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On a parabolic equation related to the p-Laplacian
Boundary Value Problems volume 2016, Article number: 78 (2016)
Abstract
Consider a parabolic equation related to the p-Laplacian. If the diffusion coefficient of the equation is degenerate on the boundary, no matter we can define the trace of the solution on the boundary or not, by choosing a suitable test function, the stability of the solutions always can be established without a boundary condition.
1 Introduction and the main results
Consider an equation related to the p-Laplacian,
where Ω is a bounded domain in \(\mathbb{R}^{N}\) with appropriately smooth boundary, \(\rho(x) = \operatorname{dist} (x,\partial\Omega)\), \(p>1\), \(\alpha >0\). Yin and Wang [1] first studied the equation
and showed that, when \(\alpha>p-1\), the solution of the equation is completely controlled by the initial value.
The author studied in cooperation with Zhan and Yuan [2] the following equation:
and had shown that, to consider the well-posedness of equation (1.3), instead of the whole boundary condition
only the partial boundary condition
is necessary. Here, \(\Sigma _{p} \subseteq\partial\Omega\) is just a portion of ∂Ω, which is determined by the first order derivative term \(\frac{\partial b_{i}(u)}{\partial x_{i}}\), \(i=1, 2, \ldots , N\). Certainly, the initial value is always necessary,
In our paper, we will consider the well-posedness of the solutions of equation (1.1). First of all, we give some basic functional spaces. For every fixed \(t\in[0, T]\), we introduce the Banach space
and denote by \(V'_{t}(\Omega)\) its dual. By \(\mathbf{W}(Q_{T})\) we denote the Banach space
\(\mathbf{W}'(Q_{T})\) is the dual of \(\mathbf{W}(Q_{T})\) (the space of linear functionals over \(\mathbf{W}(Q_{T})\)):
The norm in \(\mathbf{W}'(Q_{T})\) is defined by
Definition 1.1
A function \(u(x,t)\) is said to be a weak solution of equation (1.1) with the initial value (1.6), if
and for any function \(\varphi \in C_{0}^{\infty}({Q_{T}})\), we have
The initial value, as usual, is satisfied in the sense that
We can easily obtain the existence of the weak solution.
Theorem 1.2
Let us suppose \(1< p\), \(0<\alpha\), \(f(s,x,t)\) is a Lipschitz function. If
then equation (1.1) with initial value (1.6) has a weak solution u in the sense of Definition 1.1.
We mainly are concerned with the stability of the solutions. As in [1–3], due to the fact that the weak solution defined in our paper satisfies (1.7), when \(\alpha< p-1\), we can define the trace of u on the boundary, while for \(\alpha\geq p-1\), the obtained weak solution lacks the regularity to define the trace on the boundary. However, in the short paper, by choosing a suitable test function, we can obtain the stability of the weak solutions without the boundary condition only if \(\alpha>0\). In other words, whether the weak solution is regular enough to define the trace on the boundary is not so important.
The main result of our paper is the following theorem.
Theorem 1.3
Let u and v be two weak solutions of equation (1.1) with the different initial values \(u(x,0)=v(x,0)\), respectively. If \(\alpha>0\) and \(f(s,x,t)\) is a Lipschitz function, then
Comparing with [1], the greatest improvement lies in that we do not require any boundary condition, no matter that \(\alpha< p-1\) or \(\alpha \geq p-1\). At the same time, the nonlinear source term \(f(u,x,t)\) adds the difficulty when we use the compact convergence theorem. The proof of the existence of the weak solution is quite different from that in [1]. Moreover, we consider the following equation, which seems similar to our equation (1.1):
which has been studied thoroughly for a long time, one may refer to [4–7]. Generally, to the growth order of u in \(f(u,x,t)\) should be added some restrictions. Very recently, Benedikt et al. [8] have studied the equation
with \(0<\gamma<1\), and shown that the uniqueness of the solutions of equation (1.13) is not true. From the short comment, one can see that the degeneracy of the coefficient \(\rho^{\alpha}\) plays an important role in the well-posedness of the solutions, it even can eliminate the action from the source term \(f(u,x,t)\). By the way, the author has been interested in the boundary value condition of a degenerate parabolic equation for some time, one may refer to [9].
