Blow-Up Solutions for a Class of Semilinear Elliptic and Parabolic Equations

We study the asymptotic behavior of the solutions to the problem \[\left\{ \begin{array}{rlll} u_t-\Delta u&=a u-b(x)u^p &\mbox{in} &(0,\infty)\times \Omega,\\ \alpha u_\nu+\beta u&=0 & \mbox{on}& (0,\infty)\times \partial \Omega,\\ u(0,.)&=u_0&\mbox{in}& \Omega, \end{array}\right. \] where $p>1$, $b(x)\geq 0$ is continuous and vanishes on the closure of a nontrivial subdomain $\Omega_0$ of $\Omega\subset R^N$. This case can be regarded as a mixture of the well-understood logistic (when b(x) > 0 always) and Malthusian (when $b(x)\equiv 0$) models and has attracted much study in recent years. It follows from recent studies that the model behaves like the logistic model if the growth rate a of the species is less than some constant a₀ > 0 and it behaves differently from the logistic model once $a\geq a_0$. In this paper, we show that, when $a\geq a_0$, the model behaves like the Malthusian model on part of the domain (i.e., on $\Omega_0$ where b vanishes) and it behaves like the logistic model on the remaining part of the domain. Our study shows that the boundary blow-up problem \[ -\Delta u=au-b(x)u^p \mbox{ in} \ \Omega\setminus\overline{\Omega}_0,\ \alpha u_\nu +\beta u=0 \mbox{ on}\\partial \Omega,\ \ u=\infty \mbox{ on}\ \partial \Omega_0 \] plays a key role in understanding the dynamics of our model and that the whole theory can be described by a nice bifurcation picture involving a branch of positive solutions at "infinity."