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Asymptotic Profiles of Solutions for Structural Damped Wave Equations

  • Ryo Ikehata
  • Hiroshi TakedaEmail author
Article

Abstract

In this paper, we obtain several asymptotic profiles of solutions to the Cauchy problem for structurally damped wave equations \(\partial _{t}^{2} u - \varDelta u + \nu (-\varDelta )^{\sigma } \partial _{t} u=0\), where \(\nu >0\) and \(0< \sigma \le 1\). Our result is the approximation formula of the solution by a constant multiple of a special function as \(t \rightarrow \infty \), which states that the asymptotic profiles of the solutions are classified into 5 patterns depending on the values \(\nu \) and \(\sigma \). Here we emphasize that our main interest of the paper is in the case \(\sigma \in (\frac{1}{2},1)\).

Keywords

Nonlinear wave equation Fractional damping The Cauchy problem Critical exponent Asymptotic profile 

Mathematics Subject Classification

Primary 35L15 35L05 Secondary 35B40 

Notes

Acknowledgements

Authors wish to thank anonymous referees for their careful reading and helpful suggestions which led to an improvement of our original manuscript. The work of the first author (R. IKEHATA) was supported in part by Grant-in-Aid for Scientific Research (C)15K04958 of JSPS. The work of the second author (H. TAKEDA) was supported in part by Grant-in-Aid for Young Scientists (B)15K17581 of JSPS.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Graduate School of EducationHiroshima UniversityHigashi-HiroshimaJapan
  2. 2.Department of Intelligent Mechanical Engineering, Faculty of EngineeringFukuoka Institute of TechnologyFukuokaJapan

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