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 over a smooth bounded domain Ω: ( ut −∆u = a(x, t)u− b(x, t)u in Ω× (0, T ), u =∞ on ∂Ω× (0, T ) ∪ Ω× {0}, where T > 0 and p > 1 are constants, a and b are continuous functions, with b > 0 in Ω × [0, T ), b(x, T ) ≡ 0. We study the existence and uniqueness of positive solutions, and their asymptotic behavior near the parabolic boundary. Under the extra condition that b(x, t) ≥ c(T − t)d(x, ∂Ω) on Ω × [0, T ) for some constants c > 0, θ > 0 and β > −2, we show that such a solution stays bounded in any compact subset of Ω as t increases to T , and hence solves the equation up to t = T .