# Weighted fractional chain rule and nonlinear wave equations with minimal regularity

### Kunio Hidano

Mie University, Tsu, Mie, Japan### Jin-Cheng Jiang

National Tsing Hua University, Hsinchu, Taiwan### Sanghyuk Lee

Seoul National University, Republic of Korea### Chengbo Wang

Zhejiang University, Hangzhou, China

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## Abstract

We consider the local well-posedness for 3-D quadratic semi-linear wave equations with radial data: \begin{eqnarray*} &\Box u = a |\partial_t u|^2+b|\nabla_x u|^2,& \\ & u(0,x)=u_0(x)\in H^{s}_{\mathrm{rad}}, \quad \partial_t u(0,x)=u_1(x)\in H^{s-1}_{\mathrm{rad}}.& \end{eqnarray*} It has been known that the problem is well-posed for $s\ge 2$ and ill-posed for $s<3/2$. In this paper, we prove unconditional well-posedness up to the scaling invariant regularity, that is to say, for $s>3/2$ and thus fill the gap which was left open for many years. For the purpose, we also obtain a weighted fractional chain rule, which is of independent interest. Our method here also works for a class of nonlinear wave equations with general power type nonlinearities which contain the space-time derivatives of the unknown functions. In particular, we prove the Glassey conjecture in the radial case, with minimal regularity assumption.

## Cite this article

Kunio Hidano, Jin-Cheng Jiang, Sanghyuk Lee, Chengbo Wang, Weighted fractional chain rule and nonlinear wave equations with minimal regularity. Rev. Mat. Iberoam. 36 (2020), no. 2, pp. 341–356

DOI 10.4171/RMI/1130