Finite mass self-similar blowing-up solutions of a chemotaxis system with non-linear diffusion

For a specific choice of the diffusion, the parabolic-elliptic Patlak-Keller-Segel system with non-linear diffusion (also referred to as the quasi-linear Smoluchowski-Poisson equation) exhibits an interesting threshold phenomenon: there is a critical mass $M_c>0$ such that all the solutions with initial data of mass smaller or equal to $M_c$ exist globally while the solution blows up in finite time for a large class of initial data with mass greater than $M_c$. Unlike in space dimension 2, finite mass self-similar blowing-up solutions are shown to exist in space dimension $d?3$.


Introduction
In space dimension d = 2, the parabolic-elliptic Patlak-Keller-Segel (PKS) system is a simplified model which describes the collective motion of cells in the following situation: cells diffuse in space and emit a chemical signal, the chemo-attractant, which results in the cells attracting each other. If ρ denotes the density of cells and c the concentration of the chemo-attractant, the PKS system reads [13,19] (1)    ∂ t ρ(t, x) = div [∇ρ(t, x) − ρ(t, x)∇c(t, x)] , c(t, x) = (E 2 ⋆ ρ)(t, x) , E 2 (x) = − 1 2π ln |x| , This model may be seen as an elementary brick to understand the aggregation of cells in mathematical biology as it exhibits the following interesting and biologically relevant feature: there is a critical mass above which the density of cells is expected to concentrate near isolated points after a finite time, a property which is related to the formation of fruiting bodies in the slime mold Dictyostelium discoideum. Such a phenomenon does not take place if the density of cells is too low. More precisely, given a non-negative integrable initial condition ρ 0 with finite second moment, the system (1) has a unique maximal classical solution (ρ, c) defined on some maximal time interval [0, T ), T ∈ (0, ∞]. Its first component ρ is non-negative and the mass of ρ (that is, its L 1 -norm) remains constant through time evolution It is well-known that, if M < 8 π, the solution to (1) exists globally in time while it blows up in finite time if M > 8 π, see [3,6,11,12] and the references therein.
More recently, it was shown that there is global existence as well for the critical mass M = 8 π, the blowup occurring in infinite time with a profile being a Dirac mass of mass 8π [1]. When the mass M is above 8π, the shape of the finite time blowup is not self-similar according to asymptotic expansions computed in [5,15] (see also [10] for a related problem in a bounded domain). In addition, there is no integrable and radially symmetric blowing-up self-similar solution to (1) [18,Theorem 8].
In space dimension d ≥ 3, the system (1) seems to be less relevant from the biological point of view as blowup may occur whatever the value of M [9,17]. This means that the diffusion is too weak to balance the aggregation resulting from the chemotactic term. It is however well-known that one can enhance the effect of diffusion to prevent crowding by considering a diffusion of porous medium type which increases the diffusion of the cells when their density ρ is large. This is the generalised version of the Patlak-Keller-Segel model considered in, e.g., [2,4,22,23,24]: where m > 1, c d := 1/((d − 2) σ d ), and σ d := 2 π d/2 /Γ(d/2) denotes the surface area of the sphere S d−1 of R d . The system (2) also arises in astrophysics [4] (being then referred to as the generalised Smoluchowski-Poisson equation), and ρ and c denote the density of particles and the gravitational potential, respectively. For (2), it turns out that there is only one critical exponent of the non-linear diffusion, namely m d := 2(d − 1)/d, such that the mass plays a similar role to that in (1). Indeed, if m > m d the diffusion enhancement is too strong and the solutions always exist globally in time whereas if m < m d the diffusion is not strong enough to compensate the aggregation term and there are solutions blowing up in finite time whatever the value of the mass [22,23]. The relevant diffusion is thus achieved in the case when m = m d . In this case, it was proved in [2] that there is a unique threshold mass M c > 0 with the following properties: if the mass M = ρ 0 1 of the initial condition ρ 0 is less or equal to M c , then the corresponding solution to (2) exists globally in time, whereas given any M > M c there are initial data ρ 0 with mass M such that the corresponding solution blows up in finite time. Thus, for the peculiar choice m = m d and d ≥ 3, the system (2) exhibits the same qualitative behaviour as the PKS system (1) in space dimension 2. Still, there is a fundamental difference as the latter has no fast-decaying stationary solution with mass 8π while the former has a two-parameter family of non-negative, integrable, and compactly supported stationary solutions with mass M c for each d ≥ 3 [2, Section 3].
