Infinite-time concentration in Aggregation--Diffusion equations with a given potential

Typically, aggregation-diffusion is modeled by parabolic equations that combine linear or nonlinear diffusion with a Fokker-Planck convection term. Under very general suitable assumptions, we prove that radial solutions of the evolution process converge asymptotically in time towards a stationary state representing the balance between the two effects. Our parabolic system is the gradient flow of an energy functional, and in fact we show that the stationary states are minimizers of a relaxed energy. Here, we study radial solutions of an aggregation-diffusion model that combines nonlinear fast diffusion with a convection term driven by the gradient of a potential, both in balls and the whole space. We show that, depending on the exponent of fast diffusion and the potential, the steady state is given by the sum of an explicit integrable function, plus a Dirac delta at the origin containing the rest of the mass of the initial datum. Furthermore, it is a global minimizer of the relaxed energy. This splitting phenomenon is an uncommon example of blow-up in infinite time.


Introduction
Enormous work has been devoted over the last years to the study of mathematical models for Aggregation-Diffusion that are formulated in terms of semilinear parabolic equations combining linear or non-linear diffusion with a Fokker-Planck convection term coming either from a given potential or from an interaction potential, see [3,24,36,18,21,27,19] and the references therein, and the books [2,46]. In this paper we consider the aggregation-diffusion equation where the potential V (x) is given and 0 < m < 1, the fast-diffusion range [44]. We take as initial data a probability measure, i. e., We will find conditions on the radial initial data ρ 0 and the radial potential V so that i) we provide a suitable notion of solution of the Cauchy problem defined globally in time passing through the mass (or distribution) function, ii) as t → ∞, the solution undergoes one-point blow-up of the split form where ρ ∞ (x) > 0 is an explicit stationary solution of (P). The presence of the concentrated point measure is a striking fact that needs detailed understanding and is the main motivation of this work. Here and after we identify an L 1 function with the absolutely continuous measure it generates.
It is known that Dirac measures are invariant by the semigroup generated by the fast-diffusion equation u t = ∆u m for 0 < m < n−2 n (see [10]), but they are never produced from L 1 initial data. Here we show that the aggregation caused by the potential term might be strong enough to overcome the fast-diffusion term and produce a Dirac-delta concentration at 0 as t → ∞, in other words, infinite-time concentration.
The case of (P) with slow diffusion m > 1 was studied in [18,32], where the authors show that the steady state does not contain a Dirac delta (i.e. ρ ∞ L 1 = 1). The linear diffusion case was extensively studied in [3,36,39]. The fast diffusion range 1 > m > n−2 n with quadratic confinement potential is also well-known and its long-time asymptotics, even for Dirac initial data, is given by integrable stationary solutions, see for instance [9] and its references. See also [45] for the evolution of point singularities in bounded domains.
We will take advantage of the formal interpretation of (P) as the 2-Wasserstein flow [18,21,2] associated to the free-energy in order to obtain properties of this functional in terms of the Calculus of Variations. We also take advantage of this structure to obtain a priori estimates on the solution ρ of (P) due to the dissipation of the energy.
Main assumptions and discussion of the main results. We introduce the specific context in which point-mass concentration arises. We first examine the special stationary solutions that play a role in the asymptotics: for h ≥ 0. It is easy to check that they are solutions of (P), and they are bounded if h > 0. We now consider the class of suitable potentials. We first assume that V has a minimum at x = 0 and is smooth: V ∈ W 2,∞ loc (R n ), V ≥ 0, V (0) = 0. We are interested in radial aggregating potentials, in fact we V is radially symmetric and non-decreasing. An essential assumption in the proof of formation of a point-mass concentration is the following small-mass condition for the admissible steady states: As a simplifying assumption we will assume that B1 ρ 1+ε V (x) dx < +∞, for some ε > 0. (1.5) The bounded case in which ρ V +h1 ≤ ρ 0 ≤ ρ V +h2 with h 1 , h 2 > 0 was studied in [14] and leads to no concentration. On the contrary, we will show that there exists a class of radial initial data ρ 0 (x) ≥ ρ V (x) such that the corresponding solution converges as t → ∞ to the split measure (1.6) in the sense of mass (which will be made precise below). Moreover, under further assumptions on V , we show that µ ∞ is the global minimizer in the space of measures of the relaxation of F.
An important motivation for our paper is the current interest in the following model of aggregation diffusion with interaction potential ∂ρ ∂t = ∆ρ m + ∇ · (ρ∇W * ρ) (1.7) that has led to the discovery of some highly interesting features that have consequences for the parabolic theory and the the Calculus of Variations. Recent results [20] show that, under some conditions on W the energy minimizer of the corresponding energy functional is likewise split as The presence of the concentrated point measure is known for specific choices of W , see [16]. To the best of our knowledge, there exist no results in the literature showing that solutions of the parabolic problem actually converge to these minimisers with a Dirac delta. In this paper, we treat as a first step the where V is known. We expect similar results to hold when V is replaced by W * ρ, but the techniques will be more difficult. For example, the non-local nature of (1.7) suggests that there might not be a comparison principle.
It was shown in [7] that for very fast diffusion, m < n−2 n , then the solutions of the Fast Diffusion Equation u t = ∆u m with u 0 ∈ L 1 (R n ) ∩ L ∞ (R n ) vanish in finite time, i.e. u(t, x) = 0 for t ≥ T * . When a V < 1, we construct explicit initial data that preserve the total mass, and this holds for any m ∈ (0, 1).
The case of a ball of radius R We have a more complete overall picture when we focus on the problem posed in a ball B R , adding a no-flux condition on the boundary: ρ(0, x) = ρ 0 (x).
As a convenient assumption, we require that V R does not produce flux across the boundary We discuss this assumption on Remark 2.12. This problem is the 2-Wasserstein flow of the free energy For (P R ), we show that F R is bounded below and sequences of non-negative functions of fixed ρ L 1 (B R ) = m converge weakly in the sense of measures to This means that, if the mass a 0,R cannot be reached in the class ρ V R +h , the remaining mass is complete with a Dirac delta at 0. Notice that the mass of ρ V R +h is decreasing with h, so the largest mass is that of ρ V R .
We construct an L 1 -contraction semigroup of solutions S R of (P R ) such that, if ρ V R ≤ ρ 0 ∈ L 1 (B R ) and radially symmetric, then where m = ρ 0 L 1 (B R ) and F R is the relaxation of F R to the space of measures presented below (see [25]) . The semigroup S R is constructed as the limit of the semigroup of the regularised problems written below as (P Φ,R ). Then, we recover our results by passing to the limit in Φ and R.
The mass function. One of the main tools in this paper will be the study of the so-called mass variable, which can be applied under the assumption of radial solutions. It works as follows. First, we introduce the spatial volume variable v = |x| n |B 1 | and consider the mass function For convenience we define R v = R n |B 1 |. We will prove that M satisfies the following nonlinear diffusion-convection equation in the viscosity sense where ω n = |B 1 |. The diffusion term of this equation is of p-Laplacian type, where p = m + 1. The weight will not be problematic when v > 0, as we show in Appendix A using the parabolic theory in DiBenedetto's book [26].
Notice that the formation of a Dirac delta at 0 is equivalent to the loss of the Dirichlet boundary condition M (t, 0) = 0. Few results of loss of the Dirichlet boundary condition are known in the literature of parabolic equation. For equations of the type u t = u xx + |u x | p , it is known (see, e.g., [4]) that u x may blow up on the boundary in finite or infinite time, depending on the choice of boundary conditions. The case of infinite time blow-up was revisited in [43]. The question of boundary discontinuity in finite time, loss of boundary condition, for the so-called viscous Hamilton-Jacobi equations is studied in [6,40,41,38] and does not bear a direct relation with our results. A general reference for boundary blow-up can be found in the book [42].
Precise statement of results. In order to approximate the problem in R n , our choice of V R will be of the form and with the condition V R · x = 0 on ∂B R . We also define We will denote V = V R until Section 7.
Assume also that a 0,R > a V,R , ρ 0 radially symmetric, ρ 0 ≥ ρ V and ρ 0 ∈ L ∞ (B R \ B r1 ) for some r 1 < R. Then, the solution ρ of (P R ) constructed in Theorem 3.6 satisfies lim inf t→∞ Br (i.e., there is concentration in infinite time). Moreover, if where d 1 denotes the 1-Wasserstein distance.

