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Decay estimates for one-dimensional wave equations with inverse power potentials


Authors: O. Costin and M. Huang
Journal: Trans. Amer. Math. Soc. 367 (2015), 3705-3732
MSC (2010): Primary 35L05, 35P25, 34M37, 34M40, 35Q75
DOI: https://doi.org/10.1090/S0002-9947-2014-06345-9
Published electronically: July 29, 2014
MathSciNet review: 3314821
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Abstract: We study the one-dimensional wave equation with an inverse power potential that equals $ const.x^{-m}$ for large $ \vert x\vert$, where $ m$ is any positive integer greater than or equal to 3. We show that the solution decays pointwise like $ t^{-m}$ for large $ t$, which is consistent with existing mathematical and physical literature under slightly different assumptions.

Our results can be generalized to potentials consisting of a finite sum of inverse powers, the largest of which being $ const.x^{-\alpha }$, where $ \alpha >2$ is a real number, as well as potentials of the form $ const.x^{-m}+O( x^{-m-\delta _1})$ with $ \delta _1>3$.


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Additional Information

O. Costin
Affiliation: Department of Mathematics, The Ohio State University, Columbus, Ohio 43210

M. Huang
Affiliation: Department of Mathematics, City University of Hong Kong, Hong Kong

DOI: https://doi.org/10.1090/S0002-9947-2014-06345-9
Received by editor(s): July 19, 2013
Received by editor(s) in revised form: October 16, 2013
Published electronically: July 29, 2014
Article copyright: © Copyright 2014 American Mathematical Society