Communications on Pure and Applied Analysis (CPAA)

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

Pages: 417 - 432, Volume 7, Issue 2, March 2008      doi:10.3934/cpaa.2008.7.417

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Meng-Rong Li - Department of Mathematics, National Chengchi University, Taipei, Taiwa, Taiwan (email)

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.

Received: November 2006;      Revised: April 2007;      Available Online: December 2007.