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

The purpose of this paper is to obtain a fundamental $L^p-L^{p^{\mathrm{\prime }}}$ estimate for a class of a strongly damped wave equations where the damping operator is given by $-\delta \mathrm{\Delta }$ with $\delta \ge 0$ and the constant in the estimate is independent of the damping parameter $\delta \mathrm{.}$ 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.