Blowup analysis of a Camassa-Holm type equation with time-varying dissipation

TL;DR

This paper analyzes the blow-up behavior and wave breaking of a Camassa-Holm type equation with time-dependent dissipation, establishing criteria and universal blow-up rates.

Contribution

It extends wave breaking analysis to variable dissipation regimes and derives new blow-up criteria and rates for the equation.

Findings

Established local well-posedness using Kato's theory.

Derived blow-up criteria involving gradient and amplitude conditions.

Proved the universal blow-up rate of -2.

Abstract

This paper is concerned with the local well-posedness, wave breaking, blow-up rate for a Camassa-Holm type equation with time-dependent weak dissipation. Firstly, we obtain the local well-posedness of solutions by using Kato's theory. Secondly, by using energy estimates, characteristic methods, and comparison principles, we derive two blowup criteria involving both pointwise gradient conditions and mixed amplitude-gradient conditions, and prove the blowup rate is universally $-2$. Our results extend wave breaking analysis to physically relevant variable dissipation regimes.