A free boundary problem for a class of parabolic-elliptic type chemotaxis model

In this paper, we study a free boundary problem for a class of parabolic-elliptic type chemotaxis model in high dimensional symmetry domain \Omega. By using the contraction mapping principle and operator semigroup approach, we establish the existence of the solution for such kind of chemotaxis system in the domain \Omega with free boundary condition. Besides, we get the explicit formula for the free boundary and show the chemotactic collapse for the solution of the system.


Introduction
Understanding of the partially oriented movement of cells in response to chemical signals, chemotaxis, is of great significance in various contexts. This importance partly stems from the fact that when cells combined with the ability to produce the respective signal substance themselves, chemotaxis mechanisms are among the most primitive forms of intercellular communication. Typical examples include aggregation processes such as slime mold formation in Dictyostelium Discoideum discovered by K.B. Raper [20]. A lot of mathematicians have made efforts to develop various models and investigated them from a viewpoint of mathematics. For a broad overview over various types of chemotaxis processes, we refer the reader to the surveys and papers [4,5,6,8,9,10,16,17,19,23,24,25] and the references therein.
As we all know, in a standard setting for many partial differential equations, we usually assume that the process being described occurs in a fixed domain of the space. But in the real world, the following phenomenon may happen. At the initial state, a kind of amoeba occupy some areas. When foods become rare, they begin to secrete chemical substances on their own. Since the biological time scale is much slower than the chemical one, the chemical substances are full filled with whole domain and create a chemical gradient attracting the cells. In turn, the areas of amoeba may change ✩ This work is supported by National Natural Science Foundation of China (Grant No. 11131005) and the Fundamental Research Funds for the Central Universities (Grant No. 2014201020202).
Email addresses: chenhua@whu.edu.cn (Hua Chen), lvwenbin@whu.edu.cn (Wenbin Lv), wush8@sina.com (Shaohua Wu) according to the chemical gradient from time to time. In other words, a part of whose boundary is unknown in advance and that portion of the boundary is called a free boundary. In addition to the standard boundary conditions that are needed in order to solve the PDEs, an additional condition must be imposed at the free boundary. One then seeks to determine both the free boundary and the solution of the differential equations. The theory of free boundaries has seen great progress in the last thirty years; for the state of the field we refer to [12].
In this paper, we consider the following high dimensional free boundary problem of a chemotaxis model. Such kind of models can be found in [1,2,3,7,22], and we can give more explanations in the appendix below.
in Ω t × (0, T ), • Ω ⊂ R n is a bounded open set with smooth boundary ∂Ω and n is unit outer normal vector of ∂Ω. Besides, Ω is assumed as a symmetry domain, i.e. if x ∈ Ω then −x ∈ Ω as well; • c is a constant which will be determined in the following section; • k(x, t) = k(|x|, t) is radial symmetric and satisfying the Lipschitz condition on |x|, namely there exists a constant L > 0, such that Also, k(x, t) is bounded on t ∈ [0, +∞). In other words, there exists a constant C > 0 which depends on x, such that is an unknown function of (x, t) ∈ Ω t × (0, T ) and it stands for the density of cellular slime molds. In other words, the density u(x, t) occupies the domain Ω t , an open subset of Ω, in time t and u(x, t) = 0 in the outside of Ω t ; • v = v(x, t) is an unknown function of (x, t) ∈ Ω × (0, T ) and it stands for the concentration of chemical substances secreted by the slime molds; • Γ t : Φ(x, t) = 0 is an unknown free boundary.
For general smooth domain Ω, the system (1.1) is based on the well-known chemotaxis model with fixed boundary in Ω.
( 1.4) which was introduced by W. Jäger and S. Luckhaus [18] who first studied the conjectrues of Childress and Percus in 1992. Later, the problem (1.4) is intensively studied by many authors (see for instance [11,13,14,15,18]). Concerning the problem, classical results state that there exists a critical number C(Ω) such that c C(Ω) implies that there exists a unique, smooth positive solution to (1.4) globally in time [18]; -Ω is a disk, there exists a positive number C such that c C implies radially symmetric positive initial values can be constructed such that explosion of the solution happens in the center of the disc in finite time [18]; -Ω = R 2 , then no radial, self-similar solutions of (1.4) exist such that |x| r u(T, s)ds < ∞ as r → 0 [13,14]; • if n 3 -Ω = R 3 for any T > 0 and any constant C > 0, the radial solution (u, v) of (1.4) that is smooth for all times 0 < t < T , blows up at r = 0 and t = T , and is such that |x| r u(T, s)ds → C [13,14,15]; -Ω is a ball and some additional conditions, there exists a unique maximal classical solution with finite maximal existence time T [11].
In one dimensional case, if k is a positive constant H. Chen and S.H. Wu [3,7,22] studied the similar free boundary value problem (1.1) and established the existence and uniqueness of the solution for the system (1.1). However, to the best of our knowledge, high dimensional case for the free boundary value problem (1.1) will be more important. In view of the biological relevance of the particular case n = 3, we find it worthwhile to clarify these questions. In the present paper, we consider the system (1.1) in a high dimensional symmetry domain Ω. In addition, the condition that k is a positive constant in [3,7,22] seems too strict. So it is also worthwhile to consider the system with non-constant coefficient k.
This paper is arranged as follows. In section 2, we rewrite the model and present the main result of the paper. In section 3, we use the operator semigroup approach to establish some estimates which are essential in the proof of the main result. In section 4, we shall give the proof of the main result. In addition, we study the property of the free boundary in section 5.