Remark 1.2.
If we take a non-radial datum ρ 0 ≥ ρ 0,r with ρ 0,r radially symmetric satisfying the hypothesis of Theorem 1.1, then the corresponding solution ρ(t, x) of (P R ) constructed in Theorem 3.6 concentrates in infinite time as well, due to the comparison principle.
Through approximation as R → ∞, we will also show that Corollary 1.3 (At least infinite-time concentration of solutions of (P)). Under the hypothesis of Theorem 1.1 and suitable hypothesis on the initial data (specified in Section 7.1), we can show the existence of viscosity solutions of (M) in (0, ∞) × (0, ∞) (obtained as a limit of the problems in B R ), such that for all v > 0 and, furthermore, locally uniformly (0, ∞). We also have that Through our construction of M , we cannot guarantee in general that M (t, 0) = 0 for t finite. Producing a priori estimates, we can ensure this in some cases.
Then, the viscosity mass solution constructed in Proposition 7.1 does not concentrate in finite time, i.e. M (t, 0) = 0.
The picture for power-like V . Let us discuss the case where V is of the form In this setting, we satisfy (1.12) so concentration does not happen in finite time. The condition In fact, under this condition, ρ V ∈ L 1+ε (R n ). In addition to the behaviour at 0 and ∞, the restriction a V < 1 is a condition on the intermediate profile of V . This is sufficient to construct initial data ρ 0 (of the shape ρ D present below) so that solutions converge to µ ∞ as t → ∞, in the sense of mass. Due to (1.12), the concentration is precisely at infinite time. But we do not know that µ ∞ is a global minimiser of F. In Remark 7.8 we prove that the energy functional is bounded below whenever m > n n+λ∞ . Notice that n−λ∞ n < n n+λ∞ . Therefore, if n n + λ ∞ < m < n − λ 0 n , and a V < 1, then µ ∞ is the global minimiser in P(R n ) of the relaxation of F and it is an attractor for some initial data.
Structure of the paper. In Section 2 we write the theory in B R for a regularised problem where the fast-diffusion is replaced by a smooth elliptic non-linearity Φ. In Section 3 we construct solutions of (P R ), by passing to the limit as Φ(s) → s m the solutions of Section 2. In Section 4 we show that mass functions M of the solutions of Sections 2 and 3 are solutions in a suitable sense of Problem (M), and we prove regularity and a priori estimates. In Section 5 we construct initial data ρ 0 so that the mass M is non-decreasing in time as well a space. We show that these solutions M concentrate in the limit, a main goal of the paper. We recall that this means the formation of a jump at v = 0 for t = ∞. Section 6 is dedicated to the minimisation of F R for functions defined in B R . We prove that the minimisers are precisely of the form µ ∞,m,R described above. In Section 7, we pass to the limit as R → ∞ in terms of the mass. We show that the mass functions for suitable initial data still concentrate. We discuss minimisation of the function F. We show the class of potentials V that make F bounded below is more restrictive than for F R , and provide suitable assumptions so that µ ∞ is a minimiser. We list some comments and open problems in Section 8. We conclude the paper with two appendixes. The first, Appendix A recalls results from [26] and compacts them into a form we use for M . Appendix B is devoted to mixing partial space and time regularities into Hölder regularity in space and time.
A comment on the notation. Throughout the document, we deal with the spatial variable x, the radial variable r = |x|, and the volume variable v = |B 1 ||x| d . Since it will not lead to confusion and simplifies the notation, radial functions of x will sometimes be evaluated or differentiated in r or v. This must be understood as the correct substitution.

The regularised equation in B R
Following the theory of non-linear diffusion, we consider in general We assume that Φ ∈ C 1 and elliptic we think of the problem as Remark 2.1. Our results work in a general bounded domain Ω, where the assumption on E is that E · n(x) = 0 on ∂Ω. However, we write them in a ball of radius R since our main objective is to study the long-time asymptotics of radially symmetric solutions.
The diffusion corresponds to the flux a(u, ∇u) = Φ (u)∇u. When Φ, E are smooth and we assume Φ is uniformly elliptic, in the sense that there exist constants such that existence, uniqueness, and maximum principle hold from the classical theory. The literature is extensive: in R n this issue was solved at the beginning of the twentieth century (see [33]), in a bounded domain with Dirichlet boundary condition the result can be found in [29], and the case of Neumann boundary conditions was studied by the end of the the twentieth century (for example [1]), where the assumptions on the lower order term were later generalised (see, e.g. [47]). Following [1], we have that, if u 0 ∈ C 2 (B R ) then the solution u of (P Φ,R ) is such that Let us obtain further properties of the solution of (P Φ,R ).
For classical solutions we have that Proof. Let j be convex. We compute where F (u) = j (u)u ≥ 0 and we can pick F (0) = 0. Hence F ≥ 0. If j has a minimum at 0, then Finally, we recover By Gronwall's inequality we have that Taking the power 1/p we have (2.4) for p < ∞ and letting p → ∞ we also obtain the L ∞ estimate.
Proof. Multiplying by Φ(u) and integrating Applying Young's inequality we obtain Notice since Φ ≥ 0 we have that Ψ ≥ 0. Hence, we deduce the result.
If Ψ(u 0 ) ∈ L 1 and u 0 ∈ L 2 then the right-hand side is finite due to (2.4).
We also have, for z(t, x) = Z(u(t, x)) that (2.7) Proof. Again we we will use the notation w = Φ(u). When u is smooth, we can take w t as a test function and integrate in B R . Notice that w t = Φ (u)u t , so Since ∇w = 0 on ∂B R , then also ∇w t = 0. We can integrate by parts to recover Using assumption (2.1) the second term on the right-hand side vanishes. Integrating in [0, T ] we have Integrating by parts in time the last integral Notice that u t ∇w = Φ (u) Applying Young's inequality, we deduce (2.8) From the estimates above, we know that c 1 |∇u| ≤ Φ (u)|∇u| = |∇Φ(u)| ∈ L 2 . Similarly, the result follows. Finally, we use that using that Z = min{1, Φ }.

