March  2008, 7(2): 417-432. doi: 10.3934/cpaa.2008.7.417

Estimates for the life-span of the solutions for some semilinear wave equations

1. 

Department of Mathematics, National Chengchi University, Taipei, Taiwa, Taiwan

Received  November 2006 Revised  April 2007 Published  December 2007

In this paper we prove main result on blow-up rates, blow-up constants and some estimates for life-spans of the solutions for some initial-boundary value problems for semi-linear wave equations. Under some conditions the life-span $T\star$ can be estimated by

$\beta (k,\alpha)$: $=$ min{ $2^{3/2+1/2\alpha}\cdot\delta( k,\alpha )a(0) a'(0)^{-1}:k\in (0,1)$},

where $a(0)=\int_\Omegau_{0}(x)^{2}dx,$ $a'(0)=2\int_\Omega u_{0}( x) u_1(x) dx$ and $\delta(k,\alpha )$ is given by

$\delta(k,\alpha)$ :$=\frac{1}{k}(\frac{k^2}{1-k^2})^{\frac{\alpha }{1+2\alpha}}$ $(1-(1+(\frac{1}{ k^2}-1)^{\frac{\alpha}{1+2\alpha}})^{\frac{-1}{2\alpha} }).

Citation: Meng-Rong Li. Estimates for the life-span of the solutions for some semilinear wave equations. Communications on Pure & Applied Analysis, 2008, 7 (2) : 417-432. doi: 10.3934/cpaa.2008.7.417
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