Zeitschrift für angewandte Mathematik und Physik

, Volume 64, Issue 4, pp 1145–1159

Stability of an abstract system of coupled hyperbolic and parabolic equations

Article

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

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.

Mathematics Subject Classification (2000)

35B4047D0393D05

Keywords

Hyperbolic–parabolic equationExponentialstabilityPolynomial stabilitySemigroup

Copyright information

© Springer Basel 2012

Authors and Affiliations

  1. 1.School of Mathematical SciencesShanxi UniversityTaiyuanPeople’s Republic of China
  2. 2.LASG, Institute of Atmospheric PhysicsChinese Academy of SciencesBeijingPeople’s Republic of China
  3. 3.Department of Mathematics and StatisticsUniversity of MinnesotaDuluthUSA