Free energy and its dissipation when E = ∇V
When E = ∇V we have, again, a variational interpretation of the equation that leads to additional a priori estimates. We can rewrite equation (P Φ,R ) as Formulation (2.9) shows that this equation is the 2-Wasserstein gradient flow of the free energy Along the solutions of (P Φ,R ) it is easy to check that Also, by integrating in time we have that Finally, let us take a look at the stationary states. For any H ∈ R, the solution of Θ(u) + V = −H is a stationary state. Since Θ : [0, +∞) → R is non-decreasing, we have that H = −Θ(u(0)). We finally define u V +H := Θ −1 − (H + V ) .

Remark 2.8.
When Φ is elliptic u V +H ≤ Θ −1 (−H). In the case of the FDE we have where h = H + m 1−m . When h > 0 we have ρ V +h is bounded, but ρ V is not bounded.

Comparison principle and L 1 contraction
Let us present a class of solutions which have a comparison principle, and are therefore unique. Definition 2.9. We define strong L 1 solutions of (P Φ,R ) as distributional solutions such that 3. u t ∈ L 2 (0, T ; L 1 (B R )).
Theorem 2.10. Assume E · n(x) = 0. Let u, u be two strong L 1 solutions of (P Φ,R ). Then, we have that and, for each u 0 ∈ L 1 (B R ), there exists at most one strong L 1 solution.
Proof. We now have that w = Φ(u) − Φ(u). Let j be convex and denote p = j . We have, using the no flux condition (2.2). Expanding the divergence, we have Then, as p → sign + , we have p(w) → sign + 0 (w) = sign + 0 (u − u) and Using again that E · n(x) = 0 on ∂B R , we recover a 0 on the right hand side. This completes the proof.
Remark 2.12 (On the assumption E · x = 0 on ∂B R ). Notice that to recover the L p estimates in Theorem 2.2 (which depend on ∇ · E L ∞ ) we assume only that E · x ≥ 0 on ∂B R . However, later (as in Lemma 2.6 and Theorem 2.10) we require E · x = 0 on ∂B R . The estimates in these results do not include ∇ · E, and so it seems possible to extended the results to this setting by approximation.

The Aggregation-Fast Diffusion Equation
We start this section by providing a weaker notion of solution Definition 3.1. We say that ρ ∈ L 1 ((0, T )×B R is a weak L 1 solution of (P R ) if ρ m ∈ L 1 (0, T ; W 1,1 (B R )) and, for every ϕ ∈ L ∞ (0, T ; for a.e. t ∈ (0, T ).
If ∇V · n(x) = 0 we then have ∇ρ m · n = 0 and we can write the notion of very weak L 1 solution by integrating once more in space the diffusion term Theorem 3.2 (L 1 contraction for H 1 solutions bounded below). Assume that ρ, ρ are weak L 1 solutions of (P R ) with initial data ρ 0 and ρ 0 , ρ, ρ ∈ H 1 ((0, T ) × B R ), and ρ, ρ ≥ c 0 > 0. Then Proof. Since the solutions are in H 1 and are bounded below, then ρ m , ρ m ∈ H 1 ((0, T ) × B R ). Let p be non-decreasing and smooth. By approximation by regularised choices, let us define w = ρ m − ρ m and ϕ = p(w). Thus we deduce Proceeding as in Theorem 2.10 for Φ smooth and using (1.8) we have that and this proves the result.
We can now construct a semigroup of solutions. We begin by constructing solutions for regular data, by passing to the limit in regularised problems with a sequence of smooth non-linearities Φ k (s) → Φ(s) = s m . We consider the sequence Φ k of functions given by Φ k (0) = 0 and up to a smoothing of the interphases. We define Then, the sequence u k of solutions for and ρ is a weak L 1 solution of the problem. Moreover, we have that ρ ≥ ω(ε) > 0, In fact, ρ is the unique weak L 1 solution which is H 1 and bounded below.
Let us characterise φ as Φ(ρ). For k > m ± 1 1−m we can compute clearly min{1, Φ k } from (3.1) and hence we have Since u k are uniformly bounded in L ∞ , taking k large enough we have that Thus Z(u k ) converges pointwise to Z * . But Z is continuous and strictly increasing, so it is invertible.
We can now upgrade to strong convergence, using the uniform L ∞ bound |u k | ≤ C. Hence, together with the point-wise convergence, we can apply the Dominated Convergence Theorem to show that our chosen subsequence also satisfies Let us show that we maintain an upper and positive lower bound. The upper bound is uniform . Thus, u k ≥ ω(ε) and, therefore, so is ρ. In fact, due to this lower bound and so the convergence u k ρ is also weak in H 1 (up to a subsequence). But then ρ is the unique weak L 1 solution with this property. Since the limit is unique, the whole sequence u k converges to ρ all the senses above.

Corollary 3.4 (Approximation of the free energy). Under the hypothesis of Theorem 3.3 we have that
for a.e. t > 0.
In particular, F R [ρ(t)] is a non-increasing sequence.
Proof. Since u k → ρ converges a.e. in (0, T ) × B R , then for a.e. t > 0 we have that u k (t) → ρ(t). Since u k is uniformly bounded, then the Dominated Convergence Theorem ensures the convergence of Taking into account (2.12), then the sequence u . Therefore, up to a subsequence, it has limit ξ(x). We can write We know that ∇Φ k (u k ) + u k ∇V ∇ρ m + ρ∇V weakly in L 2 . On the other hand, since we know u k , ρ ≥ ω(ε) we can apply the intermediate value theorem to show that, up a to further subsequence, where the strong convergence L 2 follows, up to a further subsequence, from the weak H 1 convergence. Using the product of strong and weak convergence But this limit must coincide with ξ, so the limit holds also weakly in L 2 . The weak lower-continuity of the L 2 yields the result.
We are also able to deduce from these energy estimates an L 1 bound of ∇ρ m . Unlike (2.5) this bound can use only local boundedness of ∇V .

Corollary 3.5. In the hypothesis of Theorem 3.3 we have that
Proof. We therefore have that Hence, we conclude the result using Corollary 3.4, Jensen's inequality and the conservation of the L 1 norm.
. To be precise, by applying Hölder's inequality with p = 1 m > 1 we have the estimate Now we apply density in L 1 of the solutions with "good" initial data, via the comparison principle

We have L 1 comparison principle and contraction
is the limit of the solutions u k of (P Φ,R ) with (3.1) and is non-increasing and we have (3.2). Hence, it is a weak L 1 solution.

Remark 3.7.
Notice that there is no concentration in finite time. This is due the combination of the L 1 contraction with the uniform L 1+ε estimate (2.4). By the L 1 contraction, the sequence S(t) max{ρ 0 , k} is Cauchy in L 1 and hence it has a limit in L 1 . No Dirac mass may appear in finite time. In R n we do not have an equivalent guarantee that S(t)ρ 0,k ∈ L 1 (R n ) for some approximating sequence. We will, however, have this information in the space M(R n ).