Rewrite the model
In this subsection, we rewrite the model with the form of radial symmetry. We assume that the environment and solution are radially symmetric. Without loss of generality, we assume that Ω = B 1 (0) which represents a unite ball centered in origin and that u and v are radially symmetric with respect to x = 0. The free boundary can be written as r = |x| = h(t) and h(0) = b. Hence, the density u(x, 0) occupies the domain Ω 0 = B b (0) at the initial time t = 0.
Let ( u, v) denote the corresponding radial solution in B 1 (0) × (0, T ). In order to avoid confusion, we may write and A simple calculation shows that and • Reformulation of the boundary. If Γ t : Φ(r, t) = 0 ⇔ r − h(t) = 0, then the condition of the free boundary converts into Actually, substituting (2.5) and into the third and forth equations of (1.1), we can easily get • Reformulation of the equation. Substituting (2.5), (2.6) and (2.7) into the first and sixth equations of (1.1), we can easily obtain and Therefore, the model we are concerned here becomes which corresponds to the equation with normal coordinate (2.12)

Main result
Now we introduce the following space notations, which will be used in the main result. For T > 0, we define Our main results are: is radial symmetric on x and satisfies the conditions (1.2) and (1.3) and where n is the dimension of the space. If and a curve Γ t : |x| = h(t), which are the solutions of (2.12) for n < p.
Concerning the free boundary, we have the following properties: Theorem 2.2. Assume 1 < n < p, k(x, t) = a|x| and a > 0 is a constant. Then we have If M > ab n 1−b n , then h(t) is decreasing and M δ(x) as t → T * .

Some crucial estimates
In this section, we establish some crucial estimates, which will play the key roles in proving the local existence of radial symmetric solution of the system (2.12) in high dimensional case. 0 and (u, v) is the radial symmetrical solution of the system (2.12), then we have u > 0 and

Some basic properties of the solution
Proof. Since u 0 (r) > 0, by standard maximal principle of the parabolic equation, it follows that u > 0. Integrating the equation (2.9) over (0, h(t)), we have where the fifth equation of (2.11) is used. Thus one has d dt By virtue of the fact as required. Hence, the proof of the lemma is completed.

Simple calculation yields
This completes the proof of Lemma 3.2.

Some basic properties of the system
Firstly, we define where M 0 < 1 2 min{b,1−b} T is a constant. In this section, we shall establish some estimates which are important in the proof of the main result. For any fixed h(t) ∈ B, we consider the following problems and (3.14) Concerning the system (3.14), we have the following result.
Lemma 3.3. If h(t) ∈ B, then for T > 0 small enough and u ∈ X 1 u (T ), the system (3.14) admits a unique solution v ∈ X 2 v (T ) ∩ Y v (T ), and we have where C is independent of T and h(t) ∈ B.
Proof. It is obvious that the problem (3.14) has a solution v ∈ H 1,p (B 1 (0)) if and only if Moreover, we have sup From the above estimate, we can easily draw the conclusion. Now, we consider the system (3.13). Let y i = x i h(t) , u(y, t) = u(x, t), v(y, t) = v(x, t), k(y, t) = k(x, t). Then the system (3.13) is equivalent to the following system Hence, it can be reduced to the system of integral equation u(y, t) = e α(t,0)∆ u(y, 0) + Let t s > 0, e α(t,s)∆ represents the operator semigroup on L p B 1 (0) which is generated by and e α(t,s)∆ is an holomorphic semigroup on L p B 1 (0) . If h(t) ∈ B, then The operator-theoretic feature of e α(t,s)∆ necessary for the proof of this lemma is well known [21]. There exists a constant C > 0 which depends on M 0 but independent of t such that where 1 q p +∞.
Concerning the system (3.13) and (3.14), we have the following result.
Then for T small enough, the system (3.13) and (3.14) admit a unique solution where n < p.
and we define a mapping F w = u v(w) as follows Then local existence will be established via contraction mapping principle.
Step 1: F maps B(M, T ) into itself. Noticing (3.19), we have (3.20) Using the facts (3.16) and (3.17), the terms of the right-hand side of (3.20) are estimated from above by For w ∈ B(M, T ) this implies u(·, t) H 1,p C u 0 H 1,p + CT

