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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

${u}_{t}-{\Delta }u=au-b\left(x\right){u}^{p},\phantom{\rule{1.em}{0ex}}x\in {\Omega },\phantom{\rule{4pt}{0ex}}t>0$

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 ${\Omega }\subset {ℝ}^{n}$ 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

$-{\Delta }u=au-b\left(x\right){u}^{p},\phantom{\rule{1.em}{0ex}}x\in {\Omega }\setminus {\overline{{\Omega }}}_{0},\phantom{\rule{4pt}{0ex}}u=\infty \phantom{\rule{4.pt}{0ex}}\text{on}\phantom{\rule{4.pt}{0ex}}\partial {{\Omega }}_{0}$

with the same boundary conditions as $u$ on $\partial {\Omega }$ and some time is spent on the existence and uniqueness of solutions of this.

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35K60 Nonlinear initial value problems for linear parabolic equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. (PDE) 92D25 Population dynamics (general)