GLOBAL DYNAMICS OF A CHEMOTAXIS MODEL WITH SIGNAL-DEPENDENT DIFFUSION AND SENSITIVITY

. In this paper, we shall study the initial-boundary value problem of a chemotaxis model with signal-dependent diﬀusion and sensitivity as follows ∗ ) in a bounded domain Ω ⊂ R 2 with smooth boundary, where α,β,D are positive constants, θ ∈ R and ν denotes the outward normal vector of ∂ Ω. The functions χ ( v ) ,γ ( v ) and F ( v ) . We ﬁrst prove that the existence of globally bounded solution of system ( ∗ ) based on the method of weighted energy estimates. Moreover, by constructing Lyapunov functional, we show that the solution ( u,v,w ) will converge to (0 , 0 ,w ∗ ) in L ∞ with some w ∗ ≥ 0 as time tends to inﬁnity in the case of θ ≤ 0, while if θ > 0, the solution ( u,v,w ) will asymptotically converge to ( θβ , θβ , 0) in L ∞ -norm provided D > max 0 ≤ v ≤∞ θ | χ ( v ) | 2 16 β 2 γ ( v


College of Mathematics and Statistics, Shenzhen University
Shenzhen 518061, China

Introduction and main results.
To describe the motion of a slime mold ameba in response to the chemical signals, Keller and Segel [16] has proposed the following chemotaxis model where Ω ⊂ R n is a bounded domain with smooth boundary, u(x, t) denotes the cell density and v(x, t) is the concentration of chemical. The main feature of the system (1) is the motility function γ(v) > 0 and the chemotactic coefficient χ(v) are functions depending on chemical signal v, which are connected by the following relation where α denote the ratio of effective body length to step size, and γ (v) < 0(> 0) if motility decreases (increases) with concentration.
For the system (1), we can summary the results as follows.
• When γ(v) is a constant, the previous results on the system (1) are mostly focus on the global boundedness (cf. [39,22,35,31]), stabilization ( [19,36,5]) and pattern formation ( [23,31,17]). One can find more details from the review papers ( [6]). • When γ(v) is a function satisfying (2), the existing results are mostly limited to the special case χ(v) = −γ (v) > 0, and the system (1) can be rewritten as which has also been used in [7] to study the effect of density-suppressed motility. For the system (3), it has been proved that the logistic growth source can prevent the blow-up of solution in two dimensional spaces for any µ > 0 [11]. Moreover, if µ is large, the constant steady state (1, 1) is asymptotical stability. One can also find some further results on the global existence and large time behavior of solution for the parabolic-elliptic case with some weaker conditions [9]. The global existence results also been extended to the higher dimensions (n ≥ 3) for large µ [32]. When µ is small, the existence/nonexistecne of nonconstant steady states of (3) has been studied in [21] recently. When µ > 0 and γ(v) is step function, the dynamics of interface was studied in [25] in one dimension.
If the logistic growth is ignored (i.e., µ = 0), the solution behavior is complicated. More precisely, if γ(v) has a positive lower and upper bound, Tao & Winkler [30] proved the existence of global classical solution in two dimensions and global weak solution in three dimensions. If γ(v) = c 0 /v k (k > 0), it has been proved that the global classical solution exist in two dimensions [8] and in higher dimensions (n ≥ 3) provided c 0 small [40] or 0 < k < 2 n−2 [1]. If γ(v) decays faster than algebra, the solution may blow up. In fact, if γ(v) = e −χv , by constructing a Lyapunov functional, it has been proved [15,8] that there exists a critical mass m * such that the solution exists globally if Ω u 0 dx < m * and blow up if Ω u 0 dx > m * .
As recalled above, few results are known for the system (1) with more general γ(v) and χ(v). Moreover, the nutrient consume is ignored for the above discussed system. In this paper, we shall study the global dynamics of solution of system (1) with the nutrient consume. More precise, we shall study the following system under the following assumptions: In the system (4), u(x, t) denotes the cell density and v(x, t) is the chemical concentration, w(x, t) stands for the nutrient density. The parameters α, θ, β, D are constants. We should point out that the assumptions in (H2) on F (w) can be satisfied by a large class of functions including with constants λ > 0 and m > 1, which are used as the Holling type functional response function in the predator-prey systems (cf. [12,13,38,34]). Compared with the system (1), the nutrient is taken into and hence the dynamics of solution behavior may become more complicated. In fact, if θ = β = 0, F (w) = w 2 λ+w 2 and χ(v) = −γ (v), the system (4) becomes the following three-component reaction- which has been proposed in [20] to gain a quantitative understanding of the patterning process in the experiment. For the system (5), if γ(v) has lower and upper bound, it has been proved that the global classical solution exists, which will converge to (u * , u * , 0) for large D, where u * = 1 |Ω| Ω u 0 dx + α |Ω| Ω w 0 dx [14]. In this paper, we shall establish the existence of global classical solution and large time behavior of system (4) with the assumptions (H1)-(H2). First, based on the weighted energy estimates, we shall prove the existence of globally bounded solutions to the system (4) in two dimensions as follows.
Then for any β > 0, the system (4) has a unique global classical solution (u, v, w) ∈ Theorem 1.2. Let the assumptions in Theorem 1.1 hold and (u, v, w) be the classical solution of (4) obtained in Theorem 1.1. Then the following asymptotic stability results hold: (1) If θ ≤ 0, one has

