# Stochastic Allen–Cahn equation with mobility

Stochastic Allen–Cahn equation with mobility 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 ) . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Nonlinear Differential Equations and Applications NoDEA Springer Journals

# Stochastic Allen–Cahn equation with mobility

, Volume 24 (5) – Aug 14, 2017
38 pages

/lp/springer_journal/stochastic-allen-cahn-equation-with-mobility-knMjka1i9M
Publisher
Springer International Publishing
Subject
Mathematics; Analysis
ISSN
1021-9722
eISSN
1420-9004
D.O.I.
10.1007/s00030-017-0477-3
Publisher site
See Article on Publisher Site

### 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 $$d\le 3$$ 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 ) .

### Journal

Nonlinear Differential Equations and Applications NoDEASpringer Journals

Published: Aug 14, 2017

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