Optimal decay for one-dimensional damped wave equations with potentials via a variant of Nash inequality

TL;DR

This paper establishes sharp decay estimates for one-dimensional damped wave equations with potentials by employing a variant of Nash inequality and a test function method, enhancing understanding of their long-term behavior.

Contribution

It introduces a novel approach using a Nash-type inequality involving an invariant measure to derive optimal decay rates for damped wave equations with potentials.

Findings

Derived sharp decay estimates from above and below.

Connected decay properties to diffusion phenomena.

Utilized a Nash inequality variant with an invariant measure.

Abstract

The optimality of decay properties of the one-dimensional damped wave equations with potentials belonging to a certain class is discussed. The typical ingredient is a variant of Nash inequality which involves an invariant measure for the corresponding Schr\"odinger semigroup. This enables us to find a sharp decay estimate from above. Moreover, the use of a test function method with the Nash-type inequality provides the decay estimate from below. The diffusion phenomena for the damped wave equations with potentials are also considered.