Well-posedness for a model of individual clustering

We study the well-posedness of a model of individual clustering. Given p>N \geq 1 and an initial condition in W 1,p (\Omega), the local existence and uniqueness of a strong solution is proved. We next consider two specific reproduction rates and show global existence if N = 1, as well as, the convergence to steady states for one of these rates.


Introduction
In [5], a model for the dispersal of individuals with an additional aggregation mechanism is proposed. More precisely, classical models for the spatial dispersion of biological populations read ∂ t u = ∆(Φ(u)) + f (u, t, x).
where u(t, x) denotes the population density at location x and time t, and f (u, t, x) represents the population supply, due to births and deaths. The dispersal of individuals is either due to random motion with Φ(u) = u or rests on the assumption that individuals disperse to avoid crowding and Φ satisfies Φ(0) = 0, and Φ ′ (u) > 0, for u > 0. (2) No aggregation mechanism is present in this model though, as discussed in [5], the onset of clustering of individuals in a low density region might balance the death and birth rates and guarantee the survival of the colony. To account for such a phenomenon, a modification of the population balance (1) is proposed in [5] and reads where V is the average velocity of individuals, and E is the net rate of reproduction per individual at location x and time t. To complete the model, we must specify how V is related to u and E. Following [5], we assume that each individual disperses randomly with probability δ ∈ (0, 1) and disperses deterministically with an average velocity ω so as to increase his expected rate of reproduction with probability 1 − δ. The former is accounted for by a usual Fickian diffusion ∇u u while the latter should be in the direction of increasing E(u, t, x), say, of the form λ ∇E(u, t, x) with λ > 0. A slightly different choice is made in [5] and results in the following system ∂ t u = δ ∆u − (1 − δ) ∇ · (u ω) + u E(u, t, x) −ε ∆ω + ω = λ∇E(u, t, x).
As usual, v × ω is the number v 1 ω 1 + v 2 ω 2 if N = 2 and the vector field We note that the boundary condition (7) is useless if N = 1.
Summarizing, given a sufficiently smooth function E, parameters δ > 0, ε ≥ 0 and r ≥ 0, our aim in this paper is to look for (u, ω) solving the problem x ∈ Ω, t > 0 ∂ n u = 0 , ω · n = 0, In the first part of this paper, we show that, for p > N , the system (8) has a maximal solution u in the sense of Definition 2.1 where u ∈ C [0, T max ), W 1,p (Ω) ∩C (0, T max ), W 2,p (Ω) , see Theorem 2.2.
In the second part, we turn to the global existence issue and focus on space dimension 1, and two specific forms of E suggested in [5]: the "bistable case" where E(u) = (1−u)(u−a) for some a ∈ (0, 1), see Theorem 2.3, and the "monostable case" E(u) = 1 − u. In both cases, we prove the global existence of solution. In addition, in the monostable case, i.e E(u) = 1 − u, thanks to the Liapunov functional we can study the asymptotic behaviour of solutions for t large, and show that the solution u converges, when t goes to ∞, to a steady state in L 2 (−1, 1), see Theorem 2.4.
In the third part, we investigate the limiting behaviour as ε → 0. Heuristically, when ε goes to zero, the velocity ω becomes sensitive to extremely local fluctuations in E(u), and the system (8) reduces to the single equation Clearly (9) is parabolic only if δ − u E ′ (u) ≥ 0 for all u > 0. This is in particular the case when E(u) = 1 − u, see Theorem 2.6. But this limit is not well-posed in general. As a result the population distribution may become discontinuous when neighbouring individuals decide to disperse in opposite direction, that is in particular the case when E(u) = (1 − u)(u − a).

