Blowing up radial solutions in the minimal Keller–Segel model of chemotaxis

We consider the simplest parabolic–elliptic model of chemotaxis in the whole space in several dimensions. Criteria for the blowup of radially symmetric solutions in terms of suitable Morrey spaces norms are derived.


Introduction and main results
We consider in this paper solutions that blow up in a finite time for the Cauchy problem in space dimensions d ≥ 2 One motivation to study this model comes from Mathematical Biology, where Eqs. (1.1)-(1.2) are a simplified (the, so-called, minimal) Keller-Segel system modeling chemotaxis, see e.g., [4,16,23,25,26]. The unknown variables u = u(x, t) and v = v(x, t) denote the density of the population of microorganisms (e.g., swimming bacteria or slime mold), and the density of the chemical secreted by themselves that attracts them and makes them to aggregate, respectively. Another important interpretation of system (1.1)-(1.2) comes from Astrophysics, where the unknown function u = u(x, t) is the density of gravitationally interacting massive particles (micro-as well as macro-) in a cloud (of atoms, molecules, dust, stars, nebulae, etc.), and v = v(x, t) is the Newtonian potential ("self-consistent mean field") of the mass distribution u, see [1][2][3]8,17,18]. Aggregation of those particles may lead to formation of singularities (an implosion of mass phenomenon) in finite time.
Even if in applications u 0 ∈ L 1 (R d ), and then mass is conserved, we will also consider locally integrable solutions with infinite mass like the famous Chandrasekhar steady-state singular solution in [17,18] for d ≥ 3 Our results include • sufficient conditions on radial initial data which lead to a finite time blowup of solutions expressed in terms of quantities related to the Morrey space norm M d/2 (R d ) in Theorem 2.2. For instance, condition (2.14): sup T >0 T e T u 0 (0) > C(d) for some C(d) ∈ [1,2] is sufficient for the blowup of solution with the initial condition u 0 . Sufficient blowup conditions expressed in terms of the radial concentration (1.9): |||u 0 ||| > N , together with an asymptotics of the number N as d → ∞, are also in Proposition 2.6. Similar results for the system with modified diffusion operator (1.6) supplemented with the initial condition u(x, 0) = u 0 (x) ≥ 0 (1.7) will be derived and discussed in Sect. 3. In a parallel way, we have also • blowup of radial solutions with large initial data (Theorem 3.2, α ∈ (0, 2)); • a reformulation of sufficient condition for blowup of radial solutions in terms of For the proof of the main result, we revisit a classical argument of H. Fujita (applied to the nonlinear heat equation in [19]) and reminiscent of ideas in [16], which leads to a sufficient condition for blowup of radially symmetric solutions of system (1.1)-(1.2), with a significant improvement compared to [13] where local moments of solutions have been employed. Then, we derive as corollaries of condition (2.14) other criteria for blowup of solutions of (1.
Here, as usual,

Solutions blowing up in a finite time
It is well-known that if d = 2, the condition leading to a finite time blowup, i.e., is expressed in terms of mass, that is M > 8π , see e.g., [2,11,12]. If d ≥ 3, a sufficient condition for blowup for an initial condition (not necessarily radial) is that u 0 is highly concentrated, namely for some 0 < γ ≤ 2 and a (small, explicit) constantc d,γ > 0, see [9, (2.4)]. Since for some constantC d,γ > 0 and all u 0 ∈ M d/2 ∩ L 1 , see [9, (2.6)], this means that the Morrey space M d/2 norm of u 0 satisfying condition (2.1) must be (very!) large: According to [2],c d, Recently, some new results on the blowup of solutions to problem (1.1)-(1.3) appeared in [7,[11][12][13]22] with a new strategy of the proofs involving local momenta of (most frequently) radial solutions, and with improved sufficient conditions in terms of the initial datum u 0 . We will apply the classical proof of blowup in the seminal paper [19] by H. Fujita, and then improve the sufficient conditions for the blowup expressed in terms of a functional norm of u 0 .
