TY - JOUR
T1 - Allen-Cahn equation with strong irreversibility
AU - Akagi, Goro
AU - Efendiev, Messoud
N1 - Funding Information:
G. Akagi is supported in part by JSPS KAKENHI Grant Numbers JP16H03946, JP16K05199, JP17H01095, in part by the Alexander von Humboldt Foundation and in part by the Carl Friedrich von Siemens Foundation.
Publisher Copyright:
© 2018 Cambridge University Press.
PY - 2019/8/1
Y1 - 2019/8/1
N2 - 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 → +∞.
AB - 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 → +∞.
KW - Allen-Cahn equation
KW - Strongly irreversible evolution equation
KW - global attractor
KW - obstacle parabolic problem
KW - partial energy-dissipation
KW - ω-limit set
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U2 - 10.1017/S0956792518000384
DO - 10.1017/S0956792518000384
M3 - Article
AN - SCOPUS:85049937035
SN - 0956-7925
VL - 30
SP - 707
EP - 755
JO - European Journal of Applied Mathematics
JF - European Journal of Applied Mathematics
IS - 4
ER -