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Exponential decay for the KdV equation on ℝ with new localized dampings

Part of: Partial differential equations Equations of mathematical physics and other areas of application Difference equations Difference and functional equations

Published online by Cambridge University Press:  05 April 2022

Ming Wang Open the ORCID record for Ming Wang [Opens in a new window]  and
Deqin Zhou
Ming Wang
Affiliation:
Center for Mathematical Sciences, China University of Geosciences, Wuhan 430074, P.R. China ([email protected])
Deqin Zhou
Affiliation:
College of Mathematics and Statistics, Chongqing University, Chongqing 401331, P.R. China ([email protected])
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Abstract

In this paper, we prove several results on the exponential decay in $L^{2}$ norm of the KdV equation on the real line with localized dampings. First, for the linear KdV equation, the exponential decay holds if and only if the averages of the damping coefficient on all intervals of a fixed length have a positive lower bound. Moreover, under the same damping condition, the exponential decay holds for the (nonlinear) KdV equation with small initial data. Finally, with the aid of certain properties of propagation of regularity in Bourgain spaces for solutions of the associated linear system and the unique continuation property, the exponential decay for the KdV equation with large data holds if the damping coefficient has a positive lower bound on $E$, where $E$ is equidistributed over the real line and the complement $E^{c}$ has a finite Lebesgue measure.

Keywords

KdV equationexponential decaylocalized damping

MSC classification

Primary: 35Q53: KdV-like equations (Korteweg-de Vries) 35B40: Asymptotic behavior of solutions
Secondary: 39A12: Discrete version of topics in analysis
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 153 , Issue 4 , August 2023 , pp. 1073 - 1098
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Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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