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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Decay estimates for one-dimensional wave equations with inverse power potentials
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by O. Costin and M. Huang PDF
Trans. Amer. Math. Soc. 367 (2015), 3705-3732 Request permission

Abstract:

We study the one-dimensional wave equation with an inverse power potential that equals $const.x^{-m}$ for large $|x|$, 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
  • MR Author ID: 52070
  • M. Huang
  • Affiliation: Department of Mathematics, City University of Hong Kong, Hong Kong
  • Received by editor(s): July 19, 2013
  • Received by editor(s) in revised form: October 16, 2013
  • Published electronically: July 29, 2014
  • © Copyright 2014 American Mathematical Society
  • 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
  • MathSciNet review: 3314821