Qualitative Properties of Nonnegative Solutions for a Doubly Nonlinear Problem with Variable Exponents. (8th November 2018) - Chaouai, Zakariya; El Hachimi, Abderrahmane
Abstract : We consider the Dirichlet initial boundary value problem∂ t u m ( x ) - div ⁡ ∇ u p x, t - 2 ∇ u = a x, t u q ( x, t ), where the exponentsp ( x, t ) > 1, q ( x, t ) > 0, andm ( x ) > 0 are given functions. We assume thata ( 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 ifess ⁡ 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, wheness ⁡ sup ⁡ m ( x ) < ess ⁡ inf ⁡ p ( x, t ) - 1, we get the positivity of solutions for larget . 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 thatess ⁡ sup ⁡ m ( x ) < ess ⁡ inf ⁡ p ( x, t ) - 1 and eitherq ( x, t ) = m ( x ) ora ( x, t ) ≤ 0 a.e. Finally, we give a uniform localization of the support of solutions for allt > 0, in the case wherea ( x, t ) < a 1 < 0 a.e. andess ⁡ sup ⁡ q x, t < ess ⁡ inf ⁡ p ( x, t ) - 1 .