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

Abstract

We consider the Dirichlet initial boundary value problem ∂tum(x)-div⁡∇upx,t-2∇u=ax,tuq(x,t), where the exponents p(x,t)>1, q(x,t)>0, and m(x)>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 ess⁡sup⁡p(x,t)-1<ess⁡inf⁡m(x), then any weak solution will be extinct in finite time when the initial data is small enough. Otherwise, when ess⁡sup⁡m(x)<ess⁡inf⁡p(x,t)-1, 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 ess⁡sup⁡m(x)<ess⁡inf⁡p(x,t)-1 and either q(x,t)=m(x) or a(x,t)≤0 a.e. Finally, we give a uniform localization of the support of solutions for all t>0, in the case where a(x,t)<a1<0 a.e. and ess⁡sup⁡qx,t<ess⁡inf⁡p(x,t)-1.

Semidiscretization for a Doubly Nonlinear Parabolic Equation Related to the p(x)-Laplacian

“…Existence results and qualitative properties concerning the solutions of the continuous problem (1) and more general problems have been obtained by many authors in the last decade. We cote the papers [1][2][3][4][5][6]12] and the references therein.…”