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Motion of a droplet for the Stochastic mass conserving AllenCahn equationWe study the stochastic massconserving AllenCahn equation posed on a smoothly bounded domain of R2 with additive, spatially smooth, spacetime noise. This equation describes the stochastic motion of a small almost semicircular droplet attached to domain's boundary and moving towards a point of locally maximum curvature. We apply It^o calculus to derive the stochastic dynamics of the center of the droplet by utilizing the approximately invariant manifold introduced by Alikakos, Chen and Fusco [2] for the deterministic problem. In the stochastic case depending on the scaling, the motion is driven by the change in the curvature of the boundary and the stochastic forcing. Moreover, under the assumption of a su ciently small noise strength, we establish stochastic stability of a neighborhood of the manifold of boundary droplet states in the L2 and H1norms, which means that with overwhelming probability the solution stays close to the manifold for very long timescales.

On a degenerate nonlocal parabolic problem describing infinite dimensional replicator dynamicsWe establish the existence of locally positive weak solutions to the homogeneous Dirichlet problem for \[ u_t = u \Delta u + u \int_\Omega \nabla u^2 \] in bounded domains $\Om\sub\R^n$ which arises in game theory. We prove that solutions converge to $0$ if the initial mass is small, whereas they undergo blowup in finite time if the initial mass is large. In particular, it is shown that in this case the blowup set coincides with $\overline{\Omega}$, i.e. the finitetime blowup is global.