# American Institute of Mathematical Sciences

doi: 10.3934/cpaa.2021057

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

 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

Received  October 2020 Revised  February 2021 Published  April 2021

Fund Project: 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

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.

Citation: Tayeb Hadj Kaddour, Michael Reissig. Global well-posedness for effectively damped wave models with nonlinear memory. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2021057
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