Qualitative Properties of Nonnegative Solutions for a Doubly Nonlinear Problem with Variable Exponents

Abstract

We consider the Dirichlet initial boundary value problem ${\partial }_t{u}^{m(x)}-\mathrm{div}\left({\left|\nabla u\right|}^{p\left(x,t\right)-\mathrm{2}}\nabla u\right)=a\left(x,t\right)u^{q(x,t)}$, where the exponents $p(x,t)>\mathrm{1}$, $q(x,t)>\mathrm{0}$, and $m(x)>\mathrm{0}$ are given functions. We assume that $a(x,t)$ is a bounded function. The aim of this paper is to deal with some qualitative properties of the solutions. Firstly, we prove that if , then any weak solution will be extinct in finite time when the initial data is small enough. Otherwise, when , we get the positivity of solutions for large $t$. In the second part, we investigate the property of propagation from the initial data. For this purpose, we give a precise estimation of the support of the solution under the conditions that and either $q(x,t)=m(x)$ or $a(x,t)\le \mathrm{0}$ a.e. Finally, we give a uniform localization of the support of solutions for all $t>\mathrm{0}$, in the case where a.e. and .