Remark 3.8.
Notice that the construction of S(t) is unique, since for dense data it produces the unique H 1 solution bounded below (which also comes as the limit of the approximations), and then it is extended into L 1 by uniform continuity.
Proof of Theorem 3.6. We start by defining S(t)ρ 0 = ρ for the solutions constructed in Theorem 3.3. Let us construct the rest of the situations.
Taking a different ρ 0 with the same properties, and ρ 0, its corresponding approximation, again for large, 0 < ω(ε) ≤ ρ ≤ C(t). Then we have that Let → +∞ we recover the L 1 contraction. Similarly for the comparison principle.
Step 2. ρ 0 ∈ L 1 . Approximation by solutions of Theorem 3.3. We define For the solutions constructed in Step 1. we have that ρ K,ε ρ K as ε 0 and as K +∞ we have ρ K ρ. By the L 1 contraction, we have as above that the sequence are Cauchy and hence we have L 1 convergence at each stage. The contraction and comparison are proven as in Step 1.
Step 3. Item 4. Due to the L 1+ε bound, we know that On the other hand, we can select adequate regularisations of the initial datum ρ 0, ∈ H 1 such that ε ≤ ρ 0, ≤ ε −1 , and the corresponding solutions u k, of (P Φ,R ) with Φ = Φ k given by (3.1) satisfy the L 1 contraction. Integrating in (0, T ) we have that As k → ∞, by the lower semi-continuity of the norm As → ∞ we recover ρ * = S(t)ρ 0 .
Step 4. ρ 0 ∈ L 1 . Solutions in the very weak sense. Finally, let us show that the solutions satisfy the equation in the very weak sense. Since we can integrate by parts, ρ K, satisfies the very weak formulation, and we can pass to the limit to show that so does ρ K .
We have shown that ρ K ρ in L 1 . With the same philosophy, we prove that ρ K (t) ρ(t) for every t > 0 so ρ(t) ∈ L 1 (B R ) for a.e. and we can pass to the limit in the weak formulation. We only need to the deal with the diffusion term. We also have that ρ m K ρ m . Due to (3.3) and the Monotone Convergence Theorem, we deduce that ρ m ∈ L 1 ((0, T ) × B R ).
Step 5. Conservation of mass. Since all the limits above hold in L 1 , then preservation of the L 1 mass follows from the properties proved in Theorem 3.3.
Step 6. Decay of the free energy. Since all the limits above are taken monotonously and a.e., we can pass to the limit in by the Monotone Convergence Theorem. Hence, the decay of the free energy proven in Corollary 3.4 extends to L 1 solutions. We can also pass to the limit in (3.2).

An equation for the mass
The aim of this section is to develop a well-posedness theory for the mass equation (M). We will show that the natural notion of solution in this setting is the notion of viscosity solution. We will take advantage of the construction of the solution ρ of (P R ) as the limit of the regularised problems (P Φ,R ).

Mass equation for the regularised problem
If E is radially symmetric and u is the solution solution of (P Φ,R ), its mass function M satisfies for radially symmetric functions. Proof. For any λ > 0, let us consider the continuous function Notice that w → 0 as either t → +∞ or v → 0, R v . Assume, towards a contradiction that w reaches positive values. Hence, it reaches a positive global maximum at some point t 0 > 0 and v 0 ∈ (0, ∞). At this maximum At (t 0 , v 0 ), we simply write the contradictory result Let us define the Hölder semi-norm for α ∈ (0, 1) We have the following estimate Proof. Let us prove first an estimate for M t (t, ·) L 2 (0,Rv) . Since M = ∂u ∂v then ∂M ∂t = ∂u ∂v∂t . Applying Jensen's inequality Making the change of variables v = |B 1 |r n we have Due to (2.5) we recover (4.2). Finally

Aggregation-Fast Diffusion
We recall the definition of viscosity solution for the p-Laplace problem, which deals with the singular (p ∈ (1, 2)) and degenerate (p > 2) cases. We recall the definition found in many texts (see, e.g., [31,37] and the references therein).
Similarly, for our problem we define The corresponding definition of subsolution is made by inverting the inequalities. A viscosity solution is a function that is a viscosity sub and supersolution.
Remark 4.6. Since we have a one dimensional problem, we can write the viscosity formulation equivalently by multiplying by ( ∂ϕ ∂v ) 1−m everywhere, to write the problem in degenerate rather than singular form.

Remark 4.7.
Our functions M will be increasing in v. This allows to a simplification of the condition in some cases. For example, if also have a lower bound on ρ, in the sense that then we know that it suffices to take viscosity test functions ϕ such ∂ϕ ∂v ≥ c 2 . In particular, we can simplify the definition of sub and super-solution by removing the limit and the supremum.
The elements of the upper jet are usually denoted by (p, X). The lower jet J 2,− is constructed by changing the inequality above. The definition of viscosity subsolution (resp. super-) can be written in terms of the upper jet (resp. lower). . Moreover, we have the following interior regularity estimate: for any T 1 > 0 and 0 < v 1 < v 2 < R v there exists γ > 0 and α ∈ (0, 1) depending only on n, m, ∂V ∂v L ∞ (v1,v2) , v 1 , v 2 , T 1 , such that Proof.
Step 1. ε ≤ ρ 0 ≤ ε −1 and ρ 0 ∈ H 1 (B R ). Let us show that M ρ is a viscosity solution of (M) and M ρ is a weak local solution in the sense of Appendix A.
By our construction of ρ by regularised problems in Theorem 3.3, the strong L q convergence of u k to ρ ensures that , and hence (up a to a subsequence) a.e.
Through estimates (4.1), (4.2) , and Theorem B.1 we have To check that M ρ is a viscosity solution, we select v 0 ∈ (0, R v ). Taking a suitable interval (ε, R v − ε) v 0 , by the Ascoli-Arzelá theorem, a further subsequence is uniformly convergent. Since we have characterised the a.e. limit we have Due to the uniform convergence, we can pass to the limit in the sense of viscosity solutions and M ρ is a viscosity solution at x 0 .
The argument is classical and goes as follows (see [23]). Take a viscosity test function ϕ touching M ρ from above at x 0 . Then, due to the uniform convergence M u k to M ρ in a neighbourhood of x 0 , there exists points x k where ϕ touches M u k from above. We apply the definition of viscosity solution for M u k at x k , and pass to the limit.
Due to the pointwise convergence, M ρ also satisfies (4.5).
As we did in Theorem 3.6 the L 1 limit of the corresponding solutions is S(t)ρ 0 . Furthermore, the limits ε 0 and K +∞ are taking monotonically in ρ, so also monotically in M . This guarantees monotone convergence in M . With the universal upper bound 1 we have L 1 convergence.
Since the C α bound is uniform away from 0, we know that M maintains it and is continuous. Due to Dini's theorem the convergence is uniform over , and M ρ is a viscosity solution of the problem.
The value M (t, 0) = 0 is given by S(t)ρ 0 ∈ L 1 (B R ) and the value at M (t, R v ) = a 0,R by the fact that S(t)ρ 0 L 1 (B R ) = ρ 0 L 1 (B R ) = a 0,R . The uniform continuity is a direct application of Corollary A.3. We point out that, since ρ 0 ∈ L 1 (B R ), then M ρ0 is point-wise continuous, and therefore uniformly continuous over compact sets. Estimate (4.4) follows from Theorem A.1.
Let us now state a comparison principle, under simplifying hypothesis.
Since both functions are continuous, there exists ( .
With this choice, we have that For this ε and λ fixed, let us construct the variable-doubling function defined as This function is continuous and bounded above, so it achieves a maximum at some point. Let us name this maximum depending on ε, but not on λ by In particular, it holds that (4.6) Step 1. Variables collapse.
This implies that, as ε → 0, the variable doubling collapses to a single point.
We can improve the first estimate using that Since M is uniformly continuous, we have that Step 2. For ε > 0 sufficiently small, the points are interior. We show that there exists µ such that t ε , s ε ≥ µ > 0 for ε > 0 small enough. For this, since M and M are uniformly continuous we can estimate as where ω ≥ 0 is a modulus of continuity (the minimum of the moduli of continuity of M and M ), i.e. a continuous non-decreasing function such that lim r→0 ω(r) = 0. For ε > 0 such that we have ω(t ε ) > σ 4 . The reasoning is analogous for s ε . For v ε we can proceed much in the same manner And analogously for ξ ε . A similar argument holds for R v − v ε and R v − ξ ε .
Step 3. Choosing viscosity test functions. Unlike in the case of first order equations, there is no simple choice of ϕ that works in the viscosity formula. We have to take a detailed look at the jet sets. Due to [23,Theorem 3.2] applied to u 1 = M , u 2 = −M and ϕ ε (t, s, v, ξ) = |v − ξ| 2 + |s − t| 2 ε 2 + λ(s + t) for any δ > 0, there exists X and X in the corresponding jets such that where z ε = (t ε , s ε , v ε , ξ ε ) and we have In particular, this implies that the term of second spatial derivatives satisfies X 22 ≤ X 22 (see [23]). Notice that Plugging everything back into the notion of viscosity sub and super-solution Step 4. A contradiction. Substracting these two equations since v 2 n−1 n ∂V ∂v (v) = r n−1 ∂V ∂r is Lipschitz continuous and (4.7).

