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 $${u_t} - \Delta u = a(x,t)u - b(x,t){u^p}in\Omega \times (0,T),$$ $$u = \infty on\partial \Omega \times (0,T) \cup \overline \Omega \times \{ 0\} ,$$ where Ω is a smooth bounded domain, T > 0 and p > 1 are constants, and a and b are continuous functions, b > 0 in Ω × [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 $$b(x,t) \ge c{(T - t)^\theta }d{(x,\partial \Omega )^\beta } on \Omega \times \left[ {0,T} \right)$$ for some constants c > 0, θ > 0, and β > −2, 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.