Blowup and asymptotic stability of weak solutions to wave equations with nonlinear degenerate damping and source terms
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 pro...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Texas State University
2007-05-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2007/76/abstr.html |
| Summary: | 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. |
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| ISSN: | 1072-6691 |