Global existence and blow-up of solutions for a strongly damped Petrovsky system with nonlinear damping

Abstract

In this article we consider the Petrovsky equation u _tt  + Δ² u − Δu _t  + |u _t | ^m−1 u _t  = |u| ^p−1 u. We prove the global existence of the solution under conditions without any relation between m and p, and establish an exponential decay rate. We also show that the solution blows up in finite time if p > m and the initial energy is less than the potential well depth.

Keywords:

• global existence
• decay rate
• blow up in finite time
• Petrovsky system

AMS Subject Classifications::

• 35L35
• 35B40

Acknowledgements

The authors wish to express their gratitude to the anonymous referees for a number of valuable comments and suggestions. This work was partly supported by the Tianyuan Fund of Mathematics (Grant No. 11026211) and the Natural Science Foundation of the Jiangsu Higher Education Institutions (Grant No. 09KJB110005).