Dependence on initial data for a stochastic modified two-component Camassa-Holm system

We study a stochastic modified two-component Camassa-Holm equation on $R$. We establish a local well-posedness result in the sense of Hadamard, i.e. existence, uniqueness and continuous dependence on initial data, as well as blow-up criteria for pathwise solutions in the Sobolev spaces $H^s$ with $s>3/2$. Motivated by the work of Miao et al. (2024) [29], we show that the solution map $y_0\mapsto y\left(t\right)$ defined by the corresponding Cauchy problem is weakly unstable, due to either a lack of strong stability in the exiting time or the absence of uniformly continuous dependence on the initial data.