Transactions of the American Mathematical Society

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

Author(s): Mohammad A. Rammaha; Theresa A. Strei
Journal: Trans. Amer. Math. Soc. 354 (2002), 3621-3637.
MSC (2000): Primary 35L05, 35L20; Secondary 58K55
Posted: April 23, 2002
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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

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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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: 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
Posted: April 23, 2002
Additional Notes: The second author was supported in part by the National Physical Science Consortium and the National Security Agency
Copyright of article: Copyright 2002, American Mathematical Society

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