Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Existence, decay and blow-up for solutions to the sixth-order generalized Boussinesq equation

Pages: 117 - 137, Volume 35, Issue 1, January 2015      doi:10.3934/dcds.2015.35.117

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Akmel Dé Godefroy - Laboratoire de Mathematiques Appliquées,UFRMI, Université d'Abidjan Cocody, 22 BP 582 Abidjan 22, Ivory Coast (Cote D'Ivoire) (email)

Abstract: We study the existence, the decay and the blow-up of solutions to the Cauchy problem for the multi-dimensional generalized sixth-order Boussinesq equation: $$ u_{tt} - \Delta u - \Delta^{2} u- \mu \Delta ^{3} u = \Delta f(u),\; t>0, \; x \in {\mathbb{R}^{n}}, n \geq 1, $$ where $ f(u)= \gamma |u|^{p-1}u, \; \gamma \in \mathbb{R}, \; p \geq 2, \; \mu > 1/4$. We find two global existence results for appropriate initial data when $n$ verifies $1 \leq n\leq 4(p+1)/(p-1).$ On the other hand we show that if $\mu= 1/3$ and $p>13/2$, then the solution with small initial data decays in time. A blow up in finite time result is also obtained for appropriate initial data when $n$ verifies $1 \leq n\leq 4(p+1)/(p-1).$

Keywords:  Sixth-order generalized Boussinesq equation, global existence, decay, blow-up.
Mathematics Subject Classification:  35A01, 35B40, 35B44, 35Q35.

Received: July 2010;      Revised: June 2014;      Available Online: August 2014.