## 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 $\delta _1>3$.

## References

- Lars Andersson, Pieter Blue, and Jean-Philippe Nicolas,
*A decay estimate for a wave equation with trapping and a complex potential*, Int. Math. Res. Not. IMRN**3**(2013), 548–561. MR**3021792**, DOI 10.1093/imrn/rnr237 - Michael Beals,
*Optimal $L^\infty$ decay for solutions to the wave equation with a potential*, Comm. Partial Differential Equations**19**(1994), no. 7-8, 1319–1369. MR**1284812**, DOI 10.1080/03605309408821056 - P. Bizon, T. Chmaj, and A. Rostworowski,
*Anomalously small wave tails in higher dimensions*. Physical Review D (Particles, Fields, Gravitation, and Cosmology)**76**(2007), no. 12, 124035. - E. S. C. Ching, P. T. Leung, W. M. Suen, and K. Young,
*Wave propagation in gravitational systems: Late time behavior*, Physical Review D (Particles, Fields, Gravitation, and Cosmology)**52**(1995), no. 4, 2118–2132. - O. Costin and M. Huang,
*Gamow vectors and Borel summability in a class of quantum systems*, J. Stat. Phys.**144**(2011), no. 4, 846–871. MR**2826622**, DOI 10.1007/s10955-011-0276-x - O. Costin, J. L. Lebowitz, and S. Tanveer,
*Ionization of Coulomb systems in $\Bbb R^3$ by time periodic forcings of arbitrary size*, Comm. Math. Phys.**296**(2010), no. 3, 681–738. MR**2628820**, DOI 10.1007/s00220-010-1023-x - Roland Donninger and Wilhelm Schlag,
*Decay estimates for the one-dimensional wave equation with an inverse power potential*, Int. Math. Res. Not. IMRN**22**(2010), 4276–4300. MR**2737771**, DOI 10.1093/imrn/rnq038 - Piero D’ancona and Vittoria Pierfelice,
*On the wave equation with a large rough potential*, J. Funct. Anal.**227**(2005), no. 1, 30–77. MR**2165087**, DOI 10.1016/j.jfa.2005.05.013 - Piero D’Ancona and Luca Fanelli,
*$L^p$-boundedness of the wave operator for the one dimensional Schrödinger operator*, Comm. Math. Phys.**268**(2006), no. 2, 415–438. MR**2259201**, DOI 10.1007/s00220-006-0098-x - Roland Donninger, Wilhelm Schlag, and Avy Soffer,
*A proof of Price’s law on Schwarzschild black hole manifolds for all angular momenta*, Adv. Math.**226**(2011), no. 1, 484–540. MR**2735767**, DOI 10.1016/j.aim.2010.06.026 - Michael Goldberg,
*Transport in the one-dimensional Schrödinger equation*, Proc. Amer. Math. Soc.**135**(2007), no. 10, 3171–3179. MR**2322747**, DOI 10.1090/S0002-9939-07-08897-1 - W. Schlag,
*Dispersive estimates for Schrödinger operators: a survey*, Mathematical aspects of nonlinear dispersive equations, Ann. of Math. Stud., vol. 163, Princeton Univ. Press, Princeton, NJ, 2007, pp. 255–285. MR**2333215** - Robert S. Strichartz,
*A priori estimates for the wave equation and some applications*, J. Functional Analysis**5**(1970), 218–235. MR**0257581**, DOI 10.1016/0022-1236(70)90027-3 - Robert S. Strichartz,
*Convolutions with kernels having singularities on a sphere*, Trans. Amer. Math. Soc.**148**(1970), 461–471. MR**256219**, DOI 10.1090/S0002-9947-1970-0256219-1 - W Wasow,
*Asymptotic expansions for ordinary differential equations*, Interscience Publishers, 1968.

## 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