On a class of reaction-diffusion equations with aggregation

In this paper, global-in-time existence and blow up results are shown for a reaction-diffusion equation appearing in the theory of aggregation phenomena (including chemotaxis). Properties of the corresponding steady-state problem are also presented. Moreover, the stability around constant equilibria and the non-existence of non-constant solutions are studied in certain cases.

One of the motivations to study such an equation comes from the structure similarities that exist with the parabolic-elliptic Keller-Segel models for chemotaxis, i.e.
where V is the fundamental solution of Poisson equation (or some other given potential in the case of general diffusion aggregation equations).If formally the interaction potential V is replaced by a the Dirac mass δ 0 , then the above equation is reduced to In [2], the authors propose a microscopic particle model which converges at the formal level towards such a PDE.This microscopic particle model, corresponding to (1.1) with c = d = 0 and Ω = ℝ n , is the following: ) dt, X i 0 = ξ i i.i.d.random variables with distribution u 0 , where V ε (x) = ε −n V( x ε ) for ε > 0, ∫ ℝ n V(x) dx = 2b, and B i t are i.i.d.Brownian motions.It is well known (for example in [9,11] that under suitable assumptions on V, the particle model converges (when N goes to infinity) to the following intermediate (with fixed ε > 0) nonlocal problem: ∇V ε (| Xi t − y|)u ε (y, t) dy dt, Xi 0 = ξ i i.i.d.random variables with distribution u 0 , where u ε is the distribution of the i.i.d.random processes Xi t at time t.By Itô's formula one can obtain the following nonlocal partial differential equation for u ε : Furthermore, in the parabolic regime, i.e. when 0 ≤ u 0 < a 2b , it is proved in [2] that the limit of u ε satisfies equation (1.1) with c = d = 0: 3) The physical meaning of the unknown u is that of a concentration, therefore one considers only nonnegative solutions corresponding to nonnegative initial data.Furthermore (like in the case of Keller-Segel system), problem (1.1) with c = d = 0 (and the homogeneous Neumann boundary condition) possesses the following entropy structure: This entropy is a combination of a positive part from the diffusion and a negative one from the aggregation.It needs to be pointed out that here the aggregation phenomenon is much stronger than the one appearing in Keller-Segel systems because of the singular potential that appeared in (1.2)-(1.3).
As for the reaction term, it is considered to be of logistic (mono-stable) type so that (when d > 0) a significant dampening effect is exercised on the density u at those points where u becomes large.
The arrangement of the paper is the following.In Section 2, global existence and uniqueness of classical solutions are obtained for initial data such that parabolicity is expected to hold.The rest of the paper concerns cases in which parabolicity is expected to be lost at some point, so that blowup may happen.Considerations on the possible steady states and their stability as well as direct estimates of blowup are presented.Section 3 is devoted to the study of the steady states.The non-existence of nontrivial steady states is proved via Pohozaev's type arguments.Furthermore, the linear stability of constant steady states is investigated.Finally, in Section 4, blow-up (in finite time) results are presented.Two different procedures are carried out: Kaplan's method is used for the problem with Dirichlet boundary condition on one hand, and the concavity method is used for the problem with Neumann boundary condition on the other hand.In the end, we present an annex where blowup is directly observed in a class of explicit solutions linked to Barenblatt profiles, and we draw conclusions in a final section.

