# American Institute of Mathematical Sciences

October  2001, 7(4): 675-702. doi: 10.3934/dcds.2001.7.675

## Eigenfunction expansion method and the long-time asymptotics for the damped Boussinesq equation

 1 Department of Mathematics, The University of Texas at Austin, Austin, TX 78712-1082, United States

Received  September 2000 Revised  April 2001 Published  July 2001

A method of constructing solutions of semilinear dissipative equations in bounded domains is proposed. It allows to calculate the higher-order long-time asymptotics. The application of this approach is given for solving the first initialboundary value problem for the damped Boussinesq equation

$u_{t t} - 2b\Delta u_t = -\alpha \Delta^2 u+ \Delta u + \beta\Delta(u^2)$

in a unit ball $B$. Homogeneous boundary conditions and small initial data are examined. The existence of mild global-in-time solutions is established in the space $C^0([0,\infty), H^s_0(B)), s < 3/2$, and the solutions are constructed in the form of the expansion in the eigenfunctions of the Laplace operator in $B$. For $-3/2 +\varepsilon \leq s <3/2$, where $\varepsilon > 0$ is small, the uniqueness is proved. The second-order long-time asymptotics is calculated which is essentially nonlinear and shows the nonlinear mode multiplication.

Citation: Vladimir Varlamov. Eigenfunction expansion method and the long-time asymptotics for the damped Boussinesq equation. Discrete and Continuous Dynamical Systems, 2001, 7 (4) : 675-702. doi: 10.3934/dcds.2001.7.675
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