The following problem occurs in the study of population dynamics
in which and is continuous and is subject to Dirichlet or Robin boundary conditions. If is identically zero on this is a Malthusian model for population growth and if is positive it is the logistic model. In these cases the behavior of the solution is well known – if (the principal eigenvalue of the Laplacian on with the same boundary conditions as ) then all solutions with positive initial conditions decay to zero as while if then the solution either tends to the unique steady-state solution (logistic case) or blows up exponentially (Malthus case).
In this paper is assumed to vanish on a nontrivial subdomain of and so gives a mixture of these two cases. It is shown that if (with Dirichlet boundary conditions) then the solution behaves like the Malthusian model on and like the logistic model on the remaining portion of the domain. This is done by using comparison and super/sub solution methods. An important role in this analysis is played by the solution of the boundary blow-up problem
with the same boundary conditions as on and some time is spent on the existence and uniqueness of solutions of this.