Remark 1.
Our results show that all the bacterial will die and the nutrition survive if θ < 0, while if θ > 0 all the nutrition are consumed.

2.
Local existence and Preliminaries. The existence and uniqueness of local solutions of (4) can be readily proved by the Amann's theorem [3,4] (cf. also [33, Lemma 2.6]) or the fixed point theorem along with the parabolic regularity theory [11]. We omit the details of the proof for brevity.
Proof. First, applying the maximum principle to the third equation of (4), one obtains (7) directly. Moreover, multiplying the third equation of (4) by α and adding it to the first equation of (4), we have for all t ∈ (0, T max ) that Then if θ ≤ 0, we obtain (6) directly by integrating (8). On the other hand, if θ > 0, then using the Young inequality, one has which together with (8) gives Then applying Grönwall's inequality to (10) gives (6).
The following lemma will be used to show the boundedness of solution, one can see [18,Lemma 3.3] or [26,Lemma 3.4] for details.
is absolutely continuous and fulfils Then 3. Boundedness of solutions: Proof of Theorem 1.1. In this section, we are devoted to studying the existence of global classical solutions for system (4). To overcome this major obstacle that the possible degeneracy of diffusion, we employ the L 2 energy estimate directly by treating γ(v) as a weight function based on some ideas in [11]. This enables us to control the chemotactic advection term and derive a Grönwall type inequality: which, together with the facts t+τ t Ω u 2 and t+τ t Ω |∆v| 2 (see Lemma 3.1 and Lemma 3.2 ) derive the uniform-in-time bound of u(·, t) L 2 . First, we show some basic properties of solution for the system (4).
Proof. We multiply the third equation of (4) by α and add it to the first equation of (4) to have Then integrating (12) over (t, t + τ ) and using (6), one has which gives (11).
be the solution of system (4). Then there exists a constant C > 0 independent of t such that and t+τ t Ω where τ and T max are defined in Lemma 3.1.
Next, we will show the boundedness of u(·, t) L 2 based on the weighted energy estimate as follows.

Lemma 3.3.
Let Ω be a bounded domain in R 2 with smooth boundary and the hypotheses (H1)-(H2) hold. Suppose (u, v, w) is a solution of the system (4), then it holds that u(·, t) L 2 ≤ C for all t ∈ (0, T max ), (16) where the constant C > 0 is independent of t.
Proof. Testing the first equation of (4) by u and integrating the result by parts, then we use Hölder's inequality, Young's inequality and the fact F (w) ≤ F ( w 0 L ∞ ) := c 1 to obtain 1 2 for all t ∈ (0, T max ), which yields Based on the ideas in [11], we can derive that which substituted into (17) gives Then using the hypothesis (H1), one a find a constant K 1 > 0 such that which substituted into (18) gives for all t ∈ (0, T max ) that Then using the Gagliardo-Nirenberg inequality and the facts γ(v) ≤ γ(0) = c 3 and ∇v L 2 ≤ c 4 , and the Young's inequality, we have Then we substitute (20) into (19), and use the estimate c 7 u 2 L 2 ≤ 2δ Ω u 3 + c 8 to obtain d dt u 2 L 2 ≤ c 7 u 2 L 2 ∆v 2 L 2 + c 9 for all t ∈ (0, T max ).
Then we can integrate (21) over (t 0 , t), and use (22) and (23) to obtain u(·, t) 2 where C > 0 is a constant independent of t.
Proof of Theorem 1.1. From Lemma 3.4, we can find a constant c 1 > 0 such that This along with the parabolic regularity to the second and third equation of the system (4) gives Then the combination of (31), (32) and Lemma 2.1 gives Theorem 1.1 directly.

Large time behavior.
In this section, we will derive the asymptotic behavior of solutions as shown in Theorem 1.2. First, we shall improve the regularity of u, v and w by using the standard parabolic property.

