On $L^p-L^{p^{\prime }}$ estimates for a class of strongly damped wave equations

Abstract

The purpose of this paper is to obtain a fundamental $L^p-L^{p^{\prime }}$ estimate for a class of a strongly damped wave equations where the damping operator is given by $-\delta \Delta$ with $\delta \ge 0$ and the constant in the estimate is independent of the damping parameter $\delta .$ The method used here is based on an estimate of the semigroup associated to the linear equation on the Besov spaces and their elementary properties.