On the regularizing effect of nonlinear damping in hyperbolic equations
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- by Grozdena Todorova and Borislav Yordanov PDF
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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 $|u_t|^{m-1}u_t$ with $m\geq 2$ and the defocusing nonlinearity $f(u)$ is like $|u|^{p-1}u$ 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 $g(u_t)=|u_t|^{2/3}u_t$. 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
- Received by editor(s): March 11, 2013
- Received by editor(s) in revised form: May 5, 2013
- Published electronically: February 18, 2015
- © Copyright 2015
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3335409