Electron. J. Differential Equations, Vol. 2019 (2019), No. 26, pp. 1-28.

Differential inclusion for the evolution p(x)-Laplacian with memory

Stanislav Antontsev, Sergey Shmarev, Jacson Simsen, Mariza Stefanello Simsen

Abstract:
We consider the evolution differential inclusion for a nonlocal operator that involves p(x)-Laplacian,
$$ u_t-\Delta_{p(x)} u-\int_0^{t}g(t-s)\Delta_{p(x)} u(x,s)\,ds\in \mathbf{F}(u) \quad \text{in } Q_T=\Omega\times (0,T), $$
where $\Omega\subset \mathbb{R}^{n}$, $n\geq 1$, is a bounded domain with Lipschitz-continuous boundary. The exponent p(x) is a given measurable function, $p^-\leq p(x)\leq p^+$ a.e. in $\Omega$ for some bounded constants $p^->\max\{1,\frac{2n}{n+2}\}$ and $p^+<\infty$. It is assumed that $g,g'\in L^2(0,T)$, and that the multivalued function $\mathbf{F}(\cdot)$ is globally Lipschitz, has convex closed values and $\mathbf{F}(0)\neq\emptyset$. We prove that the homogeneous Dirichlet problem has a local in time weak solution. Also we show that when $p^->2$ and $u\mathbf{F}(u)\subseteq \{v\in L^2(\Omega): v\leq \epsilon u^2\text{ a.e. in }\Omega\}$ with a sufficiently small $\epsilon>0$ the weak solution possesses the property of finite speed of propagation of disturbances from the initial data and may exhibit the waiting time property. Estimates on the evolution of the null-set of the solution are presented.

Submitted June 20, 2018. Published February 13, 2019.
Math Subject Classifications: 35R70, 35B99, 35K92, 45K05.
Key Words: Evolution p(x)-Laplacian; nonlocal equation; differential inclusion; finite speed of propagation; waiting time.

Show me the PDF file (403 KB), TEX file for this article.

Stanislav Antontsev
CMAF-CIO, University of Lisbon
Lisbon, Portugal.
email: antontsevsn@mail.ru
Sergey Shmarev
Department of Mathematics
University of Oviedo
c/Calvo Sotelo s/n, 33007
Oviedo, Spain
email: shmarev@uniovi.es
Jacson Simsen
Instituto de Matemática e Computação
Universidade Federal de Itajubá, Av. BPS n.1303
Bairro Pinheirinho, 37500-903
Itajubá, MG, Brasil
email: jacson@unifei.edu.br
Mariza Stefanello Simsen
Instituto de Matemática e Computação
Universidade Federal de Itajubá, Av. BPS n.1303,
Bairro Pinheirinho, 37500-903
Itajubá, MG, Brasil
email: mariza@unifei.edu.br

Return to the EJDE web page