Asymptotic estimates for an integral equation in theory of phase transition

In this paper, we study the asymptotic behavior of solutions of an integral equation of the Allen-Cahn type in R-n u(x)=(l) over right arrow +C*integral(Rn)u(y)(1-|u(y)|(2))|1-|u(y)|(2)|(p-2)/|x-y|(n-alpha)dy , when |x| -> infinity. Here u:R-n -> R-k is uniformly continuous, and k >= 1, n >= 2, alpha is an element of (0, n) and p-1>n/n-alpha. In addition, (l) over right arrow is an element of R-k C-* is a real constant. If 1-|u|(2)is an element of L-s(R-n) s is an element of [1, infinity), we know that |u| -> 1 when |x| -> infinity. Furthermore, we prove that if 1-|u|(2)is an element of L-s(R-n) s is an element of[1,n/alpha(p-1)) <(l)over right arrow> x| -> infinity, and hence (l) over right arrow|=1 1-|u|2 is an element of Ls(Rn) s is an element of[1,n alpha(p-2))

Automated summary

This paper investigates the asymptotic behavior of solutions of an integral equation of the Allen-Cahn type as |x| approaches infinity, with certain conditions on uniform continuity and integrability. The study reveals the limits of the solutions under various scenarios.