, Volume 64, Issue 4, pp 1145-1159

Stability of an abstract system of coupled hyperbolic and parabolic equations

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Abstract

In this paper, we provide a complete stability analysis for an abstract system of coupled hyperbolic and parabolic equations $$\begin{array}{ll}\;\;u_{tt} = -Au + \gamma A^{\alpha} \theta,\\ \quad \theta_t = -\gamma A^{\alpha}u_t - kA^{\beta}\theta,\\ u(0) = u_0, \quad u_t(0) = v_0, \quad \theta(0) = \theta_0\end{array}$$ where A is a self-adjoint, positive definite operator on a Hilbert space H. For ${(\alpha,\beta) \in [0,1] \times [0,1]}$ , the region of exponential stability had been identified in Ammar-Khodja et al. (ESAIM Control Optim Calc Var 4:577–593,1999). Our contribution is to show that the rest of the region can be classified as region of polynomial stability and region of instability. Moreover, we obtain the optimality of the order of polynomial stability.

Jianghao Hao: Research supported by NNSFC (No. 60974034).