Logistic type attraction-repulsion chemotaxis systems with a free boundary or unbounded boundary. I. Asymptotic dynamics in fixed unbounded domain

The current series of research papers is to investigate the asymptotic dynamics in logistic type chemotaxis models in one space dimension with a free boundary or an unbounded boundary. Such a model with a free boundary describes the spreading of a new or invasive species subject to the influence of some chemical substances in an environment with a free boundary representing the spreading front. In this first part of the series, we investigate the dynamical behaviors of logistic type chemotaxis models on the half line $\mathbb{R}^+$, which are formally corresponding limit systems of the free boundary problems. In the second of the series, we will establish the spreading-vanishing dichotomy in chemoattraction-repulsion systems with a free boundary as well as with double free boundaries.


LIANZHANG BAO AND WENXIAN SHEN
and to study the asymptotic dynamics of where ν > 0 in (1) is a positive constant, and in both (1) and (2), χ i , λ i , and µ i (i = 1, 2) are nonnegative constants, and a(t, x) and b(t, x) satisfy the following assumption, Chemotaxis is the influence of chemical substances in the environment on the movement of mobile species.This can lead to strictly oriented movement or to partially oriented and partially tumbling movement. The movement towards a higher concentration of the chemical substance is termed positive chemotaxis and the movement towards regions of lower chemical concentration is called negative chemotaxis. The substances that lead to positive chemotaxis are chemoattractants and those leading to negative chemotaxis are so-called repellents.
One of the first mathematical models of chemotaxis was introduced by Keller and Segel ( [13], [14]) to describe the aggregation of certain type of bacteria. A simplified version of their model involves the distribution u of the density of the slime mold Dyctyostelum discoideum and the concentration v of a certain chemoattractant satisfying the following system of partial differential equations complemented with certain boundary condition on ∂Ω if Ω is bounded, where Ω ⊂ R N is an open domain, ≥ 0 is a non-negative constant linked to the speed of diffusion of the chemical, χ represents the sensitivity with respect to chemotaxis, and the functions G and F model the growth of the mobile species and the chemoattractant, respectively. Since their publication, considerable progress has been made in the analysis of various particular cases of (3) on both bounded and unbounded fixed domains (see [1], [3], [4], [6], [10], [12], [18], [24], [25], [26], [29], [30], [31], [32], [33], [34], [36], and the references therein). Among the central problems are the existence of nonnegative solutions of (3) which are globally defined in time or blow up at a finite time and the asymptotic behavior of time global solutions. When > 0 (3) is referred to as the parabolic-parabolic Keller-Segel model and = 0, which models the situation where the chemoattractant diffuses very quickly, is the case of parabolic-elliptic Keller-Segel model. The reader is referred to [7,8] for some detailed introduction into the mathematics of KS models.
When the cells undergo random motion and chemotaxis towards attractant and away from repellent [17] on a fixed domain, we have a chemoattraction-repulsion process, which combined with proliferation and death of cells leads to the following parabolic-elliptic-elliptic differential equations, where χ 1 and χ 2 are nonnegative constants. Note that, when χ 2 = 0, the first two equations in (4) are decoupled from the third equation and form system (3) with χ = χ 1 and v = v 1 . Compared to the studies of (3), the global existence of classical solutions on bounded or unbounded domain, and the stability of equilibrium solutions of (4) are also studied in many papers (see [5,9,11,15,16,17,22,27,28,35,37] and the references therein).
System (1) describes the movement of a mobile species with population density u(t, x) in an environment with a free boundary subject to a chemoattractant with population density v 1 (t, x), which diffuses very quickly, and a repellent with population density v 2 (t, x), which also diffuses very quickly. Due to the lack of first principles for the ecological situation under consideration, a thorough justification of the free boundary condition is difficult to supply. As in [2], we present in the following a derivation of the free boundary condition in (1) based on the consideration of "population loss" at the front and the assumption that, near the propagating front, population density is close to zero. Then, in the process of population range expansion, on one hand, the individuals of the species are suffering from the Allee effect near the propagating front. On the other hand, as the front enters new unpopulated environment, the pioneering members at the front, with very low population density, are particularly vulnerable. Therefore it is plausible to assume that as the expanding front propagates, the population suffers a loss of κ units per unit volume at the front.
By Fick's first law, for a small time increment ∆t, during the period from t to t + ∆t, the number of individuals of the population that enter the region (through diffusion, or random walk) bounded by the old front x = h(t) and new front where d is some positive constant. The population loss in this region is approximated by So the average density of the population in the region bounded by the two fronts is given by As ∆t → 0, the limit of this quantity is the population density at the front, namely u(t, h(t)), which by assumption is 0. This implies that

