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

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.