Global Existence
In this section, the global existence and uniqueness of a solution is obtained thanks to Leray-Schauder fixed point theorem, under the condition that the initial datum belongs to the parabolic region.
and using the notation v := u − a 2b , (1.1) can be rewritten as (2.1) It can be expected that global existence holds in the case when the parabolicity can be kept in the evolution (that is, when for all time v < 0, or equivalently u < a 2b ).At the same time, the logistic term u (c − du) and the expected nonnegativity of u imply that the estimate 0 ≤ u ≤ c d should also hold.Therefore, a natural sufficient condition for getting global existence for equation ( Proof.We first observe that we can take ε 0 > 0 small enough in such a way that max x∈Ω u 0 (x) ≤ a 2b − ε 0 and c d ≤ a 2b − ε 0 .Then we will prove the existence and uniqueness of a solution u of the problem, which satisfies the estimate 0 ≤ u(x, t) ≤ a 2b − ε 0 for all x ∈ Ω, t ≥ 0.
For any fixed T > 0, we will use the Leray-Schauder fixed point theorem to prove the existence.Let We define an operator as follows: for given w ∈ X and σ ∈ [0, 1], let u := T(w, σ) be the C 2+α,1+ α 2 ( Ω × [0, T]) solution (see [8,Chapter V,Theorem 7.4] for the existence and uniqueness of the solution) of the following problem: In order to build up the map T, we have to show that 0 ≤ u(x, t) ≤ a 2b − ε 0 for all (x, t) ∈ Ω × [0, T].For σ = 0, it is obvious that 0 ≤ u ≤ a 2b − ε 0 , since in that case u satisfies the heat equation.For σ ∈ (0, 1], we first prove that u ≥ 0. To this end, let ε > 0 be small and let u ε be the solution of The solution More precisely, we take here t 1 ≥ 0 as the last time before the solution takes some negative value.If x 1 ∈ Ω, we have ∇u ε (x 1 , t 1 ) = 0.If x 1 ∈ ∂Ω, in the case of the homogeneous Neumann boundary condition, we also have ∇u ε (x 1 , t 1 ) = 0. Then we get (still in the case of Neumann boundary condition) which is a contradiction.Therefore, u ε ≥ 0. If x 1 ∈ ∂Ω (and u ε (x, t 1 ) > 0 for all x ∈ Ω), in the case of the homogeneous Dirichlet boundary condition, we can prove (see the sequel of the proof) that there exists a sequence x n ∈ Ω satisfying x n → x 1 and such that lim This limit, together with the fact that u ε is smooth and that the tangential derivative of u ε vanishes because of the homogeneous Dirichlet boundary condition, shows that ∇u ε | (x 1 ,t 1 ) = 0. Thus we can follow the same argument as above.In order to show the limit (2.3), we consider t n = t 1 + 1 n a sequence that converges to t 1 and x n (n ≥ 2) one of the minimal points of u ε (x, t n ) such that x n → x 1 (note that if several points x 1 ∈ ∂Ω satisfy (2.2), one at least can be selected in such a way that the construction above makes sense).Then because of the continuity of u ε .We get therefore Then the mean value theorem implies that there exists a sequence where t * * n ∈ (t 1 , t * n ) and the last line comes again from the homogeneous Dirichlet boundary condition.As a consequence, since we are working with bounded classical solutions, the limit vanishes.
On the other hand, ρ ε = u ε − u satisfies the following linear problem: where all the coefficients are uniformly bounded in ε.Therefore by the maximum principle, we have Then we have ∂ t u(x 0 , t 0 ) ≥ 0, and moreover x 0 ∈ Ω if we consider the Dirichlet boundary condition, so that In the case of the Neumann boundary conditions, x 0 might appear on the boundary, but in this case, we still have ∇u(x 0 , t 0 ) = 0, therefore the above argument also works.This implies a 2b − ε 0 < c d , which is a contradiction with the assumption Thus the map T : X × [0, 1] → X is well defined.Due to the compact embedding from C 2+α,1+ α 2 to C α, α 2 , we know that T( ⋅ , σ) : X → X is a compact operator.
Next we show that the map T is continuous in w and σ.For all w ∈ X and σ ∈ [0, 1], let w j ∈ X be a sequence such that ‖w j − w‖ C α, α 2 → 0 as j → ∞, and let σ j ∈ [0, 1] be a sequence such that |σ j − σ| → 0. Let u j = T(w j , σ j ), the Schauder estimates show that ‖u j ‖ 2+α,1+ α 2 ≤ C uniformly in j.Notice that ρ j = u j − u satisfies the following linear problem: where Using Schauder's theory for linear parabolic equations, we get the estimate Hence, T is continuous in w and σ.Furthermore, it is obvious that T(w, 0) = 0. Additionally, for any fixed point of T(u, σ) = u, the uniform estimates for quasilinear parabolic equation show ([8, Chapter V, Theorem 7.2]) that there exists a constant M depending only on a 2b , c, Therefore, by Leray-Schauder's fixed point theorem, there exists a fixed point to the map T( ⋅ , 1), i.e. u is a solution of the following problem: which is equivalent to equation (1.1).The uniqueness of classical solutions follows directly from comparison principles.

Steady States
In this section, two results concerning stationary states are given.One of them shows that nontrivial nonnegative solutions do not exist for equation (1.1) with Dirichlet boundary condition.The other one has to do with the linear instability of constant steady states to equation (1.1) with homogeneous Neumann boundary condition.

