The parabolic logistic equation with blow-up initial and boundary values

In this article, we investigate the parabolic logistic equation with blow-up initial and boundary values u(t) - Delta u = a(x, t)u - b(x, t)u(p) in Omega x (0, T), u = infinity on partial derivative Omega x (0, T) boolean OR (Omega) over bar x {0}, where Omega is a smooth bounded domain, T > 0 and p > 1 are constants, and a and b are continuous functions, b > 0 in Omega x [0, T) and b(x, T) = 0. We study the existence and uniqueness of positive solutions and their asymptotic behavior near the parabolic boundary. We show that under the extra condition that b(x, t) = c(T - t)(theta)d(x, partial derivative Omega)(beta) on Omega x [0, T) for some constants c > 0, theta > 0, and beta > -2, such a solution stays bounded in any compact subset of Omega as t increases to T, and hence solves the equation up to t = T.