Optimal energy decay for the wave-heat system on a rectangular domain

TL;DR

This paper establishes the precise polynomial decay rate of energy for solutions to a coupled wave-heat system on a rectangular domain, showing it decays like t^{-2/3} and that this rate is optimal.

Contribution

It proves the sharp decay rate of energy for the wave-heat system, improving upon general estimates and employing novel direct estimates and separation of variables techniques.

Findings

Energy decays like t^{-2/3} as t→∞

Decay rate is proven to be sharp

General estimates are suboptimal for this system

Abstract

We study the rate of energy decay for solutions of a coupled wave-heat system on a rectangular domain. Using techniques from the theory of $C_0$-semigroups, and in particular a well-known result due to Borichev and Tomilov, we prove that the energy of classical solutions decays like $t^{-2/3}$ as $t\to\infty$. This rate is moreover shown to be sharp. Our result implies in particular that a general estimate in the literature, which predicts at least logarithmic decay and is known to be best possible in general, is suboptimal in the special case under consideration here. Our strategy of proof involves direct estimates based on separation of variables and a refined version of the technique developed in our earlier paper for a one-dimensional wave-heat system.