the Hölder inequality, we have
Meanwhile, by the Lebesgue dominated convergence theorem,
Due to the fact \(|\nabla \rho |\leq c\), \(\alpha _{1}\geq p\), we have
and
If \(1< p<2\), then \(p'>2\), and if \(\alpha _{1}\geq p\), when \(\rho <1\), then \(\rho ^{\frac{\alpha _{1}-1}{p-1}}\leq \rho ^{\frac{\alpha _{1}}{p}}\). When \(1\leq \rho \leq D=\sup_{x\in \overline{\varOmega }}\rho (x)\), it is obvious that
Thus, \(\rho ^{\frac{\alpha _{1}-1}{p-1}}\leq c\rho ^{ \frac{\alpha _{1}}{p}}\) is always true, and then we have
If \(p\geq 2\), then \(p'<2\), and for \(\alpha _{1}\geq 2\), by the Hölder inequality,
By (3.6)–(3.14), after letting \(\varepsilon \rightarrow 0\), we let \(\lambda \rightarrow 0\) in (3.3). Then
where \(q<1\). Now,
where \(\zeta \in (v,u)\).
If for any \(s\geq \tau \)
is true, then
clearly holds.
If there is an \(s_{0}\geq \tau \) such that
then by (3.16)
where \(\zeta \in (v,u)\), \(M=\max \{\|u\|_{L^{\infty }(Q_{T})}, \|v\|_{L^{\infty }(Q_{T})} \}\).
By (3.15), (3.16), and (3.19),
Then
Since
by (3.20), we have
Here \(m=\min \{\|u\|_{L^{\infty }(Q_{T})}, \|v\|_{L^{\infty }(Q _{T})} \}\). Inequality (3.21) implies
This inequality contradicts assumption (3.18).
Thus, for any \(s,\tau \in [0,T)\), (3.17) is always true. By the arbitrariness of τ, we have
The proof is complete. □
Since ρ satisfies (1.6), from (3.1), we can deduce Theorem 1.3.
4 The proof of Theorem 1.4
Proof
For any given positive integer n, let \({g_{n}}(s)\) be an odd function, and
Clearly,
and
where c is independent of n.
Let \(u_{\varepsilon }\) and \(v_{\varepsilon }\) be the asymptotic solutions of u and v, respectively. They satisfy the asymptotic problem (2.1)–(2.3). Since the weak solution of Eq. (1.7) with the initial value (1.8) is unique, we have
We can choose \(\chi _{[\tau ,s]}{g_{n}}(u_{\varepsilon } - v_{\varepsilon })\) as the test function. Then
At first, by (4.4), we have
Secondly, we have
Here, we have used two facts. The first one is, by (4.6),
The second one is, since \((u_{\varepsilon }-v_{\varepsilon })\rightarrow (u-v)\), a.e. in Ω,
using (4.6),
Thirdly, we have
Moreover, since
we have
Then
Based on it, we are able to prove that
In details, the limitation (4.9) is established by the following calculations:
Due to (1.12),
where \(\xi \in (v,u)\).
In (4.10), let \(n\rightarrow \infty \). If \(\{ x \in \varOmega :|u - v| = 0\}\) is a set of zero measure, by (4.11), then
If the set \(\{ x \in \varOmega :|u - v| = 0\}\) has positive measure, then
Therefore, in both cases, the right-hand side of inequality (4.10) goes to 0 as \(n\rightarrow \infty \). So (4.9) is true.
Now, letting \(n\rightarrow \infty \) in (4.5),
Then, by (4.6), (4.7), (4.8), (4.9) and (4.12), we have
It implies that
Theorem 1.4 is proved. □
5 Conclusions
The equation considered in this paper comes from many reaction–diffusion problems. If the diffusion coefficient not only depends on the unknown solution u, but also on the spatial variable x, the degeneracy of the equation becomes more complicated. If the diffusion coefficient is degenerate on the boundary, the usual Dirichlet boundary value condition seems redundant completely. The uniqueness of the weak solution is proved without any boundary value conditions. Based on this fact, the stability of weak solutions can also be proved without any boundary value conditions.
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Zhan, H. Uniqueness and stability of solutions without the boundary condition. Bound Value Probl 2019, 21 (2019). https://doi.org/10.1186/s13661-019-1139-7
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DOI: https://doi.org/10.1186/s13661-019-1139-7