Abstract
We introduce a class of stochastic Allen–Cahn equations with a mobility coefficient and colored noise. For initial data with finite free energy, we analyze the corresponding Cauchy problem on the d-dimensional torus in the time interval [0, T]. Assuming that \(d\le 3\) and that the potential has quartic growth, we prove existence and uniqueness of the solution as a process u in \(L^2\) with continuous paths, satisfying almost surely the regularity properties \(u\in C([0,T]; H^1)\) and \(u\in L^2([0,T];H^2)\).
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The present work was financially supported by PRIN 20155PAWZB “Large Scale Random Structures”.
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Bertini, L., Buttà, P. & Pisante, A. Stochastic Allen–Cahn equation with mobility. Nonlinear Differ. Equ. Appl. 24, 54 (2017). https://doi.org/10.1007/s00030-017-0477-3
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DOI: https://doi.org/10.1007/s00030-017-0477-3