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

TL;DR

This paper establishes decay rates for solutions to one-dimensional wave equations with inverse power potentials, showing solutions decay like t^{-m} for large t, extending previous results to more general potentials.

Contribution

It provides new decay estimates for wave equations with inverse power potentials, including sums of inverse powers and more precise asymptotic potentials.

Findings

Solution decays like t^{-m} for large t

Results extend to potentials with multiple inverse powers

Applicable to potentials with higher-order decay terms

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 (see e.g. Bizon, Chmaj, and Rostworowski, 2007; Donninger and Schlag, 2010; Schlag, 2007). 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 $\delta_1>3$.