Main results
Throughout this paper and unless otherwise stated, we assume that We first define the notion of solution to (8) to be used in this paper.
Definition 2.1. Let T > 0, p > N , and an initial condition where, for all t Our first result gives the existence and uniqueness of a maximal solution of (8) in the sense of Definition 2.1.
Theorem 2.2. Let p > N and a nonnegative function u 0 ∈ W 1,p (Ω). Then there is a unique maximal solution u ∈ C [0, T max ), W 1,p (Ω) ∩ C (0, T max ), W 2,p (Ω) to (8) in the sense of Definition 2.1, for some T max ∈ (0, ∞]. In addition, u is nonnegative. Moreover, if for each T > 0, there is C(T ) such that The proof of the previous theorem relies on a contraction mapping argument.
We now turn to the global existence issue and focus on the one dimensional case, where E(u) has the structure suggested in [5]. In the following theorem we give the global existence of solution to (8) in the bistable case, that is when E(u) = (1 − u)(u − a), for some a ∈ (0, 1). Theorem 2.3. Assume that u 0 is a nonnegative function in W 1,2 (−1, 1), and E(u) = (1 − u)(u − a) for some a ∈ (0, 1). Then (8) has a global nonnegative solution u in the sense of Definition 2.1.
The proof relies on a suitable cancellation of the coupling terms in the two equations which gives an estimate for u in L ∞ (L 2 ) and for ω in L 2 W 1,2 .
Next, we can prove the global existence of a solution to (8) in the monostable case, that is, when E(u) = 1 − u, and we show that the solution converges as t → ∞ to a steady state. More precisely, we have the following theorem Theorem 2.4. Assume that u 0 is a nonnegative function in W 1,2 (−1, 1), and E(u) = 1 − u. There exists a global nonnegative solution u of (8) in the sense of Definition 2.1 which belongs to L ∞ [0, ∞); W 1,2 (−1, 1) . In addition, if r = 0, and if r > 0 the solution u(t) converges either to 0 or to 1 in L 2 (−1, 1) as t → ∞.
In contrast to the bistable case, it does not seem to be possible to begin the global existence proof with a L ∞ (L 2 ) estimate on u. Nevertheless, there is still a cancellation between the two equations which actually gives us an L ∞ (L log L) bound on u and a L 2 bound on ∂ x √ u.

(13)
So that u is a solution to our model.
When E(u) = 1 − u, the limit ε → 0 is formally justified and (8) takes the qualitative form of (1) with Φ(u) = δu + 1 2 u 2 . In this example though, since E ′ < 0, the individuals dispersing so as to maximise E would seek isolation, and there is clearly no mechanism capable of producing aggregation of individuals. This observation is actually consistent with Remark 2.5.
Theorem 2.6. Assume that u 0 is a nonnegative function in W 1,2 (−1, 1), and that E(u) = 1 − u. For ε > 0 let u ε be the global solution to (8) given by Theorem 2.4. Then, for all T > 0, where u is the unique solution to Since δ + u > 0 for u ≥ 0, the previous equation (15) is uniformly parabolic and has a unique solution u, see [6] for instance. The proof of Theorem 2.6 is performed by a compactness method.

Preliminaries
We first recall some properties of the following system, where f ∈ (L p (Ω)) N and Ω is a bounded open subset of R N , N = 2, 3. Let us first consider weak solutions of (16). For that purpose, we define W 1 = {v ∈ (H 2 (Ω)) N ; v · n = 0, and ∂ n v × n = 0 on ∂Ω} and take W as the closure of W 1 in (H 1 (Ω)) N . If f ∈ (L 2 (Ω)) N , the weak formulation for (16) is We recall some results about the existence, regularity and uniqueness of solution for (17), see [3,4].
In other words, the strong solution has the same regularity as elliptic equations with classical boundary conditions. We finally recall some functional inequalities: in several places we shall need the following version of Poincaré's inequality with arbitrary p ≥ 1 and q ∈ [1, p]. Also, we will frequently use the Gagliardo-Nirenberg inequality which holds for all p ≥ 1 satisfying p (N − 2) < 2 N and q ∈ [1, p).

