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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 value problems for linear parabolic equations 35Q35 PDEs in connection with fluid mechanics 35B40 Asymptotic behavior of solutions of PDE 26A33 Fractional derivatives and integrals (real functions) 35S10 Initial value problems for pseudodifferential operators