Blow-up of Solutions of Semilinear Parabolic Equations

Merle, 1992 [30]: Consider the problem
u t = u xx + u p , |x| < 1, p > 1,
u = 0, |x| = 1,
u(x, 0) = u 0 (x), |x| ≤ 1.
Given any positive integer k and −1 < x 1 < · · · < x k < 1, there is u 0 such
that u blows up at t = T < ∞ and B = {x 1 , . . . , x k }.
Giga and Kohn, 1989 [18]: Consider the problem
u t = ∆u + u p , x ∈ Ω,
u = 0,
x ∈ ∂Ω,
u(x, 0) = u 0 (|x|) ≥ 0, x ∈ Ω,
where Ω = {|x| < R} ⊂ R N , p > 1 and (N − 2)p < N + 2. Then there is u 0
such that u blows up on a sphere B = {|x| = R 0 }, 0 < R 0 < R.
Velázquez, 1993 [36]: Consider the equation
u t = ∆u + u p , x ∈ R N , p > 1, (N − 2)p < N + 2.
Since (N − 1)-dimensional Hausdorff measure is finite in every compact sets,
the dimension of B is at most (N − 1) in every compact sets if u ≢ C(T −
t) − 1
p−1 .
3 Blow-up rate
In this section, we consider the blow-up rate of solutions for
Define M(t) := max Ω u(·, t), then
u t = ∆u + |u| p−1 u.
M ′ (t) ≤ M p (t), M(t) ≥ ((p − 1)(T − t)) − 1
p−1 .
Definition 2. Blow-up is of type I if (T − t) − 1
p−1 ‖u(·, t)‖∞ ≤ C, otherwise
blow-up is of type II.
Weissler, 1985 [38]: Assume the condition
(N − 2)p < N + 2, Ω = {|x| < R} ⊂ R N , u − radial, u r , u rr ≤ 0, u t ≥ 0.
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