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

We refine the asymptotic estimates for minimizers of a class of nonlocal energy functionals of the form $$\frac{1}{4}{\iint }_{{\mathbb{R}}^{2n}\setminus ({\mathbb{R}}^n\setminus \mathrm{\Omega }{)}^2}\text{snr}{u(x)-u(y)}^{2}K(x-y)dxdy+{\int }_{\mathrm{\Omega }}W(u(x))dx,$$ 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.