Electronic Journal of Differential Equations,
Vol. 2007(2007), No. 76, pp. 1-10.

Title: Blowup and asymptotic stability of weak solutions to wave
       equations with nonlinear degenerate damping and source terms

Authors: Qingying Hu   (Henan Univ., Zhengzhou, China)
         Hongwei Zhang (Henan Univ., Zhengzhou, China)

Abstract:
 This article  concerns the blow-up and asymptotic
 stability of weak solutions to the wave equation 
 $$
 u_{tt}-\Delta u  +|u|^kj'(u_t)=|u|^{p-1}u
 \quad \hbox{in }\Omega \times (0,T),
 $$
 where $p>1$ and $j'$  denotes the  derivative of a $C^1$ convex 
 and real value function $j$.
 We prove that every weak solution  is asymptotically stability,
 for every $m$ such that $0<m<1$, $p<k+m$ and the the initial energy is
 small; the solutions blows up in finite time, whenever $p>k+m$ and
 the initial data is positive, but  appropriately bounded.
  
Submitted February 27, 2007. Published May 22, 2007.
Math Subject Classifications: 35B40.
Key Words: Wave equation; degenerate damping and source terms;
           asymptotic stability; blow up of solutions.