Local well-posedness
Throughout this section, we assume that Proof of Theorem 2.2. We fix p > N , R > 0, and define for T ∈ (0, 1) the set which is a complete metric space for the distance We then define Λ(u) by where e t (δ ∆) denotes the semigroup generated in L p (Ω) by δ ∆ with homogeneous Neumann boundary conditions. We now aim at showing that Λ maps X R (T ) into itself, and is a strict contraction for T small enough. In the following, (C i ) i≥1 and C denote positive constants depending only on Ω, δ, r, ε, E, p and R.
We first recall that there is C 1 > 0 such that and for all v ∈ W 1,p (Ω). Indeed, (23) follows from the continuous embedding of W 1,p (Ω) in L ∞ (Ω) due to p > N while (24) is a consequence of the regularity properties of the heat semigroup. Consider u ∈ X R (T ), and t ∈ [0, T ]. It follows from (24) that Thanks to (23), we have Therefore, using elliptic regularity (see Theorem 3.2) and (25), we obtain Using again (25) along with (26) we find Since u ∈ X R (T ), using (25) we can see that which gives that by (26) and (23), we use once more (25) and obtain that Combining (27) and (29) we get Choosing R = 2 C 1 ||u 0 || W 1,p and T ∈ (0, 1) such that we obtain that sup It follows that Λ maps X R (T ) into itself.
• Step 2. We next show that Λ is a strict contraction for T small enough.
Let u and v be two functions in X R (T ). Using (24) we have Note that, by (25) and (28), we have and it follows from Theorem 3.2 and (25) that Combining (32) and (31) we obtain Since u and v are bounded by (25), we have Then, we get Substituting (33) and the above inequality in (30) we conclude that Using again (24), we have Since the mapping is bilinear and continuous due to p > N , we deduce from (26) and (32) that Thus, On the other hand, due to (25) and the embedding of Then Therefore, Choosing T ∈ (0, 1) such that T 1 2 C 16 < 1 we obtain that Λ is indeed a strict contraction in X R (T ) and thus has a unique fixed point u.
• Step 3. Thanks to the analysis performed in Steps 1 and 2, the existence and uniqueness of a maximal solution follows by classical argument, see [1] for instance.

