DYNAMICS IN A PARABOLIC-ELLIPTIC CHEMOTAXIS SYSTEM WITH GROWTH SOURCE AND NONLINEAR SECRETION

. In this work, we are concerned with a class of parabolic-elliptic chemotaxis systems with the prototype given by with nonnegative initial condition for u and homogeneous Neumann boundary conditions in a smooth bounded domain Ω ⊂ R n ( n ≥ 2), where χ,b,κ > 0, a ∈ R and θ > 1. First, using diﬀerent ideas from [9, 11], we re-obtain the boundedness and global existence for the corresponding initial-boundary value problem under, either Next, carrying out bifurcation from ”old multiplicity”, we show that the corre- sponding stationary system exhibits pattern formation for an unbounded range of chemosensitivity χ and the emerging patterns converge weakly in L θ (Ω) to some constants as χ → ∞ . This provides more details and also ﬁlls up a gap left in Kuto et al . [13] for the particular case that θ = 2 and κ = 1. Finally, for θ = κ + 1, the global stabilities of the equilibria (( a/b ) 1 κ ,a/b ) and (0 , 0) are comprehensively studied and explicit convergence rates are computed out, which exhibits chemotaxis eﬀects and logistic damping on long time dynamics of solutions. These stabilization results indicate that no pattern formation arises for small χ or large damping rate b ; on the other hand, they cover and extend He and Zheng’s [6, Theorems 1 and 2] for logistic source and linear secretion ( θ = 2 and κ = 1) (where convergence rate estimates were shown) to generalized logistic source and secretion.

Next, carrying out bifurcation from "old multiplicity", we show that the corresponding stationary system exhibits pattern formation for an unbounded range of chemosensitivity χ and the emerging patterns converge weakly in L θ (Ω) to some constants as χ → ∞. This provides more details and also fills up a gap left in Kuto et al. [13] for the particular case that θ = 2 and κ = 1. Finally, for θ = κ + 1, the global stabilities of the equilibria ((a/b) 1 κ , a/b) and (0, 0) are comprehensively studied and explicit convergence rates are computed out, which exhibits chemotaxis effects and logistic damping on long time dynamics of solutions. These stabilization results indicate that no pattern formation arises for small χ or large damping rate b; on the other hand, they cover and extend He and Zheng's [6, Theorems 1 and 2] for logistic source and linear secretion (θ = 2 and κ = 1) (where convergence rate estimates were shown) to generalized logistic source and nonlinear secretion.
ways (see the review articles [2,7,8] for instance). Due to its important applications in biological and medical sciences, chemotaxis research has become one of the hottest topics in applied mathematics nowadays and tremendous theoretical progress has been made in the past few decades. This work is devoted to the global dynamics including boundedness, pattern formation and long time behavior for the following parabolic-elliptic chemotaxis system with nonlinear production of signal and growth source: x ∈ Ω, t > 0, ∂u ∂ν = ∂v ∂ν = 0, x ∈ ∂Ω, t > 0, u(x, 0) = u 0 (x), x ∈ Ω, (1.1) where Ω ⊂ R n (n ≥ 1) is a bounded domain with smooth boundary ∂Ω and ν denotes the outward normal vector of ∂Ω, u(x, t) and v(x, t) denote the cell density and chemical concentration, respectively. The chemotactic sensitivity coefficient χ(> 0) measures the strength of chemotaxis and the kinetic term f (u) describes cell proliferation and death (simply referred to as growth source) and g(u) accounts for the chemical secretion by cells. The parabolic-elliptic chemotaxis system (1.1) could be physically justified when the chemicals diffuse much faster than cells do; indeed, this simplified system was first introduced for the case f (u) = 0 and g(u) = u (minimal model) in [10] and thereafter various variants of (1.1) have been studied by many other authors (e.g. see [5,9,10,21,22,30] and the references therein). Based on the commonly used choices for f and g in the literature [5,9,21,24,25], throughout this paper, we assume that f is smooth in [0, ∞) satisfying f (0) ≥ 0 and there are a ≥ 0, b > 0 and θ > 1 such that f (u) ≤ a − bu θ for all u ≥ 0 (1.2) and, the secretion function g is continuous in [0, ∞) and there are β > 0 and κ > 0 such that g(u) ≤ βu κ for all u ≥ 0. (1.3) This project originates