ABSTRACT
We begin with the celebrated results on blow-up by Fujita (1966) for 1 < p < p0 [134, 135] for 1 < p < p0 and Hayakawa (1973) [200] (1973) for the critical
and
case p = p0, N = 1, 2; see [240, 9] for first extensions to any N ≥ 1. In the last fifty years, these pioneering results were extended in several ways to various semilinear and quasilinear PDEs (see a list of monographs to be given below shortly). Fujita and Hayakawa were the first, to establish existence of a very special critical exponent
p = p0 = 1 + 2 N (1)
for the semilinear heat equation, with a parameter p > 1,
ut = Δu+ |u|p−1u in IRN × IR+, u(x, 0) = u0(x) in IRN , (2) where initial data u0 ∈ L1 ∩ L∞ are, typically and for simplicity, assumed to decay exponentially fast at infinity. Namely:
(I) in the subcritical range
1 < p ≤ p0 = 1 + 2N , (3) all the nontrivial nonnegative solutions (i.e., for data u0 ≥ 0) blow up in finite time, while
(II) for p > p0, there exists a class of “small” solutions that are global in time.
