A Generalized Interpolation Inequality and its Application to the Stabilization of Damped Equations

TL;DR

This paper introduces a generalized interpolation inequality for weighted spaces with non-homogeneous weights and applies it to derive explicit decay rates for the stabilization of damped wave equations, including pointwise damping models.

Contribution

It presents a new generalized interpolation inequality and demonstrates its application to stabilize damped wave equations with explicit decay estimates.

Findings

Established a generalized Hölder's inequality for weighted spaces.

Derived explicit decay rates for damped wave equations.

Applied results to pointwise damping in 1D models.

Abstract

In this paper, we establish a generalized H{\"o}lder's or interpolation inequality for weighted spaces in which the weights are non-necessarily homogeneous. We apply it to the stabilization of some damped wave-like evolution equations. This allows obtaining explicit decay rates for smooth solutions for more general classes of damping operators. In particular, for $1-d$ models, we can give an explicit decay estimate for pointwise damping mechanisms supported on any strategic point.