  This paper is concerned with a fully nonlinear variant of the Allen-Cahn
equation with strong irreversibility, where each solution is constrained to be
non-decreasing in time. Main purposes of the 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 behaviors of solutions. More precisely, by deriving
\emph{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 $t \to +\infty$.
