On the behavior of positive solutions of semilinear elliptic equations in asymptotically cylindrical domains

Abstract

The goal of this note is to study the asymptotic behavior of positive solutions for a class of semilinear elliptic equations which can be realized as minimizers of their energy functionals. This class includes the Fisher-KPP and Allen–Cahn nonlinearities. We consider the asymptotic behavior in domains becoming infinite in some directions. We are in particular able to establish an exponential rate of convergence for this kind of problems.