Heat–wave interaction in 2–3 dimensions: Optimal rational decay rate

Abstract

In this paper, we consider a simplified version of a fluid–structure PDE model which has been of longstanding interest within the mathematical and biological sciences. In it, a n-dimensional heat equation replaces the original Stokes system, so as to ultimately have a vector-valued heat equation and vector-valued wave equation compose the coupled PDE system under study. The coupling between the two disparate PDE components occurs across a boundary interface. As such, the entire PDE dynamics manifests features of both hyperbolicity and parabolicity. For this heat–structure system, our main result of uniform stability is as follows: Given smooth initial data – i.e., data in the domain of the underlying semigroup generator of the coupled PDE system – the corresponding solutions decay at the rate of o(t−1). This establishes the long-conjectured optimal rate. The problem of obtaining sharp rational decay rates for the heat–wave PDE, under present consideration, has been a much considered problem, with the modus operandi of earlier efforts taking place within the time domain. By contrast, we adopt here a frequency domain approach which is based upon a recent resolvent criterion, and which was initiated in our prior effort, wherein we obtained the rate of decay o(1t). The present optimal improvement o(t−1) is achieved by employing an additional tool in our analysis – a microlocal analysis argument – to estimate a critical term involving two problematic boundary traces.