Open Access Research

Existence and blow up of solutions to a Petrovsky equation with memory and nonlinear source term

Faramarz Tahamtani* and Mohammad Shahrouzi

Author Affiliations

Department of Mathematics, College of Sciences, Shiraz University, Shiraz, 71454, Iran

For all author emails, please log on.

Boundary Value Problems 2012, 2012:50  doi:10.1186/1687-2770-2012-50

Published: 26 April 2012

Abstract

We consider the semilinear Petrovsky equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M1">View MathML</a>

in a bounded domain and prove the existence of weak solutions. Furthermore, we show that there are solutions under some conditions on initial data which blow up in finite time with non-positive initial energy as well as positive initial energy. Estimates of the lifespan of solutions are also given.

Mathematics Subject Classification (2000): 35L35; 35L75; 37B25.

Keywords:
viscoelasticity; existence; blow-up; life-span; negative initial energy; positive initial energy