from our two years ago's preprint [23], which aims at extending the fundamental boundedness of Tello and Winkler [21] for logistic source and linear secretion to more general growth source and nonlinear secretion term. During the last two years, new progresses on variants and extensions of (1.1) have been obtained in [5,9,11]. In the starting work [21], for linear secretion g(u) = βu and logistic source θ = 2, Neumann heat semigroup type arguments are used to obtain the global boundedness under This fundamental global boundedness has been extended extensively in a sequel of works, cf. [3,5,9,22,30], for a system with nonlinear diffusion, nonlinear chemosensitivity, generalized logistic source or nonlinear production. Here, in this work, beyond boundedness, we wish to provide a full picture about other dynamical behaviors of solutions on the interactions between nonlinear cross-diffusion, generalized logistic source and signal production for (1.1) such as the ability of pattern formations, the asymptotical behavior for large χ and the large time behavior of bounded solutions. For these purposes, we will stick to the parabolic-elliptic chemotaxis system (1.1) and, we will not go into further generality as done in [5,9] instead. Therefore, we only mention the following direction of extensions for comparison: for some θ > 1, κ ≥ 1, a ∈ R, b > 0, χ > 0, x ∈ Ω, t > 0, (1.6) its boundedness and global existence for (1.6) are guaranteed in the non-borderline cases that [5]. (1.7) Under further conditions like a = b, θ ≥ κ + 1 and b > 2χ, the latter work extends the comparison argument in [21] to show that the constant equilibrium (1, 1) is globally stable and obeys (1.5).
The boundedness and global existence of Tello-Winkler were first extended for (1.6) to the borderline case by Kang and Stevens [11] under In the same year as the work [11], Hu and Tao [9] extended the boundedness and global existence for (1.6) in [5] to the borderline case that Here, we notice that (1.8) and (1.9) impose restrictions like κ ≥ 1 and n ≥ 3, and that the methods in [11,9] are relatively implicit or indirect. Here, for completeness, we wish to employ a simpler argument to show the borderline boundedness of solutions to (1.1). We now sketch our main results and give some comments for the motivation of our study: (C1) Bounded classical solutions. In Section 4, by fully making use of the L κn/2+ -boundedness criterion obtained in Lemma 2.3, we establish the boundedness and global existence of classical solutions to (1.1) with f and g satis- The case of (1.10) is quite simple and its proof is short and can be readily adapted from existing approaches in literature, cf. [3,5,22,30]. The idea used to prove boundedness under (1.11) is first to prove the L κn/2boundedness of u and then to use G-N interpolation inequality to prove its L κn/2+ -boundedness, which is different from (more direct) the existing methods in [9,11]. For completeness and consistency, we include it to make the flow of the proof of (1.11) more smooth. The precise results are provided in Theorems 3.1 and 3.2, Corollary 1 and Remark 2. Here, we especially note that the logistic damping effect is always kept in force even when κn − 2 = 0, that is, the premise b > 0 is always required. For instance, when κn − 2 = 0, 258 TIAN XIANG our result ensures that the following chemotaxis-growth system with Neumann boundary condition has no blowup solutions for any n ≥ 2, χ > 0, a ∈ R and b > 0; while, it is well-known that (1.12) does possess blowup solutions for a = b = 0, n = 2, cf. [8,10]. (C2) Pattern formations and their limiting behavior for large chemosensitivity. In Section 5, we first use energy method to study the regularity and then perform Leray-Schauder index formula and bifurcation from "old multiplicity" [17,18] to show the existence of non-constant steady states of (1.1) for an unbounded range of χ, which not only covers the results of Tello and Winkler [21] with logistic source (θ = 2) and linear secretion (κ = 1), but also provide more verifiable conditions for the existence of pattern formations (Theorem 4.3). Furthermore, we investigate the asymptotic behavior of stationary solutions as χ → ∞ in certain parameter regimes, which demonstrates that the emerging patterns will converge