A blow-up dichotomy for semilinear fractional heat equations
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Mathematische Annalen
A blow-up dichotomy for semilinear fractional heat equations Robert Laister1
· Mikołaj Sier˙zega ˛ 2
Received: 18 December 2019 / Revised: 14 August 2020 © The Author(s) 2020
Abstract We derive a blow-up dichotomy for positive solutions of fractional semilinear heat equations on the whole space. That is, within a certain class of convex source terms, we establish a necessary and sufficient condition on the source for all positive solutions to become unbounded in finite time. Moreover, we show that this condition is equivalent to blow-up of all positive solutions of a closely-related scalar ordinary differential equation. Mathematics Subject Classification 35A01 · 35B44 · 35K58 · 35R11
1 Introduction In this paper we investigate the local and global existence properties of positive solutions of fractional semilinear heat equations of the form u t = α u + f (u),
u(0) = φ ∈ L ∞ (Rn ),
(1.1)
where α = − (−)α/2 denotes the fractional Laplacian operator with 0 < α ≤ 2 and f satisfies the monotonicity condition
Communicated by Y.Giga.
B
Robert Laister [email protected] Mikołaj Sier˙ze˛ga [email protected]
1
Department of Engineering Design and Mathematics, University of the West of England, Bristol BS16 1QY, UK
2
Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland
123
R. Laister, M. Sier˙z e˛ ga
(M) f : [0, ∞) → [0, ∞) is locally Lipschitz continuous, non-decreasing and f (0) = 0. We present a new dichotomy result for convex nonlinearities f satisfying the ODE blow-up criterion ∞ 1 du < ∞, (B) f (u) 1 together with an additional, technical assumption (S) (see Sect. 2). Specifically, for this class of nonlinearities, we show that all positive solutions of (1.1) blow-up in L ∞ (Rn ) in finite time if and only if 0+
f (u) du = ∞. u 2+α/n
(1.2)
Furthermore, we establish an equivalence between finite time blow-up of all positive solutions of (1.1) and finite time blow-up of all positive solutions of the scalar, nonautonomous ODE n x, x(t0 ) = x0 > 0. (1.3) x = f (x) − αt To the best of our knowledge this kind of blow-up equivalence, between the PDE (1.1) and a scalar ODE such as (1.3), has not been established before. We will refer to the phenomenon of blow-up in finite time of all non-negative, nontrivial solutions of (1.1) simply as the ‘blow-up property’. We will also identify the phrase ‘non-negative, non-trivial solution’ synonymously with ‘positive solution’. For the case of classical diffusion (α = 2) it has long been known that for f convex and sufficiently large initial data φ, blow-up in (1.1) occurs; see [15, Theorem 17.1] for bounded domains and the whole space alike. The central question then was whether diffusion could prevent blow-up for initial data sufficiently small. For general continuous sources f , this problem is highly non-trivial and remains open. However, under further restrictions on the form of the nonlinearity there has been significant progress, for example when f is the power l
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