Existence of concentrating solutions
When we now take F : (0, ∞) → (0, ∞) We will prove that with this initial data we have U = ∂M ∂t ≥ 0 by showing it satisfies a PDE with a comparison principle and U (0, ·) ≥ 0. First, we prove an auxiliary result for the regularised problem.
Theorem 5.1 (Solutions of (P Φ,R ) with increasing mass). Let h ∈ R, F be such that F ≤ 1, F ≥ 0, F (0) = 0, u be the solution of (P Φ,R ) and M be its mass. Then, we have that and , and this is a stationary solution. Hence this inequality holds for u(t) as well, due to Theorem 2.10. Thus (5.4) holds. Since u ∈ C 1 ((0, T ); C(B R )), we can consider . This can be justified in the weak local sense.
we can simply take ϕ = ∂ψ ∂t and integrating by parts in time to recover Since u is C 1 then ∂ ∂t (Φ (u)) = Φ (u) ∂U ∂t is a continuous function. Operating with the derivatives of ψ, we recover that We now show that U is a solution in the weak sense, incorporating the boundary conditions. Since U is continuous and U (t, 0) = U (t, R v ) = 0, for any ψ suitably regular we can use an approximating Fix Ψ 0 smooth and let Ψ the solution of If Ψ is a classical interior solution, then taking as a test function ψ(t, x) = Ψ(T − t, x) we have that Notice that A(0) = 0. Substituting A by the uniformly elliptic diffusion A(T − t, v) + δ, δ > 0, and letting δ 0, for any Ψ 0 ≥ 0, we can construct a non-negative solution of (5.5). Therefore, since U (0) ≥ 0 we have that U ≥ 0 in (0, T ) × (0, R v ), and the proof is complete.
Before we continue, we point out that M ρ F , lies between M ρ V and one of its upward translations On the other hand, integrating backwards from R v we have

(5.6)
Now we move to considering suitable initial data for (P R ). We make the following construction The function F D can be taken as the limit of functions F ε ∈ C 1 in the assumptions of Theorem 5.1. Take F ε (0) = 0 and where given 0 < b 1 < b 2 < V (R) and ε < b2−b1 4 and 0 < D ≤ b 1 , we can always select Notice that F ε (s) > 0 for s > 0 and F ε ≤ 1 and F ε ∈ C 1 . This form is rather elaborate, so we pick the limit as ε 0. Notice that inf F ε → −∞ as ε 0.
Proof of Lemma 5.3. We start by pointing out that B R ρ F is continuous in all parameters. Taking Notice that ρ D ∈ L 1+ε (B R ) due to the assumption (1.5). We sketch the profile in Figure 1.

Theorem 5.4 (Solutions of (P R ) with increasing mass). Under the hypothesis of Theorem 3.3, let ρ D be given by (5.7). Then, the mass M of ρ(t) = S(t)ρ D constructed in Theorem 3.6 is such that
In particular, ρ(t, ·) (a 0,R − a V,R )δ 0 + ρ V weak-in the sense of measures.
Proof of Theorem 5.4. Step 1. Properties by approximation. Since ρ D ∈ L 1+ε , looking at how we constructed S(t)ρ 0 in Theorems 3.3 and 3.6, it can approximated by S k (t)ρ 0 where S k is the semigroup of (P Φ,R ) with Φ k given by (3.1). Notice that, the associated Θ k given by (2.10) is

Hence, we recover
Taking h = m 1−m in (5.2) we have initial data u 0,k such that and M u k non-decreasing in t. This corresponds to an interval of the form v ∈ [ε k , R v − δ k ]. Let us denote u k = S k (t)u 0,k . Due to the L 1 contraction we have that Hence, by Theorem 3.6 we infer that S k (t)u 0,k → S(t)ρ D in L 1 (B R ) for a.e. t > 0. This guarantees the a.e. convergence of the masses. Hence, the mass function M , which is already a viscosity solution of (M) and C α regular, also inherits the point-wise estimate from M u k in (5.4). M is also non-decreasing in t and v. Moreover, due to (5.6) and Theorem 4.9 due to Equation (4.4), we conclude that Step 2. Uniform convergence of M (t, ·) as t → +∞. Since M is point-wise non-decreasing in t and bounded above by a 0,R , we know there exists a function M ∞ such that By the estimate (4.4) we know that M ∞ belongs to C α loc ((0, R v )) and hence continuous in interior points. On the other hand, (5.8) implies Hence, by the sandwich theorem, M ∞ (R v ) = a 0,R and it is continuous at R v (due to the explicit formulas we can actually show rates). Since M ∞ is non-decreasing and M ∞ ≥ 0, due to (5.9), there exists a limit lim Due to (5.4) and our choice of h, we have that Step 3. Characterisation of M ∞ as a viscosity solution. Let us check that M ∞ is a viscosity solution of Due to our lower bound (5.10), ∂M∞ ∂v is bounded below. We define the sequence of masses M n : n, v). These are viscosity solutions for (M) due to Theorem 4.9. We also know that By standard arguments of stability of viscosity solutions, M ∞ is also a solution of (M). Since it does not depend on t, we can select spatial viscosity test functions, and hence it is a solution of (5.11). Since we have removed the time dependency, we dropped also the spatial weight (nω n v n−1 n ) 2 .
Step 4a. Lipschitz regularity Since M ∞ is non-decreasing, at the point of contact of a viscosity test function touching from below, we deduce Hence, M ∞ is a viscosity super-solution of −∆M = 0. Due to [30], we have that M is also a distributional super-solution of −∆M = 0. Distributional super-solutions are concave. Since Step 4b. Higher regularity by bootstrap. Now we can treat the right-hand side as a datum Applying the regularisation results in [12] we recover that Step By the comparison principle, which holds due to (5.10), we conclude the equality Due to (5.10), the singularity at 0 is incompatible with h > 0. Thus h = 0.
To check the convergence in Wasserstein distance, we must write the convergence of the masses in L 1 in radial coordinates. Let µ ∞,R = (a 0,R − a V,R )δ 0 + ρ V , then we have that due to the fact that the optimal transport between radial densities is radial and the characterisation of d 1 in one dimension (see [46]). Since we have shown in the proof above that Br ρ(t) dx ≤ µ ∞ (B r ) Due to the monotone convergence Br ρ(t, x) µ ∞,R (B r ) for r ∈ (0, R], the right-hand goes to 0 as t → +∞.

