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The asymptotic solution of the Cauchy problem for a generalized Boussinesq equation. (English) Zbl 1124.35335
Summary: We consider the solution of an initial value problem for the generalized damped Boussinesq equation $$u_{tt}- au_{ttxx}- 2bu_{txx}= -cu_{xxxx}+ u_{xx}- p^u+ \beta(u^2)_{xx},$$ where $x\in\Bbb R^1$, $t>0$, $a$, $b$ and $c$ are positive constants, $p\ne0$, and $\beta\in\Bbb R^1$. For the case $a+c> b^2$ corresponding to damped oscillations with an infinite number of oscillation cycles, we establish the well-posedness theorem of the global solution to the problem and derive a large time asymptotic solution.

35Q35PDEs in connection with fluid mechanics
35B40Asymptotic behavior of solutions of PDE
35C20Asymptotic expansions of solutions of PDE
35G25Initial value problems for nonlinear higher-order PDE
93C15Control systems governed by ODE
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