Optimal energy decay in a one-dimensional coupled wave-heat system

TL;DR

This paper analyzes a one-dimensional coupled wave-heat system to determine the optimal energy decay rate of solutions, utilizing semigroup theory and resolvent estimates for a simplified and precise approach.

Contribution

It provides a sharp estimate for energy decay in a coupled wave-heat system using a novel application of semigroup resolvent bounds, simplifying previous methods.

Findings

Established the optimal decay rate of energy in the system.

Demonstrated the effectiveness of resolvent-based estimates for decay analysis.

Provided a simpler alternative to existing techniques in similar problems.

Abstract

We study a simple one-dimensional coupled wave-heat system and obtain a sharp estimate for the rate of energy decay of classical solutions. Our approach is based on the asymptotic theory of $C_0$-semigroups and in particular on a result due to Borichev and Tomilov (Math. Ann., 2010), which reduces the problem of estimating the rate of energy decay to finding a growth bound for the resolvent of the semigroup generator. This technique not only leads to an optimal result, it is also simpler than the methods used by other authors in similar situations.