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On the regularizing effect of nonlinear damping in hyperbolic equations


Authors: Grozdena Todorova and Borislav Yordanov
Journal: Trans. Amer. Math. Soc. 367 (2015), 5043-5058
MSC (2010): Primary 35L70, 35B65; Secondary 35L05, 35B33
DOI: https://doi.org/10.1090/S0002-9947-2015-06173-X
Published electronically: February 18, 2015
MathSciNet review: 3335409
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Abstract: Global well-posedness in $ H^2(\mathbb{R}^3)\times H^1(\mathbb{R}^3)$ is shown for nonlinear wave equations of the form $ \Box u+f(u)+g(u_t)=0,$ where $ t\in \mathbb{R}_+.$ The main assumption is that the nonlinear damping $ g(u_t)$ behaves like $ \vert u_t\vert^{m-1}u_t$ with $ m\geq 2$ and the defocusing nonlinearity $ f(u)$ is like $ \vert u\vert^{p-1}u$ with $ p\geq 2.$ The result also applies to certain exponential functions, such as $ f(u)=\sinh u.$ It is observed that the nonlinear damping gives rise to a new monotone quantity involving the second-order derivatives of $ u$ and leading to a priori estimates for initial data of any size.

Global well-posedness in $ H^1(\mathbb{R}^3)\times L^2(\mathbb{R}^3)$ is shown for the same equation in the critical case $ f(u)=u^5$ and $ g(u_t)=\vert u_t\vert^{2/3}u_t$. The main tool is a new estimate for the solution of the nonlinear equation in $ L^4(\mathbb{R}_+,L^{12}(\mathbb{R}^{3})).$


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Additional Information

Grozdena Todorova
Affiliation: Department of Mathematics, University of Tennessee, Knoxville, Tennessee, 37996
Email: todorova@math.utk.edu

Borislav Yordanov
Affiliation: Department of Mathematics, University of Tennessee, Knoxville, Tennessee, 37996
Email: yordanov@math.utk.edu

DOI: https://doi.org/10.1090/S0002-9947-2015-06173-X
Keywords: Nonlinear wave equation, supercritical power, nonlinear damping, regularity
Received by editor(s): March 11, 2013
Received by editor(s) in revised form: May 5, 2013
Published electronically: February 18, 2015
Article copyright: © Copyright 2015 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.