First, we note a general property of potentials of radial functions Thus, for the radial function ∇v(x) · x |x| and |x| = R, we obtain the required identity Now, we proceed to apply the classical idea of blowup proof in [19]. Proof. For a fixed T > 0 consider the weight function G = G(x, t), x ∈ R d , t ∈ [0, T ), which solves the backward heat equation with the unit measure as the final time condition Clearly, we have a (unique nonnegative) solution , (2.5) defined by the Gauss-Weierstrass kernel, satisfying G(x, t) dx = 1, so that Define for a solution u of (1.1)-(1.2) which exists on [0, T ] the moment Since G decays exponentially fast in x as |x| → ∞, the moment W is well defined (at least) for solutions u = u(x, t) which are polynomially bounded in x. The evolution of the moment W is governed by the identity where we used the radial symmetry of the solution u in (2.8), Lemma 2.1 and, of course, the radial symmetry of G.
Expressing W in the radial variables we obtain Now, applying the Cauchy inequality to the quantity (2.9), we get dr.
Returning to the time derivative of W in identity (2.8), we arrive at the differential inequality . Recalling (1.11), we denote Thus, we finally obtain which, after an integration, leads to (2.13) Now, it is clear that if then lim sup t T W (t) = ∞ which means: lim sup t T, x∈R d u(x, t) = ∞, a contradiction with the existence of a locally bounded solution u on (0, T ]. , For other results on blowup rates (e.g., a faster blowup, the so-called, type II blowup), see [20,25,26].
REMARK 2.4. For d = 2 and radially symmetric nonnegative measures u 0 , by identity (2.2) and C(2) = 2, condition (2.14) after the integration by parts reads for the radial distribution function M of the initial condition u 0 . This means: sup r >0 M(r ) > 8π , and the well-known blowup condition for radially symmetric solutions in R 2 is recovered.
Observe that the equality in the Cauchy inequality (2.10) holds if and only if with some A(t) ≥ 0.
Then inequality (2.13) reads then the solution blows up not later than T . This holds exactly when A(0) ≥ 4σ d since (1.11). This solution (cf. [ , twice the singular stationary solution, when t T so that the density of this solution becomes infinite at the origin for t = T .
Clearly, for this solution u and the corresponding initial density u 0 we have for We express below a sufficient condition (2.14) for blowup in terms of the radial concentration.
blows up in a finite time} .

Then the asymptotic relation
holds as d → ∞. Therefore, if u 0 ≥ 0 is such that |||u 0 ||| > N , then the solution with u 0 as initial datum blows up in a finite time.
The inequality |||u 0 ||| > N means that the radial distribution function corresponding to such Proof. Here and in the sequel, due to the scaling properties of system (1.1)-(1.2), we may consider R = 1 which does not lead to loss of generality.
for all r > 0 and the solution u with u 0 as the initial datum blows up in a finite time. Indeed, this is an immediate consequence of the averaged comparison principle, i.e., [10,Theorem 2.1] or the comparison principle for Eq. (2.17), see also [5] in the case d = 2 then u blows up in a finite time. To check that . (2.20) Note that, by the above computations, there exist radial initial data u 0 ∈ L 1 (R d ) ∩ L ∞ (R d ) with W (0) as close to 1 as we wish. In other words, we have C(d) ∈ [1, 2).
To calculate the asymptotics of the number N observe that the quantity sup t>0 te t u 0 (0) in (2.18) for the normalized Lebesgue measure σ −1 d dS on the unit sphere S d−1 is equal to Therefore, by (1.11) and the Stirling formula for the Gamma function This improves the estimate of N dσ d in [13,Section 8].
We give below some other examples of initial data leading to a finite time blowup of solutions.
REMARK 2.7. Observe that for each initial condition u 0 ≡ 0 there is N > 0 such that condition (2.14) is satisfied for N u 0 .