Minimisation of F R
It is very easy to see that the free energy F R is bounded below, in particular 1) due to (3.3) and that V ≥ 0. Therefore, there exists a minimising sequence. The problem is that the functional setting does not offer sufficient compactness to guarantee its minimiser is in L 1 (B R ). However, we can define its extension to the set of measures as This is the unique extension of F R to M + (B R ) that is lower-semicontinuous in the weak-topology (see [25] and related results in [11]).
Since we work on a bounded domain, tightness of measures is not a limitation. For convenience, let us define for ρ ∈ L 1 (B R ), Let us denote the set of non-negative measures of fixed total mass m in B R as We have the following result Remark 6.2 (Lieb's trick). Given a radially decreasing ρ ≥ 0, ρ q ∈ L 1 (B R ) for some q > 0 (for any R ≤ ∞), using and old trick of Lieb's (see [34,35]) we get, for |x| ≤ R, Hence, we deduce the point-wise estimate It is easy to see that (6.3) is not sharp. However, it is useful to prove tightness for sets of probability measures. Similarly, if additionally V ρ ∈ L 1 (B R ), and V ≥ 0 we can estimate so we recover the point-wise estimate Proof of Theorem 6.1. The second equality in (6.2) is due to the weak-density of L 1 + (B R ) in the space of non-negative measures, and the construction of F R (see [25]). Let us consider a minimising sequence. Let us show that we can replace it by a radially-decreasing minimising sequence. Let ρ j ∈ L 1 + (B R ) with ρ j L 1 = m. By standard rearrangement results Since V ≥ 0 and radially symmetric and non-decreasing then Hence, there exists minimising sequence ρ j ∈ L 1 (B R ) that we can assume radially non-increasing. Since ρ j ∈ P m (B R ), by Prokhorov's theorem, this minimising sequence must have a weak-limit in the sense of measures, denoted by µ ∞,m .
We use the following upper and lower bounds that follow from (3.3) Due to (6.4) we have a uniform bound in Let us now characterise this measure. For ϕ ∈ C ∞ c (R n ) we take For ϕ fixed, there is ε 0 > 0 such that for ε < ε 0 , µ ∞,m + εψ ∈ P m (R n ) and, hence, Hence, we get the expression We write Since we have the estimate we recover by the Dominated Convergence Theorem Thus, as ε → 0 the following inequality holds Applying the same reasoning for −ψ (which corresponds to taking −ϕ instead of ϕ), we deduce the reversed inequality, and hence the equality to 0. This means that Since mδ 0 is not a minimiser (see Remark 6.3) then ρ ∞ ≡ 0. As ϕ concentrates to a point, we recover for a.e. x either Notice that the right hand of the second term is a constant. Since ρ ∞ is radially decreasing then there exists R ∞ > 0 such that where, by evaluating close to 0 we deduce that h = −C[ρ ∞ ] ≥ 0. Notice that ρ ∞ is the minimiser of the two variable function under the total mass constraint that It is not a difficult exercise to check that the minimum is achieved with R ∞ = R and h as small as possible. When m > a V,R (which corresponds to h = 0) we have to add a Dirac Delta at the origin, with the difference of the masses m − a V,R .
To check this, first we point out that We deduce that f ≤ 0 and increasing the integration domain decreases f , i.e ∂f ∂τ < 0 for all τ, h > 0. On the other hand, since ∂ρ V +h ∂h < 0 for r, h > 0 we have that Hence, the derivative is not achieved at interior points. We look at the boundaries of the domain: 1. The segment (τ, h) ∈ {0} × [0, +∞), where f = 0. These are all maximisers.

The segment
If m > a V,R , then h 0 = 0. Using the derivative respect to h, the minimum in this segment is achieved at (R, h 0 ).

A segment
If m > a V,R , then R 0 = R. Using the derivative respect to τ , the minimum in this segment is achieved at (R 0 , 0).
Hence, if m ≥ a V,R , the minimum is achieved at τ = R and h = 0. The remaining mass is completed with a Dirac. Lastly, if m < a V,R there is an extra part of the boundary, where the mass condition is achieved with equality 4. The curve (τ, h(τ )) such that Notice that this segment contains the minima of the other segments. Taking a derivative respect to τ we deduce that Therefore, using Leibniz's rule again we recover Finally, the minimum is achieved for the lasted τ , so again the minimum is (R, h 0 ). 0 is not a minimiser). Let ρ ∈ L 1 + (B R ) smooth be fixed and let us consider the dilations ρ s (x) = s n ρ(sx) for s ≥ 1. Notice that ρ s → δ 0 as s → +∞ in the weak-of M(B R ). As s → ∞ we can compute

Remark 6.3 (mδ
It is not difficult see that F R takes negative values, so this is not a minimiser. In [14] the authors prove that in R n if ρ V +h1 ≤ ρ 0 ≤ ρ V +h2 then ρ(t) → ρ V +h of the same initial mass. This shows that µ ∞,m = ρ V +h is attractive in the cases without Dirac Delta concentration at the origin.
We have constructed initial data ρ 0 > ρ V such that ρ(t) → µ ∞,m in the sense of their mass functions. Furthermore, we show that Lemma 6.4 (Minimisation of F R through solution of (P R )). Assume ρ V ≤ ρ 0 , (1.11), a V,R < a 0,R = ρ 0 L 1 (B R ) and let ρ be constructed in Theorem 1.1. Then Proof. From the gradient flow structure we know F[ρ(t)] is non-increasing. First, we prove ρ(t) → ρ V in L 1 (B R \ B ε ) for some ε small. We know that ρ(t) ≥ ρ V so as t → ∞ due to Theorem 1.1. Now we can explicitly compute Due to the L 1 convergence, we can extract a sequence t k → ∞ such that ρ(t k ) → ρ V a.e. in B R \ B ε . For this subsequence, due to Fatou's lemma and ρ(t) ≥ ρ V we have Collecting the above estimates, we conclude that for any ε > 0. Letting ε → 0 we recover that lim sup k is actually a lim k , and it is equal to 0. Since F[ρ(t)] is non-increasing, we recover the limit as t → ∞.

