Weighted fractional chain rule and nonlinear wave equations with minimal regularity

Abstract

We consider the local well-posedness for 3-D quadratic semi-linear wave equations with radial data:

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.