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

• Main Authors: Hongwei Zhang, Qingying Hu
• Format: Article
• Language: English
• Published: Texas State University 2007-05-01
• Series: Electronic Journal of Differential Equations
• Subjects: Wave equation; degenerate damping and source terms; asymptotic stability; blow up of solutions
• Online Access: http://ejde.math.txstate.edu/Volumes/2007/76/abstr.html

This article concerns the blow-up and asymptotic stability of weak solutions to the wave equation $$ u_{tt}-Delta u +|u|^kj'(u_t)=|u|^{p-1}u quad hbox{in }Omega imes (0,T), $$ 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.