DYNAMICS OF SPIKE IN A KELLER-SEGEL’S MINIMAL CHEMOTAXIS MODEL

. The dynamics are studied for the Keller-Segel’s minimal chemotaxis model on a bounded interval with homogeneous Neumann boundary conditions, where τ (cid:62) 0 and k (cid:29) 1 are parameters and the total mass of u is scaled to be one. In general, the dynamics can be divided into three stages: the ﬁrst stage is very short in which u quickly becomes a delta like function with mass concentrated near the point of global maximum of v ; in the second stage, the point of the global maximum of v drifts towards the boundary of the domain and reaches it at the end of the second stage; in the third stage, the proﬁle of the solution evolves to a steady state proﬁle. This paper considers a special case in which the relaxation parameter τ is set to be zero, so the ﬁrst stage takes no time. A free boundary problem describing the second stage is presented. Rigorous asymptotic behavior is proven for the third stage evolution.


1.
Introduction. Chemotaxis is movement of an organism in response to a chemical stimulus. Somatic cells, bacteria, and other single-cell or multicellular organisms direct their movements according to certain chemicals in their environment. This is important for bacteria to find food by swimming toward the highest concentration of food molecules, or to flee from poisons. Positive chemotaxis occurs if the movement is toward a higher concentration of the chemical in question; negative chemotaxis occurs if the movement is in the opposite direction.
Various PDE models for chemotaxis have been extensively studied; see the survey papers by Horstmann [23,24], Hillen and Painter [21], and the references therein. One of the most important phenomenon about chemotaxis is cell aggregation, which typically is modeled by spiky steady states. The pioneering papers that prove the existence of such steady states are Lin, Ni and Takagi [28,32]; see also [37] for a brief survey on spiky steady states.
One of the simplest and prominent model for chemotaxis is the following Keller and Segel's [27] (see also Patlak [34]) minimal chemotaxis model, which in its dimensionless form, can be written as where Ω = (0, ), t relates the time, x the space, u the cell density, and v the chemo-attractant density; here τ 0 and k > 0 are parameters; the total mass of the cell is scaled to be 1. First observed by Childress and Perkus [12], the minimal chemotaxis model possesses rich and interesting properties. The well-posedness of (1) is established by Nagai [29], Osaki and Yagi [33], and Hillen and Potapov [22]; see also Section 6 of this paper for a simple proof of key a priori estimates needed for global existence.
Note that a steady state of (1) is a solution of the non-local ode Since any steady state can be extended evenly and periodically, we can assume without loss of generality that Ω is half-period and v x > 0 in Ω. In [22], Hillen and Potapov constructed asymptotically, as k → ∞, a boundary spike steady state solution, i.e., a family of steady state solutions {(u(k, x), v(k, x))} k>0 satisfying lim k→∞ Ω u(k, x)ζ(x) dx = ζ( ) ∀ ζ ∈ C(Ω).
The first term of the expansion of the boundary spike solution of (2) is also derived in [26] by Kang, Kolokolnikov, and Ward. In [37], Wang and Xu proved rigorously the existence of a boundary spike solution of (2). Using a phase plane analysis, Chen, Hao, Wang, Wu, and Zhang established in [8] the existence, uniqueness, and local exponential stability of the spike steady state solution, together with rigorous arbitrary high order internal and boundary layer asymptotic expansions. To obtain more detailed information on the stability of the spike solution, we reinvestigated in [38] the associated eigenvalue problem in a general setting with a systematic method.
In this paper, we focus on the dynamical aspect of the minimal chemotaxis model (1). According to numerical simulations (e.g. [22]), we divide the evolution into three stages 1 : 1. The first stage generates the spike. It is very short in which u quickly becomes a delta like function with mass concentrated near the points of the global maximum of v(k, ·, t). After the generation, where z(t) is the point of global maximum of v(k, ·, t). In particular, if the relaxation parameter τ is taken to be zero, then u = e kv / Ω e kv dx, so this first stage takes no time. 2. Assume for simplicity that v(k, ·, t) admits a unique point of global maximum.
