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