Existence and Blow-up of solutions for Stochastic Modified Two-component Camassa-Holm System

TL;DR

This paper investigates the stochastic modified two-component Camassa-Holm system, establishing local existence, uniqueness, and how noise influences solution blow-up or global existence.

Contribution

It proves local existence and uniqueness of solutions and demonstrates how noise can prevent blow-up or cause finite-time blow-up depending on initial conditions.

Findings

Strong noise prevents blow-up with probability 1

Weak noise conditions lead to global existence or finite-time blow-up

Probabilistic conditions for solution behavior based on initial data

Abstract

In this paper, we consider the modified two-component Camassa-Holm System with multiplicative noise. For these SPDEs, we first establish the local existence and pathwise uniqueness of the pathwise solutions in Sobolev spaces $H^{s}\times H^{s}, s>\frac{3}{2}$. Then we show that strong enough noise can actually prevent blow-up with probability 1. Finally, we analyse the effects of weak noise and present conditions on the initial data that lead to the global existence and the blow-up in finite time of the solutions, and their associated probabilities are also obtained.