| Abstract: |
| This talk concerns a one-dimensional Allen-Cahn equation on the whole line with the positive-part function, which constrains the growth of each solution to be non-decreasing. The study on such constrained evolution equations are motivated from phase-field models in Damage Mechanics. Indeed, evolution of damage is supposed to be monotone due to the nature of damaging. In this talk, we shall discuss traveling wave dynamics, which has been well studied for classical Allen-Cahn equations, for the constrained one. More precisely, we shall start with constructing a one-parameter family of \emph{degenerate} traveling wave solutions (identified when coinciding up to translation) and investigate their properties. Furthermore, the traveling wave dynamics turns out to be relevant to a free boundary problem with a peculiar motion equation for the boundary through an analysis on a regularity issue for the constrained Allen-Cahn equation, and then, such a viewpoint enables us to prove exponential stability of degenerate traveling waves with some basin of attraction, although they are unstable in a usual sense. This talk is based on a joint work with C.~Kuehn (M\unchen) and K.-I.~Nakamura (Kanazawa). |
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