Mathematical Challenges in the Theory of Chemotaxis

Abstract We consider the simplest parabolic-elliptic model of chemotaxis in the whole space and in several space dimensions. Criteria either for the existence of radial global-in-time solutions or their blowup in terms of suitable Morrey spaces norms are discussed. This is an extended version of the lecture presented at the University of Silesia on January 12, 2018, commemorating Professor Andrzej Lasota-great scholar, master of fine mathematics and applications to real world.

Another and even earlier important interpretation of system (1.1)-(1.2) stems from astrophysics, where the unknown function u = u(x, t) is the density of gravitationally interacting massive particles in a cloud (of molecules, stars, nebulae, etc.), and v = v(x, t) is the Newtonian potential ("mean field") of the mass distribution u, see [34,35,5,7,6,18]. Note that similar mean field models, with + sign in equation (1.2) replaced by − sign have been used for more than one century to model migration of electrically charged particles in electrolytes, plasma and semiconductors, see e.g., [37] and [16] for further references.
The initial data (1.3) are nonnegative integrable functions u 0 ∈ L 1 (R d ). The total mass M = u 0 (x) dx = u(x, t) dx ∈ [0, ∞) is conserved during the evolution. Further, we will also consider solutions with infinite mass like the famous Chandrasekhar steady state singular solution in [34] related to black holes (1.4) u C (x) = 2(d − 2) |x| 2 .
Another class of biologically relevant models appear with the following choice for the first equation where D(s) = h(s) − sh (s). The literature on these subjects is abundant and fast growing. We will concentrate on the simplified system (1.1)-(1.2) which deserves a deep analysis by mathematicians since this features many interesting behaviors of solutions. This is a review paper, a potpourri of some old results, some new, with no proofs -except for a sketch of one being an application of a classical idea of H. Fujita [38] to radial solutions of chemotaxis systems.
The system (1.1)-(1.2) has a variational structure, so that the quantity (of a clear physical origin of "entropy" or "free energy") However, unlike the authors of [3,32,30], we had not used that subtle property in the proofs of our results presented here. Finally, it should be noted that similar phenomena take place and can be proved for nonnegative solutions (not necessarily radial) of the nonlinear heat equation A general reference is the monograph [61], and recent results are in [63,10,11].

The 8π-problem in the two-dimensional case
Let us now describe previous results which motivated us to start this study and we limit ourselves to those publications, which are directly related to that topic.
We begin with the classical case of d = 2 where the value M = 8π of mass plays a crucial role. Namely, if u 0 is a nonnegative measure of mass M < 8π (the subcritical case), then there exists a unique solution which is globalin-time and bounded (see, e.g., [3,32,28]) and its asymptotics is essentially selfsimilar in space-time. These results have been known previously for radially symmetric initial data, see [21,22,9,24,25] for recent presentations.
On the other hand, if M > 8π (the supercritical case), then this solution cannot be continued to a global-in-time regular one, and a finite time blowup occurs The first proof of blowup was in [47], then [27,18,6,9,56,52] appeared, and constructions of blowing up radial solutions have been presented in [41,42]. The radial blowup is accompanied by the concentration of mass equal to 8π at the origin.
The book [64] is devoted to a fine description of solutions at the blowup time, in particular there is a quantization of mass property presented: the local singularities of blowing up solutions eventually grow to integer multiples of 8π.
These phenomena are closely related to the question of local solvability of the Cauchy problem for system (1.1)-(1.2) under minimal regularity on the initial data u 0 ≥ 0 in (1.3). Namely, if u 0 is a nonnegative measure then local in time solution exists if and only if all the atoms of u 0 have mass less than 8π, see [3] and [28] for much simpler argument.
The critical case M = 8π is rich in fine asymptotic behavior results (see [31,30]) even in the radially symmetric case. And the case of a ball [49,21] is quite different from the case of the whole plane in [22]. The study of radially symmetric solutions of system (1.
They are locally asymptotically stable which is shown by considering relative entropy functionals as in [22]. But the global dynamics picture is much more complicated; there are solutions which diffuse mass to infinity, so that M (r, t) → 0 as t → ∞, solutions with an infinite time blowup that concentrate at the origin: M (r, t) → 8π for all r > 0 as t → ∞, and solutions that oscillate ("bounce") between two different steady states M b , see [58]. The doubly parabolic case of Keller-Segel system, i.e., equation (1.1) supplemented with the linear parabolic diffusion equation for ϕ (2.5) εϕ t = ∆ϕ + u instead of (1.2), is even more difficult to study, especially when blowup questions are considered. A striking difference of its behavior is, e.g., result in [15]. Namely, selfsimilar solutions satisfying scaling property The role of consumption term γϕ in the modified equation (2.5) for the chemoattractant εϕ t = ∆ϕ − γϕ + u is discussed in [17] for two-dimensional doubly parabolic model (together with the dependence on diffusivity coefficient ε > 0) and in [14] in the parabolic-elliptic case (ε = 0). Namely, for each initial condition there is ε 0 = ε(u 0 ) such that for ε ≥ ε 0 (γ 0 = γ(u 0 ) and for γ ≥ γ 0 , resp.) solution with u 0 as the initial datum is global in time.

