A Blow-up Dichotomy for Semilinear Fractional Heat Equations

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.


Introduction
In this paper we investigate the local and global existence properties of positive solutions of fractional semilinear heat equations of the form 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, non-trivial solutions of (1.1) simply as the 'blow-up property'. We will also identify the phrase 'nonnegative, non-trivial solution' synonomously 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-trival 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 law nonlinearity f (u) = u p . In [3] a threshold phenomenon was established, whereby the (Fujita) critical exponent, given by p F = 1+2/n, separated two regimes: for 1 < p < p F (1.1) has the blow-up property, whereas for p > p F it is possible to find small initial conditions φ evolving into global-in-time solutions. Non-existence of positive global solutions in the delicate critical case p = p F was later established in [6] for the case n ≤ 2 and subsequently by [17] for all n ≥ 1. Thus was obtained the first blow-up dichotomy for (1.1): in the special 2 case f (u) = u p and α = 2, (1.1) has the blow-up property if and only if 1 < p ≤ p F . Some slight generalisations can also be found in [4,5].
Subsequently it was shown in [9], in the special case of classical diffusion (α = 2), that An important aspect of this paper is that we demonstrate (via an explicit construction) that, for all α ∈ (0, 2], there exist monotone, convex f for which (1.1) has the blow-up property, but for which the results in [17] and [9]) do not apply.
The remainder of the paper is organised as follows. In Section 2 we prove that, for a suitable class of sources f , (1.2) is sufficient for the ODE (1.3) to have the blow-up property.
In Section 3 we show for this class that if the ODE (1.3) possesses the blow-up property 3 then so too does (1.1). In Section 4 we present a construction which demonstrates that our assumption (S) (stated below) is strictly weaker than (B.3) of [9] in the case α = 2. We then establish in Section 5 the necessity of (1.2) for (1.1) to have the blow-up property and conclude with some remarks in Section 6.
The particular, homogeneous, Fujita-critical case where g(µ) = µ p and p = p F was considered in [9, (B.3)] on a strictly larger λ-µ region than appears in (S); i.e., the condition imposed upon f in [9] is a more restrictive one than that in (S).
We mention also that a condition such as f (u)/u p being non-decreasing was used in [1], although there the condition at infinity was relevant rather than near zero. 4 (ii) It is easy to verify that if 0 = f ∈ C 1 satisfies (M) and (C) and the condition then there exists a p > 1 such f (u)/u p is non-decreasing near zero. Consequently f satisfies (S) by (i) above. Note that for 0 = f ∈ C 1 satisfying (M) and (C), we If Suppose first that x is bounded away from zero, i.e., there exists ε > 0 such that x(t) ≥ ε for all t ≥ t 0 . By monotonicity of L, L(x(t)) ≥ L(ε) > 0 for all t ≥ t 0 . Hence there exists For such t we have and so by (B) x blows up in finite time, a contradiction. Now suppose that x does not remain bounded away from zero. We then claim that For suppose this is not the case, so that there exists t 2 ≥ t 0 such that Since x is C 1 and not bounded away from zero, there exists t 3 > t 2 such that and so by the monotonicity Now set y(t) = t n/α x(t) so that y satisfies the ODE By (2.4), y is clearly increasing and so For τ > t 0 sufficiently large we can ensure that y(t) ≥ µ 0 and and so Letting t → ∞ and using (2.3) we again obtain a contradiction, on recalling the integrability of 1/g in (S).

Blow-up of the PDE
In this section we show that blow-up of the ODE (2.1) implies blow-up of the PDE (1.1).
where K α is the (positive) fractional heat kernel. As is commonplace in the study of semilinear problems, we may then study (1.1) via the variation of constants formula It is well known that for any non-negative initial condition φ ∈ L ∞ (R n ) there is a T φ > 0 such that (1.1) has a unique non-negative solution u which is bounded on R n × [0, T ] for  Proof. We proceed as in the proof of the main theorem in [17,Section 4]. We briefly outline the initial steps of that proof for the reader's convenience.
Suppose, for contradiction, that u is a non-negative, global solution of (1.1). Then u satisfies the integral equation Clearly u > 0 for all t > 0 and so, by translating in time if necessary, we may assume without loss of generality that φ > 0.
Using the integral formulation (3.3), positivity of the solution and standard properties of K α , one can then show that there exist constants c > 0, τ 0 > 0 and t 0 > 0 such that [17, p. 48]). It follows that Clearly v(t) ∈ L ∞ (R n ) for all t > 0 since u is assumed to be in L ∞ (R n ) for all t > 0. Now set Evidently z(t) is positive and finite for all t > 0. Multiplying (3.4) by K α (x, t), integrating over R n and using the semigroup property of K α , gives where k = k(n, α, c) is a positive constant. Now using the scaling property of K α (see e.g., [17, p. 46-47]) and the fact that K α (x, t) is decreasing in |x|, we have for Hence, by Jensen's inequality, for all t ≥ t 1 , κ < 2 −n/α min{1, k} and t 1 > t 0 sufficiently large. Here we point out that It now follows from (3.5) that for t > t 1 , z is a supersolution of the ODE w ′ = κf (w) − n αt w.

