Global existence and blow-up for a weakly dissipative $\mu$DP equation

TL;DR

This paper investigates a weakly dissipative version of the periodic Degasperis-Procesi equation, establishing local well-posedness, analyzing blow-up scenarios, and providing explicit examples of solutions that either exist globally or blow up.

Contribution

It introduces a weakly dissipative variant of the DP equation, proves local well-posedness in certain function spaces, and characterizes blow-up behavior with explicit solution examples.

Findings

Established local well-posedness in $H^s$ for $s>3/2$

Characterized blow-up scenarios for $s=3$

Provided explicit examples of global solutions and blow-up cases

Abstract

In this paper, we study a weakly dissipative variant of the periodic Degasperis-Procesi equation. We show the local well-posedness of the associated Cauchy problem in $H^s(\S)$, $s>3/2$, and discuss the precise blow-up scenario for $s=3$. We also present explicit examples for globally existing solutions and blow-up.