weakly in L θ (Ω) to some constants as χ → ∞, cf. Theorem 4.4. This part provides more details and clarifies a vague point made in Kuto et al [13] for the special cases f (u) = au − bu 2 and g(u) = βu, see Remark 4. (C3) Large time behavior of bounded-in-time solutions. In Section 6, instead of using the comparison arguments [21,5], we apply the energy functional method from [1,6] to undertake a comprehensive analysis for the global asymptotic stabilities of the parabolic-elliptic system (1.6) with θ = κ+1. Under explicit conditions, the global stabilities of the equilibria ((a/b) 1 κ , a/b) and (0, 0) are obtained, which implies no pattern formations could arise for small chemosensitivity χ or large damping rate b. Moreover, we calculate their respective exponential and algebraic convergence rates explicitly in terms of the model parameters, which exhibits chemotaxis effects and logistic damping on long time dynamics of solutions, cf. Theorem 5.1. For logistic source (θ = 2) and linear secretion (κ = 1), convergence rate estimates were derived but not explicitly stated in [6, Theorems 1 and 2]. In a word, our stability results extend the result of [6, Theorems 1 and 2] for logistic source and linear secretion (where convergence rate estimates were shown) to generalized logistic source and nonlinear nonlinear secretion (θ > 1 and κ > 0) and, refine the uniform convergence in [21] and [5, m = 1] to exponential convergence under different set of conditions. Finally, we mention that various variants of (1.1) or its fully parabolic version have been investigated to understand the interplay of (nonlinear) diffusion, the chemotactic sensitivity and the cell kinetic in enforcing boundedness and stabilization toward constant equilibria, as well as more unexpected behavior witnessing a certain strength of chemotactic destabilization etc (e.g. see [20,24,25,27,28] and the references therein).
2. Preliminaries and a boundedness criterion for the chemotaxis system. For convenience, we quote the well-known Gagliardo-Nirenberg interpolation inequality below and state the local well-posedness of the chemotaxis-growth system (1.1).
The local-in-time existence of classical solutions to the chemotaxis-growth system (1.1) is quite standard; see similar discussions in [3,21,22,30].
Proof. As mentioned above, the assertions concerning the local-in-time existence of classical solutions to the initial-boundary value problem (1.1) and the criterion (2.3) are well-studied. Since f (0) ≥ 0, the maximum principle asserts that both u and v are nonnegative, as shown in [21,28]. Integrating the u-equation in (1.1) and using (1.2), one can easily deduce that where c = max{a − bu θ + u : u ≥ 0} < ∞ thanks to the fact that θ > 1. Solving this standard Gronwall's inequality shows that L 1 -norm of u is uniformly bounded.
For the chemotaxis model without growth, we know that the total cell mass u(t) L 1 is conservative. This is no long true for the chemotaxis model with growth, but u(t) L 1 is still uniformly bounded (cf. Lemma 2.2). However, the uniform boundedness of u(t) L 1 is not sufficient to prevent the blow-up of solutions in finite/infinite time (see [8,26]). By [2,19,28], it is quite known that the hard task of proving the (L ∞ , W 1,∞ )-boundedness of (u, v) can be reduced to proving only the L p -boundedness of the u-component for suitably finite p. Since the existing results (cf. [2,19,28]) don't give us the precise information that we need in the sequel, we here supply the following convenient criterion along its proof (the idea is essentially quite known) which says that the uniform boundedness of L p -norm of u(t) for some p > κn/2 can rule out blowing up of solutions. then (u(·, t), v(·, t)) is uniformly bounded in L ∞ (Ω) × W 1,∞ (Ω) for all t ∈ (0, T m ), and so T m = ∞; that is, the solution (u, v) exists globally with uniform-in-time bound.
Proof. For any p ≥ 2, multiplying the u-equation in (1.1) by u p−1 and integrating over Ω by parts, using Young's inequality and the growth condition (1.2), we conclude that 1 p which, upon the substitution w = u p 2 , reads as 1 p where and hereafter ϑ = θ − 1 > 0. Below, we shall apply the Gagliardo-Nirenberg interpolation inequality (2.1) to control the second integral on the right-hand side of (2.4).
