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Fractional derivatives of composite functions and the Cauchy problem for the nonlinear half wave equation

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Abstract

We show new results of well-posedness for the Cauchy problem for the half wave equation with power-type nonlinear terms. For the purpose, we propose two approaches on the basis of the contraction-mapping argument. One of them relies upon the \(L_t^q L_x^\infty \) Strichartz-type estimate together with the Ginibre–Ozawa–Velo type chain rule of fairly general fractional orders. This chain rule has a significance of its own. Furthermore, in addition to the weighted fractional chain rule established in Hidano et al. (Weighted fractional chain rule and nonlinear wave equations with minimal regularity. Preprint, arXiv:1605.06748v3 [math.AP], 2018), the other approach uses weighted space-time \(L^2\) estimates for the inhomogeneous equation which are recovered from those for the second-order wave equation. In particular, by the latter approach we settle the problem left open in Bellazzini et al. (Math Ann 371(1–2):707–740, 2018) concerning the local well-posedness in \(H^{s}_{\mathrm{rad}}({\mathbb R}^n)\) with \(s>1/2\).

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Acknowledgements

The second author,Chengbo Wang is grateful to Professor Yoshio Tsutsumi for helpful discussion and valuable comments on the fractional chain rule. The authors thank Professors Piero D’Ancona and Kiyoshi Mochizuki for helpful discussions on the resolvent and smoothing estimates. Thanks also goes to the referee for pointing out [23, Theorem 14.3.2]. Kunio Hidano was supported in part by JSPS KAKENHI Grant Nos. JP15K04955 and JP18K03365. Chengbo Wang was supported in part by National Support Program for Young Top-Notch Talents.

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  1. Department of Mathematics, Faculty of Education, Mie University, 1577 Kurima-machiya-cho, Tsu, Mie, 514-8507, Japan

    Kunio Hidano

  2. School of Mathematical Sciences, Zhejiang University, Hangzhou, 310027, China

    Chengbo Wang

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  1. Kunio Hidano
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Correspondence to Chengbo Wang.

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Hidano, K., Wang, C. Fractional derivatives of composite functions and the Cauchy problem for the nonlinear half wave equation. Sel. Math. New Ser. 25, 2 (2019). https://doi.org/10.1007/s00029-019-0460-4

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