Blowup of solutions for nonlinear nonlocal heat equations

Blowup analysis for solutions of a general evolution equation with nonlocal diffusion and localized source is performed. Sufficient conditions for blowup are expressed in terms of some Morrey space norms. A comparison of these with recent results on global-in-time solutions is discussed.


Introduction
We consider here nonnegative solutions u = u(x, t) ≥ 0 of the Cauchy problem with the linear nonlocal diffusion operator defined by the convolution of u with a nonnegative radially symmetric function J satisfying R d J (x) dx = 1, and with the nonlinearity (a localized source) defined by Similar evolution equations extending the classical nonlinear heat equation thoroughly presented in [17] u t = u + |u| p−1 u, x ∈ R d , t > 0, (5) and the equations with nonlocal diffusion operators defined by fractional powers of the Laplacian, α ∈ (0, 2) and even more general nonlinearities, have been studied in, e.g., [11] (the linear case), and e.g., [1,13,20] (the nonlinear case). Equations of the type (1) are related to the differential and integrodifferential equations (5) and (6) by their long time asymptotic behavior determined frequently by the linear equations (12)- (13) below, and studied in, e.g., [1,11,13].
There are plenty of results on closely related questions on conditions on the initial data guaranteeing the local-in-time existence of solutions to Eq. (5) and sufficient conditions leading to finite time blowup of solutions, see [2,3,14], the latter reference dealing with a general nondecreasing but not necessarily convex nonlinearity.
A standard theory of the existence of solutions for problem (1)-(2) in [1,11,13] is developed in the framework of mild solutions, i.e., those satisfying the Duhamel formula Here, u ∈ C([0, T ), L 1 (R d )) is required when F ≡ 0, i.e., in the case of linear equations (1). Note that the semigroup e tA is strongly continuous, so that the initial condition u 0 is attained in the sense of L 1 -limit as t → 0. In the case of nonlinear equations with, say, |F(u)| ≤ C(1 +|u| p ), assumption u 0 ∈ L 1 ∩ L ∞ (R d ) guarantees the local-in-time well-posedness of the Cauchy problem (1)- (2) with see [11,13]. In both the cases mild solutions are weak solutions so that, in particular, holds for t < T and each function φ , which is completely analogous to the concept of weak solutions of nonlinear heat equations with either the Laplacian (5) or fractional Laplacian (6).
However, it is convenient to adopt here a slightly more general definition of locally bounded solutions admitting unbounded initial data which are merely in L 1 ∩ L ∞ loc (R d ). This is motivated by the fact that solutions to problem (1)- (2) with data in L 1 ∩ L q loc (R d ) for some particular q may exist, see [14] in the case of equations like (5), (6) with general nonlinearities. Note that this definition permits to consider locally bounded in space but unbounded solutions as in [4,12].
Remark that unlike the case of the Eqs. (5) and (6), we cannot expect that solutions are smooth for t > 0, i.e., an instantaneous regularization effect of the semigroup e tA generated by bounded diffusion operators A on solutions is absent.
The phenomenon of a blowup of a solution is understood here in the local L ∞ sense, i.e., u is a blowing up solution not later than at t = T > 0 if for some R > 0 the relation holds. This definition is consistent with the above meaning of solutions requiring for t > 0. Theorem 1 on local-in-time solutions to Eq. (1) that cannot be continued to globalin-time ones is one of the main results in this paper. Proposition 3 interprets a general condition in Theorem 1 in terms of the Morrey space norms related to approximative scaling properties of the problem. There are also subsidiary results on the size of global-in-time solutions compared to blowing up ones in the case of Eqs. (5) and (6).
The main idea here is that we are looking for a single quantity = (u 0 ) (a functional norm) which decides on the blowup versus global existence. Unfortunately, we do not have a dichotomic partition of the set of admissible initial data but weaker results like: (u 0 ) < c implies the global existence while (u 0 ) > C (with C > c) leads to a blowup of solutions in Theorem 5. Quite often the condition (u 0 ) < ∞ is necessary, but not always sufficient, for the local-in-time existence of solutions, see [4,18] for the cases of Eqs. (5) and (6).