Now, by assumption u(t) L r is bounded, it follows that g(u(t)) L r/κ is bounded due to the fact that g(u) ≤ βu κ . Then a simple application of the elliptic W 2,qestimate to the v-equation in (1.1) shows that v(t) W 2,r/κ is bounded. This in turn entails by Sobolev embedding that v(t) W 1,q is bounded with by the choice of r in (2.5). Then we obtain from Hölder inequality that A use of the Gagliardo-Nirenberg inequality (2.1) to (2.6) gives Notice that r > κn/2, a simple calculation from (2.8) shows that δ ∈ (0, 1) as long as Hence, for any p ≥ 2 fulfilling (4.22), the estimate (2.7) holds. Then applying Young's inequality, we conclude from (2.7) that for any 1 , 2 > 0 and some constant C depending on 1 , 2 . By Young's inequality, one has Then substituting (2.11) into (2.10), we have, for any 1 , 2 > 0, Thus, for p satisfying (4.22), by taking 1 , 2 > 0 in (2.12) such that which, together with the fact max aw for some possibly large constant C. The substitution of w = u p 2 then yields 1 p Solving this Gronwall's inequality, we deduce that u(t) L p is bounded for p satisfying (4.22) and our stipulation p ≥ 2. Now, the point-wise elliptic W 2,q -estimate applied to the v-equation in (1.1) shows that v(t) W 2,p/κ is bounded, which is embedded in C 1 (Ω) by choosing p such that p/κ > n. This shows that v(t) W 1,∞ are uniformly bounded with respect to t ∈ (0, T m ). As such, we can perform the well-known Moser iteration technique to show that the u(t) L ∞ is bounded uniformly in time t; see details in [28, p. 4290-4292]. Accordingly, the extension criterion (2.3) implies T m = ∞ and hence global existence follows. Moreover, u(t) L ∞ and v(t) W 1,∞ are uniformly bounded with respect to t ∈ (0, ∞). Remark 1. The boundedness criterion obtained in Lemma 2.3 holds also for the fully parabolic version of (1.1).

Remark 2.
Even in the absence of growth source, the assumption g(u) ≤ βu κ with κ < 2 n induces that (u(·, t), v(·, t)) is bounded in L ∞ (Ω) × W 1,q (Ω) for some q > n, cf. [2,15]. The point here is that the uniform spatial L 1 -boundedness of u is sufficient to prevent blowup of solutions. This is not usually the case as noted in the beginning of this section.
It is known from [26] that, even for a simpler chemotaxis-growth model than (1.1) with κ = 1, blow-up is still possible despite logistic dampening. Hence, there is a need to give an equivalent characterization of Lemma 2.3 in terms of blowup solutions. This means, for any blowup solution (u, v) of (1.1), u blows up not only in L ∞topology but also in L p -topology for any p > κn/2, and v blows up in W 1,∞ -topology.
Proof. If v(t) W 1,∞ is bounded, then the crucial inequality (2.6) is valid. Then one can readily see from the proof of the Lemma that u(·, t) L p is bounded.
3. The L κn/2+ -boundedness of u and global existence. In this section, we use the criterion established in Lemma 2.3 to study the boundeness and global existence for (1.1). This idea is different from [9,11]. To make our presentation self-contained, we would like to provide necessary details to make the flow of the proof of (1.11) more smooth. Also, we would like to rewrite (1.1) here for purpose of reference.
x ∈ Ω. (3.1) The for some a ≥ 0, b > 0 and θ > 1, and the production term g ∈ C 1 ((0, ∞)) and satisfies If κ < 2 n , then the boundedness for (3.1) is ensured by Corollary 1 and Remark 2. Therefore, we will consider the case κ ≥ 2 n only in the rest of this section. Then the unique classical global solution (u(·, t), v(·, t)) of the minimal chemotaxis- Proof. For any p > 1, we multiply the u-equation in (3.1) by pu p−1 and integrate the result over Ω by parts to deduce that Testing the v-equation in (3.1) against u p , we end up with Substituting (3.6) into (3.5) and using (3.2) and (3.3) yield Thanks to the relation (3.4), we conclude Then it follows from (3.7) that which, upon a use of Gronwall's inequality, yields that for any p > 1 and for any t ∈ (0, T m ). As a consequence, the L κn/2+ -boundedness criterion provided by Lemma 2.3 immediately guarantees that T m = ∞ and, furthermore, u(t) L ∞ and v(t) W 1,∞ are uniformly bounded for t ∈ (0, ∞).