M.
If we take M > 0 as large as and then take T > 0 as small as Step 2: F is a contract mapping. For w 1 , w 2 ∈ B(M, T ), we can construct v 1 , v 2 as above and it holds that Similar to the proof of Lemma 3.3, we can obtain that for T > 0 small enough Let u 1 , u 2 denote the corresponding solution of (3.19) respectively. Then the difference (u 1 − u 2 )(y, 0) = 0, y ∈ B 1 (0).
We have For w 1 , w 2 ∈ B(M, T ) this implies Taking T > 0 as small as we obtain F is a contract mapping. This completes the proof of the lemma.

The existence of the solution locally in time
Having at hand the preliminary material collected above, we are now prepared to prove our main result on local in time existence of solutions. We prove Theorem 2.1 by using Lemma 3.3 and Lemma 3.4.
For each h(t) ∈ B, by Lemma 3.4 we know that there exists a pair which is the solution of the system (3.13) and (3.14).
The following, we will use the contraction mapping principle to show the existence of the free boundary r = h(t). Set Define G : h(t) → g(t), therefore G maps B into itself. Next we will demonstrate that G is contractive. Then the fixed point theorem yields that there exist a pair (u, v) and a curve Γ : r = h(t) which are the solution of (2.11).
For h 1 (t), h 2 (t) ∈ B, let (u 1 , v 1 ), (u 2 , v 2 ) represent the corresponding solutions of (2.12) respectively, then The first term (I 1 ) on the right-hand side are estimated from above by By Lemma 3.2, the second term (I 2 ) on the right-hand side are estimated from above by Hence the fixed point theorem yields that there exist a pair (u, v) and a curve Γ : r = h(t) ∈ B which are the solution of (2.11).

Proofs of Theorem 2.2 and Theorem 2.3
In the section, we study the property of the free boundary and give the proofs for Theorem 2.2 and Theorem 2.3.
) and a curve Γ : r = h(t) ∈ B be the solution of (2.11). We have According to the assumption and Lemma 3.2, we get Let s(t) = h n (t), then it holds that A direct calculation gives the following unique solution: Hence, we obtain

Appendix
In this section, let us recall the construction of the system. All of the material here can be found in [7].
Let Ω ⊂ R n be a bounded open domain and Ω 0 ⊂ Ω be an open sub-domain. Assume a population density u(x, 0) occupying the domain Ω 0 , and in the outside of Ω 0 the population density u(x, 0) ≡ 0 and the external signal v occupying Ω. For t > 0, u(x, t) spreads to domain Ω t ⊂ Ω. Let ∂Ω t denote the boundary of Ω t and n t denote the outer normal vector of ∂Ω t , then Γ t = ∂Ω t × (0, T ) is the free boundary.
The spatial diffusion of species is related to the free boundary of Ω t at the time t 0. Observe the flux is increasing with respect to the density of the species, so it would be reasonable to suppose that flux is proportional to the density. Thus we have following flux condition on ∂Ω t , −∇u · n t = k(x, t)u on ∂Ω t where k(x, t) is positive function, and 1 k(x,t) > 0 is mass flow ratio. On the other hand, noticing that the total flux on ∂Ω t is j = −∇u · n t + χu∇v · n t , By conservation of population, one has uv nt = −∇u · n t + χu∇v · n t on ∂Ω t , (6.23) where v nt is the normal diffusion velocity of ∂Ω t . Assume Γ t : Φ(x, t) = 0, then v nt = dx 1 dt , dx 2 dt , · · ·, dx n dt · n t = dx 1 dt , dx 2 dt , · · ·, dx n dt · ∇Φ |∇Φ| , (6.24) where x = (x 1 , x 2 , · · ·, x n ) and ∇ = ( ∂ ∂x 1 , ∂ ∂x 2 , · · ·, ∂ ∂xn ). Notice that ∂Φ ∂t  where Γ t : Φ(x, t) = 0 is the free boundary.