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

Meng-Rong Li

doi:10.3934/cpaa.2008.7.417

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2008, Volume 7, Issue 2: 417-432. Doi: 10.3934/cpaa.2008.7.417

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

Meng-Rong Li^1,

1.
Department of Mathematics, National Chengchi University, Taipei, Taiwa

Received: October 31, 2006

Revised: March 31, 2007

Published: February 2008

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Abstract

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} }).

Keywords:

Wave equation,
blow-up,
estimate.

Mathematics Subject Classification: 35R30, 35L80, 35L70.

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