8
By rescaling time (t → κt) we see that z(t) ≥ x(κt), where x is the solution of the ODE By assumption x (and hence z) blows-up in finite time, yielding the required contradiction to our earlier statement that z(t) is finite for all t > 0.
By Theorem 2.1 and Theorem 3.1 we obtain the following blow-up result for (1.1).

A Distinguishing Example
In this section we present an example of a function f which satisfies the hypotheses of  Let α ∈ (0, 2]. Step 1. Define the monotonically decreasing sequences Step 2. We now modifyf to create a function f satisfying (M), (C) and (B).
Fix p and θ > 1 such that and set v i = θu i+1 . It is easily verified that θu i+1 < u i for all i sufficiently large, and so for all such i i so that (4.1) holds. We now choose a i and b i to ensure that f is continuous, i.e. such that the line y = b i u−a i passes through the points (u i+1 , σ i+1 u pα i+1 ) and By construction f is also increasing and Lipschitz on [0, δ].
In order that f be convex on (by comparing the gradient of f at the endpoints of the intervals), or equivalently . continuity and also such that (B) holds.
Step 3. Next we show that f satisfies the remaining hypotheses of Corollary 3.2. By

Remark 2.1(i) it suffices to show that
(i) f (u)/u p is non-decreasing on (0, δ), and For (i) let F (u) := f (u)/u p . This continuous, piecewise differentiable function is given explicitly by Hence F ′ ≥ 0 on J i if and only if Now, recalling (4.6), we have as i → ∞. Hence, by (4.5), (4.8) holds for all i sufficiently large. Thus F is non-decreasing on (0, δ).
Step 4. Finally we show that f fails to satisfy assumption [9, (B3)] when α = 2. In fact we establish a more general result: for any α ∈ (0, 2], we find a sequence λ i → 0 such that Consequently (4.3) fails in the special case α = 2.
To achieve this we show that there is a sequence recalling that σ i = e −i 2 .
Fix 1/2 < q < 1 and let λ i = v q i . Clearly λ i > v i since v i < 1 and q < 1. It is also easily verified that λ i = θ q u q i+1 < u i for i sufficiently large, recalling (4.4). Hence λ i ∈ M i for such i. Next, Hence in order to show that λ 2 i > v i+1 , it suffices to show that u 2q i+1 > θu i+2 . This is readily verified for large i, recalling (4.4). It follows that λ 2 i ∈ M i+1 , as required.

Remark 4.2.
It is reasonable to speculate whether the analogous condition to (4.2), with the power law µ p F replaced by µ pα , might provide the basis for similar results to those in [9] for the fractional diffusion case 0 < α < 2. However, the f constructed above satisfies for any α ∈ (0, 2]. Consequently, the f constructed above pre-empts any improvemts that might possibly be obtained in this way, at least within the class of convex source terms.

Global Existence
In this section we consider the issue of global continuation of locally bounded solutions of (1.1). We set Q T = R n × (0, T ) and write · q for the norm in L q (R n ).
Definition 5.1. Let T > 0. We say that a non-negative, measurable, finite almost everywhere function w : Q T → R is an integral supersolution of (1.1) on Q T if w satisfies We recall the following well-known smoothing estimate for the fractional heat semigroup for 1 ≤ q ≤ r ≤ ∞ and φ ∈ L q (R n ) (see e.g., [13, Lemma 3.1]): where C = C(n, α, q, r).
We are now in a position to state our main result.  (c) 0 + f (u) u 2+α/n du < ∞.

Concluding remarks
We have established a new blow-up dichotomy for positive solutions of fractional semilinear heat equations, extending those of [3,6,9,17]. In particular, for a class of convex (t/s) −n/α f (x(s)) ds.
The similarity arises when considering the decay rate of the operator norm of S α (t) : , which is given (via the smoothing estimate (5.1)) by It is intriguing that this formal similarity manifests itself as an equivalence with respect to the blow-up property.
It would be interesting to know whether the technical hypothesis (S) can be removed in Theorem 2.1 and consequently in Theorem 5.2. This would yield a sharper and perhaps more natural result. However, recalling Remark 2.1 (i), we suspect that the stronger (but more easily verified) assumption that f (u)/u p be non-decreasing near zero for some p > 1, will prove more useful in applications. Similarly we would like to better understand the rôle of the convexity assumption on f . It is this convexity that permits us to show, via Jensen's inequality, that blow-up of the ODE implies blow-up of the PDE. It remains open whether our blow-up equivalence result can be obtained without the convexity assumption and without assuming (S).
Finally, we mention that the analysis of fractional semilinear parabolic equations such as (1.1) is intimately related to the study of symmetric α-stable processes in probability theory. It seems reasonable to hope that our work might have parallels in that domain and provide new insights for such processes.