2 The proof of Theorem 1.2
By [10] and [11], we have the following lemma.
Lemma 2.1
Let \(q\geq1\). If \(u_{\varepsilon}\in L^{\infty }(0,T;L^{2}(\Omega))\cap\mathbf{W}(Q_{T})\), \(\| u_{\varepsilon t}\| _{\mathbf{W}'(Q_{T})}\leq c\), \(\|\nabla(|u_{\varepsilon }|^{q-1}\times u_{\varepsilon})\|_{p,Q_{T}}\leq c\), then there is a subsequence of \(\{u_{\varepsilon}\}\) which is relatively compact in \(L^{s}(Q_{T})\) with \(s\in(1,\infty)\).
To study equation (1.1), since f is Lipschitz, without loss the generality, we may assume that \(f\in C^{1}\). We consider, as e.g. in [12] by Ragusa, the associated regularized problem
where \({\rho_{\varepsilon}} = \rho\ast\delta_{\varepsilon}+ \varepsilon\), \(\varepsilon > 0\), \(\delta_{\varepsilon}\) is the mollifier as usual, \({u_{\varepsilon ,0}} \in{C^{\infty}_{0} }(\Omega)\) and \(\rho_{\varepsilon}^{\alpha}{ \vert {\nabla{u_{\varepsilon,0}}} \vert ^{p}}\in {L^{1}}(\Omega)\) is uniformly bounded, and \({u_{\varepsilon,0}}\) converges to \(u_{0}\) in \(W_{0}^{1,p}(\Omega)\). It is well known that the above problem has a unique classical solution [12–14].
Lemma 2.2
There is a subsequence of \({u_{\varepsilon}}\) (we still denote it as \({u_{\varepsilon}}\)), which converges to a weak solution u of equation (1.1) with the initial value (1.5).
Proof
By the maximum principle, there is a constant c only dependent on \({\Vert {{u_{0}}} \Vert _{{L^{\infty}}(\Omega)}}\) but independent on ε, such that
Multiplying (2.1) by \(u_{\varepsilon}\) and integrating it over \(Q_{T}\), we get
By (2.4), (2.5), and the assumption that f is \(C^{1}\), we have
For small enough \(\lambda>0\), let \(\Omega_{\lambda}=\{x\in\Omega: \operatorname{dist}(x,\partial\Omega)>\lambda\}\). Since \(p>1\), by (2.6),
Now, for any \(v\in\mathbf{W}(Q_{T})\), \(\|v\|_{W(Q_{T})}=1\),
by the Young inequality, we can show that
then
Now, let \(\varphi\in C_{0}^{1}(\Omega)\), \(0\leq\varphi\leq1\), such that
Then
we have
and so
By Lemma 2.1, \(\varphi u_{\varepsilon}\) is relatively compact in \(L^{s}(Q_{T})\) with \(s\in(1,\infty)\). Then \(\varphi u_{\varepsilon }\rightarrow\varphi u\) a.e. in \(Q_{T}\). In particular, due to the arbitrariness of λ, \(u_{\varepsilon}\rightarrow u\) a.e. in \(Q_{T}\).
Hence, by (2.4), (2.6), (2.8), there exist a function u and an n-dimensional vector function \(\overrightarrow{\zeta}= ({\zeta_{1}}, \ldots,{\zeta_{n}})\) satisfying
and
In order to prove u satisfies equation (1.1), we notice that, for any function \(\varphi \in C_{0}^{\infty}({Q_{T}})\),
and \(u_{\varepsilon} \rightarrow u\) is almost everywhere convergent, so \(f(u_{\varepsilon},x,t)\rightarrow f(u,x,t)\) is true. Then
Now, similar to [4] or [15], we can prove
for any function \(\varphi \in C_{0}^{\infty}({Q_{T}})\), we omit the details here. Thus u satisfies equation (1.1).