Next, we explore the borderline case θ − κ = 1. In this case, we will see that the L κn/2+ -boundedness criterion in Lemma 2.3 plays a crucial role.
then the unique classical global solution (u(·, t), v(·, t)) of the chemotaxis-growth Proof. Due to Lemma 2.3, it suffices to prove that u(t) L κn/2+ is uniformly bounded for some sufficiently small > 0. To this end, for any p ≥ 2, we apply Let us first treat the strict inequality case of (3.9); that is This allows us to fix a small > 0 in such a way that Setting p = κn/2 + and using (3.12), one has [bp − (p − 1)βχ] > 0. The fact κ > 0 then ensures which combined with (3.11) leads us to This immediately shows that u(t) L κn/2+ is uniformly bounded for t ∈ (0, T m ).
Let us now examine the borderline case of (3.10). In this case, the premise b > 0, cf. (3.2), entails κn 2 > 1; then, for any p ∈ (1, κn 2 ] (nonempty), we have Accordingly, we infer from (3.11) and (3.13) that Now, we apply the Gagliardo-Nirenberg interpolation inequality (2.2), L 1 -boundedness of u and Young's equality with epsilon to derive that A combination of (3.14) and (3.15) gives rise to which coupled with (3.15) once more implies Solving this differential inequality immediately yields In the sequel, we shall prove that u L p 0 is also uniformly bounded for some p 0 > κn/2. To this end, for any p ∈ ( κn 2 , κn 2 + 1), the Hölder inequality along with (3.17) yields that which in conjunction with (3.11) allows us to conclude that where we have substituted the value of b in (3.10).
In the sequel, we wish to bound the first term on the right-hand side of (3.18) in terms of the dissipation term on its left-hand side. Case I: n > 2. In this case, Hölder's inequality shows Then we infer from the Sobolev embedding W 1,2 → L 2n n−2 , (3.15) and (3.17) that Here, we emphasize that the constant C in (3.19) is independent of p since we used only the Sobolev embedding W 1,2 → L 2n n−2 and (3.15) with η = 1.
Case II: n = 1, 2. In this case, we set q := 3p, which implies q > p + κ by the choice p > κn 2 ; then we choose A use of the Hölder inequality leads to Then we conclude from the G-N interpolation inequality (2.2) and (3.17) that where we have utilized the following facts the latter equality is due to (3.20) and (3.22). Observe that 2κn which implies that the constant C in (3.21) can be uniformly bounded in p ∈ ( κn 2 , κn 2 + 1) and then can be chosen independent of such p. That is, (3.19) is also valid in the case of II.
To sum up our discussion, we have shown that for any p ∈ ( κn 2 , κn 2 + 1), where the constant C is independent of such p. Now, we fix a Finally, we take η = 2(κn−2) κn in (3.19), and then we deduce from (3.23), (3.24), (3.25) and (3.18) a Gronwall differential inequality for u L p 0 : trivially yielding that u(t) L p 0 is uniformly bounded. Thanks to the fact p 0 > κn/2 by (3.24), the assertions of Theorem 3.2 follow as a consequence of Lemma 2.3.

Remark 3.
From the discussion in Section 3 and the work of [24] on sub-quadratic dampening enforcing the existence of global"very weak" solutions, we are led to speculate that no blow-up would occur for the minimal-chemotaxis-growth model (3.1) whenever If this turned out to be true, then it would be a significant improvement of Theorems 3.1 and 3.2 and hence of existing results (cf. [3,21,22,30]). In particular, under additional smallness assumptions, this has been verified in [24] for the KS system (3.1) with g(u) = u (or κ = 1) and f satisfying f (u) ≤ a − bu θ for all u ≥ 0 and for some a ≥ 0, b > 0 and We are unable to obtain such a sharp conjectured result via the approach described above. Innovative ways should be found to explore this speculation.
