Regularity analysis for an abstract system of coupled hyperbolic and parabolic equations

In this paper, we provide a complete regularity analysis for the following abstract system of coupled hyperbolic and parabolic equations{utt=−Au+γAαw,wt=−γAαut−kAβw,u(0)=u0,ut(0)=v0,w(0)=w0, where A is a self-adjoint, positive definite operator on a complex Hilbert space H, and (α,β)∈[0,1]×[0,1]. We are able to decompose the unit square of the parameter (α,β) 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 is sharp, under proper conditions.