Finally, we can prove (1.9) as in [11], we also omit the details here. Then Lemma 2.2 is proved. □
3 The stability of the solutions
As we have said before, by choosing a suitable test function, we can prove the stability of the solutions without any boundary value condition only if \(\alpha>0\).
Proof of Theorem 1.3
For any given positive integer n, let \({g_{n}}(s)\) be an odd function, and
Clearly,
and
where c is independent of n.
Let \(\beta\leq\frac{\alpha}{p}\) and
By taking the limit, we can choose \({g_{n}}(\phi(u - v))\) as the test function, then
Thus
Let \(n\rightarrow\infty\). If \(\{ x \in\Omega:\rho^{\beta}|u - v| = 0\} \) is a set with 0 measure, then
If \(\{ x \in\Omega:\rho^{\beta}|u - v| = 0\}\) is a set with positive measure, then
Both cases lead to the right hand side of (3.8) going to 0 as \(n\rightarrow\infty\). Meanwhile,
Now, let \(n\rightarrow\infty\) in (3.5). Then
It implies that
Theorem 1.3 is proved. □
References
Yin, J, Wang, C: Properties of the boundary flux of a singular diffusion process. Chin. Ann. Math., Ser. B 25(2), 175-182 (2004)
Zhan, H, Yuan, H: A diffusion convection equation with degeneracy on the boundary. J. Jilin Univ. Sci. Ed. 53(3), 353-358 (2015) (in Chinese)
Zhan, H: The boundary value condition of an evolutionary \(p(x)\)-Laplacian equation. Bound. Value Probl. 2015, 112 (2015). doi:10.1186/s13661-015-0377-6
Zhao, JN: Existence and nonexistence of solutions for \({u_{t}} =div({| {\nabla u} |^{p - 2}}\nabla u) + f(\nabla u,u,x,t)\). J. Math. Anal. Appl. 172(1), 130-146 (1993)
Wang, J, Gao, W, Su, M: Periodic solutions of non-Newtonian polytropic filtration equations with nonlinear sources. Appl. Math. Comput. 216, 1996-2009 (2010)
Lee, K, Petrosyan, A, Vazquez, JL: Large time geometric properties of solutions of the evolution p-Laplacian equation. J. Differ. Equ. 229, 389-411 (2006)
Yin, J, Wang, C: Evolutionary weighted p-Laplacian with boundary degeneracy. J. Differ. Equ. 237, 421-445 (2007)
Benedikt, J, Girg, P, Kotrla, L, Takáč, P: Nonuniqueness and multi-bump solutions in parabolic problems with the p-Laplacian. J. Differ. Equ. 260, 991-1009 (2016)
Zhan, H: The solutions of a hyperbolic-parabolic mixed type equation on half-space domain. J. Differ. Equ. 259, 1449-1481 (2015)
Antontsev, SN, Shmarev, SI: Anisotropic parabolic equations with variable nonlinearity. Publ. Mat. 53, 355-399 (2009)
Antontsev, SN, Shmarev, SI: Parabolic equations with double variable nonlinearities. Math. Comput. Simul. 81, 2018-2032 (2011)
Ragusa, MA: Cauchy-Dirichlet problem associated to divergence form parabolic equations. Commun. Contemp. Math. 6(3), 377-393 (2004)
Gu, L: Second Order Parabolic Partial Differential Equations. The Publishing Company of Xiamen University, Xiamen (2002) (in Chinese)
Taylor, ME: Partial Differential Equations III. Springer, Berlin (1999)
Zhan, H: The solution of convection-diffusion equation. Chin. Ann. Math. 34(2), 235-256 (2013)
Acknowledgements
The paper is supported by NSF of China (no. 11371297) and supported by NSF of Fujian Province (no. 2015J01592), China.
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Zhan, H. On a parabolic equation related to the p-Laplacian. Bound Value Probl 2016, 78 (2016). https://doi.org/10.1186/s13661-016-0587-6
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DOI: https://doi.org/10.1186/s13661-016-0587-6
MSC
- 35K55
- 46E35
- 35R35
Keywords
- p-Laplacian
- boundary degeneracy
- stability