*(English)*Zbl 0959.35065

The following problem occurs in the study of population dynamics

in which $p>1$ and $b\left(x\right)>0$ is continuous and $u$ is subject to Dirichlet or Robin boundary conditions. If $b$ is identically zero on ${\Omega}$ this is a Malthusian model for population growth and if $b$ is positive it is the logistic model. In these cases the behavior of the solution is well known – if $a<{\lambda}_{i}\left({\Omega}\right)$ (the principal eigenvalue of the Laplacian on ${\Omega}$ with the same boundary conditions as $u$) then all solutions with positive initial conditions decay to zero as $t\to \infty $ while if $a>{\lambda}_{1}$ then the solution either tends to the unique steady-state solution (logistic case) or blows up exponentially (Malthus case).

In this paper $b$ is assumed to vanish on a nontrivial subdomain ${{\Omega}}_{0}$ of ${\Omega}\subset {\mathbb{R}}^{n}$ and so gives a mixture of these two cases. It is shown that if $a<{\lambda}_{1}\left({{\Omega}}_{0}\right)$ (with Dirichlet boundary conditions) then the solution behaves like the Malthusian model on ${{\Omega}}_{0}$ 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 $u$ on $\partial {\Omega}$ and some time is spent on the existence and uniqueness of solutions of this.