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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 \(s>1/2\).

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