Parabolic-elliptic model in higher dimensions
In view of results in the two-dimensional case mentioned above (when the parameter of total mass M plays decisive role in the temporal behavior of solutions), for d ≥ 3 we are looking for a critical quantity˜ =˜ (u 0 ) which decides about the blowup. More precisely, do there exist constants leads to a finite time blowup of solution? We will give an answer to that dichotomy question in Corollary 5.4, showing that is close to the radial concentration -and thus equivalent to the Morrey norm in the space M d/2 (R d ). A generalization to the case of the dissipation defined by a fractional power of Laplacian (−∆) α/2 with α ∈ (1, 2) is in a forthcoming paper [20], showing that˜ is close to the Morrey norm in the space M d/α (R d ). Here, the radial concentration of a locally integrable radial function u ≥ 0 is defined by The homogeneous Morrey spaces of measures on R d are defined by their norms

Main new results
Our main results obtained recently include: • global-in-time existence of radially symmetric solutions with initial data in the critical Morrey space M d/2 (R d ) whose initial conditions are uniformly below the singular solution u C in (1.4) in an averaged sense, Theorem 4.1 below, together with their convergence to 0 as t → ∞; • sufficient conditions on the 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 ), Theorem 5.1; for instance, condition (5.7): here, e T ∆ denotes the heat semigroup on R d ) is sufficient for the blowup of solution with the initial condition u 0 ; • a perturbation result on convergence of a solution u(t) to u C as t → ∞ studied via the hypercontractivity property of the semigroup linearized at u C in high dimensions d ≥ 15 (in the radial case), and for d ≥ 17 (with neither symmetry nor sign assumptions) in Section 6. The proof of the first result involves a pointwise argument, a powerful tool used in different contexts such as free boundary problems and fluid dynamics, cf. also [10] for the case of a nonlinear heat equation. A sufficient condition for the global-in-time existence is, in fact, an estimate of the Morrey space M d/2 (R d ) norm of the initial condition (modulo a mild regularity assumption).
For the proof of the second result, we revisit a classical argument of H. Fujita (applied to the nonlinear heat equation in [38]) and reminiscent of ideas in [33]. This leads to a sufficient condition for blowup of radially symmetric solutions of system (1.1)-(1.2), with a significant improvement compared to [26] where local moments have been employed. Then, we derive as corollaries of condition (5.7) other criteria for blowup of solutions of (1.1)-(1.3). Remark 5.3 deals with the initial trace of a nonnegative solution of the Keller-Segel system, and again the Morrey space M d/2 (R d ) norm enters as a critical quantity which measures the minimal regularity of the initial data needed for the existence of a local-in-time solution of that system.

