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

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.