Wave breaking of periodic solutions to the Fornberg-Whitham equation

TL;DR

This paper investigates wave breaking and blow-up phenomena for periodic solutions of the Fornberg-Whitham equation, establishing conditions under which solutions develop singularities in finite time.

Contribution

It provides new results linking finite lifespan of solutions to wave breaking and demonstrates finite-time blow-up for specific initial conditions.

Findings

Finite maximal lifespan implies wave breaking.

Certain initial profiles lead to finite-time blow-up.

Wave breaking occurs under specific well-posedness conditions.

Abstract

Based on recent well-posedness results in Sobolev (or Besov spaces) for periodic solutions to the Fornberg-Whitham equations we investigate here the questions of wave breaking and blow-up for these solutions. We show first that finite maximal life time of a solution necessarily leads to wave breaking. Second, we prove that for a certain class of initial wave profiles the corresponding solutions do indeed blow-up in finite time.