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 $$d\le 3$$ d ≤ 3 and that the potential has quartic growth, we prove existence and uniqueness of the solution as a process u in $$L^2$$ L 2 with continuous paths, satisfying almost surely the regularity properties $$u\in C([0,T]; H^1)$$ u ∈ C ( [ 0 , T ] ; H 1 ) and $$u\in L^2([0,T];H^2)$$ u ∈ L 2 ( [ 0 , T ] ; H 2 ) .
Nonlinear Differential Equations and Applications NoDEA – Springer Journals
Published: Aug 14, 2017
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