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2018 Qualitative Properties of Nonnegative Solutions for a Doubly Nonlinear Problem with Variable Exponents
Zakariya Chaouai, Abderrahmane El Hachimi
Abstr. Appl. Anal. 2018: 1-14 (2018). DOI: 10.1155/2018/3821217

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 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 ) < a 1 < 0 a.e. and ess sup q x , t < ess inf p ( x , t ) - 1 .

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Zakariya Chaouai. Abderrahmane El Hachimi. "Qualitative Properties of Nonnegative Solutions for a Doubly Nonlinear Problem with Variable Exponents." Abstr. Appl. Anal. 2018 1 - 14, 2018. https://doi.org/10.1155/2018/3821217

Information

Received: 18 July 2018; Revised: 3 October 2018; Accepted: 11 October 2018; Published: 2018
First available in Project Euclid: 14 December 2018

zbMATH: 07029286
MathSciNet: MR3877884
Digital Object Identifier: 10.1155/2018/3821217

Rights: Copyright © 2018 Hindawi

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