Local-and global-in-time solutions
It is well-known that problem (1.1)-(1.3) has a unique local-in-time mild solution u ∈ C([0, T ); L p (R d )) for every u 0 ∈ L p (R d ) with p > d/2, see [5,48,51]. For solvability results in other functional spaces like weak Lebesgue (Marcinkiewicz), Morrey and Besov spaces, see also [13,36,48,53] where the classical Fujita-Kato iterations procedure for construction of mild solutions is used. Mild solutions are those which are weakly continuous in t: ) and satisfy the Duhamel formula Above, e t∆ denotes the heat semigroup on R d , and the bilinear form B is defined by Here, we consider arbitrary sign changing and not necessarily radially symmetric solutions. A new result in this spirit is below, with its proof similar to reasonings in [5, Proposition 1, Theorem 1] based on the Morrey space norms counterparts of the L p − L q estimates for the heat semigroup in, e.g., [65] and [39]. Note that, in general, we have only weak convergence of e t∆ u 0 to the initial data u 0 ∈ M p (R d ) while u(t) is norm continuous for t ∈ (0, T ). Thus, we are obliged to consider weakly continuous (C w ) instead of more natural norm continuous (C) functions. (ii) Moreover, if u 0 ∈ M d/2 (R d ) is sufficiently small then T can be chosen arbitrarily large so that the solution is global-in-time: and enjoys the decay and regularity property sup t>0 t β | |u(t)| | M r < ∞ for r > d and β = d The second assumption u 0 ∈ M p , with some p > d/2, is a kind of regularity assumption that rules out local singularities stronger than or equal to 1 Similarly as in previous works, the existence of global-in-time solutions with small data is shown in critical spaces, i.e., those which are scale-invariant under the natural scaling (see, e.g., [5,13,48,53]) A much more technical reasoning, that involves Morrey spaces modelled on weak-Lebesgue (i.e. Marcinkiewicz) spaces (see [53]), shows that small initial conditions in M d/2 (R d ) also lead to global in time solutions. Caution: even local solutions cannot evolve from (very) big data with singularities in the space M d/2 (R d ), and for data of intermediate size in M d/2 (R d ) the Cauchy problem is ill-posed. For instance, there is no property of continuity with respect to the initial data, see [26,29].
We skip the discussion of an interesting structure of steady states and selfsimilar solutions referring the readers to [7,5,8] and to a synthesis in [23,Appendices].

Continuation of radially symmetric solutions with moderate size in the critical Morrey space
The main result in this direction is 26]). If a radially symmetric initial condition , can be continued to a global-in-time one, satisfying the bound For d ≥ 6 and |||u 0 ||| < 2σ d , the L 2 estimate: d dt u 2 2 +µ ∇u 2 2 ≤ 0 holds. If, additionally, |||u(t)||| ≤ 2σ d for some 0 < < d−2 d , then also lim t→∞ u(t) p = 0 The assumption on u 0 in Theorem 4.1 reads in terms of the concentration (and thus the Morrey space M d/2 (R d ) norm): |||u 0 ||| < 2σ d for some ∈ (0, 1). Notice that | |u C | | M d/2 = |||u C ||| = 2σ d for the Chandrasekhar solution (1.4) so that this singular solution has a regular, bounded (even the constant one) auxiliary function z(r, t) = r 2−d M (r, t).
Here, the radial distribution function M = M (r, t) of a radial solution u = u(x, t) is defined in (2.2) so that M satisfies the equation (equivalent to equation (2.3)) cf. e.g. [18], and for the radial function u the equality u(x) = 1 Note that for d = 2 this theorem gives a nonoptimal result: the global-intime existence for M < 4π rather than for the optimal range M < 8π.

