Note on stability of an abstract coupled hyperbolic-parabolic system: singular case

TL;DR

This paper completes the stability analysis of a coupled hyperbolic-parabolic system by identifying polynomial stability in a specific parameter region, considering singularities at zero, extending previous results.

Contribution

It introduces a detailed stability characterization in the region where eta < 2 ext{alpha} - 1, accounting for singular behavior at zero, which was not previously analyzed.

Findings

Polynomial stability in the region S_3

Identification of a singularity at zero

Extension of stability results to new parameter range

Abstract

In this paper we try to complete the stability analysis for an abstract system of coupled hyperbolic and parabolic equations $$ \left\{ \begin{array}{lll} \ds u_{tt} + Au - A^\alpha w = 0, \\ w_t + A^\alpha u_t + A^\beta w = 0,\\ u(0) = u_0, u_t(0) = u_1, w(0) = w_0, \end{array} \right. $$ where $A$ is a self-adjoint, positive definite operator on a complex Hilbert space $H$, and $(\alpha, \beta) \in [0,1] \times [0,1]$, which is considered in \cite{Amk}, and after, in \cite{liu1}. Our contribution is to identify a fine scale of polynomial stability of the solution in the region $ S_3: = \left\{(\alpha,\beta) \in [0,1] \times [0,1]; \, \beta < 2\alpha -1 \right\}$ taking into account the presence of a singularity at zero.