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arXiv · 2604.06721

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

Francesco De Pas·Serena Dipierro·Enrico Valdinoci
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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.

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Francesco De Pas, Serena Dipierro, Enrico Valdinoci. 2026-04-08. Optimal decay of heteroclinic solutions of the fractional Allen-Cahn equation with a degenerate potential. https://arxiv.org/abs/2604.06721

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