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

In this paper, we study a free boundary problem for a class of parabolic 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.


1.
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 [17]. Then many mathematicians have made efforts to develop various models and investigated the problems from mathematical point of view. For a broad overview over various types of chemotaxis processes, we refer the reader to the survey [1,3,4,11,12,23,25,26] 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 occupied 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 domains and create a chemical gradient attracting the cells. In turn, the areas of amoeba may change 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 [9].

HUA CHEN, WENBIN LV AND SHAOHUA WU
In this paper, we consider the following high dimensional free boundary problem of a chemotaxis model. Such kind of models can be found in [2,5,24], and we can give more explanations in the appendix below.
in Ω t × (0, T ), u = 0, in Ω × (0, T ) \ Ω t × (0, T ), −∇u · ∇Φ |∇Φ| = k(x, t)u, on ∂Ω t × (0, T ), u ∂Φ ∂t = ∇u · ∇Φ − u∇v · ∇Φ, on ∂Ω t × (0, T ), in Ω × (0, T ), ∂v ∂n = 0, on ∂Ω × (0, T ), v(x, 0) = v 0 (x), in Ω, (1.1) where we assume • 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. • 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) occupying the domain Ω t , an open subset of Ω, in time t and u(x, t) = 0 in the outside of Ω 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 E.F. Keller and L.A. Segel [14]. The problem (1.4) is intensively studied by many authors (see for instance [6,7,8,15,18,21,22]). The initial functions u 0 ∈ C 0 (Ω) and v 0 ∈ C 1 (Ω) are assumed to be nonnegative. Within this framework, classical results state that • if n = 1 then all solutions of (1.4) are global in time and bounded (see [16]); • if n = 2 then in the case Ω u 0 (x)dx < 4π, the solution will be global and bounded (see [10,20]), whereas for any m > 4π satisfying m ∈ {4kπ | k ∈ N } there exist initial data (u 0 , v 0 ) with m = Ω u 0 (x)dx such that the corresponding solution of (1.4) blows up either in finite or infinite time, provided Ω is simply connected (see [13,19]); • if n ≥ 3 given any q > n 2 and p > n one can find a bound for u 0 in L q (Ω) and for ∇v 0 in L p (Ω) guaranteeing that (u, v) is global in time and bounded; on the other hand, if Ω is a ball then for arbitrarily small mass m > 0 there exist u 0 and v 0 having Ω u 0 (x)dx = m such that (u, v) blows up either in finite or infinite time (see [21]). In one dimensional case, if k is a positive constant H. Chen and S.H. Wu [2,5,24] 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) on a high dimensional symmetry domain Ω. In addition, the condition that k is a positive constant in [2,5,24] seems too strict, 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.

