Global unbounded solutions of the Fujita equation in the intermediate range

Abstract

We consider the Fujita equation u t = ∆u + u p on R N with N ≥ 3.

Blowup versus global in time existence of solutions for nonlinear heat equations

“…p−1 = ∞, then solutions of (1)-(2) cannot be global in time. On the other hand, the authors of[20] showed that for some u 0 (x) ∼ |x| − 2 p−1 global in time solutions of (1)-(2) exist and are unbounded as t → ∞.…”

Localized Solutions of a Semilinear Parabolic Equation with a Recurrent Nonstationary Asymptotics

“…By (3.13), statement (v) applies to u 0 for each k. Sincet k → ∞, by (i) and (vi), we see that the solution u(·, ·, u 0 ) is global and Lemma 2.5 then gives the bound (3.1). Relations However, to prove the existence of such global unbounded solutions, one can use a more direct and simpler proof; see [33].…”

“…As in that paper, we take advantage of the threshold behavior of solutions which is characteristic for the range of exponents p S < p < p JL (see section 2 for details). Compared to [33], the construction here is more involved as we need to select the initial condition more carefully to guarantee that the solution u is bounded and its ω-limit set does not contain nontrivial equilibria. For the latter we will employ intersection-comparison (zero number) arguments.…”

“…Our construction of a function u 0 with the desired properties shares some steps with [33], where our goal was to prove the existence of global unbounded solutions of the same equation. As in that paper, we take advantage of the threshold behavior of solutions which is characteristic for the range of exponents p S < p < p JL (see section 2 for details).…”

Threshold and strong threshold solutions of a semilinear parabolic equation

“…Global solutions with large-time behaviors just mentioned have mostly been known (see [21] in the case of (4) and p S < p < p JL , for example), but it is not clear whether those solutions are strong threshold solutions. On the other hand, we often use those solutions or the methods of proofs of their existence in order to prove the existence of a GSTS with the same large-time behavior.…”