Optimal decay of heteroclinic solutions of the fractional Allen-Cahn equation with a degenerate potential

TL;DR

This paper refines asymptotic estimates for minimizers of nonlocal energy functionals related to the fractional Allen-Cahn equation with degenerate potentials, establishing optimal bounds and analyzing decay properties of heteroclinic solutions.

Contribution

It improves previous bounds on minimizers of nonlocal energies with degenerate potentials, demonstrating the optimality of these bounds for fractional Allen-Cahn type problems.

Findings

Refined asymptotic estimates for minimizers.

Proved the optimality of decay bounds.

Analyzed heteroclinic solutions with degenerate potentials.

Abstract

We refine the asymptotic estimates for minimizers of a class of nonlocal energy functionals of the form \[ \frac{1}{4} \iint_{\R^{2n} \setminus (\R^n \setminus \Omega)^2} \snr{u(x) - u(y)}^2 K(x - y) \,dx\,dy + \int_\Omega W(u(x)) \,dx, \] as originally studied in~\cite{DPDV}, and we prove the optimality of our improved bounds. Here, $W$ denotes a possibly \emph{degenerate} oscillatory double-well potential, satisfying a polynomial control on its second derivative near the wells. The kernel~$K$ belongs to a broad class of measurable functions and is modeled on the one of the fractional Laplacian.