Z. Naturforsch. 68a, 343 – 349 (2013)
Blow-Up of Solutions for a System of Petrovsky Equations with an Indirect Linear Damping
Wenjun Liu
College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China
Received August 6, 2012 / revised November 7, 2012 / published online January 23, 2013
Reprint requests to: W. L.; E-mail: wjliu@nuist.edu.cn
In this paper, we consider a coupled system of Petrovsky equations in a bounded domain with clamped boundary conditions. Due to several physical considerations, a linear damping which is distributed everywhere in the domain under consideration appears only in the first equation whereas no damping term is applied to the second one (this is indirect damping). Many studies show that the solution of this kind of system has a polynomial rate of decay as time tends to infinity, but does not have exponential decay. For four different ranges of initial energy, we show here the blow-up of solutions and give the lifespan estimates by improving the method of Wu (Electron. J. Diff. Equ. 105, 1 (2009)) and Li et al. (Nonlin. Anal. 74, 1523 (2011)).

From the applications point of view, our results may provide some qualitative analysis and intuition for the researchers in other fields such as engineering and mechanics when they study the concrete models of Petrovsky type.

Key words: Petrovsky Systems; Blow-Up; Indirect Damping; Lifespan Estimates.
Mathematics Subject Classification 2000: 35L05; 35L57; 35B44
Full-text PDF