Critical exponent for damped wave equations with nonlinear memory

Abstract

We consider the Cauchy problem in ${\mathbb{R}}^n,$ $n\ge 1,$ for a semilinear damped wave equation with nonlinear memory. Global existence and asymptotic behavior as $t\to \infty$ of small data solutions have been established in the case when $1\le n\le 3.$ Moreover, we derive a blow-up result under some positive data in any dimensional space.