Remarks on asymptotic order for the linear wave equation with the scale-invariant damping and mass with $L^r$-data
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- by Takahisa Inui and Haruya Mizutani PDF
- Proc. Amer. Math. Soc. 149 (2021), 3473-3484 Request permission
Abstract:
In the present paper, we consider the linear wave equation with the scale-invariant damping and mass. It is known that the global behavior of the solution depends on the size of the coefficients in front of the damping and mass at initial time $t=0$. Indeed, the solution satisfies the similar decay estimate to that of the corresponding heat equation if it is large and to that of the modified wave equation if it is small. In our previous paper, we obtained the scattering result and its asymptotic order for the data in the energy space $H^1\times L^2$ when the coefficients are in the wave regime. In fact, the threshold of the coefficients relies on the spatial decay of the initial data. Namely, it varies depending on $r$ when the initial data is in $L^r$ ($1\leq r < 2$). In the present paper, we will show the scattering result and the asymptotic order in the wave regime for $L^r$-data, which is wider than the wave regime for the data in the energy space. Moreover, we give an improvement of the asymptotic order obtained in our previous paper for the data in the energy space.References
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Additional Information
- Takahisa Inui
- Affiliation: Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan
- MR Author ID: 1094227
- Email: inui@math.sci.osaka-u.ac.jp
- Haruya Mizutani
- Affiliation: Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan
- MR Author ID: 917770
- ORCID: 0000-0002-2685-048X
- Email: haruya@math.sci.osaka-u.ac.jp
- Received by editor(s): May 19, 2020
- Received by editor(s) in revised form: December 22, 2020
- Published electronically: May 18, 2021
- Additional Notes: The first author was supported by JSPS KAKENHI Grant-in-Aid for Early-Career Scientists JP18K13444 and the second author was supported by JSPS KAKENHI Grant-in-Aid for Young Scientists (B) JP17K14218 and, partially, for Scientific Research (B) JP17H02854.
- Communicated by: Tanya Christiansen
- © Copyright 2021 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 149 (2021), 3473-3484
- MSC (2020): Primary 35L05, 35B40, 47A40
- DOI: https://doi.org/10.1090/proc/15481
- MathSciNet review: 4273150