Blow-up solutions for a class of semilinear elliptic and parabolic equations

We study the asymptotic behavior of the solutions to the problem [GRAPHICS] where p > 1, b(x) greater than or equal to 0 is continuous and vanishes on the closure of a nontrivial subdomain Ohm(0) of Ohm subset of R-N. This cas e 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 stu dy in recent years. It follows from recent studies that the model behaves l ike the logistic model if the growth rate a of the species is less than som e constant a(0) > 0 and it behaves differently from the logistic model once a greater than or equal to a(0). In this paper, we show that, when a great er than or equal to a(0), the model behaves like the Malthusian model on pa rt of the domain (i.e., on Ohm(0) where b vanishes) and it behaves like the logistic model on the remaining part of the domain. Our study shows that t he boundary blow-up problem -Delta u = au - b(x)u(p) in Omega \ <(Ohm)over bar>(0), alpha u nu + beta u = 0 on partial derivative Ohm, u = infinity on partial derivative Ohm(0) plays a key role in understanding the dynamics of our model and that the wh ole theory can be described by a nice bifurcation picture involving a branc h of positive solutions at "infinity."