Optimal estimates for blowup rate and behavior for nonlinear heat equations

We first describe all positive bounded solutions of ∂w ∂s = ∆w − 1 2 y.∇w − w p− 1 + w . where (y, s) ∈ RN × R, 1 < p and (N − 2)p ≤ N + 2. We then obtain for blow-up solutions u(t) of ∂u ∂t = ∆u+ u uniform estimates at the blow-up time and uniform space-time comparison with solutions of u′ = up.