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.