Then in the second stage, the position, z(t), of the point of the global maximum of v(k, ·, t) drifts towards the boundary of the domain; see Figure 1. The second stage ends when z(t) reaches the boundary. 3. In the third stage, the profile of the solution evolves to the steady state profile, i.e., the solution of (2); see Figure 1.  It is well-known that when space dimension is bigger than one, blow-ups may occur; here a blow up refers to the formation of Dirac mass like singularities of the solution in finite time (cf. [4,11,12,17,18,19,20,25,29,30,31,35]). It is also well-known that in the one-dimensional case blow-up never happen, i.e., global in time classical solution always exists (cf. [33]). The Dirac mass used in this paper is obtained as the limit when the chemotaxis intensity parameter k → ∞. Thus, the Dirac mass used here is fundamentally different from the typical blow-ups phenomena that numerous research has focus on. Nevertheless the k = ∞ limit is related to initial value being a Dirac mass, our solution of the limit problem indeed describes the movement of the location of the Dirac mass.
For mathematical analysis, under the current notation with an assumption that is equivalent here to 1 and k 1, Kang, Kolokolnikov and Ward [26] formally derived that after a certain amount of time in the second stage, a quasi-steady state for v is reached: where G(x, y) is the Green's function associated with the operator −∂ xx + 1 with homogeneous Neumann boundary condition; more precisely, Remarkably, they derived the law of the motion of the spike, i.e., equation (2.45) in [26], which, under the current notation, can be expressed as When 1, for each ξ ∈ Ω, G(·, ξ) is a "quasi" steady state, so the spike moves slowly. For rigorous analysis of slow (meta-stable) dynamics, a successful example is for the Allen-Cahn equation by Fife and McLeod [14], Fife and Hsiao [13], Carr and Pego [6], Fusco [15], and Fusco and Hale [16]; see also Chen [7] for the evolution of all stages, i.e., generation, evolution, annihilation, and subsequent motion to equilibrium. Similar analysis can also be found for the (viscous) Cahn-Hilliard equation [1,5,2,3,36] and Gierer-Meinhartdt system [9,10].
We are interested in rigorous verification of the formal derivation of the dynamics. Since problem (1) is a rather complicated system, we shall consider only the simple case when the relaxation parameter τ = 0, so the first stage takes no time. Hence, (1) with τ = 0 and supplement of an initial condition can be written as, for v = v(k, x, t), We shall assume that v 0 is a non-constant function that does not depend on k. We study the asymptotic behavior of the solution as k → ∞.
Extending the formal analysis in [26] where 1, here we consider the case when is not necessarily large. When is not large, G(·, z(t)) is no longer a quasisteady state, so the approximation (3) does not hold. We can formally derive (with derivation omitted here) the following: 1. Assume that v 0 has exactly one local maximum, located at z 0 ∈ (0, ). Then where w, together with z and T , form the solution of the following free boundary problem: We expect to prove rigorously that when 1, the function w(x, t) := G(x, z(t)) with z(t) being the solution of (5) with initial condition z(0) = z 0 approximates the solution of (7) up to T at which z reaches the boundary. 2. Assume for simplicity that T < ∞ and that w(·, T − ) is increasing. Then where w is the solution of the initial boundary value problem A numerical simulation confirming the convergence of the solution of (1) to the solution of (7)+(8) is shown in Figure 2. 2 We shall prove the evolution limit system (7) in a subsequent paper. In this paper we prove only the limit system (8). For this, we assume for simplicity that T = 0 and w(·, T − ) = v 0 is a non-constant increasing function. We shall prove that In our proof, we discover a new L ∞ (0, ∞; W 1,p (Ω)) estimate. As an application, we shall use this estimate to provide a simple argument to establish a time independent a priori estimate for the solution of (1), improving those from [33,22]. The rest of the paper is organized as follows. In Section 2 we state our main result about the convergence of v to w as k → ∞, along with the idea of the proof. In Section 3, we prove the local in time convergence, using a potential analysis for an L ∞ ((0, ∞); W 1,p (Ω)) estimate. Observed that both (6) and (8) are gradient flows, we study in Section 4 the energy functionals and related estimates. We complete the proof of the convergence of the solution of (6) to the solution of (8) in Section 5. Finally in Section 6, we use our L ∞ ((0, ∞); W 1,p (Ω)) estimate to establish a time independent a prior estimate for the solution of (1), a key estimate needed for the well-posedness of (1).

