Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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


Authors: Hao Wu and Songmu Zheng
Journal: Quart. Appl. Math. 64 (2006), 167-188
MSC (2000): Primary 35B40, 35Q99
DOI: https://doi.org/10.1090/S0033-569X-06-01004-0
Published electronically: January 24, 2006
MathSciNet review: 2211383
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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

DOI: https://doi.org/10.1090/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
Published electronically: 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.
Article copyright: © Copyright 2006 Brown University


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