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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Global existence and nonexistence for nonlinear wave equations with damping and source terms


Authors: Mohammad A. Rammaha and Theresa A. Strei
Journal: Trans. Amer. Math. Soc. 354 (2002), 3621-3637
MSC (2000): Primary 35L05, 35L20; Secondary 58K55
Published electronically: April 23, 2002
MathSciNet review: 1911514
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Abstract: We consider an initial-boundary value problem for a nonlinear wave equation in one space dimension. The nonlinearity features the damping term $\left\vert u\right\vert^{m-1}u_t$ and a source term of the form $\left\vert u\right\vert^{p-1}u$, with $m,\,p>1$. We show that whenever $m\geq p$, then local weak solutions are global. On the other hand, we prove that whenever $p>m$ and the initial energy is negative, then local weak solutions cannot be global, regardless of the size of the initial data.


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Additional Information

Mohammad A. Rammaha
Affiliation: Department of Mathematics and Statistics, University of Nebraska-Lincoln, Lincoln, Nebraska 68588-0323
Email: rammaha@math.unl.edu

Theresa A. Strei
Affiliation: 7210 C Eden Brook Drive, #204, Columbia, Maryland 21046
Email: tastrei@yahoo.com

DOI: http://dx.doi.org/10.1090/S0002-9947-02-03034-9
PII: S 0002-9947(02)03034-9
Keywords: Wave equations, weak solutions, blow-up
Received by editor(s): May 25, 2001
Published electronically: April 23, 2002
Additional Notes: The second author was supported in part by the National Physical Science Consortium and the National Security Agency
Article copyright: © Copyright 2002 American Mathematical Society