Time-fractional Allen-Cahn equations versus powers of the mean curvature. (English) Zbl 1540.34016
Summary: We show by a formal asymptotic expansion that level sets of solutions of a time-fractional Allen-Cahn equation evolve by a geometric flow whose normal velocity is a positive power of the mean curvature. This connection is quite intriguing, since the original equation is nonlocal and the evolution of its solutions depends on all previous states, but the associated geometric flow is of purely local type, with no memory effect involved.
MSC:
| 34A08 | Fractional ordinary differential equations |
| 34D05 | Asymptotic properties of solutions to ordinary differential equations |
| 34E05 | Asymptotic expansions of solutions to ordinary differential equations |
| 26A33 | Fractional derivatives and integrals |
| 53E99 | Geometric evolution equations |
Keywords:
Allen-Cahn equation; memory effect; time-fractional derivative; geometric flow; power of the mean curvatureReferences:
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