Local criteria for blowup in two-dimensional chemotaxis models

We consider two-dimensional versions of the Keller--Segel model for the chemotaxis with either classical (Brownian) or fractional (anomalous) diffusion. Criteria for blowup of solutions in terms of suitable Morrey spaces norms are derived. Moreover, the impact of the consumption term on the global-in-time existence of solutions is analyzed for the classical Keller--Segel system.

Here the unknown variables u = u(x, t) and v = v(x, t) denote the density of the population and the density of the chemical secreted by the microorganisms, and the given consumption (or degradation) rate of the chemical is denoted by γ ≥ 0. The (anomalous) diffusion operator is described either by the usual Laplacian (α = 2) or by a fractional power of the Laplacian (−∆) α/2 with α ∈ (0, 2). The initial data are finite measures u 0 ∈ M(R 2 ) of the total mass these results consists in using local properties of solutions instead of a comparison of the total mass and moments of a solution as was done in e.g. [13], [10], [9] and [3].
Many previous works have dealt with the existence of global in time solutions with small data in critical Morrey spaces (i.e. those which are scale-invariant under a natural scaling of the chemotaxis model, cf. e.g. [2] and [12]). Our criteria for a blowup of solutions with large concentration can be expressed by Morrey space norms (see Remark 3 below for more details) and have found that the size of such a norm is critical for the global in time existence versus finite time blowup. The analogous question for radially symmetric solutions of the d-dimensional Keller-Segel model with d ≥ 3 has been studied in [4].

Statement of results
First, we formulate some new sufficient conditions for blowup of solutions of (1.1)-(1.2). (iii) Let α ∈ (0, 2) and γ ≥ 0 (the Keller-Segel model with fractional diffusion). If there exist x 0 ∈ R 2 and R > 0 such that for some explicit constants: small ν > 0 and big C > 0, then the solution u blows up in a finite time.
Remark 1. The result (i) for α = 2 and γ = 0 is, of course, well known, but the proof below slightly differs from the previous ones.
Remark 2. The case (ii) α = 2 and γ > 0 has been considered in [10] but the sufficient conditions for blowup were expressed in terms of globally defined quantities: i.e. mass M > 8π and the moment |x| 2 u 0 (x) dx.
The first condition in (2.2) is equivalent to sufficiently large Morrey norm of u 0 in the for every x 0 and R > 0, but also there is  Remark 5. Let us note that the reference [9, Theorem 2.9] provides us with precise conditions on radial kernels K leading to a blowup of solutions of general diffusive aggregation equations with the Brownian diffusion of the form u t − ∆u + ∇ · (u(∇K * u)) = 0. They are strongly singular, i.e. they have the singularity at 0: lim sup x→0 x · ∇K(x) < 0, and are of moderate growth at ∞: |x · ∇K(x)| ≤ C|x| 2 .
Of course, the Bessel kernel K γ of (−∆ + γ) −1 solving equation (1.2) is strongly singular in the sense of [9] as it is seen from the global one-sided bound Moreover, the Bessel kernel K γ satisfies the following asymptotic properties Notations. In the sequel, C's are generic constants independent of t, u, z, ... , which may, however, vary from line to line. Integrals with no integration limits are meant to be calculated over the whole plane.

Blowup of solutions
In this section we prove Theorem 2.1 using the method of truncated moments which is reminiscent of that in the papers [14], [11]. First, we define the "bump" function ψ and its rescalings for R > 0 The function ψ is piecewise C 2 (R 2 ), with supp ψ = {|x| ≤ 1}, and satisfies We will use in the sequel the following fact that ψ is strictly concave in a neighbourhood of x = 0.  Proof For every ξ ∈ R 2 the following identity holds Thus, by the Schwarz inequality, we have ξ ⊥ Hψξ ≤ 4|ξ| 2 3|x| 2 − 1 .
Next, we recall a well-known property of concave functions.
Lemma 3.2. For every function Ψ : R 2 → R which is strictly concave on a domain Ω ⊂ R 2 we have for all x, y ∈ R 2 where θ > 0 is the constant of strict concavity of Ψ on Ω, i.e. satisfying HΨ ≤ −θ I.
Proof By the concavity, we obtain Summing this inequality with its symmetrized version (with x, y interchanged) leads to the claim.
We have the following scaling property of the fractional Laplacian and we notice the following boundedness property of (−∆) α/2 ψ. Proof For α = 2, this is an obvious consequence of the explicit form of ψ, hence we assume α ∈ (0, 2).
where θ is the constant of the strict concavity of Ψ introduced in Lemma 3.2, and g γ is a radially symmetric continuous function, such that x |x| 2 g γ (|x|).
We are in a position to prove our main blowup result.
Proof of Theorem 2.1. We consider the quantity denote mass of the distribution u at the moment t contained in the ball {|x| < R}.
We estimate the first integral on the right-hand side of (3.15) in the following way In the above inequalities we used the fact that g γ is a continuous decreasing function and 0 ≤ g γ ≤ 1.
Next, since we have the inclusion  (3.17) where C = sup |∇K γ (z) · z| D 2 ψ ∞ . Finally, estimates (3.14)-(3.17) as well as inequality (3.12) applied to equation (3.13) lead to the inequalities Now, notice that the linear function of w R in the parentheses on the right-hand side of (3.18) is monotone increasing. Thus, if at the initial moment t = 0 the right hand side of (3.18) is positive, then w R (t) will increase indefinitely in time.
Consequently, after a moment T = O R α M wR(0) − 1 the function w R (t) will become larger than the total mass M . This is a contradiction with the global-intime existence of a nonnegative solution u since it conserves mass (1.4). Now, let us analyze the cases when the right-hand side of inequality (3.18) is strictly positive.