Extremal Solutions and Instantaneous Complete Blow-up for Elliptic and Parabolic Problems

whenever g > 0 is increasing and convex on R and superlinear at +∞ in the sense (7) stated below. Under these assumptions, the extremal parameter of (2) satisfies 0 < λ∗ < ∞. Here and throughout the paper, Ω is a smooth bounded domain of R . For each 0 ≤ λ < λ∗, the minimal solution of (2) is classical. Their limit as λ ↑ λ∗ is the extremal solution u∗ and it may be singular (i.e. unbounded) for some nonlinearities and domains. When g(u) = e, it is known that u∗ ∈ L∞(Ω) if N ≤ 9 (for every Ω), while u∗(x) = log(1/|x|) if N ≥ 10 and Ω = B1. Brezis and Vázquez [BV] raised the question of determining the