2.
Main result and idea of the proof. To present our result, we introduce, for p 1 and k > 0, Note that Our main result is the following: Theorem 2.1. Let v 0 ∈ H 1 (Ω) be a given non-constant increasing function and w be the solution of For each k > 0, let v(k, ·, ·) be the solution of problem (6). Then Remark 1. The estimate (12) is optimal since w x − v x = 1 at x = .
Now we outline the proof of Theorem 2.1. We first establish the local in time convergence: By a potential analysis, this suffices to show that Ω e kv(k,y,t) dy approaches a delta function concentrated at . This is established by finding a positive lower bound of v x , using maximum and comparison principles.
For global in time convergence, we first use the fact that (6) and (11) are gradient flows with energies E k and E defined in (9) and (10), respectively. Indeed, we have the identities Next by solving the minimization problems of the associated energy functionals of increasing functions, we find that where Finally, we notice that w * is the unique equilibrium of (11), so Then from (15), (16), and (18) we find that It then follows from (17) that v(k, ·, t) approaches w * in H 1 (Ω) as k and t approach infinity. Using the boundedness of v x in L p+1 and the local convergence, we then obtain (12). The rest assertions of Theorem 2.1 can be proven by using energy identities and the convergence of energies.
3. Local in time convergence. In this section, we prove the local in time convergence, i.e., (15). We shall first use Green's formula to show that the convergence of v to w is equivalent to the convergence of u to the delta function concentrated on the boundary x = . Then we use comparison to establish a positive lower bound of v x and show the smallness of u in [0, − η] × [η 2 , η −2 ] for any fixed small positive η. Finally, we establish (15). In the meanwhile, we establish a W 1,p bound of v, global in time.
3.1. The Green's representation. Denote by Γ(x, y, t) the Green's function associated with the linear differential operator ∂ t −∂ xx +1 equipped with the Neumann boundary condition: for each y ∈Ω, where δ is the delta function. Then by Green's formula, the solutions of (6) and (11) are given by Since Ω u(k, y, s)dy = 1, we find that Hence, to show the convergence v → w, we need only show that for each s > 0, u(k, x, s) approaches a delta function with mass concentrated at x = , i.e., u(k, ·, s) is small in [0, − η] for any fixed small positive η and k 1.

3.2.
Non-degeneracy of the maximum of v at x = .
Proof. Note that v x solves the "linear" initial boundary value problem It then follows by the maximum principle that One finds thatv t −v xx +v = kuv x > 0 in Ω × (0, ∞) and v = 0 on the parabolic boundary of Ω × (0, ∞). Hence, by the maximum principle, v > 0 in Ω × (0, ∞). Finally, applying the maximum principle to the equation for v 0 x , we find that v 0 x > 0 in Ω × (0, ∞). The assertion of the lemma thus follows.
3.3. The function u. Here we show that for each fixed t > 0, u(k, ·, t) approaches a delta function as k → ∞. Since u 0, Ω u(k, y, t)dy = 1, and u x = kuv x 0, we need only estimate the smallness of u(k, − η, t) for any fixed small positive η.
In the general case Ω = (0, ), for x ∈ [0, ] and y ∈ [0, ], we can express Γ as where H is the solution of the heat equation H t = H xx in Ω × (0, ∞) subject to the initial and boundary conditions It is easy to see that H is a smooth function with bounded derivatives of arbitrary order. Hence, for each p ∈ [1, ∞), we have Consequently, After a similar estimate for the L p (Ω) norm of ζ, we obtain the following: Later on we shall use, with q = 1, the estimate where D 2 represents all second order partial derivatives of Γ(x, y, s) with respect to x and y. The proof is analogous to that for (23).
3.5. The local in time W 1,p (Ω) convergence. Integrating the differential equation (v − w) t − (v − w) xx + (v − w) = u over Ω and using the boundary condition w x ( , t) = 1 = Ω udx, one can derive that Thus, to show the local in time convergence of v to w in the W 1,p (Ω) norm, by Lemma 3.2, we need only show the L 1 convergence of v x to w x . For this, we fix t and define ζ(x) = v(k, x, t) − w(x, t) as in the previous subsection. For any fixed small positive η, we write ζ x = ζ 1 + ζ 2 + ζ 3 where  (23) and (24), This implies that there exists a positive constant C( ) such that when t ∈ [0, η −2 ], First sending k → ∞ then sending η 0 we then obtain from (21) that From this L 1 convergence, the L p+1 estimate in Lemma 3.2 and the Hölder inequality we then obtain the following: To complete the proof, we need only estimate the difference of It then follows from Lemma 3.3 that for each T > 0, Sending η → 0, we then obtain the assertion of Lemma 3.4.

