Almost global existence for Kirchhoff equations around global solutions

TL;DR

This paper extends the understanding of the lifespan of solutions to Kirchhoff equations, showing it tends to infinity near global solutions and providing lower bounds for specific initial data classes.

Contribution

It proves the lower semicontinuity of the lifespan and offers estimates for initial data close to known global solutions.

Findings

Lifespan tends to infinity near global solutions.

Lifespan is lower semicontinuous with respect to initial data.

Provides explicit lower bounds for solutions with special initial data classes.

Abstract

It is well-known that the life span of solutions to Kirchhoff equations tends to infinity when initial data tend to zero. These results are usually referred to as almost global existence, at least in a neighborhood of the null solution. Here we extend this result by showing that the life span of solutions is lower semicontinuous, and in particular it tends to infinity whenever initial data tend to some limiting datum that originates a global solution. We also provide an estimate from below for the life span of solutions when initial data are close to some of the classes of data for which global existence is known, namely data with finitely many Fourier modes, analytic data and quasi-analytic data.