Heat equation with a nonlinear boundary condition and growing initial data

We discuss the solvability and the comparison principle for the heat equation with a nonlinear boundary condition ∂tu = Δ x ∈ Ω t > 0; ∇u .v(x) = up; x ∈ ∂ Δ t > 0;(x; 0) ∝ = (x) x ∈ Ω where N ≥ 1, p > 1, is a smooth domain in RN and '(x) = O(e^λd(x)2 ) as d(x) → 1 for some λ ≥ 0. Here, d(x) = dist (x; ∈ Ω). Furthermore, we obtain the lower estimates of the blow-up time of solutions with large initial data by use of the behavior of the initial data near the boundary ∈ Ω.