| Abstract: |
| In this talk, we shall discuss energy-dissipation phenomena and smoothing effect of solutions for an Allen-Cahn equation with nondecreasing constraint. More precisely, we shall treat the Cauchy-Dirichlet problem for the equation
$$
u_t = \Big( \Delta u - W`(u) \Big)_+,
$$
where $W(\cdot)$ is a double-well potential and $(\cdot)_+$ is the positive-part function. Hence solutions are constrained to be nondecreasing. Such a constraint prevents emergence of the energy-dissipation and smoothing effect, which are completely realized for classical Allen-Cahn equation. As a result, one can prove non-existence of global attractor in any $L^p$-spaces (and hence, in any Sobolev spaces). On the other hand, this equation still involves a gradient structure, and hence, energy-dissipation and smoothing effect emerge in an incomplete way. Main purpose of this talk is to explain how to extract such an incomplete emergence of energy-dissipation and smoothing effect for evolution equations with nondecreasing constraint from a functional analytic point of view. This talk is based on a joint work with M. Efendiev (M\"unchen). |
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