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

TL;DR

This paper analyzes the energy decay rate in a one-dimensional coupled wave-heat-wave system using semigroup theory, extending previous work with sharper estimates based on resolvent growth bounds.

Contribution

It provides a sharp estimate for the energy decay rate in a coupled wave-heat-wave system by applying advanced semigroup asymptotic theory, extending prior related results.

Findings

Established a growth bound for the resolvent of the semigroup generator

Derived a sharp decay rate estimate for the system's energy

Extended previous theoretical frameworks to a more complex coupled system

Abstract

Harnessing the abstract power of the celebrated result due to Borichev and Tomilov (Math.\ Ann.\ 347:455--478, 2010, no.\ 2), we study the energy decay in a one-dimensional coupled wave-heat-wave system. We obtain a sharp estimate for the rate of energy decay of classical solutions by first proving a growth bound for the resolvent of the semigroup generator and then applying the asymptotic theory of $C_0$-semigroups. The present article can be naturally thought of as an extension of a recent paper by Batty, Paunonen, and Seifert (J.\ Evol.\ Equ.\ 16:649--664, 2016) which studied a similar wave-heat system via the same theoretical framework.