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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 \[ u_t-\Delta u=au-b(x)u^p,\quad x\in\Omega,\;t>0 \] 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 \(\Omega\) 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 \(a < \lambda_i(\Omega)\) (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 \(a < \lambda_1(\Omega_0)\) (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 \[ -\Delta u = au - b(x)u^p,\quad x \in \Omega\setminus\overline{\Omega}_0,\;u =\infty\text{ on }\partial\Omega_0 \] 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, boundary and initial-boundary value problems for linear parabolic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
92D25 Population dynamics (general)
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