It is then tempting to figure out whether this striking difference extends above the critical mass M c and this leads us to investigate the existence of blowing-up (or backward) self-similar solutions with finite mass. More precisely, since mass remains unchanged throughout time evolution, we look for solutions (ρ, c) to (2) with m = m d and d ≥ 3 of the form and c(t, x) = 1 As a consequence of Theorem 1, we realize that non-negative, integrable, and radially symmetric self-similar blowing-up solutions to (2) with a non-increasing profile only exist below a threshold mass. Another by-product of our analysis is the existence of non-negative and non-integrable self-similar blowing-up solutions to (2), see Proposition 8 below.

Blowing-up self-similar profiles
From now on, and we look for a solution (ρ, c) to (2) of the form and We further assume that Φ enjoys the following properties: is radially symmetric and non-negative, Inserting the ansatz (4) in (2) gives that (Φ, Ψ) solves for y ∈ R d . Since Ψ = E d ⋆ Φ, the radial symmetry of Φ ensures that of Ψ and, introducing the profiles (ϕ, ψ) of (Φ, Ψ) By [14, Theorem 9.7, Formula (5)], we have for r ≥ 0. We can also write the equation for ϕ as for r ∈ (0, ∞). Since we are looking for an integrable profile, we formally conclude that In particular, J is constant on any connected component of P ϕ . But, if C is a connected component of P ϕ , we have either Remark 2. If we additionally assume that the profile ϕ is non-increasing then P ϕ has only one connected component which is necessarily of the form (9).
Owing to the assumed integrability of Φ, the function r → r d−1 ϕ(r) belongs to L 1 (0, ∞) and it follows from (6) that the function r → r d−2 ψ(r) is bounded in C. Therefore (10) only complies with the integrability of Φ if R s < ∞ which implies the boundedness of C. Introducing Ξ := ϕ (d−2)/d and taking the Laplacian of both sides of (10) yield that Ξ is a positive solution to A final change of scale, namely , leads us to the following boundary-value problem for η: either We have thus reduced our study to one or several boundary-value problems (depending on the number of connected components of P ϕ ) for a nonlinear second order differential equation. The purpose of the next section is then a precise study of this ordinary differential equation. However, before going on, let us point out that (11) is not equivalent to (10). Indeed, since (7), the fact that Ξ is a solution to (11) only guarantees that ∂ r (r d−1 ∂ r J(r)) = 0 for r ∈ C. Consequently, there are constants C 1 and C 2 such that it is yet unclear whether the boundary conditions (13) might imply this property. On the other hand, if C = (0, R s ), the boundary conditions (12) ensure that ∂ r J(0) = 0 and thus C 1 = 0. We shall only deal with this case in the remaining of this paper and thus focus on the non-increasing profiles ϕ.
3. An auxiliary ordinary differential equation For a ∈ R, let u(., a) ∈ C 1 ([0, r max (a))) denote the maximal solution to the Cauchy problem  for i ≥ 1.
According to (14), we are interested in finding solutions to the initial value problem (16) which are positive and vanish at a finite value of r. We thus focus on the case a > 0 and investigate the positivity properties of u(., a).
Consider now a ∈ N 0 . Then U(x) := u(|x|, a) is a radial positive solution to the homogeneous Dirichlet-Neumann free boundary problem ∆U + U p − 1 = 0 in B(0, R(a)) with U = ∂ ν U = 0 on ∂B(0, R(a)). According to [20, Theorem 3 (iii)], there is only one value of a for which this solution has a positive radial solution and it is unique. Consequently, there is a unique a c > 0 such that N 0 = {a c }.