The problem in R n
We start by showing the existence of a viscosity solution of the mass equation (M), by letting R → +∞. As R → ∞ we can modify V R only on (R − 1) < |x| < R to have ∇V R (x) · x = 0 for |x| = R. Fix ρ 0 ∈ L 1 (R n ) radially symmetric. Let M R be the solution of the mass equation with this data. Consider the extension where, as above, we denote We can carry the estimate in Proposition 7.1. Assume V ∈ W 2,∞ loc (R n ) is radially symmetric, strictly increasing, V ≥ 0, V (0) = 0 and the technical assumption (1.5). Let ρ 0 ∈ L 1 (R n ) be radially symmetric such that ρ 0 L 1 = 1. Then, there exists M ∈ C loc ([0, +∞] × (0, +∞)) a viscosity solution of (M) in (0, ∞) × (0, ∞) that satisfies the initial condition We also have the C α loc interior regularity estimate (4.4) with R v = ∞.
Notice that, at this point, we do not check that M (t, 0) = 0, and hence concentration in finite time may, in principle, happen in R n . We also do not show, at this point, that M (t, ∞) = 1. There could, in principle, be loss of mass at infinity. Remark 7.2 (Conservation of total mass if m ∈ ( n−2 n , 1)). For this we use the following comparison. We consider u k the solution of the pure-diffusion equation Then the associated mass satisfies the equation If u 0 ≥ 0 is radially decreasing, then so is ∂M k ∂v = u. Therefore, in the viscosity sense Let u be the solution of (P Φ,R ). Due to Theorem 4.9 we have that Recalling the limit through Φ k given by (3.1) and the limit R → ∞, the mass constructed in Proposition 7.1 we have the estimate where u is the solution of u t = ∆u m in R n . When m ∈ ( n−2 n , 1) we know that R n u(t, x) dx = R n u 0 (x) dx and, hence M (t, ∞) = 1.

At least infinite-time concentration of the mass
Let assume a V < 1 and that ρ 0 is such that there exists F with the following properties Remark 7.3. For example, this covers the class of initial data that satisfy the following three assumptions: In this setting, we can take a suitable initial datum ρ D as in the case of balls, and we are reduced to a problem in [0, v 0 ], since the upper and lower bound guarantee that This is a Dirichlet boundary condition for the mass.
Again, there exists a point-wise limit As in Theorem 5.4, M ∞ preserves the C α loc estimates, using Dini's theorem we can prove uniform convergence in intervals [ε, ε −1 ]. Thus M ∞ is a viscosity solution of (5.11). Due to the sandwich theorem and monotonicity It is easy to characterise M ∞ as we have done in the case of balls.
Remark 7.4 (Convergence of ρ R as R → ∞). Since we do not have any L q bound for ρ for q > 1, we do not have any suitable compactness. We can extend ρ R (t) by 0 outside B R and we do know that ρ R (t) M(R n ) ≤ 1. If we assume that (7.2) and that V (x) ≥ c|x| α for c, α > 0. The properties can be inhereted to ρ R so For ρ 0 in a suitable integrability class, we have tightness, and hence a weakly convergent subsequence such that We also know that ρ m R is uniformly integrable. However, since we cannot assure ρ m R (µ ac ) m , we cannot characterise µ as a solution of (P). This remark is still valid for radial initial data.

Minimisation of the free energy
Following the arguments in [5,13,20], we have an existence and characterisation result for the minimiser. In R n the free-energy of the FDE u t = ∆u m with 0 < m < 1, is not bounded below, and u(t) → 0 as t → ∞. In fact, the mass of solutions escapes through ∞ in finite time if m < n−2 n . We need to ask further assumptions on V so that the formal critical points ρ V +h are in fact minimisers.
We show below that it suffices that V is not critical in the sense of constants, i.e.
We provide an example of V where this property holds below. As in B R , we define an extension of F to the space of measure as where µ ac is the absolutely continuous part of the measure µ. Notice that, since we choose V (0) = 0, Proposition 7.5. Assume V ≥ 0 and V (0) = 0 and (7.2). Then, we have the following: 1. There exists a constant C > 0 such that If, furthermore V is radially symmetric and non-decreasing then 2. There exists µ ∞ ∈ P(R n ) such that

We have that
Proof of Proposition 7.5. Due to the lower bound, we have that On the other hand, we get Finally, we recover This completes the proof of Item 1.
Clearly, we have that Hence, the infimum of F is finite. As in the proof of Theorem 6.1, we can consider a minimising sequence ρ j . As in Theorem 6.1 we may assume that ρ j are radially symmetric and non-increasing.
Let us prove Item 2. As in Theorem 6.1, the second equality of (7.3) is due to the weak-density of L 1 (R n ) in the set of measures and the construction of F. For our minimising sequence we know hence that Using Lieb's trick in Remark 6.2, we obtain that ρ j ≤ C min{|x| −n , |x| −n/m }. Integrating outside of any ball B R , we can estimate Since m < 1, this is a tight sequence of measures. By Prokhorov's theorem, there exists a weaklyconvergent subsequence in the sense of measures. Let its limit be µ ∞ .
For the proof of Item 3, we proceed as in Theorem 6.1. Notice that we still have the estimate Since V is strictly increasing, this is an L ∞ (R n \ B κ ) of any κ > 0, and we can repeat the argument in B R .
Let us illustrate the previous theorem by giving sufficient conditions on V satisfying the main assumption of Proposition 7.5. We extend the argument in [15] to show a family of potentials V for which (7.2) holds. Theorem 7.6. Assume that, for some α ∈ (0, m) we have that Then, (7.2) holds for any ε ∈ (0, 1).

Remark 7.7.
If the function r → V (r) − α 1−m r n is non-increasing, then the integral criterion for series and the change of variable show that the condition becomes We are requesting that ρ m−δ V ∈ L 1 for some δ ∈ (0, m). This is only slightly more restrictive than simply that ρ V gives a finite quantity in either term of F.
Proof of Theorem 7.6. We look first at the integral on B 1 . Due to Hölder's inequality, we have that On the other hand, since V, ρ ≥ 0 we know that B1 V ρ dx ≥ 0. Hence, we only need to care about the integration on R n \ B 1 . We define, for j ≥ 1 First, we point out that Due to Jensen's inequality Applying the triple Hölder inequality with exponents p = (1 − m) −1 , q = α −1 , r = (m − α) −1 we recover Lastly, using Young's inequality we have, for any ε > 0 This completes the proof.
Remark 7.8 (The power-type case V (x) = C|x| λ for |x| ≥ R 0 ). In this setting, (7.4) becomes m > n n+λ (equivalently n(1−m) m < λ), and in this case can take any α such that n(1−m) λ < α < m. This condition is sharp. Let us see that, otherwise, F is not bounded below. We recall the following computation, which can be found in [15,Theorem 15] following the reasoning in [17,Theorem 4.3].
Assume m < n n+λ . We can construct densities ρ where the energy attains −∞. Let where β > 0 is a constant we will choose later, and j 0 is such that 2 j0 > R 0 . We can explicitly compute ∞ j=j0 2 −jβ This is a finite number whenever β > λ. On the other hand This number is infinite if mβ < n(1 − m). Hence, The case of the equality m = n n+λ is, as usual, more delicate due to the scaling. However, we still prove that As in the proof of [20, Proposition 4], we can take the following functions: It is a direct computation that For any α k > 0, we have that the rescaling ρ k (x) = α n k ρ k (α k x) is such that For any sequence b k which is yet to be determined, we can pick α k so that R n ρ n n+λ k = b k by taking Then, passing to the notation m = n n+λ , we recover that if pick the sequence b k so that b

Infinite-time concentration if V is quadratic at 0
Our aim in this section is to compare the solutions of (P) with the solutions of the pure-aggregation problem ∂ρ ∂t = ∇ · (ρ∇ V ), (7.6) where V is a different potential. The equation for the mass can be written in radial coordinates as We will show that infinite-time aggregation happens for (7.6) if and only if Clearly, a sufficient condition that ∂ V ∂r ≤ Cr near 0. This is the so-called Osgood condition used to distinguish infinite from finite time blow-up in aggregation equations [8].