Non-existence of Nontrivial Steady States for Dirichlet Boundary Conditions
The steady states corresponding to (1.1) satisfy the equation We write down a non-existence result (based on Pohozaev method, see [6]) which works for general elliptic problems, and explain how to use it specifically in the case of equation (3.1).
Theorem 2. Let n > 2, let Ω be a star shaped domain of ℝ n with respect to the origin and suppose that g, h are C 1 functions defined on ℝ + such that nF(s Then the problem This last condition is satisfied in particular when c and d are negative and 0 < ad(n − 6) − bc(n + 6) < 12 √ abcd or ad(n − 6) − bc(n + 6) < 0. Note also that since h(0) = 0, the homogeneous Dirichlet boundary condition u = 0 on ∂Ω implies that h(u) = 0 on ∂Ω, so that the first part of Theorem 2 can be applied.
Proof.By testing (3.3) with x ⋅ ∇h(u), we get Integrating over Ω and applying the divergence lemma to the left-hand side, we get so that using (3.4), we obtain We first compute where Integrating by parts, we get For the second term, using problem (3.3), we get For the last term in (3.5), we compute so that Using the Dirichlet boundary condition, we see that |∇h(u)| ∂Ω = |ν ⋅ ∇h(u)| ∂Ω , so that on ∂Ω, we have Thus, the above relation becomes Since Ω is star shaped, there exists α ≥ 0 such that and relation (3.9) yields Therefore, a sufficient condition for the non-existence of (nontrivial, nonnegative, classical) solutions is As stated in the theorem, this happens when for example, c, d < 0 and ad(n − 6) − bc(n + 6) < 12 √ abcd.
Remark 3. We obtain (3.10) from (3.9) by neglecting the first boundary term (since Ω is starshaped).Now we keep this first boundary integral (the second boundary integral in (3.9) is 0 because of the boundary conditions) and compute where we have used the geometry of the domain, Cauchy-Schwarz inequality, |∂Ω| = 1, the divergence lemma and the problem itself.Then relation (3.9) yields Therefore, we can get a more precise description for the non-existence of solutions, since now we need to check the less stringent inequality Remark 4. We present here a computation related to the linear stability of steady states for the Neumann boundary condition.We denote by {λ k , e k } ∞ k=1 the solution of the eigenvalue problem for the Laplacian with homogeneous Neumann boundary condition, with λ k ≥ 0 for k = 1, 2, . . .and 0 = λ 1 < λ 2 ≤ λ 3 ≤ ⋅ ⋅ ⋅ .
We assume that c, d > 0. Then the equilibrium c d for equation (1.1) with homogeneous Neumann boundary condition is asymptotically linearly stable if and only if c d ≤ a 2b .Indeed, we set u = c d + εw, so that the problem is transformed into Thus By projecting the equation onto the k-th eigenspace (and by using the notation w k (t) := ⟨w(t, ⋅ ), e k ⟩), we obtain The condition for linear asymptotic stability of the steady state c d is therefore, for all k ∈ ℕ, whence the result.

Blow-Up Results
In this section, we present blow-up results (for different boundary conditions).Namely, we show that the solution to equation (1.1) blows up, under appropriate conditions, for both Dirichlet and Neumann boundary conditions, by using two different classical methods, i.e.Kaplan's and concavity method.

A Sufficient Blow-Up Condition via Kaplan's Method for Dirichlet Boundary Conditions
The problem under consideration in this subsection is Let (μ, ϕ) be the solution to the eigenvalue problem where μ is the first eigenvalue and Ω is a connected bounded domain.Then μ > 0 and ϕ is strictly positive and bounded in Ω.For convenience, we also impose the normalization condition ∫ Ω ϕ(x) dx = 1.The main result in this subsection is: Assume that Ω is a bounded smooth domain of ℝ n and let u 0 ∈ L 1 (Ω; ϕ dx) satisfy where (μ, ϕ) is the solution to (4.2).Then any nonnegative solution to problem (4.1) blows up in finite time in L 1 (Ω; ϕdx).Remark 6.When c = d = 0, the above blow-up condition on the initial data can be roughly translated as u 0 > Cst a b , which is coherent with our global existence result, and with the assumption of Theorem 7. Note also that the homogeneous Dirichlet boundary condition u = 0 on ∂Ω could be replaced in the theorem above by the less stringent condition (a − bu) u = 0 on ∂Ω.
Proof.We begin, motivated by [5], with testing (4.1) with the eigenfunction ϕ, and set where we have used problem (4.2).Next we recall that μb > d.
After applying Jensen's inequality, we remind that ∫ Ω ϕ(x) dx = 1, we get from which the blowup of the solution can be obtained.Namely, by using the change of variables or (as long as Ξ(t) > 0, remembering that Ξ(0) = A(0) > 0) When c − μa > 0, we see that Ξ −1 (t) > 0 cannot remain true for 0) , so that blowup occurs before time t * .When c − μa < 0, a similar computation shows that a blowup also occurs, under the extra assumption A 0 > μa−c μb−d .

