Electronic Journal of Differential Equations (Feb 2019)
Differential inclusion for the evolution p(x)-Laplacian with memory
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 $p^+2$ 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.