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 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] Our results include Similar results for the system with modified diffusion operator u t + (−∆) α/2 u + ∇ · (u∇v) = 0, x ∈ R d , t > 0, (1.5) ∆v + u = 0, x ∈ R d , t > 0, (1.6) supplemented with the initial condition (1.7) u(x, 0) = u 0 (x) ≥ 0 will be derived and discussed in Section 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 (1.1)-(1.3). .
By a direct calculation, we have Here, as usual, denotes the area of the unit sphere S d−1 in R d .
The relation f ≈ g means that lim s→∞ f (s) g(s) = 1 and f g means: lim sup s→∞

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. lim sup 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 [22,11,12,7,13] 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 Proof. By the Gauss theorem, we have for the distribution function M of u 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 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 G dr (2.10) dr.
Returning to the time derivative of W in identity (2.8), we arrive at the differential . Recalling (1.11), we denote Clearly, C(2) = 2, and C(d) < 2 for d ≥ 3, since we have Thus, we finally obtain which, after an integration, leads to then lim sup tրT W (t) = ∞ which means: lim sup tրT, 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]. 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. [16, (33)]) satisfies identity (2.15) with W (0) = C(d) T , and it is, in a sense, a kind of the minimal smooth blowing up solution. So, we have whose density approaches 4(d−2) |x| 2 = 2u C (x), i.e. twice the singular stationary solution, when t ր 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 each We express below a sufficient condition (2.14) for blowup in terms of the radial concentration. Then the asymptotic relation 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 equation (2.17) below, see also [5] in the case d = 2 Thus, from equation (2.16) we know that if then u blows up in a finite time. To check that . (2.20) Note that, by the above computations, there exist radial initial data 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 and N 4σ d π(d − 2) hold.
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 Nu 0 .
Moreover, for each η > 2 and sufficiently large R = R(η) > 1 the bounded initial condition of compact support u 0 = η1 I {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 [10, Theorem Remark 2.8 (Equivalent qualitative conditions for blowup).
The quantityl ≥ ℓ • sup t>0 t e t∆ u 0 ∞ ≫ 1, {|y−x|<r} u 0 (y) dy ≫ 1, are mutually equivalent, however, with comparison constants depending on d. 3) is in L ∞ for t ∈ [t 1 , t 2 ] with some 0 ≤ t 1 < t 2 . 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 tց0 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 Then u C (x) = s(α,d) |x| α is a distributional, radial, stationary solution to system (1.5)-(1.6).
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 Section 2): lim sup tրT, x∈R d u(x, t) = ∞. In fact, some L p norms (with p > d α ) of u(t) blow up together with the L ∞ -norm. blows up in L ∞ not later than t = T .
The proof of Theorem 3.2 below does not apply to the case d = 1 which is studied by completely different methods in [15].
with some functions f t,α (λ) ≥ 0 independent of d. In fact, the subordinators f t,α satisfy e −ta α = ∞ 0 f t,α (λ)e −λa dλ, so that they have selfsimilar form f t,α (λ) = t − 1 α f 1,α λt − 1 α . 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 functon R decaying algebraically, together with its derivatives R ′ , R ′′ , . . . , Here r = |x| and ̺ = r . This is normalized so that Proof. As in the original reasoning of Fujita in [19] applicable to the case α = 2 (in Section 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 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 Using the Cauchy inequality as in Section 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) d̺, the ordinary differential inequality obtained from (3.11) and (3.12) leads to the estimate 1 Thus, a sufficient condition for the blowup becomes Indeed, if (3.14) holds, then and lim tրT W (t) = ∞, and therefore lim sup tրT 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 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 I [1,∞) (r) for some constantk(α) > 0 independent of d, d → ∞, similarly as was in computations of (3.15), with the use of the Stirling formula (2.22). Now, we need an asymptotic lower bound for the quantity L α (d). Observe that