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obstacle parabolic problem

European Journal of Applied Mathematics (1)

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1 results

Allen–Cahn equation with strong irreversibility
GORO AKAGI, MESSOUD EFENDIEV
Journal:

European Journal of Applied Mathematics / Volume 30 / Issue 4 / August 2019

Published online by Cambridge University Press:

16 July 2018, pp. 707-755
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This paper is concerned with a fully non-linear variant of the Allen–Cahn equation with strong irreversibility, where each solution is constrained to be non-decreasing in time. The main purposes of this paper are to prove the well-posedness, smoothing effect and comparison principle, to provide an equivalent reformulation of the equation as a parabolic obstacle problem and to reveal long-time behaviours of solutions. More precisely, by deriving partial energy-dissipation estimates, a global attractor is constructed in a metric setting, and it is also proved that each solution u(x,t) converges to a solution of an elliptic obstacle problem as t → +∞.