Notation The homogeneous Morrey spaces
with the convention M s Here, B(x, R) denotes the ball centred at x of radius R: {y : |y − x| < R}, and 1 I B(x,R) is its characteristic function.
The asymptotic relation f ≈ g as s → s 0 (with either s 0 = 0 or s 0 = ∞) means that lim s→s 0 f (s) g(s) = 1, and f g is used whenever lim s→s 0 f (s) g(s) ∈ (0, ∞). We will use the notation g 1 if the quantity g (depending on some parameters) is supposed to be sufficiently large (in terms of those parameters).

Blowup for a general nonlinear source, nonlocal diffusion model
Our aim in this paper is to give a simple proof of blowup of solutions for the Cauchy problem (1)-(2) based on the classical idea of Fujita [12]. We believe that this proof is simpler than monotonicity arguments given in [1]. Moreover, this argument applies to a class of initial data much larger than in [1, Theorem 2.4], giving explicit and rather precise general sufficient conditions on functions u 0 in (2) in order to solutions of (1)-(2) blow up in a finite time, together with estimates of the blowup time, cf. also [20] for Eq. (6).
Let us briefly recall some facts from Alfaro [1] and Chasseigne et al. [11]. The linear nonlocal diffusion operator A : (3) is bounded and generates the semigroup of linear operators e tA . This semigroup is represented as Typical and the most interesting examples of functions J are those with their Fourier transforms J satisfying with A > 0, α = 2 (corresponding to, e.g., the case of smooth, compactly supported functions J ), and those with α ∈ (0, 2), cf. [11,Sec. 1]. Note that for α ∈ (0, 2) such J do not have finite second moment. As it was studied in [11], the long time asymptotics of solutions of the linear Cauchy problem is then determined either by that of the classical heat equation or the fractional heat equation see (22) below for a precise statement from Chasseigne et al. [11,Theorem 1]. Other examples of functions J with the Fourier transform J like J (ξ ) − 1 ≈ A|ξ | 2 log |ξ | as |ξ | → 0, are mentioned in [11,Th. 5.1], and then J (x) 1 |x| d+2 as |x| → ∞. Thus, they are examples of operators A whose kernels have "heavy tails": J (x) 1 |x| n as |x| → ∞ with some n > d, discussed in [1,11]. For them, if n ∈ (d, d + 2), then α = n − d ∈ (0, 2) holds. Their semigroup kernels have also heavy tails unlike the Gauss-Weierstrass kernel of the heat semigroup for α = 2. Besides bounded diffusion operators (3) studied here, the proof of Theorem 1 below applies also to unbounded operators and −(− ) α/2 as was in classical papers [12,20]. Following the idea in [12], let us compute the time derivative of W T (t) by the symmetry property of the function J , so that the symmetry of the semigroup, and the Jensen inequality in the last line. Integrating this from 0 to t, we obtain If initially W T (0) > h −1 (T ) holds, then taking into account the property lim w 0 h(w) = ∞, we arrive at Finally, if lim t T W T (t) = ∞ then evidently lim sup t T sup |x|<R u(x, t) = ∞ for every R > 0 which means that u blows up not later than at t = T . Indeed, we have lim t T sup |x|>R k T −t (x) = 0 for every R > 0.

Remark 2
In some particular cases, under more restrictive assumptions than condition (4), the sufficient condition for blowup of solutions of (1)-(2) can be described in a more explicit way than condition considered in Theorem 1. For instance, for the nonlinear heat equation (5) with Au = u and F(u) = u p , so that for e t defined with the Gauss-Weierstrass kernel, the condition (17) is equivalent to This is a sufficient condition for the blowup of (5) derived in [12], and this has been interpreted in a recent paper [4] in terms of the Morrey space norm M d( p−1)/2 (R d ).
We have, in this direction, the following general