Solutions blowing up in a finite time
We will revisit the classical proof of blowup for the nonlinear heat equation in the seminal paper [38] by H. Fujita, and improve the sufficient conditions for the blowup mentioned before in Section 2, cf. [29].
The key observation is that for a radially symmetric function u ∈ L 1 loc (R d ) Thus, for the radial function ∇v(x) · x |x| and |x| = R, we obtain the required identity Now, we proceed to apply a classical idea of blowup proof of Fujita. Clearly, we have a (unique nonnegative) solution defined by the Gauss-Weierstrass kernel, satisfying G(x, t) dx = 1, and moreover, Define for a solution u of (1.1)-(1.2), which is supposed to exist on [0, T ), the moment Since G decays exponentially fast in x as |x| → ∞, the moment W is well defined even for a wider class of solutions u = u(x, t) polynomially bounded in x. The evolution of the moment W is governed by the differential identity where we used the radial symmetry of the solution u in (5.2), identity (5.1) and, of course, the radial symmetry of G.
Expressing the moment W in the radial variables we obtain Now, applying the Cauchy inequality to the quantity (5.3), we get dr.
Returning to the time derivative of W in equation (5.2), we arrive at the differential inequality where = r 2(T −t) 1/2 . Recalling (2.4), 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 and completes the proof.
Note that the blowup rate is such that lim inf t T (T − t)W (t) > 0. For other results on blowup rates (e.g., a faster blowup, i.e., of the II type), see [40,55,54].
Observe that the equality in the Cauchy inequality (5.4) holds if and only if with some A(t) ≥ 0. Consequently, inequality (5.6) becomes then the solution blows up not later than T . This holds exactly when A(0) ≥ 4σ d . This solution (cf. [33, (33)]) satisfies identity (5.8) with W (0) = C(d) T , and is, in a sense, a kind of the minimal smooth blowing up solution, i.e., it gives a lower bound on blowing up solutions. So, we have an explicit example of blowing up solution with infinite mass whose density approaches 4(d−2) |x| 2 = 2u C (x), i.e., twice the singular stationary solution, when t T , so that the density of the solution becomes infinite at the origin for t = T . The corresponding initial density is, of course, We give below some other examples of initial data leading to a finite time blowup of solutions.
Remark 5.2. Observe that for each initial condition u 0 ≡ 0 there is N > 0 such that condition (5.7) is satisfied for N u 0 .
Clearly, by |||u C ||| = | |u C | | M d/2 = 2σ d , for each η > 2 the solution with the initial condition u 0 = ηu C blows up. 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 (5.7). 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 (3.2).
On the other hand, the initial data like min{1, u C } + ψ with a smooth nonnegative, compactly supported function ψ and a sufficiently small > 0 (they are somewhere above the critical u C pointwisely) lead to global-in-time solutions according to Theorem 4.1.

Sufficient conditions for blowup for radial
are mutually equivalent. Note that, however, some of these equivalences are rather nontrivial and the comparison constants for pairs of those quantities strongly depend on d. Thus, our results for radially symmetric solutions (which we suppose to exist) can be summarized in the following dichotomy result.

Hypercontractivity properties
A (proto)typical result in this direction is the following.
for all t > 0 and a number C(p, d) independent of t and u.
The question of the existence of solutions with 0 ≤ u(x, t) ≤ u C (x) is nontrivial, and the proof involves an approximation procedure by solutions like those in Proposition 3.1 and Theorem 4.1.
To prove Theorem 6.1 we need a few ingredients including the hypercontractivity of the semigroup of linear operators e −tL describing the evolution in vicinity of the singular solution u C , and a perturbation result, cf. an analogous scheme in [60,59] in the case of nonlinear heat equations.
In the following, we study properties of the operator L, see an analogous approach in [59,60] for nonlinear heat equations. for all radial functions w ∈ H 1 (R d ). (ii) Assume that d ≥ 17. Then inequality (6.3) holds for another λ > 0 and each w ∈ H 1 (R d ).
(i) This is a consequence of the identity obtained by multiplying equation w t + Lw = 0 by w and integrating by parts.
(ii) An alternative approach to the property of hypercontractivity of the linearization operator L is obtained by estimating the middle term in formula (6.4)

4(d − 2)
x |x| 4 · ∇ϕ w dx ≤ 4(d − 2) Then, e −tL is shown to be a holomorphic semigroup on L 2 (R d ). The other ingredients of the proof are technical and include global existence for quadratic perturbation (6.1), decay estimates based on the properties of the semigroup e −tL , etc.