Moreover, for each η > 2 and sufficiently large R = R(η) > 1 the bounded initial condition of compact support u 0 = η1 {1≤|x|≤R} u C leads to a blowing up solution, see (2.14). The singularity of that solution at the blowing up time is 1 |x| 2 at the origin. It seems that the latter result cannot be obtained applying previously known sufficient criteria for blowup like (2.1).
On the other hand, the initial data like min{1, u C } + εψ with a smooth nonnegative, compactly supported function ψ and a sufficiently small ε > 0 (somewhere they are above the critical u C pointwisely) still lead to global-in-time solutions according to [ are mutually equivalent, however, with comparison constants depending on d. This assumption means that all local singularities of such data are strictly weaker than 1 |x They enjoy an instantaneous regularization property: u(t) ∈ L ∞ (R d ) for each t > 0. More precisely, sup 0<t≤T t d 2 p u(t) ∞ < ∞ for such solutions. The Chandrasekhar locally unbounded solution u C is a threshold in the following sense: all solutions with the initial data strictly below u C are global and locally bounded, see for precise statement [10, Theorem 2.1]. Therefore, the above criteria for blowup apply to solutions with data in has a singularity at the origin strong enough, in the sense that lim sup for some largeC(d), then the existence time for suitable truncations of u 0 : 1 {|y|>R n } u 0 , R n → 0, tends to 0. Therefore, a phenomenon of discontinuity of solutions with respect to the initial data occurs in L ∞ . There is no local mild solution emanating from u 0 that enjoy instantaneous L ∞ regularization effect. To see this, recall from [13] an estimate for the existence time of solutions of (1.1)-(1.3). In the case of small R > 0 in [13, Theorem 2.9] inequality [13, (8.13) Thus, w R (T ) > M and blowup occurs for T R 2 when R → 0. In fact, if u 0 satisfies relation (2.24) then we see that any mild (hence weak) solution of system (1.1)-(1.2) does not regularize to L ∞ . Indeed, suppose a contrario that a solution u with u 0 as the initial data ( Assuming t 1 has been chosen sufficiently small, by weak continuity u 1 = u(·, t 1 ) satisfies the blowup condition (2.24) on a ball of fixed small radius R > 0. Thus, this solution blows up before T R 2 , so that u itself blows up before t 1 + T . Since we can choose sufficiently small R > 0, there exists arbitrarily small t * > 0 such that u is not in L ∞ for 0 < t < t * .
Further results on the existence of global small solutions, the well-posedness (and also ill-posedness) of system (1.1)-(1.3) in Besov type spaces can be found, e.g., in [21].
Note that, there is no nonnegative initial condition u 0 with the Morrey space norm | |u(t)| | M d/2 blowing up. Indeed, one can prove that each nonnegative local-in-time solution of system (1.5)-(1.6) satisfies the condition lim sup r →0, x∈R d r 2−d {|y−x|<r } |u(y, t)| ≤ J (d) < ∞ for all t ∈ (0, T ) and a universal constant J (d), cf. [6]. The analogue of this condition for d = 2 has a clear meaning: the atoms of admissible nonnegative initial data u 0 , u 0 = lim t 0 u(t) in the sense of weak convergence of measures, are strictly smaller than 8π , see [14].
Our results for radially symmetric solutions in [10] and the present paper can be summarized in the dichotomy.

Blowup of solutions of system with fractional diffusion
Here, we generalize results for the Brownian diffusion case α = 2 to the case of the system of nonlocal diffusion-transport equations (1.5)-(1.7) generalizing the classical Keller-Segel system of chemotaxis to the case of the diffusion process given by the fractional power of the Laplacian (− ) α/2 with α ∈ (0, 2), a nonlocal operator, as was in [7,13].
This discontinuous solution u C ∈ M d/α (R d ) is, in a sense, a critical one which is not smoothed out by the diffusion operator in system (1.5)-(1.6).