2.1.
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).
Let ( u, v) denote the corresponding radial solution in B 1 (0) × (0, T ). In order to avoid confusion, we may write A simple calculation shows that and v xixi (x, t) = v rr (r, t) 3) • Reformulation of the boundary.
If Γ t : Φ(r, t) = 0 ⇔ r − h(t) = 0, then the condition of the free boundary convert into k h(t), t u h(t), t + u r h(t), t = 0, for k(r, t) = k(x, t), Actually, substituting (2.1) and into the third and forth equations of (1.1), we can easily get • Reformulation of the equation. Substituting (2.1), (2.2) and (2.3) into the first and sixth equations of (1.1), we can easily obtain Therefore, the model we are concerned here becomes which corresponds to the equation with normal coordinate Main result. Now we introduce the following space notations, which will be used in the main result here. For t 0 > 0, we define Our main result is: is radial symmetric on x and satisfying the conditions (1.2) and (1.3). If are radial symmetric on x, where 0 < b < 1 and b is a constant. Then there exist t 0 > 0 small enough, a radial symmetric pair and a curve Γ t : |x| = h(t), which are the solutions of (2.8) for each 1 < < 2, 3. 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 system (2.8) in high dimensional case.
3.1. A basic property 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 (2.5) over (0, h(t)), we have where the fifth equation of (2.7) is used. Thus one has d dt Integrating the equation of (2.6) over (0, 1), we have d dt where the eighth and ninth equations of (2.7) is used. Through simple calculation, we can get The proof of the lemma 3.1 is completed.
3.2. Some basic properties of the system. Firstly, we define 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)
where C depends on M 0 but is independent of t 0 and h(t) ∈ B.
Proof. By scalar coordinate transform, the system (3.1) is equivalent to the system
In particular, we have ∆uds.
So we can obtain  If h(t) ∈ B, then The operator semigroup feature of T t (t) necessary for the proof of this lemma is well known. There is a constant C > 0 which is dependent on M 0 but independent of t such that where 1 ≤ q ≤ p ≤ +∞. Using the facts (3.7) and (3.8), the terms of the right-hand side of (3.6) are estimated from above by Thus, for t small enough, it holds that In case of t = 0, then for each 0 ≤ t 2 ≤ t 0 , we have sup 0≤s≤t2 u H ,p ≤ C u(·, 0) H 2,p + Ct From the two estimates above, we can easily deduce the conclusion.
Proof. It is obvious that the problem (3.2) has a unique solution The terms of the right-hand side are estimated from above by Thus, for t 0 small enough, it holds that The result of Lemma 3.3 is proved.
Then for t 0 small enough, the system (3.1) and (3.2) admit a unique solution
Proof. Firstly, we consider w ∈ X u (t 0 ) ∩ Y u (t 0 ) and w(x, 0) = u(x, 0). Let v = v(w) denote the corresponding solution of the equation (3.10) Secondly, for the solution v of the equation (3.10), we define u = u v(w) to be the corresponding solution of (3.11) Actually we have introduced a mapping F w = u v(w) . Now we take M = 2C u(·, 0) H 2,p (B b (0)) and a ball where the constant C is given by (3.3). Thus the local existence of the solution will be established via contraction mapping principle. In fact, Lemma 3.2 shows that F maps B M into itself. Actually, we have by (3.3) and (3.9). Thus, for t 0 small enough, it holds that Next, we can prove that for t 0 small enough, F is a contract mapping. For w 1 , w 2 ∈ B M , let u 1 , u 2 denote the corresponding solution of system (3.11) respectively. Then the difference u 1 − u 2 satisfies Take scalar coordinate transform and set Similar to the proof of Lemma 3.2, one can obtain that for each 1 < < 2 and t 0 > 0 small enough where C depends on M 0 but is independent of t 0 and h(t) ∈ B.
On the other hand, we have Similar to the proof of Lemma 3.3, one can obtain that for t 0 > 0 small enough which is the solution of the system (3.1) and (3.2). Now we will use Schauder theorem to prove the result of Theorem 2.1. Set Then we have  Let M 1 denote the constant at the right hand of (4.1). If t 0 is small enough, then We choose M 0 = M 1 in B, then it is clear that B ⊂ C[0, t 0 ] is a compact and convex set.
Define G : h(t) → g(t), therefore G maps B into itself. Next we will demonstrate that G is continuous. Then the Schauder theorem yields that there exist a pair (u, v) and a curve Γ t : r = h(t) which are the solution of (2.7).
For h 1 (t), h 2 (t) ∈ B, let (u 1 , v 1 ), (u 2 , v 2 ) represent the corresponding solutions of (2.8) respectively and ( u 1 , v 1 ), ( u 2 , v 2 ) represent the corresponding solutions of (2.7) respectively, then The terms on the right-hand side are estimated from above by On the following, we will focus on the term (I 2 ). Thus, we have On the other hand, we have Let h(t) = max{h 1 (t), h 2 (t)} and h(t) = min{h 1 (t), h 2 (t)}, then we have from Lemma 3.4 and It is trivial that sup 0≤τ ≤t0 As h 1 − h 2 C[0,t0] converges to zero, (I 1 ) and (I 2 ) converge to zero. From this we can get that sup 0≤t≤t0 |G(h 1 ) − G(h 2 )| also converges to zero, which shows that the map G is continuous on C[0, t 0 ]. Now the Schauder theorem yields that there exist a pair (u, v) and a curve Γ t : r = h(t) which are the solution of (2.8).

5.
Appendix. In this section, let us recall the construction of the problem. All of the material here can be found in [5].
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 ∂Ω t × (0, T ) is the free boundary.