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.