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Front-type solutions of fractional Allen-Cahn equation. (English) Zbl 1160.35442

Summary: Super-diffusive front dynamics have been analysed via a fractional analogue of the Allen-Cahn equation. One-dimensional kink shape and such characteristics as slope at origin and domain wall dynamics have been computed numerically and satisfactorily approximated by variational techniques for a set of anomaly exponents \(1<\gamma <2\). The dynamics of a two-dimensional curved front has been considered. Also, the time dependence of coarsening rates during the various evolution stages was analysed in one and two spatial dimensions.

MSC:

35K57 Reaction-diffusion equations
35K55 Nonlinear parabolic equations
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35Q35 PDEs in connection with fluid mechanics
35B40 Asymptotic behavior of solutions to PDEs
26A33 Fractional derivatives and integrals
35S10 Initial value problems for PDEs with pseudodifferential operators
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