Wellposedness, Spectral Analysis and Asymptotic Stability of a Multilayered Heat-Wave-Wave System

TL;DR

This paper establishes the well-posedness and asymptotic stability of a multilayered heat-wave system modeled as a fluid-structure interaction PDE, using semigroup theory and spectral analysis.

Contribution

It proves the well-posedness of a coupled heat-wave PDE system and demonstrates its strong stability through spectral analysis, extending understanding of fluid-structure interaction models.

Findings

The system generates a $C_0$-semigroup on a finite energy space.

Solutions tend asymptotically to zero for all initial data.

The spectrum of the generator does not intersect the imaginary axis.

Abstract

n this work we consider a multilayered heat-wave system where a 3-D heat equation is coupled with a 3-D wave equation via a 2-D interface whose dynamics is described by a 2-D wave equation. This system can be viewed as a simplification of a certain fluid-structure interaction (FSI) PDE model where the structure is of composite-type; namely it consists of a \textquotedblleft thin\textquotedblright\ layer and a \textquotedblleft thick\textquotedblright\ layer. We associate the wellposedness of the system with a strongly continuous semigroup and establish its asymptotic decay. Our first result is semigroup well-posedness for the (FSI) PDE dynamics. Utilizing here a Lumer-Phillips approach, we show that the fluid-structure system generates a $C_0$-semigroup on a chosen finite energy space of data. As our second result, we prove that the solution to the (FSI) dynamics generated by the $C_0$-semigroup tends asymptotically to the zero state for all initial data. That is, the semigroup of the (FSI) system is strongly stable. For this stability work, we analyze the spectrum of the generator $\mathbf A$ and show that the spectrum of $\mathbf A$ does not intersect the imaginary axis. \vskip.3cm \noindent \textbf{Key terms:} Fluid-structure interaction, heat-wave system, well-posedness, semigroup, strong stability.