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Proceedings of the American Mathematical Society

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Blow up of solutions of the Cauchy problem for a wave equation with nonlinear damping and source terms and positive initial energy

Authors: Howard A. Levine and Grozdena Todorova
Journal: Proc. Amer. Math. Soc. 129 (2001), 793-805
MSC (1991): Primary 35L15, 35Q72
Published electronically: September 19, 2000
MathSciNet review: 1792187
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In this paper we consider the long time behavior of solutions of the initial value problem for semi-linear wave equations of the form

\begin{equation*}u_{tt}+ a\vert u_t\vert^{m-1}u_t - \Delta u = b\vert u\vert^{p-1}u\qquad \quad \text{in} [0,\infty )\times R^n. \qquad \qquad \end{equation*}

Here $a,b>0.$

We prove that if $p>m \ge 1,$ then for any $\lambda>0$ there are choices of initial data from the energy space with initial energy $\mathcal{E}(0)=\lambda^2,$ such that the solution blows up in finite time. If we replace $b\vert u\vert^{p-1}u $ by $b\vert u\vert^{p-1}u -q(x)^2u$, where $q(x)$ is a sufficiently slowly decreasing function, an analogous result holds.

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Additional Information

Howard A. Levine
Affiliation: Department of Mathematics, Iowa State University, Ames, Iowa 50011

Grozdena Todorova
Affiliation: Institute of Mathematics, Bulgarian Academy of Sciences, Boul. Acad. Bonchev bl.8, Sofia 1113, Bulgaria
Address at time of publication: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455

Received by editor(s): May 11, 1999
Published electronically: September 19, 2000
Additional Notes: The first author was supported in part by NATO grant CRG-95120. The second author was supported by the Institute for Theoretical and Applied Physics at Iowa State University.
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2000 American Mathematical Society