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

We study the asymptotic behavior of the solutions to the problem \[\{ {array}{rlll} u_t-Δ u&=a u-b(x)u^p &in &(0,∞)× Ω,\\ α u_ν+β u&=0 & on& (0,∞)× ∂ Ω,\\ u(0,.)&=u_0&in& Ω, {array}. \] where $p>1$, b(x)≥ 0 is continuous and vanishes on the closure of a nontrivial subdomain Ω₀ of Ω⊂ RN. This case can be regarded as a mixture of the well-understood logistic (when b(x) > 0 always) and Malthusian (when b(x)≡ 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 a 0 > 0 and it behaves differently from the logistic model once a≥ a₀. In this paper, we show that, when a≥ a₀, the model behaves like the Malthusian model on part of the domain (i.e., on Ω₀ 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 \[ -Δ u=au-b(x)u^p in \ ΩΩ̄_0,\ α u_ν +β u=0 on\∂ Ω,\ \ u=∞ on\ ∂ Ω_0 \] 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."