Blow-up of Solutions of Semilinear Parabolic Equations

(i) N = 1 f is locally Lipschitz (Fila, 1992 [6]),
(ii) N = 2,|f ′ (u)| ≤ Ke |u|q , 0 < q < 2 (Fila, 1992 [6]),
(iii) N > 3, f is locally Lipschitz, |f(u)| ≤ C(1 + |u| p ), (N − 2)p < N + 2
(Cazenave and Lions, 1984 [4]),
(iv) N > 2, f(u) = u p , u ≥ 0, p > N+2,
p < 1 + 6 if N > 10, Ω = B N−2 N−10 R =
{|x| < R}, u is radial (Galaktionov and Vázquez, 1994 [14]),
(v) 3 ≤ N ≤ 9, f(u) = e u , Ω = B R , u is radial (Fila and Poláčik, 1999 [9]),
(vi) N > 2, f ∈ C 1 f(u)
, f ≥ 0, lim u→∞ = 1, p > N+2,
Ω = B u p N−2 R, u is radial,
p < p JL if N > 10 (Chen, Fila and Guo).
The answer is “Yes”, if
(vii) f(u) = u p , N > 2, p = N+2
N−2 , Ω = B R, u is radial, (Ni, Sacks and
Tavantzis, 1984 [34], Galaktionov and Vázquez, 1994 [14]),
(viii) f(u) = 2(N − 2)e u , N > 9, Ω = B 1 (Lacey and Tzanetis, 1987 [27]).
Explanations
(vii) Define
λ ∗ := sup{λ > 0; u(·, t; λϕ) → 0 as t → ∞}
for ϕ ∈ L ∞ (Ω), ϕ ≥ 0, ϕ ≢ 0. Then u(·, t; λ ∗ ϕ) is a global weak solution.
Galaktionov and Vázquez, 1997 [15], showed that blow-up in finite time
is complete. This implies that u(·, t; λ ∗ ϕ) is a global classical solution. If
u(·, t; λ ∗ ϕ) is bounded, u(·, t; λ ∗ ϕ) → 0 as t → ∞ by Corollary 1, but 0 is
stable, which contradicts the definition of λ ∗ . It follows that u(·, t; λ ∗ ϕ) is a
classical unbounded solution.
Pohožaev identity, 1965 [35] (for a ball)
If u is a nontrivial solution of
{
urr + N−1 u
r r + |u| p−1 u = 0, 0 < r < 1,
(S)
u ′ (0) = u(1) = 0,
then
(
N − 2 − 2N ) ∫ 1
p + 1
0
r N−1 |u| p+1 dr = −u 2 r(1).
Corollary 1. If p ≥ N+2 , N ≥ 2, then there is no nontrivial solution of (S).
N−2
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