A degenerate parabolic logistic equation

Abstract

We analyze the behavior of positive solutions of parabolic equations with a class of degenerate logistic nonlinearity and Dirichlet boundary conditions. Our results concern existence and strong localization in the spatial region in which the logistic nonlinearity cancels. This type of nonlinearity has applications in the nonlinear Schrödinger equation and the study of Bose-Einstein condensates. In this context, our analysis explains the fact that the ground state presents a strong localization in the spatial region in which the nonlinearity cancels.