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. By comparison with recent results on global-in-time solutions, a dichotomy result is obtained.

The main result of this paper is Theorem 1 on local-in-time solutions that cannot be continued to global in time ones.
Notation. The homogeneous Morrey spaces M s q (R d ) modeled on the Lebesgue space L q (R d ), q ≥ 1, are defined for u ∈ L q loc (R d ) and 1 ≤ q ≤ s < ∞, by their norms (7) {|y−x|<R} |u(y)| q dy

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 in [11]. 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], gives explicit and rather general sufficient conditions on functions u 0 in (2) in order to solutions of (1)-(2) blow up in a finite time, as well as estimates on the blowup time, cf. also [18].
The linear nonlocal diffusion operator A : generates the semigroup of linear convolution operators e tA with kernels k t ∈ L 1 (R d ), normalized so that R d k t (x) dx = 1, which are defined by the inverse Fourier transform 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. [10]. As it was studied in [10], the long time asymptotics of solutions of the linear Cauchy problem is then determined either by that of the classical heat equation or by the fractional heat equation as |x| → ∞).
These are examples of operators A whose kernels have "heavy tails": J(x) ≈ c |x| n as |x| → ∞ with some n > d, and they are discussed in [1] and [10]. 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.
Theorem 1. Suppose that u 0 ≥ 0 satisfies for some T > 0 the condition Here the decreasing function h(w) = Proof. By definition (14), we have W T (t) = k T −t * u(., t)(0) and, of course, z(., t) = k T −t solves the backward diffusion equation by the symmetry property of the semigroup, and the Jensen inequality in the last line.
Integrating this from 0 to t and passing to the limit t ր T , we obtain Remark 2. In some particular cases, under more restrictive assumption than condition (3), the sufficient condition for blowup of solutions of (1)-(2) can be described in a more explicit way than condition (16) sup in Theorem 1.
For the nonlinear heat equation (4) the condition (13) implies a sufficient condition for the blowup of (4) which was derived in [11], and has been analyzed in a recent paper [3].
We have, in this direction, the following Proposition 3. (i) If the infinitesimal generator A of the semigroup e tA satisfies (10) for an α ∈ (0, 2], F (u) = cu p with some p > 1 and c > 0, is a sufficient condition of blowup of solution of problem (1)- (2). Condition (18) 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.
Therefore, the sufficient condition for blowup (16) becomes By the translational invariance of equation (1), condition (20) for u 0 ≥ 0 is equivalent to On the other hand, according to [10,Theorem 1] and large t > 0 the semigroup e tA applied to u 0 can be well approximated by Finally, condition (20) is equivalent to which, in turn, is equivalent for u 0 ≥ 0 to condition (19) by above remarks.
Note that for e tA = e −t(−∆) α/2 the assumptions on the initial data u 0 ∈ L 1 (R d ) with Of course, condition (23)  (ii) Rewriting the quantity in (23) as we see that for each p < 1 + α d and u 0 1 > 0, the upper bound equals ∞ as claimed; remember relation (21). Above, the kernel of the semigroup e −t(−∆) α/2 has selfsimilar form, and is given by with a smooth function R. This satisfies the bound and, moreover, its gradient satisfies following from standard estimates for the kernels of linear fractional heat equations, cf. [10,14].
The proof of (ii) for p = p F and e tA = e −t(−∆) α/2 , α ∈ (0, 2), is in [18]. A rather simple new proof of the result (ii) for α = 2 and p = p F is in [3]. . For analogous questions for radial solutions of chemotaxis systems, see also [4].
Corollary 4 (dichotomy). There exist two positive constants c(α, d, p) and C(α, d, p) such , implies that problem (5) 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 (5) 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. [7]. One of the results in this direction is [7, Theorem 2.6]: if withs > d(p − 1)/α, lim x→0 |x| α p−1 u(x, t) = lim x→∞ |x| α p−1 u(x, t) = 0, uniformly in t ∈ (0, T )), then u can be continued to a global in time solution which still satisfies the bound This is a natural extension of properties of the Cauchy problem (4), (6) studied in, e.g., [15,16] and [3].

Estimates of discrepancy
Similarly to the considerations in [8] 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 the Morrey norm of the initial data for the model problem (5)-(6).
Theorem 5. (i) For each α ∈ (0, 2] and p > 1+ α d there exists a constant ν α,p independent of the dimension d such that if N > ν α,p , then each solution of the Cauchy problem (5)- (6) in R d with the initial data u 0 (x) ≥ Nu ∞ (x) blows up in a finite time.
(ii) For α = 2 and p > 1 + 2 d there exists a constant κ 2,p independent of d such that if the d(p−1) 2 -radial concentration of u 0 ≥ 0 defined by with some κ > κ 2,p then each solution of (5)-(6) in R d blows up in a finite time.
Using relations (36), (40) and the comparison principle, we see that suffices to a finite time blowup, thus (i) follows since the bound for K α,p (d) is d-independent.
(ii) Now, let us compute the asymptotics of the expression in condition (20) for α = 2 and the normalized Lebesgue measure dS on the unit sphere S d−1 and then relation (37) is used.
(iii) For α ∈ (0, 2), instead of (31) an analogous sufficient condition is of different order than for α = 2, namely (42) with a constant κ α,p independent of d. Indeed, for the normalized Lebesgue measure dS on the unit sphere S d−1 we have an upper bound for the quantity since representation (39), formulas (26) and (37) hold.
For an asymptotic lower bound on the quantity L α,p (d), begin with the observation that for β = d From formulas (43) and (44) we infer Therefore L α,p (d) ≍ 1 σ d d − α 2(p−1) holds. This is an estimate of optimal order and different from its counterpart for α = 2. Now, it is clear that a sufficient condition for blowup is satisfied if NL α,p (d) > c α,p with either N = κσ d d 1/(p−1) if α = 2 or N = κσ d d α/2(p−1) if α ∈ (0, 2).