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Linking type solutions for elliptic equations with indefinite nonlinearities up to the critical growth
Eigenfunction expansion method and the longtime asymptotics for the damped Boussinesq equation
1.  Department of Mathematics, The University of Texas at Austin, Austin, TX 787121082, 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 globalintime 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 secondorder longtime asymptotics is calculated which is essentially nonlinear and shows the nonlinear mode multiplication.
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