# Revista Matemática Iberoamericana

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**Published online first: 2019-11-25**

**DOI: 10.4171/rmi/1130**

Weighted fractional chain rule and nonlinear wave equations with minimal regularity

Kunio Hidano^{[1]}, Jin-Cheng Jiang

^{[2]}, Sanghyuk Lee

^{[3]}and Chengbo Wang

^{[4]}(1) Mie University, Tsu, Mie, Japan

(2) National Tsing Hua University, Hsinchu, Taiwan

(3) Seoul National University, Republic of Korea

(4) Zhejiang University, Hangzhou, China

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.

*Keywords: *Glassey conjecture, fractional chain rule, nonlinear wave equations, generalized Strichartz estimates, unconditional uniqueness

Hidano Kunio, Jiang Jin-Cheng, Lee Sanghyuk, Wang Chengbo: Weighted fractional chain rule and nonlinear wave equations with minimal regularity. *Rev. Mat. Iberoam.* Electronically published on November 25, 2019. doi: 10.4171/rmi/1130 (to appear in print)