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Convergence to equilibrium for the damped semilinear wave equation with critical exponent and dissipative boundary condition

Author(s): Hao Wu; Songmu Zheng
Journal: Quart. Appl. Math. 64 (2006), 167-188.
MSC (2000): Primary 35B40, 35Q99
Posted: January 24, 2006
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Abstract: This paper is concerned with the asymptotic behavior of the solution to the following damped semilinear wave equation with critical exponent:

$\displaystyle u_{tt} + u_t -\Delta u + f(x,u) = 0, \qquad (x,t) \in \Omega \times \mathbb{R}^+$ (1)

subject to the dissipative boundary condition

$\displaystyle \partial_\nu u+ u + u_t = 0, \qquad t > 0, x \in \Gamma$ (2)

and the initial conditions

$\displaystyle u\vert _{t=0} = u_0(x),\quad u_t\vert _{t=0}=u_1(x), \qquad x \in \Omega,$ (3)

where $ \Omega$ is a bounded domain in $ \mathbb{R}^3$ with smooth boundary $ \Gamma$ , $ \nu$ is the outward normal direction to the boundary, and $ f$ is analytic in $ u$. In this paper convergence of the solution to an equilibrium as time goes to infinity is proved. While these types of results are known for the damped semilinear wave equation with interior dissipation and Dirichlet boundary condition, this is, to our knowledge, the first result with dissipative boundary condition and critical growth exponent.

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Additional Information:

Hao Wu
Affiliation: Institute of Mathematics, Fudan University, 200433 Shanghai, P.R. China
Email: haowufd@yahoo.com

Songmu Zheng
Affiliation: Institute of Mathematics, Fudan University, 200433 Shanghai, P.R. China
Email: songmuzheng@yahoo.com

PII: S0033-569X-06-01004-0
Keywords: Semilinear wave equation, critical growth exponent, dissipative boundary condition, Simon-Lojasiewciz inequality
Received by editor(s): July 18, 2005
Posted: January 24, 2006
Additional Notes: The authors are supported by the NSF of China under grant No. 10371022 and by the Ministry of Education in China under grant No. 20050246002, and by Key Laboratory of Mathematics for Nonlinear Sciences in Fudan University sponsored by the Ministry of Education in China.
Copyright of article: Copyright 2006, Brown University


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