Stability analysis for two coupled second order evolution equations

Abstract

In this paper, we provide a stability analysis for the following abstract system of two coupled second order evolution equations$\{\begin{array}{c}{u}_{tt}=-a{A}^{\gamma }u+b{A}^{\alpha }{y}_t,\\ y_{tt}=-Ay-b{A}^{\alpha }{u}_t-k{A}^{\beta }{y}_t,\\ u\left(0\right)=u_0,u_t\left(0\right)=v_0,y\left(0\right)=y_0,y_t\left(0\right)=z_0.\end{array}$ where A is a self-adjoint, positive definite operator on a complex Hilbert space H, and parameters $(\alpha ,\beta )\in [0,\frac{\gamma +1}{2}]\times [0,1]$ with $\gamma \in [\frac{1}{2},2]$. We are able to completely divide the parameter region into subsets where the semigroup associated with the system is (i) exponentially stable, (ii) polynomially stable of optimal order, and (iii) merely strong stable.