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

 

On nonlinear wave equations with degenerate damping and source terms


Authors: Viorel Barbu, Irena Lasiecka and Mohammad A. Rammaha
Journal: Trans. Amer. Math. Soc. 357 (2005), 2571-2611
MSC (2000): Primary 35L05, 35L20; Secondary 58J45
Published electronically: March 1, 2005
MathSciNet review: 2139519
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Abstract: In this article we focus on the global well-posedness of the differential equation $u_{tt}- \Delta u + \vert u\vert^k\partial j(u_t) = \vert u\vert^{ p-1}u \, \text{ in } \Omega \times (0,T)$, where $\partial j$ is a sub-differential of a continuous convex function $j$. Under some conditions on $j$ and the parameters in the equations, we obtain several results on the existence of global solutions, uniqueness, nonexistence and propagation of regularity. Under nominal assumptions on the parameters we establish the existence of global generalized solutions. With further restrictions on the parameters we prove the existence and uniqueness of a global weak solution. In addition, we obtain a result on the nonexistence of global weak solutions to the equation whenever the exponent $p$ is greater than the critical value $k+m$, and the initial energy is negative. We also address the issue of propagation of regularity. Specifically, under some restriction on the parameters, we prove that solutions that correspond to any regular initial data such that $u_0\in H^2(\Omega)\cap H^1_0(\Omega)$, $u_1 \in H^1_0(\Omega)$ are indeed strong solutions.


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Additional Information

Viorel Barbu
Affiliation: Department of Mathematics, University “Al. J. Cuza", 6600 Iasi, Romania
Email: vb41@uaic.ro

Irena Lasiecka
Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22904-4137
Email: il2v@virginia.edu

Mohammad A. Rammaha
Affiliation: Department of Mathematics, University of Nebraska-Lincoln, Lincoln, Nebraska 68588-0130
Email: rammaha@math.unl.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-05-03880-8
PII: S 0002-9947(05)03880-8
Keywords: Wave equations, damping and source terms, generalized solutions, weak solutions, blow-up of solutions, sub-differentials, energy estimates.
Received by editor(s): March 31, 2003
Published electronically: March 1, 2005
Article copyright: © Copyright 2005 American Mathematical Society