2.
We have M (t, 0) = 0 for all t > 0, i.e. there is no concentration in finite time.
Proof of Proposition 7.10. Equation (7.7) is a first order linear PDE that we can solve by characteristics. We can look at the characteristic curves of constant mass M (t, r c (t, r 0 )) = M (0, r 0 ). Taking a derivative we recover drc dt (t) = − ∂ V ∂r (r c (t)). These are the same characteristics obtained when applying the method directly to (7.6). Clearly r c (t, r 0 ) ≤ r 0 . Since V ∈ C 2 (R n ), these characteristics exists for some time t(r 0 ) > 0, and are unique up to that time. Hence, let Concentration will occur if r c (t, r 0 ) = 0 for some r 0 > 0 and t < ∞, which is incompatible with (7.8).
Notice that since 0 < r c (t, r 0 ) ≤ r 0 , these functions are defined for all t > 0. Let us check that r c (t, r 0 ) do not cross, and hence can be used as characteristics. If two of them cross at time t, we have that ds.
As ∂ V ∂r > 0 outside 0, then r 0 = r 1 and the characteristics are the same. Due to the regularity of V , there is continuous dependence and, since the characteristics point inwards and do not cross, they fill the space [0, +∞) × [0, +∞).
Finally, notice also that ∂ V ∂r (0) = 0 and positive otherwise, then for any r 0 > 0 we have that lim t→+∞ r c (t, r 0 ) = 0. Since V is C 2 , then we have ∂ V /∂r(0) = 0 so r c (t, 0) = 0, i.e. M (t, 0) = 0. Proposition 7.11. Let ρ be a solution by characteristics of the aggregation equation (7.6), and let r 0 (t, r) the foot of the characteristic through (t, r). Then In particular, if ρ is a decreasing solution and V ∈ C 2 (R n ) with ∆ V (0) = 0, then Remark 7.12. For V (r) = r 2 then ∆ V is constant, and we only have the last term, so all solutions with decreasing initial datum are decreasing. If ∆ V is non-increasing, then in (7.10) we have −∆ V (r) + ∆ V (r 0 ) ≤ 0 and all solutions are decreasing. This is the case for V (r) = γr γ with λ ∈ (0, 2]. When V (r) = γr λ with λ > 2, let us show that decreasing solutions of (7.6) are not L 1 (R n ). Hence, any decreasing integrable initial data produces a solution that losses monotonicity. Indeed, if V (r) = r λ then ∆ V = (n + λ − 2)r λ−2 and integrating in (7.11) we recover ρ 0 ≥ Cr −(n+λ−2) which is not integrable for λ > 2.
The support of ρ 0 is a ball. Fixing a value a value of r ∈ supp ρ 0 we have that Letting t → +∞, since r c (t, r) → 0, ∆ V is continuous and ∆ V (0) = 0, we recover (7.11). This completes the proof.
Now we have the tools to show that concentration does not happen in finite time if ∂V ∂r ≤ C v r close to 0. We construct a super-solution using the pure-aggregation equation.
Proof of Theorem 1.4. Take Obtain M as the solution by characteristics of (7.7) constructed in Proposition 7.10. Due the definition of V , we know that it satisfies the hypothesis of Proposition 7.11 and we have ∆ V = nC V ≥ 0. Thus, (7.10) shows that ρ(t, ·) is decreasing, and non-negative. Therefore, it holds that, in the viscosity sense ∂M ∂v ≥ 0 and ∂ 2 M ∂v 2 ≥ 0. Hence, still in the viscosity sense Since characteristics retract, supp ∂M ∂v ⊂ B R V so the last term is non-negative by the assumption, because either ∂M ∂v = 0 or C V r − ∂V ∂r ≤ 0. Thus, using the comparison principle in B R for R ≥ R V given in Theorem 4.10 we have that M R ≤ M for all t ≥ 0, v ∈ [0, R v ] Since M is constructed by letting R → ∞, we conclude M ≤ M for t, v ≥ 0.

Final comments
1. Blow-up is usually associated in the literature to superlinear nonlinearities, both in reaction diffusion or in Hamilton-Jacobi equations, cf. instance [42,28] and its many references. Here it is associated to sublinear diffusion, notice that (1.13) implies, at least, 0 < m < 1. This might seem surprising but it is not, due to two facts. First, recall that 0 < m < 1 means that the diffusion coefficient mu m−1 is large when u is small, and small when u is large. This translates into fast diffusion of the support but slow diffusion of level sets with high values (see e.g. [22] for a thorough discussion). This explains why δ 0 may not be diffused for m small (see [10]). Secondly, the confinement potential V needs to be strong enough at the origin to compensate the diffusion and produce a concentration. In B R , this is translated in the assumption B R ρ V < 1 (recall that, for V (x) = |x| λ0 , this implies 0 < m < n−λ0 n < 1). In R n we need to deal with the behaviour at infinity, as mentioned in the introduction.
2. Formation of a concentrated singularity in finite time is a clear possibility in this kind of problem. In this paper, we do not consider the case V / ∈ W 2,∞ loc (R n ) (e.g. V (x) = |x| λ with λ < 2). So long as ∂V ∂v is continuous (e.g. λ ≥ 1), it makes sense to use the theory of viscosity solutions of the mass equation (M). In principle, there could be concentration in finite time, even in (P R ). Notice that, in our results, the estimate for ρ(t) ∈ L q (B R ) depends on ∆V L ∞ (B R ) . For more general V , better estimates for ρ are needed in order to pass the limits Φ k (s) → s m and R → ∞. Some of these issues will be studied elsewhere.
3. For ρ 0 ∈ L 1 + (B R ), S R (t)ρ 0 is constructed extending the semigroup through a density argument. We do not know whether it is the limit of the solutions u k of (P Φ,R ) with (3.1). Furthermore, this question can be extended to initial data so that F R [ρ 0 ] < ∞.
4. Non-radial data. We provide a well-posedness theory in B R when ρ 0 > 0, but not in R n . In B R , as mentioned in Remark 1.2, we can show concentration in some non-radial cases, but the exact splitting of mass in the asymptotic distribution is still unknown. The asymptotic behaviour in the non-radial case is completely open.

A Recalling some classical regularity results
The equation for the mass of the solution of u t = ∇ · (∇Φ(u) + u∇V ) is given by Let us prove local regularity of bounded solutions by applying the results in [26]. To match the notation of [26], in this appendix we choose the notation x = v, u = M , and a 0 (x) = (nω The standard hypothesis set in [26] are that for some p > 1 we have a(x, t, u, Du) · Du ≥ C 0 |Du| p − ϕ 0 (t, x), |a(x, t, u, Du)| ≤ C 1 |Du| p−1 + ϕ 1 (t, x), |b(x, Du)| ≤ C 2 |Du| p + ϕ 2 (t, x).
This completes the proof. where C depends only on the norms of u in the spaces above.