Asymptotic estimates for an integral equation in theory of phase transition

Abstract

In this paper, we study the asymptotic behavior of solutions of an integral equation of the Allen–Cahn type in , when |x| → ∞. Here is uniformly continuous, and k ⩾ 1, n ⩾ 2, α ∈ (0, n) and . In addition, is a constant vector and C * is a real constant. If for some s ∈ [1, ∞), we know that |u| → 1 when |x| → ∞. Furthermore, we prove that if for some , then when |x| → ∞, and hence . When for some , then there exists some positive constant C such that |1 − |u(x)|2| ⩽ C|x| α−n for large |x|. Here the Harnack type estimate and the regularity lifting lemma come into play in those proofs.