The bistable case:
In this case, the system (8) now reads for a some a ∈ (0, 1).
To prove Theorem 2.3 we show that, for all T > 0 and t ∈ [0, T ] ∩ [0, T max ), u(t) is bounded in W 1,2 (−1, 1). We begin the proof by the following lemmas which give some estimates on u and ϕ.
Lemma 5.1. Let the same assumptions as that of Theorem 2.3 hold, and u be the nonnegative maximal solution of (35). Then for all T > 0 there exists C 1 (T ), such that u and ϕ satisfy the following estimates Proof. Multiplying the first equation in (35) by u(t) and integrating it over (−1, 1), we obtain d dt Multiplying now the second equation in (35) by ϕ and integrating it over (−1, 1) we obtain At this point we notice that the cubic terms on the right hand side of (39) and (40) cancel one with the other, and summing (40) and (39) we obtain We integrate by parts and use Cauchy-Schwarz inequality to obtain (a + 1) On the other hand, u 2 E(u) ≤ 0 if u / ∈ (a, 1) so that The previous inequalities give that Therefore, for all T > 0 there exists C 1 (T ) such that (36), (37) and (38) hold.
Lemma 5.2. Let the same assumptions as that of Theorem 2.3 hold, and u be the nonnegative maximal strong solution of (35). For all T > 0, there is C ∞ (T ) such that Proof. The estimates (36) and (37) and the Gagliardo-Nirenberg inequality (20) yield that there exists C 2 (T ) such that Using Hölder's inequality , we obtain Introducing the bound being a sequence of (43), we integrate (44) and find ||u(t)|| q q ≤ ||u 0 || q q e (q−1) φ(t) + 2 q r t 0 e (q−1) (−φ(s)+φ(t)) ds ≤ (||u 0 || q q + 2 q r) T e q C(T ) , ||u(t)|| q ≤ (||u 0 || q q + 2 q r) T Consequently, by letting q tend to ∞, we see that there exists C ∞ (T ) such that Lemma 5.3. Let the same assumptions as that of Theorem 2.3 hold, and u be the nonnegative maximal strong solution of (35). For all T > 0, there is C 4 (T ) such that Proof. We multiply the first equation in (35) by (−∂ 2 xx u) and integrate it over (−1, 1) to obtain Using Cauchy-Schwarz inequality and Lemma 5.2 we obtain, Using (38) and Sobolev embedding theorem we obtain the following estimate Since (38) and (47) hold, then it follows from (46) after integration that It remains to prove Theorem 2.3.
Since E ∈ C 2 (R), Theorem 2.2 ensures that there is a maximal solution u of (48) in In contrast to the previous case, it does not seem to be possible to begin the global existence proof with an L ∞ (L 2 ) estimate on u. Nevertheless, there is still a cancellation between the two equations which actually gives us an L ∞ (L log L) bound on u and a L 2 bound on ∂ x √ u. Integrating (48) over (0, t) × (−1, 1) and using the nonnegativity of u, we first observe that, To prove Theorem 2.4 we need to prove the following lemmas: Lemma 5.4. Let the same assumptions as that of Theorem 2.4 hold, and let u be the maximal solution of (48). Then for all T > 0, there exists a constant C 1 (T ) such that Proof. The proof goes as follows. On the one hand, we multiply the first equation in (48) by (log u + 1) and integrate it over (−1, 1).
On the other hand, we multiply the second equation in (48) by ϕ and integrate it over (−1, 1) to obtain Adding (53) and (52) Finally, (50) and (51) are obtained by a time integration of (54).
Lemma 5.5. Let the same assumptions as that of Theorem 2.4 hold, and let u be the maximal solution of (48). Then for all T > 0, there exists a constant C 2 (T ) such that Proof. A simple computation shows that, since Using Cauchy-Schwarz inequality, Gagliardo-Nirenberg inequality (20), Young inequality and (49) we obtain that for all T > 0, We substitute the previous inequality in (57) to obtain Integrating (58) in time, and using (51) yield that there exists C 3 (T ) such that (55) which together with (56), implies that Thanks to this estimate, we now argue as in the proof of Lemma 5.2 and Lemma 5.3 to get that Thus, the maximal solution u of (48) cannot explode in finite time.
To complete the proof of Theorem 2.4, it remains to prove the asymptotic behaviour of u when t → ∞. We note that we have the following lemma which controls the L 1 (−1, 1) norm of u. For f ∈ L 1 (−1, 1), we set Lemma 5.6. Let the same assumptions as that of Theorem 2.4 hold, and let u be the nonnegative global solution of (48). For r > 0, there exists a constant C 0 > 0 such that and if r = 0 Proof. We note that if r = 0, d dt < u >= 0, so that whence < u(t) >≤ max {1, < u 0 >}.
Next we turn to the existence of a Liapunov functional for (48) which is the cornerstone of our analysis. Lemma 5.7. Let the same assumptions as of that Theorem 2.4 hold, and let u be the nonnegative global solution of (48). There exists a constant C 1 such that Proof. Let us define the following functional L and show that it is a Liapunov functional. Indeed Combining (63) and (53) we obtain that Since u 0 and u are nonnegative , we have Therefore, (65) yields there exists C 1 > 0 such that From (66), we see that (61) holds true. In addition, inequality (61) together with Sobolev's embedding theorem give (62).
Lemma 5.9. Let the same assumptions as that of Theorem 2.4 hold, and let u be the nonnegative global solution of (48). There is C 2 such that Proof. We multiply the first equation in (48) by ∂ t u and integrate it over (−1, 1) to obtain Using Young and Cauchy-Schwarz inequalities we obtain which gives d dt where F (u) = r − u 2 2 + u 3 3 ≥ − r 6 . Next we integrate the above inequality in time, and use (62), (61) and Lemma 5.8 to obtain We have thus proved (75).
To end the proof of Theorem 2.4, our aim now is to look at the large time behaviour of the solution.
We first prove that Indeed for each s ∈ (−1, 1) The right hand side goes to zero as n → ∞ by Lemma 5.9. Letting n → ∞ in the above inequality gives (76).