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

Abstract EN

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 esssupp(x,t)-1

Otherwise, when esssupm(x)

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 esssupm(x)

Finally, we give a uniform localization of the support of solutions for all t>0, in the case where a(x,t)

and esssupqx,t