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

TL;DR

This paper establishes a fundamental $L^p-L^{p'}$ estimate for a class of strongly damped wave equations, with a damping operator involving the Laplacian, using semigroup estimates on Besov spaces, independent of the damping parameter.

Contribution

It provides an $L^p-L^{p'}$ estimate for strongly damped wave equations with damping operator $- abla^2$, independent of the damping coefficient, using semigroup methods on Besov spaces.

Findings

The estimate is independent of the damping parameter $ abla^2$.

The method employs semigroup estimates on Besov spaces.

The results apply to a broad class of strongly damped wave equations.

Abstract

The purpose of this paper is to obtain a fundamental $L^p-L^{p'}$ estimate for a class of a strongly damped wave equations where the damping operator is given by $-\delta \Delta$ with $\delta \geq 0$ 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.