Regularity Analysis for an Abstract System of Coupled Hyperbolic and Parabolic Equations

TL;DR

This paper conducts a comprehensive regularity analysis of a coupled hyperbolic-parabolic system in a Hilbert space, identifying parameter regions with different semigroup regularity properties and establishing the sharpness of Gevrey class orders.

Contribution

It provides a complete decomposition of parameter space into regions with analytic, Gevrey, and non-smoothing semigroup behaviors, including sharpness results for Gevrey orders.

Findings

Parameter space partitioned into three regularity regions.

Identified sharp Gevrey class orders under certain conditions.

Established non-smoothing behavior in specific parameter regions.

Abstract

In this paper, we provide a complete regularity analysis for an abstract system of coupled hyperbolic and parabolic equations in a complex Hilbert space. We are able to decompose the unit square of the parameters into three parts where the semigroup associated with the system is analytic, of specific order Gevrey classes, and non-smoothing, respectively. Moreover, we will show that the orders of Gevrey class are sharp, under proper conditions.