| 概要 |
This talk is concerned with a strongly irreversible Allen-Cahn equation (irAC),
which is a variant of the Allen-Cahn equation and describes a non-decreasing evolution
of a constrained gradient flow. Such a unidirectional evolution appears in the
study of Damage Mechanics, where a phase parameter is introduced to describe the
degree of damage in material (hence the evolution of the phase parameter is naturally
supposed to be monotone, and moreover, the evolution is often described by a sort of
gradient flow). In this talk, we shall discuss asymptotic behavior of solutions to
the Cauchy-Dirichlet problem for (irAC) and (Lyapunov) stability of equilibria as well
as fundamental issues such as existence and uniqueness of solutions, by introducing
two equivalent forms to (irAC). Particularly for stability analysis, we emphasize
that equilibria are accumulating (in a proper energy space), and hence, one cannot
expect asymptotic stability and we need to overcome some difficulty arising from
the accumulation of equilibria. This talk is based on a joint work with Messoud Efendiev (Munchen).
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