Abstract
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 Hs with s>3/2. Motivated by the work of Miao et al. (2024) [29], we show that the solution map y0↦y(t) 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.
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