Breakdown for the Camassa-Holm Equation Using Decay Criteria and Persistence in Weighted Spaces

TL;DR

This paper establishes decay-based criteria for finite-time blowup and persistence of solutions in weighted spaces for the Camassa-Holm equation, highlighting the influence of initial data decay rates on solution behavior.

Contribution

It introduces decay conditions at infinity that determine wave breaking and provides persistence results in weighted spaces for solutions with slower decay.

Findings

Wave breaking occurs when initial data decay faster than solitons.

Persistence of solutions in weighted spaces for data decaying slower than solitons.

Explicit asymptotic profiles demonstrate the optimality of decay conditions.

Abstract

We exhibit a sufficient condition in terms of decay at infinity of the initial data for the finite time blowup of strong solutions to the Camassa--Holm equation: a wave breaking will occur as soon as the initial data decay faster at infinity than the solitons. In the case of data decaying slower than solitons we provide persistence results for the solution in weighted $L^p$-spaces, for a large class of moderate weights. Explicit asymptotic profiles illustrate the optimality of these results.