Almost global existence for some semilinear wave equations with almost critical regularity

TL;DR

This paper establishes almost global existence for certain 2D semilinear wave equations with cubic derivatives, using advanced Strichartz estimates, extending results to higher dimensions and nonlinearities.

Contribution

It introduces an endpoint generalized Strichartz estimate in a specific mixed space, enabling almost global well-posedness for subcritical regularity initial data.

Findings

Proves almost global existence for 2D cubic derivative wave equations.

Develops endpoint Strichartz estimates in mixed space.

Extends analysis to higher dimensions and nonlinearities.

Abstract

For any subcritical index of regularity $s>3/2$, we prove the almost global well posedness for the 2-dimensional semilinear wave equation with the cubic nonlinearity in the derivatives, when the initial data are small in the Sobolev space $H^s\times H^{s-1}$ with certain angular regularity. The main new ingredient in the proof is an endpoint version of the generalized Strichartz estimates in the space $L^2_t L_{|x|}^\infty L^2_\theta ([0,T]\times \R^2)$. In the last section, we also consider the general semilinear wave equations with the spatial dimension $n\ge 2$ and the order of nonlinearity $p\ge 3$.