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On a nonlinear coupled system with internal damping. (English) Zbl 0962.35002
This paper studies an initial-boundary value problem for the following coupled hyperbolic-parabolic system: \[ u_{tt}-\mu\Delta u+\sum_{i=1}^n{\partial\theta\over \partial x_i} +\gamma |u|^\rho u =0, \quad \theta_t -\Delta\theta +\sum^n_{i=1} {\partial^2 u\over\partial t\partial x_i} =0,\quad \text{in }\Omega\times (0,T), \] together with Dirichlet boundary conditions for both \(u\) and \(\theta\), and prescribed initial data, where \(\Omega\in \mathbb{R}^n\) is a smooth bounded domain, \(\mu\) is a positive function of \(t\), \(\gamma\) and \(\rho\) are positive constants. The case of \(\gamma =0\) has been investigated by H. R. Clark, L. P. San Gil Jutuca and M. Milla Miranda [Electron. J. Differ. Equ. 1998, Paper 4 (1998; Zbl 0886.35043)]. Under the assumption that \(\mu\in W^{1,1}(0,\infty)\) and \(\mu'\leq 0\), \(\rho\leq {n\over n-1}\) for \(n\geq 3\) and \(\rho\) is arbitrary but fixed for \(n\leq 2\), the authors prove the existence and uniqueness of global strong and weak solutions; and moreover, the exponential stability of the total energy associated to the strong and weak solutions is obtained. The main ingredients in the proof are the use of the Galerkin method, energy estimates and Lions-Aubin’s compactness theorem, and the construction of a suitable Lyapunov functional.

MSC:
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35L70 Second-order nonlinear hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
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