We finally argue as in [16,Lemma 15] to show that there is A > 0 such that (A, ∞) ⊂ N .
Lemma 5. If a > a c , there is a unique z(a) ∈ (0, R(a)) such that In addition, u(z(a), a) > 1 and the ratio ϑ(., a)/u(., a) is a decreasing function of r on (0, R(a)).
Proof of Lemma 5. Since the proof follows rather closely that of [25] and [7, Lemma 2.1], we sketch it briefly for the sake of completeness. Fix a > a c and set u = u(., a) and ϑ = ϑ(., a) to simplify notations. We first argue as in [16,Lemma 17] to show that ϑ vanishes at least once in the interval (0, z 1 (a)), where z 1 (a) denotes the unique zero in (0, R(a)) of u − 1. Indeed, (16) also reads and (u(r) p − 1)/(u(r) − 1) ≤ p u(r) p−1 for r ∈ [0, z 1 (a)). It then follows from Sturm's comparison theorem that ϑ vanishes at least once in the interval (0, z 1 (a)). Let z ∈ (0, z 1 (a)) denote the first zero of ϑ.
We are now in a position to state and prove some properties of the map a → R(a). The monotonicity of a → R(a) is shown in Figure 3. According to numerical simulations, the function a → a (p−1)/2 R(a) also seems to be a decreasing function of a ∈ [a c , ∞), see Figure 3. Proof of Proposition 6. By Lemma 4, u ′ (R(a), a) < 0 for all a ∈ (a c , ∞) and the implicit function theorem warrants that R ∈ C 1 ((a c , ∞)) with dR da (a) = − ϑ(R(a), a) u ′ (R(a), a) .
Since ϑ(R(a), a) < 0 by Lemma 5, the previous formula implies the strict monotonicity of a → R(a). We next define If R l > R(a c ), there is ̺ ∈ (R(a c ), R l ) such that u(̺, a c ) > 0 by Lemmata 3 and 4. Then, there is δ > 0 such that R(a) > ̺ for a ∈ (a c , a c + δ). It then follows from the continuous dependence of u(., a) with respect to a and the monotonicity of u(., a) with respect to r that 0 = u(R(a c ), a c ) = lim Owing to (16), v(., a) solves In addition, for a > a c by Lemma 4. Since a −p −→ 0 as a → ∞, we have where w denotes the unique solution to By [8], there is z 1 > 0 such that Owing to (33), there is δ > 0 such that w(r) < 0 for r ∈ (z 1 , z 1 + δ). It then follows from (31) that, given r ∈ (z 1 , z 1 + δ), v(r, a) < 0 for a large enough (depending on r), whence a (p−1)/2 R(a) ≤ r for a large enough by (30). Letting r → z 1 guarantees that lim sup a→∞ a (p−1)/2 R(a) ≤ z 1 .
Combining the above two inequalities completes the proof of Proposition 6.
The above information allow us to estimate from above and from below a specific integral of u(., a). Recalling that w is the solution to (32) and z 1 is its first positive zero, we have in [21] for n ∈ (0, 3/2). It is likely that, given M ∈ (M c , M 2 ], there is only a unique radially symmetric and non-increasing self-similar blowing-up solution with mass M and Figure 4 provides some numerical evidence of this fact. Besides this uniqueness question, the question of stability of these blowing-up solutions is also of interest. Another challenging question is the existence (or non-existence) of integrable profiles ϕ with a non-connected positivity set as discussed in Section 2. Figure 5 provides numerical evidence that, if a > a c is large enough, u(., a) may have several zeroes and each positive "hump" actually corresponds to a solution of (15) for suitable values of R i and R s . Whether the additional constraint (10) may be satisfied does not seem to be clear.