A Sufficient Blow-Up Condition by the Concavity Method for Neumann Boundary Condition
As has been stated in the beginning of Section 2, after the transformation v = u − a 2b , the equation can be rewritten into ).
In this subsection, we consider the following more general equation with homogeneous boundary condition, and after giving a result about the blowup for the above general problem, we explain how (and under which conditions) it applies to problem (2.1).We refer the interested reader to [3,4,7,10].The main result of this subsection is the following: Theorem 7. Suppose that m > 1, and that h is a continuous real function such that for all s ≥ 0, one has s m h(s) ≥ 2H(s), where H(s) := ∫ s 0 mt m−1 h(t) dt.We assume that v := v(x, t) is a smooth nonnegative solution to problem Then there exists t * > 0 (depending only on m, b, h and v(0, ⋅ )) such that T < t * .In other words, a blowup occurs before t * .More precisely, Remark 8.Note that the function h is not assumed to be nonnegative.Actually, Theorem 7 still holds when h is negative, or when it changes sign.Coming back to the original unknown u (instead of v), Theorem 7 states that (under assumption (4.6)) the (smooth) solutions to (2.1) which are such that u ≥ a 2b is pointwise true, cannot exist globally.Note that a significant limitation of this result is related to the assumption that u ≥ a 2b pointwise.Indeed, this estimate is propagated at the formal level by the equation only in very special cases in which a, b, c, d are linked by some equality, like when c = ad 2b ≥ 0.
Proof.The proof is given by contradiction argument.Assume that the solution is global, and define The idea of the concavity method is to find an α > 0 and a t 0 ≥ 0 such that Ψ −α is a concave function on [t 0 , +∞[.Then, from the concavity property of Ψ −α written at time t 0 in a differential way, we get Using this inequality together with the fact that Ψ  (t) ≥ 0 for all t ≥ 0, we obtain an upper bound t * for the blow-up time (that is, the first time t * such that Ψ −α (t * ) = 0) To prove this concavity property, we compute from which it can be deduced that a sufficient condition for Ψ −α to be concave (on In fact, we start by computing the derivative of the functional and its second derivative Next, we test (4.3) with (m + 1)v m , and get so that substituting (4.8) into (4.7),we obtain We now test (4.3) with (v m ) t , and get where we recall that From (4.10), we can also deduce that If the initial energy is strictly positive, namely E(0) > 0, (4.12) which is assumption (4.4) in Theorem 7, then we see that for all t ≥ 0, E(t) > 0, thanks to (4.11).
As observed at the beginning of the proof, the above inequality implies that we cannot extend the solution for all times, since (4.5) holds at a some point t * > 0.

Annex: Self-Similar Solutions Blowing Up in the Whole Space
In this annex, we provide a few explicit computations concerning problem (1.1) in the case when c = d = 0: and we still consider only the nonnegative solutions.
We recall that it is equivalent to studying the following problem: with u ≥ 0 if and only if v ≥ − a 2b .This equation is a (reverse in time) porous medium equation, for which explicit solutions of Barenblatt type [1] can be computed (on a given time interval [0, T * ) for the first type given below): ) It is possible to take linear combinations of those solutions and still get solutions, though the equation is nonlinear, as long as the support of those solutions remain separate (more precisely, when each two solutions have support with empty intersections during the time of existence of the solutions).

Remark 9 .
When we consider problem (2.1) with homogeneous Neumann boundary condition, we are led to use m = 2 and h(s) := (c − d a 2b − ds)(s + a 2b ) in Theorem 7. The condition s m h(s) ≥ 2H(s) (for all s ≥ 0) becomes (remember that a, b > 0, but the sign of c, d is not fixed) and v(x, t) = −1 b(T * + t) ((T * + t)