  In this paper, we verify the Glassey conjecture in the radial case for all
spatial dimensions, which states that, for the nonlinear wave equations of the
form $\Box u=|\nabla u|^p$, the critical exponent to admit global small
solutions is given by $p_c=1+\frac{2}{n-1}$. Moreover, we are able to prove the
existence results with low regularity assumption on the initial data and extend
the solutions to the sharp lifespan. The main idea is to exploit the trace
estimates and KSS type estimates.

Analysis of PDEs