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The Strichartz estimates for the damped wave equation and the behavior of solutions for the energy critical nonlinear equation

  • Takahisa InuiEmail author
Article
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Abstract

For the linear damped wave equation (DW), the \(L^p\)\(L^q\) type estimates have been well studied. Recently, Watanabe (RIMS Kôkyûroku Bessatsu B 63:77–101, 2017) showed the Strichartz estimates for DW when \(d=2,3\). In the present paper, we give Strichartz estimates for DW in higher dimensions. Moreover, by applying the estimates, we give the local well-posedness of the energy critical nonlinear damped wave equation (NLDW) \(\partial _t^2 u - \Delta u +\partial _t u = |u|^{\frac{4}{d-2}}u\), \((t,x) \in [0,T) \times {\mathbb {R}}^d\), where \(3 \le d \le 5\). Especially, we show the small data global existence for NLDW. In addition, we investigate the behavior of the solutions to NLDW. Namely, we give a decay result for solutions with finite Strichartz norm and a blow-up result for solutions with negative Nehari functional.

Keywords

Damped wave equation Dissipation Strichartz estimates Energy critical 

Mathematics Subject Classification

35L71 35A01 35B40 35B44 

Notes

Acknowledgements

The author would like to express deep appreciation to Professor Masahito Ohta and Professor Yuta Wakasugi for many useful suggestions, valuable comments and warm-hearted encouragement. The author was partially supported by JSPS Grant-in-Aid for Early-Career Scientists JP18K13444.

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Graduate School of ScienceOsaka UniversityToyonakaJapan

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