Abstract
Consider the diffusion equation ut − Δu = |u|p − 1u (where p>1+2Nand (N − 2) p < N + 2) on the space ℝN. We prove that either ∥u(t)∥∞ blows-up in finite time or ∥u(t)∥∞ goes to zero like t−1/(p − 1) as t → +∞. We give also a new proof to the fact that when u(t) ≧ 0 and 1<p≦1+2N then ∥u(t)∥∞ blows-up in finite time. Sufficient conditions for global existence or blow-up are given, and the case where instead of ℝN one has a cone like domain is also studied.
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