Zeitschrift für angewandte Mathematik und Physik

, Volume 64, Issue 4, pp 1145–1159

Stability of an abstract system of coupled hyperbolic and parabolic equations


  • Jianghao Hao
    • School of Mathematical SciencesShanxi University
    • LASG, Institute of Atmospheric PhysicsChinese Academy of Sciences
    • Department of Mathematics and StatisticsUniversity of Minnesota

DOI: 10.1007/s00033-012-0274-0

Cite this article as:
Hao, J. & Liu, Z. Z. Angew. Math. Phys. (2013) 64: 1145. doi:10.1007/s00033-012-0274-0


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.

Mathematics Subject Classification (2000)



Hyperbolic–parabolic equationExponentialstabilityPolynomial stabilitySemigroup
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