4.1.
Case of θ ≤ 0. In this subsection, we will study the large time behavior of solutions in the case of θ ≤ 0. In fact, if θ ≤ 0, we can directly obtain ∞ 0 Ω u 2 < ∞, which combined with the relative compactness of (u(·, t)) t>1 in C(Ω) (see Lemma 4.1) implies some decay information for u and hence the decay properties of v from the second equation of (4). More precisely, we have the following results.
Proof. Multiplying the third equation of (4) by α and adding it to the first equation of (4), we have due to θ ≤ 0 and u > 0. Then from (39), we obtain Next, we shall use some ideas in [29, Lemma 3.10] to prove (37) by the argument of contradiction. In fact, suppose that (37) is wrong, then for some constant c 4 > 0, one can find some sequences (x j ) j∈N ⊂ Ω such that (t j ) j∈N ⊂ (0, ∞) satisfying t j → ∞ as j → ∞ such that u(x j , t j ) ≥ c 2 , for all j ∈ N.
From Lemma 4.1, we know u is uniformly continuous in Ω × (1, ∞). Then for any j ∈ N it holds that for some r > 0 and T 1 > 0, for all x ∈ B r (x j ) ∩ Ω and t ∈ (t j , t j + T 1 ).
Noting the smoothness of ∂Ω, one can find a constant c 3 > 0 such that Then combining (41) and (42), for all j ∈ N, we have tj +T1 tj Ω However, noting the fact t j → ∞ as j → ∞, from (40) one has tj +T1 tj Ω which contradicts (43). Hence (37) holds by the argument of contradiction.
Next, we show (38) holds to complete the proof of this lemma. In fact, from the second equation of (4), we have which combined with the fact Ω u → 0 as t → ∞ gives Then using the fact ∇v L 4 ≤ c 6 and applying the Gagliardo-Nirenberg inequality, one has v L ∞ ≤ c 7 ∇v which combined with (44) gives (38).
where w * ≥ 0 is a constant determined by w * = 1 Proof. We can rewrite the third equation of (4) as wheref (t) = 1 |Ω| Ω f . Then applying the variation-of-constants formula to (47) and using the fact w(·, t) L ∞ ≤ c 1 , one has which combined with the decay property of u in (37) gives We integrate the third equation of (4) over Ω × (0, t) to havē Then the combination of (49) and (50) gives which yields (46). Next, we shall show that w * > 0 in the case of θ < 0. Using the boundedness of u, w and the properties of F (w), we can derive that Then suppose thatw(x, t) is the solution of the problem x ∈ Ω, Then by using the comparison principle,w(x, t) is a sub-solution of w(x, t) and hence Then from [10, Lemma 3.1], we know that there a constant Γ 0 > 0 such that for all which together with (52) gives Then we multiply the third equation of the system (4) by 1 w , and integrate by parts with respect to x ∈ Ω to have Then integrating (54) by parts over (1, t), we obtain On other hand, from the system (4), we have which implies t 0 Ω u ≤ c 8 in the case of θ < 0. Then the combination of (53), (55) and the fact t 0 Ω u ≤ c 8 gives Ω ln w(x, t) ≥ −c 9 , for all t ≥ 1, which together with the fact (46) implies w * > 0.

4.2.
Case of θ > 0. In this subsection, we will study the large time behavior of solution for the system (4) with θ > 0. To this end, we shall show the following energy functional can act as Lyapunov functional under some conditions on the parameters based on some ideas in [11].
Then we can derive that , then there exists a positive constant β such that for all t > 0 where Proof. The non-negativity of E(t) can be verified by noting φ(u) := u− θ β −ln βu θ , u > 0 is non-negativity. In fact applying the Taylor's formula to φ(u) to gives which implies E(t) ≥ 0. Next, we prove (57) to complete the proof of this lemma. In fact, from the system (4), we have Then letting we can rewrite I 1 = −Θ T 1 A 1 Θ 1 where Θ T 1 denotes the transpose of Θ 1 . Then it is easy to check that A 1 is non-negative definite and hence I 1 ≤ 0 if and only if Similarly, we can also rewrite I 2 as One can check that A 2 is positive definite if and only if Then if D > max 0≤v≤∞ θ|χ(v)| 2 16β 2 γ(v) there exists a positive constant δ such that (60) and (61). Hence from (59), we can find a constant ζ > 0 such that (57) holds. Then we complete the proof of this lemma.
On the other hand, we can use the Gagliardo-Nirenberg inequality to find a constant c 1 > 0 such that w(·, t) L ∞ ≤ c 1 ∇w where δ 1 > 0 is arbitrary, c 2 > 0 is a constant depends on δ. Noting that ∇w L 4 ≤ c 3 , then from (69) we can derive