Energy estimates.
In this section we first establish energy identities for solutions of (6) and (11). Then we find minimizers of the associated energy functionals. Finally, we estimate the H 1 (Ω) distance of a generic function to the minimizer in terms of the distance of their energies.

Energy identities.
Assume that v is the solution of (6). Define E k [v] as (9). Then Similarly, for solution w of (11), one can show by maximum principle that w x > 0 in Ω × (0, ∞). Hence, maxΩ w(·, t) = w( , t). Consequently, using w x = 1 at x = we obtain Thus, we have the following
5. Global in time estimates and proof of Theorem 2.1. In this section, we first establish the longtime behavior of the solution w of (11). Then we use local in time convergence and the energy dissipation to show that E[v(·, k, T )] approaches its minimum as T and k approaches infinity. Then we use Lemma 4.3 to obtain the global in time convergence. Finally, we use energy identity and energy convergence to show the convergence of v t to w t in L 2 .

Asymptotic behavior of
It then follows from the Gronwall's inequality that Hence, we have the following: From Lemmas 3.3 and 3.4, there exists K ε > 0 such that when k K ε , It then follows that As the energy is decreasing in t, we obtain from Lemmas 4.1 and 4.3 that for every t T ε , v(k, ·, t) − w * 2 It then follows from (28) that This implies that sup t 0 v(k, ·, t) − w(·, t) H 1 (Ω) → 0 as k → ∞. Finally using Hölder inequality and the W 1,p estimate in Lemma 3.2, we then obtain the first assertion (12) of Theorem 2.1. The above proof also shows that the energy estimate (13) holds. To completes the proof of Theorem 2.1, it remains to show the convergence of v t to w t .
Fix any 0 < η min{ /3, 1/3}. By local parabolic estimate, we have Sending k → ∞ and using (21) and (12) we then obtain Next fix T > 0. Using Lemma 4.1 and Hence, Similarly, Using energy identity and convergence of energy, we also have Finally, using (29), (30), and In conclusion, First sending η → 0 and then T → ∞ we then obtain This completes the proof of Theorem 2.1.
6. An a priori estimate. In this section we establish an a priori estimate for the solution of (1) with τ > 0. We begin with the following lemma. We denote by C α,α/2 (Ω × [0, ∞)) the Hölder space.
In summary, we have the following: Theorem 6.2. Assume that u 0 , v 0 ∈ L ∞ (Ω), u 0 0 and Ω u 0 (x)dx = 1. Let (u, v) be the solution of (1) subject to (u, v)| t=0 = (u 0 , v 0 ). Then, for any p > 1, Remark 2. By the maximum principle, we have, for each t > 0, Remark 3. First using (32) and then using standard local Hölder estimates for scalar parabolic equations and a bootstrap argument, one can show that under the conditions of Theorem 6.2, for any positive integer m and real number ε > 0, We omit the details of the proof.
In [22], using a semi-group theory, Hillen and Potapov established the existence of a solution in where s ∈ (0, 1) and ps > 1. Their estimates for the bounds of u(t) and v(t) depend on t.