A generalisation of the Cauchy-Kovalevskaia theorem

TL;DR

This paper extends the Cauchy-Kovalevskaia theorem by showing solutions to linear evolution equations are sectorial or holomorphic depending on initial data regularity, with optimal results for many PDEs including KdV.

Contribution

It generalizes the classical theorem to broader classes of linear evolution equations, establishing sectorial solutions under mild assumptions.

Findings

Solutions are sectorial with size depending on initial data regularity

Holomorphic solutions occur with sufficiently regular initial data

The results are proven to be optimal for several PDE systems including KdV

Abstract

I prove, under mild assumptions, that solutions to linear evolution equations admit sectorial solutions. The size of the sector depends on the regularity of the initial data. If it is regular enough the solution is holomorphic and unique otherwise it is sectorial. I also prove that the result is optimal for many partial differential systems (which includes KdV and other examples).