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Scattering for critical wave equations with variable coefficients

Part of: Hyperbolic equations and systems

Published online by Cambridge University Press:  30 April 2021

Shi-Zhuo Looi  and
Mihai Tohaneanu
Shi-Zhuo Looi
Affiliation:
Department of Mathematics, University of Kentucky, Lexington, KY40506, USA (shizhuo.looi@uky.edu; mihai.tohaneanu@uky.edu)
Mihai Tohaneanu
Affiliation:
Department of Mathematics, University of Kentucky, Lexington, KY40506, USA (shizhuo.looi@uky.edu; mihai.tohaneanu@uky.edu)
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Abstract

We prove that solutions to the quintic semilinear wave equation with variable coefficients in ${{\mathbb {R}}}^{1+3}$ scatter to a solution to the corresponding linear wave equation. The coefficients are small and decay as $|x|\to \infty$, but are allowed to be time dependent. The proof uses local energy decay estimates to establish the decay of the $L^{6}$ norm of the solution as $t\to \infty$.

Keywords

scatteringdefocusingvariable coefficientsnonlinear wave equationhyperbolic equationpower-type nonlinearitypower-type non-linearitysemilinear wave equations

MSC classification

Primary: 35L05: Wave equation
Secondary: 35L71: Semilinear second-order hyperbolic equations
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 64 , Issue 2 , May 2021 , pp. 298 - 316
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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