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Blow-up solutions for a class of semilinear elliptic and parabolic equations. (English) Zbl 0959.35065

The following problem occurs in the study of population dynamics

u t -Δu=au-b(x)u p ,xΩ,t>0

in which p>1 and b(x)>0 is continuous and u is subject to Dirichlet or Robin boundary conditions. If b is identically zero on Ω 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<λ i (Ω) (the principal eigenvalue of the Laplacian on Ω with the same boundary conditions as u) then all solutions with positive initial conditions decay to zero as t while if a>λ 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 Ω 0 of Ω n and so gives a mixture of these two cases. It is shown that if a<λ 1 (Ω 0 ) (with Dirichlet boundary conditions) then the solution behaves like the Malthusian model on Ω 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

-Δu=au-b(x)u p ,xΩΩ ¯ 0 ,u=onΩ 0

with the same boundary conditions as u on Ω and some time is spent on the existence and uniqueness of solutions of this.

MSC:
35J65Nonlinear boundary value problems for linear elliptic equations
35K60Nonlinear initial value problems for linear parabolic equations
35B05Oscillation, zeros of solutions, mean value theorems, etc. (PDE)
92D25Population dynamics (general)