LIANZHANG BAO AND WENXIAN SHEN
with ν = d/κ, and the free boundary condition in (1) is then derived. Consider (1), it is interesting to know whether the species will spread into the whole region [0, ∞) or will vanish eventually. Formally, (2) can be viewed as the limit system of (1) as h(t) → ∞. The study of the asymptotic dynamics of (2) plays an important role in the characterization of the spreading-vanishing dynamics of (1) and is also of independent interest. The objective of this series is to investigate the asymptotic dynamics of (2) and the spreading and vanishing scenario in (1).
In this first part of the series, we investigate the asymptotic dynamics of (2) as well as the asymptotic dynamics of the following chemotaxis system on the whole line, Formally, (5) can be viewed as the limit of the following free boundary problem with double free boundaries as g(t) → −∞ and h(t) → ∞. The investigation of the asymptotic dynamics of (5) then plays a role in the characterization of the spreading-vanishing dynamics of (6) and is also of independent interest.
In the second of the series, we will establish spreading and vanishing dichotomy scenario in (1) and (6).
In the following, we state the main results of this paper. Let is uniformly continuous and bounded on R + } with norm u ∞ = sup x∈R + |u(x)|, and is uniformly continuous and bounded on R} with norm u ∞ = sup x∈R |u(x)|. Define and Let (H1)-(H3) be the following standing assumptions.
The main results of this first part are stated in the following theorems. Theorem 1.1 (Global existence). Consider (2). If (H1) holds, then for any t 0 ∈ R and any nonnegative function Theorem 1.2 (Persistence). Consider (2).
(3) (5) with χ 2 = 0 is a special cases of the parabolic-elliptic chemotaxis model with space-time dependent logistic sources on R N studied in [20] and [21]. Theorem 1.4 in the case χ 2 = 0 is proved in [20] and [21] (see [ (4) Logistic type attraction-repulsion chemotaxis systems on a half space are studied for the first time. The results stated in Theorems 1.1-1.3 are similar to those stated in Theorem 1.4 for logistic type attraction-repulsion chemotaxis systems on the whole space. Several existing techniques developed for the study of logistic type attraction-repulsion chemotaxis systems on a whole space are applied for the study of (2) with certain modifications. But, due to the presence of the boundary x = 0 as well as the unboundedness of the domain, such modifications are nontrivial and some other technical difficulties also arise in the study of (2).
The rest of this paper is organized in the following way. In section 2, we present some preliminary lemmas to be used in the proofs of the main results. We prove the main results of the paper in section 3. In this section, we present some lemmas to be used in the proof of the main results in later sections.
2. Preliminary lemmas. The first lemma is on the local existence of solutions of (2) and (5).
For any t 0 ∈ R and any nonnegative func- (2) Consider (2). For any t 0 ∈ R and any nonnegative function Proof. (1) It follows from the similar arguments used in the proof of [23, Theorem 1.1]. For the reader's convenience and for the proof of (2), we outline the proof in the following.
First, let T (t) be the semigroup generated by ∂ xx − I on C b unif (R). Then for any By [23, Lemma 3.2], T (t)∂ x can be extended to C b unif (R), and for any u ∈ C b unif (R), there holds By [23,Lemma 3.3], for any u ∈ C b unif (R), By the similar arguments as those in [23, Theorem 1.1], there is τ > 0 such that (2) It can be proved by the arguments in (1). To be more precise, first, letT (t) be the semigroup generated by ∂ xx − I on C b unif (R + ) with Neumann boundary at 0. Then for any for every x ∈ R + , whereũ(z) = u(|z|). Hence by the arguments in (1),T (t)∂ x can be extended to C b unif (R + ), and for any u ∈ C b unif (R + ), there holds We can then apply the arguments in [23, Theorem 1.1] to prove that there is Next, by the standard extension arguments, there is T max > 0 such that (2) has a unique solution (u(t, The second lemma is on the estimate of where M and C 0 (u 0 ) are as in (7) and (11), respectively.
respectively. Then where K is as in (8). Proof.
(1) It can be proved by some similar arguments as those in [22,Theorem A].
For the completeness, we provide a proof for (2). It can be proved similarly for (5).