As usual for nonlinear evolution equations of parabolic type, blowup of a solution u at t = T means (as in Sect. The proof of Theorem 3.2 does not apply to the case d = 1 which is studied by completely different methods in [15].
Informally speaking, this sufficient condition for blowup is equivalent to | |u 0 | | M d/α 1. Indeed, for radially symmetric nonnegative functions the condition sup T >0 This fact can be proved using fine estimates of the kernel of the semigroup e −t (− ) α/2 restricted to radial functions, as was in the case α = 2. Further, the quantity sup T >0 with some functions f t,α (λ) ≥ 0 independent of d. In fact, the subordinators f t,α satisfy Therefore, the kernel P t,α of e −t (− ) α/2 is also of selfsimilar radial form, and can be expressed as with a positive function R decaying algebraically, together with its derivatives R , R , . . . , = ∂ ∂ : Here r = |x| and = This is normalized so that with (3.6) For α = 2 we have, of course, Proof. As in the original reasoning of Fujita in [19] applicable to the case α = 2 (in Sect. 2) and in [27] for a nonlinear fractional heat equation with power sources and α ∈ (0, 2), here we consider the moment with the weight function G = G(x, t) solving the backward linear fractional heat equation on (0, T ) It is clear that G has the selfsimilar radially symmetric form with the same function R as above.
For radially symmetric functions W in (3.7) becomes |y| d+α dy must be finite. Further, we have by Lemma 2.1 that ∇v(x) · x = − 1 σ d |x| 2−d {|y|≤|x|} u(y) dy, and therefore by selfadjointness of the operator (− ) α/2 Using the Cauchy inequality as in Sect. 2 we estimate dr. (3.12) Note that the function 1−d R ( ) is strictly decreasing as the product of two strictly decreasing positive functions so that the denominator of the integrand in (3.13) is strictly positive. Now, with the definition of C α (d) (3.13) the ordinary differential inequality obtained from (3.11) and (3.12) leads to the estimate Thus, a sufficient condition for the blowup becomes (3.14) Indeed, if (3.14) holds, then and lim t T W (t) = ∞, and therefore lim sup t T u(x, t) = ∞.
Next, we express condition (3.14) in terms of the d α -concentration (1.10) of u 0 , as was for α = 2 in Proposition 2.6. Again, by scaling properties of system (1.5)-(1.6), it is sufficient to consider R = 1. Proof. Let us compute for the kernel P t,α of the semigroup e −t (− ) α/2 the quantity  (3.17) so that and the right-hand side inequality in estimate (3.16) follows. Now, we will test the normalized Lebesgue measure σ −1 d dS on the unit sphere S d corresponding to the radial distribution function 1 [1,∞) then the solution with u 0 as the initial condition blows up in a finite time again by the comparison principle [13,Th. 2.4]. Therefore, by (3.16), we obtain that N = C α (d) REMARK 3.4. (Examples of blowing up solutions) Observe that for any initial condition u 0 ≡ 0 there is N > 0 such that (3.14) is satisfied for N u 0 .
Similarly as was in Remark 2.7 for α = 2, if u 0 (x) = ηu C (x), then the sufficient condition for blowup (3.14) is satisfied for large d whenever η > 1 k(α) 2d d−2 . Indeed, it suffices to have η > C α (d) K α (d) , and by relation (3.16) asymptotically C α (d) More generally than in Remark 2.7, for each such η and sufficiently large R = R(η) > 1, the bounded initial condition of compact support u 0 = η1 {1≤|x|≤R} u C leads to a blowing up solution. It seems that this result cannot be obtained applying previous sufficient criteria for blowup involving moments in [9] and in [13,Th. 2.9].
Taking into account [11, Theorem 2.1], all the above remarks on the critical Morrey space and the d α -radial concentration, we formulate the following dichotomy result This, together with (3.19), shows that for α ∈ (1, 2) the discrepancy between bounds of the d α -radial concentration sufficient for either global-in-time existence or the finite time blowup, i.e., C(α,d) c (α,d) , is of order d