Proposition 3 (i) If the infinitesimal generator
A of the semigroup e tA satisfies (11) for an α ∈ (0, 2], F(u) u p with some p > 1 as s → 0, is a sufficient condition of blowup of solution of problem (1)- (2). Condition (19) is equivalent to a large value of the Morrey space norm of u 0 (ii) Moreover, if 1 < p < p F where p F = 1 + α d is the so-called Fujita exponent, then each nontrivial nonnegative solution u ≡ 0 blows up in a finite time.
z → ∞. Therefore, the sufficient condition for blowup (17) becomes By the translational invariance of Eq. (1), condition (21) for u 0 ≥ 0 is equivalent to On the other hand, according to [11,Theorem 1], for u 0 ∈ L 1 (R d ) with u 0 ∈ L 1 (R d ) and large t > 0, the semigroup e tA applied to u 0 can be well approximated by the semigroup e −t(− ) α/2 generated by the fractional power of the Laplacian where B −κ ∞,∞ is the homogeneous Besov space of order −κ < 0. The above condition (23) ∈ (0, 2).
Finally, condition (21) is equivalent to which is, in turn, equivalent for u 0 ≥ 0 to condition (20) by the above remarks. Note that for e tA = e −t(− ) α/2 the assumptions on the initial data Of course, condition (24) is quite general, and involves one free parameter T > 0. Particular examples of initial data considered in [1,Th. 2.3] leading to blowup of solutions do satisfy (24).
(ii) Rewriting the quantity in (24) as we see that for each p < 1 + α d and u 0 1 > 0, the upper bound equals ∞ as claimed; remember relation (22). Above, the kernel of the semigroup e −t(− ) α/2 has the selfsimilar form P t,α ( with a smooth function R. This kernel satisfies the bound 0 < P t,α (x) ≤ C see for instance [11,16].
These are counterparts of results in [4, Remark 7, Theorem 2] for the classical nonlinear heat equation. These, together with results of [8,Proposition 2.3], lead to the following result, similarly as was in [4,Corollary 11] for the Cauchy problem (6) with (2). For analogous questions for radial solutions of chemotaxis systems, see also [5]. c(α, d, p) and C(α, d, p) such that if p > 1 + α d then

Corollary 4 There exist two positive constants
, implies that problem (6) with data (2) has a global in time, smooth solution which, moreover, satisfies the time decay estimate u(t) C(α, d, p) implies that each nonnegative solution of problem (6) with the initial condition (2) blows up in a finite time.
It is of interest to estimate the discrepancy of these constants c(α, d, p) and C(α, d, p) compared to the Morrey space norm of the singular stationary solution u ∞ > 0 of (6) which exists for p > 1 + α d−α ; here is the area of the unit sphere S d−1 in R d . This singular stationary solution u ∞ > 0 is homogeneous, see [8,Prop. 2.1], with the constant Note that asymptotically with constants c α, p independent of d.
Of course, there are many interesting behaviors of solutions (and still open questions) for the initial data of intermediate size satisfying and/or suitable pointwise estimates comparing the initial condition u 0 with the singular solution u ∞ , see e.g., [8]. One of the results in this direction is [8,Theorem 2.6 t) is a solution of problem (6) with (2) and T )), then u can be continued to a global in time solution which still satisfies the bound 0 ≤ u(x, t) ≤ u ∞ (x).
Once again, here it should be stressed on the fact that conditions on initial data guaranteeing local-in-time existence of solutions of Eq. (5) derived in [2] but motivated by [3], and then interpreted in [4] as a bound on the Morrey space M d( p−1)/2 (R d ) norm, are qualitatively close to those above even if they involve M s q (R d ) spaces with any q > 1, see also [4,Remark 3.4], [18,Proposition 6.1].

Estimates of discrepancy
Similarly to the considerations in [9] on blowup for radial solutions of chemotaxis systems, we determine asymptotic (with respect to the variable of dimension d → ∞) discrepancy between bounds in sufficient conditions for blowup either in terms of multiple of the singular solution or in terms of critical value of the radial concentration (and therefore of the Morrey norm) of the initial data for the model problem (6) with data (2).
Proof A more detailed analysis of condition (21) reveals that is a sufficient condition for blowup, with some constant c α, p > 0 independent of d.
First, we compute The last two lines follow using the relation see e.g. [21], which is an immediate consequence of the Stirling formula (z + 1) ≈ √ 2π z z z e −z as z → ∞.