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We prove a short time existence result for a system consisting of a geometricevolution equation for a hypersurface and a parabolic equation on this evolvinghypersurface. More precisely, we discuss a mean curvature flow scaled with aterm that depends on a quantity defined on the surface coupled to a diffusionequation for that quantity. The proof is based on a splitting ansatz, solvingboth equations separately using linearization and a contraction argument. Ourresult is formulated for the case of immersed hypersurfaces and yields auniform lower bound on the existence time that allows for small changes in theinitial value of the height function.

Short Time Existence for Coupling of Scaled Mean Curvature Flow and Diffusion