Regular solutions of wave equations with super-critical sources and exponential-to-logarithmic damping

Lorena Bociu; Petronela Radu; Daniel Toundykov

doi:10.3934/eect.2013.2.255

Article Contents

2013, Volume 2, Issue 2: 255-279. Doi: 10.3934/eect.2013.2.255

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Regular solutions of wave equations with super-critical sources and exponential-to-logarithmic damping

1.
NC State University, Department of Mathematics, 3236 SAS Hall, Raleigh, NC 27695-8205
2.
Department of Mathematics, University of Nebraska-Lincoln, Avery Hall 239, Lincoln, NE 68588
3.
Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588

Received: January 2013

Revised: February 2013

Published: June 2013

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Abstract

We study regular solutions to wave equations with super-critical source terms, e.g., of exponent $p>5$ in 3D. Such sources have been a major challenge in the investigation of finite-energy ($H^1 \times L^2$) solutions to wave PDEs for many years. The wellposedness has been settled in part, but even the local existence, for instance, in 3 dimensions requires the relation $p\leq 6m/(m+1)$ between the exponents $p$ of the source and $m$ of the viscous damping.
We prove that smooth initial data ($H^2 \times H^1$) yields regular solutions that do not depend on the above correlation. Local existence is demonstrated for any source exponent $p\geq 1$ and any monotone damping including feedbacks growing exponentially or logarithmically at infinity, or with no damping at all. The result holds in dimensions 3 and 4, and with some restrictions on $p$ in dimensions $n\geq 5$. Furthermore, if we assert the classical condition that the damping grows as fast as the source, then these regular solutions are global.

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Mathematics Subject Classification: Primary: 35L05; Secondary: 35A01, 35L20, 35B33, 46E30.

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