Cauchy problem for a higher order generalized Boussinesq-type equation with hydrodynamical damped term

Abstract

We consider the existence and nonexistence of global weak solutions to the Cauchy problem for a higher order generalized Boussinesq-type equation with hydrodynamical damped term in -dimensional space. The existence of global weak solutions is proved under the assumptions that the nonlinear term is polynomial growth order, either , , the constant , or the initial data belongs to the potential well. Moreover, we establish two finite-time blow up results for any weak solutions with negative initial energy or nonnegative initial energy using the concavity method.

Keywords:

• Boussinesq-type equation
• Cauchy problem
• higher order
• existence of solutions
• blow up
• hydrodynamical damped term

AMS Subject Classifications:

• 35L30
• 35L82
• 35D40
• 35B44

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is supported by the NSF of China [grant number 11401122], [grant number 40890150]; the Scientific Program [grant number 2008B080701042] of Guangdong Province.