Global existence and pointwise estimates of solutions for the multidimensional generalized Boussinesq-type equation

Miao Liu; Weike Wang

doi:10.3934/cpaa.2014.13.1203

Article Contents

2014, Volume 13, Issue 3: 1203-1222. Doi: 10.3934/cpaa.2014.13.1203

This issue Previous Article Non-smooth critical point theory on closed convex sets Next Article Disconjugacy and extremal solutions of nonlinear third-order equations

Global existence and pointwise estimates of solutions for the multidimensional generalized Boussinesq-type equation

Miao Liu^1, and
Weike Wang^2,

1.
Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240
2.
Department of Mathematics, Shanghai Jiao Tong University, 800 Dong Chuan Road, 200240, Shanghai

Received: May 31, 2013

Revised: October 31, 2013

Published: April 2015

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Abstract

In this paper, we study the Cauchy problem for the Boussinesq-type equation \begin{eqnarray} \partial^2_t u-\varepsilon \partial_t \Delta u=-\Delta ^2 u+\Delta u+\Delta g(u), \end{eqnarray} where $g(u)=O(u^\rho),$ $\rho \geq 2.$ By means of long wave-short wave decomposition, Green's function method and energy method, we show that the Cauchy Problem admits a global classical solution in multi dimension. We also show the pointwise estimate of the time asymptotic shape of the solutions in odd dimensional space.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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