Global well-posedness for effectively damped wave models with nonlinear memory

Tayeb Hadj Kaddour; Michael Reissig

doi:10.3934/cpaa.2021057

Article Contents

2021, Volume 20, Issue 5: 2039-2064. Doi: 10.3934/cpaa.2021057

This issue Previous Article A generalized complex Ginzburg-Landau equation: Global existence and stability results Next Article Multiplicity of solutions for a class of fractional elliptic problems with critical exponential growth and nonlocal Neumann condition

Global well-posedness for effectively damped wave models with nonlinear memory

Tayeb Hadj Kaddour^1,2, and
Michael Reissig^3, ,

1.
Department of Mathematics, Faculty of exact sciences and informatics, University of Chlef, P.O. Box 50, 02000, Ouled-Fares, Chlef, Algeria
2.
Laboratory of mechanic and energetic, University of Chlef, Algeria
3.
Faculty for Mathematics and Computer Science, TU Bergakademie Freiberg, Prüferstr. 9, 09596, Freiberg, Germany

^* Corresponding author
^* Corresponding author

Received: October 2020

Revised: February 2021

Early access: April 2021

Published: May 2021

The research of this paper is supported by DAAD, Erasmus+ Project between the Hassiba Benbouali University of Chlef and TU Bergakademie Freiberg, 2015-1-DE01-KA107-002026, during the stay of the first author at Technical University Bergakademie Freiberg within the periods April 2016 to June 2016, and a stay of one month April 2017 supported by Hassiba Benbouali University

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Abstract

In this paper, we study the Cauchy problem for a special family of effectively damped wave models with nonlinear memory on the right-hand side. Our goal is to prove global (in time) well-posedness results for Sobolev solutions. Due to the effective dissipation the model is parabolic like from the point of view of energy decay estimates of the corresponding linear Cauchy problem with vanishing right-hand side. For this reason there appears a Fujita type exponent as a threshold. Applying modern tools from Harmonic Analysis we prove several results by taking into consideration different regularity properties of the data.

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Mathematics Subject Classification: Primary: 35G25, 35B40; Secondary: 35B33.

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References

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