The Glassey conjecture with radially symmetric data

TL;DR

This paper confirms the Glassey conjecture for radially symmetric data across all dimensions, establishing the critical exponent for global solutions and extending solutions with low regularity initial data.

Contribution

It proves the conjecture in the radial case for all dimensions, with new low-regularity existence results and lifespan extensions.

Findings

Verification of the Glassey conjecture in radial case for all dimensions

Existence of solutions with low regularity initial data

Extension of solutions to sharp lifespan

Abstract

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.