LIANZHANG BAO AND WENXIAN SHEN
We then have Similarly, we can prove that (1) then follows.
(1) For given t 0 ∈ R and u 0 ∈ C b unif (R + ) with u 0 ≥ 0 and u 0 = 0, assume that u(t, x; t 0 , u 0 ) exists on [t 0 , ∞) and lim sup t→∞ u(t, ·; t 0 , u 0 ) < ∞. Let By the assumption,ū < ∞. Then for every ε > 0, there is T ε > 0 such that Hence, it follows from comparison principle for elliptic equations, that By similar arguments as those in Lemma 2.2, we have for t ≥ t 0 + T ε and then for t ≥ t 0 + T ε . By (16) and comparison principle for parabolic equations, where U ε (t) is the solution of It then follows that The lemma is thus proved.
Before we state the next lemma, let a 0 = a inf 3 and L > 0 be a given constant.
Note also that u(t, x) = e σ L t φ L (x) is a solution of (17). Let u(t, x; u 0 ) be the solution of (17) with u 0 ∈ C([−L, L]). Then for all κ ∈ R. Moreover, we have that φ L (x) satisfies and (19) also holds when u(t, x; κφ L ) is the solution of with u(0, x; κφ L ) = κφ L (x) for x ∈ [0, L].
In the following, fix T 0 > 0 and let L 0 0 be such that σ L 0 > 0. Note that σ L is increasing as L increases. Choose N 0 ∈ N and α 1 > 1 such that Then for any L ≥ L 0 .
Let C 0 = C 0 (u 0 ) be as in Lemma 2.2. For any give 0 < T < T max , let endowed with the norm Consider the subset E of E T defined by It is clear that Moreover, E is a closed bounded and convex subset of E T . We shall show that u(t 0 + ·, ·; t 0 , u 0 ) ∈ E.
To this end, for any given u ∈ E, let v i (t, x; u) be the solution of Let U (t, x; u) be the solution of the initial value problem By Lemma 2.2, we have Hence for t ∈ (t 0 , t 0 + T ], Observe that U ≡ C 0 is a super-solution of (39). Hence by comparison principle for parabolic equations, we have Therefore, U (·, ·; u) ∈ E. By the similar arguments as those in [19,Lemma 4.3], the mapping E u → U (·, ·; u) ∈ E is continuous and compact, and then by Schauder's fixed theorem, it has a fixed point u * . Clearly (u * (·, ·), v 1 (·, ·; u * ), v 2 (·, ·; u * )) is a classical solution of (2). Thus, by Lemma 2.1, we have Since 0 < T < T max is arbitrary, by Lemma 2.1 again, we have T max = ∞ and Theorem 1.1 then follows.

3.2.
Persistence. In this subsection, we prove Theorem 1.2 on the persistence of solutions of (2) with strictly positive initial functions.
Proof of Theorem 1.2.
If suffices to prove that m 0 ≤ u ≤ū ≤ M 0 . By (1), u > 0. Using the definition of limsup and liminf, we have that for every 0 < ε < u, there is T ε > 0 such that Hence, it follows from comparison principle for elliptic equations, that We then have for t ≥ t 0 + T ε . This together with comparison principle for parabolic equations implies that . .
It then follows that .
(ii) It can be proved by combing the arguments in (i) and properly modified arguments in [21, Theorem 1.5]. We do not provide the proof in this paper.
Proof of Theorem 1.4. (1) It follows from the similar arguments as those in Theorem 1.1.
(2) It follows from the similar arguments as those in Theorem 1.2.
(3) It follows from the similar arguments as those in Theorem 1.3(1).
(4) It follows from the similar arguments as those in Theorem 1.3 (2).