Stability and regularity of an abstract coupled hyperbolic system: A case of zero spectrum

In this paper, we analyze the decay rate and regularity of a coupled hyperbolic system given by {u_tt=−aA^γu+bA^αy_t,y_tt=−Ay−bA^αu_t−kA^βy_t,u(0)=u₀,u_t(0)=v₀,y(0)=y₀,y_t(0)=z₀, where A is a positive definite self-adjoint operator and a,b,k are positive constants. The system is parameterized by three parameters (α,β,γ) and we consider the case where α>[Formula presented], which presents a singularity at zero. Based on frequency domain analysis, we identify the optimal decay rate and the sharp order of Gevrey class of the semigroup associated with the system. Especially, we propose a clear relationship between the choice of parameters and the decay rates (or the orders of Gevrey class) for the system. Our analysis complements the stability analysis of the system previously discussed in [6], which was restricted to the region {(α,β,γ)|0<α≤[Formula presented],0≤β≤1,[Formula presented]≤γ≤2}. We extend the analysis to the high order coupling case where α>[Formula presented] and also add the region 0<γ<[Formula presented]. A complete analysis for the decay rate and regularity of the system is given. We also provide some examples to illuminate our results.