4. Steady states for the K-S model. In this section, we study the steady states of the minimal chemotaxis-growth model (3.1):  Then where K is the largest zero point of f . Furthermore, the W 2, θ κ -norm of v is uniformly bounded in χ. In particular, if f (u) = cu − bu θ , then maxΩ u ≥ K = (c/b) (θ−1) −1 .
Proof. Integrating the u-equation and using the fact f (u) ≤ a − bu θ , we have which directly gives the first two inequalities in (4.2). Then integrating the vequation, using g(u) ≤ βu κ and Hölder inequality, we arrive at the last desired inequality in (4.2).
Notice that then the elliptic regularity applied to the v-equation in (4.1) yields the stated W 2, θ κestimate for v. Especially, for f (u) = cu − bu θ , if maxΩ u < K, then f (u) > 0 on Ω and so Ω f (u) > 0, which is a contradiction.
Proof. (i) The elliptic counterpart of (3.7) is In the case θ − 1 > κ, a simple application of Young's inequality with to (4.5) shows that Ω u p+θ−1 is bounded for any p > 1; while, in the case θ − 1 = κ, it follows from (4.5) that which immediately implies u ∈ L p+κ (Ω) for any p < βχ (βχ−b) + . Then multiplying the v-equation by v q , integrating by parts and using (3.3) and Young's inequality, we deduce which, coupled with the integrability of u, yields that v ∈ L q+1 (Ω) for any q < βχ κ(βχ−b) + .
(iii) The W 2,p -elliptic regularity applied to Then the Sobolev embedding says v ∈ L ∞ (Ω).
leading to the desired upper bound for u.
(iv) Let (u, v) be a solution of (4.1). Then we test (4.8) by The positivity of f on (0, ( a b ) In a similar way, testing (4.8) by In what follows, we study the capability of the system (4.1) to form patterns. We perform Leray-Schauder index formula (The possibility of realization of such method was mentioned in [13] but not carried out even for a simpler model than (4.1)) to show that, for each equilibrium state, the stationary system (4.1) admits an increasing sequence of {χ k } ∞ k=1 such that it has at least one nonconstant solution whenever χ ∈ (χ 2k−1 , χ 2k ), k = 1, 2, · · · . More precisely, we have the following existence result for pattern formations.
the stationary chemotaxis-growth system (4.1) has at least one nonconstant solution.
Before presenting the proof, we want to remark that Theorem 4.3 not only gives the existence of non-constant solutions for (4.1) which is a generalization of the model considered in [21] where f is of logistic type, but also provides more explicit conditions which are cleaner and easier to verify. Our proof is the consequence of bifurcation from "eigenvalues" of odd multiplicity.
By definition, ((u, v), µ) is an eigen-pair of (−∆ + I) (4.14) By the idea of eigen-expansion, we let Substituting (4.15) into (4.14) and using the completeness of eigenfunctions {e j }, we obtain an algebraic system in u j and v j as follows.
which has a nonzero solution (u j , v j ) for some j if and only if Solving (4.16) and comparing (4.11), we find that the eigenvalues of (−∆+I) −1 A(χ) are (4.17) Recall that (λ + (O ± k ) ∪ λ − (O ± k )) ∩ Σ = ∅, and so 1 is not an eigenvalue of (−∆ + I) −1 A(χ) for χ ∈ O ± k . Then the Leray-Schauder index formula gives where γ ± k is the sum of the algebraic multiplicities of the real eigenvalues of (−∆ + I) −1 A(χ), χ ∈ O ± k which are greater than 1. In the case of f (ũ) < 0, since λ − (χ) < σ j for any j ≥ 1 and χ >χ 1 , we conclude from (4.17) and the properties of λ + that Here the notation (σ k ) denotes the finite algebraic multiplicity of σ k . While, in the case of f (ũ) ≥ 0, since λ − (χ) < σ j for any j ≥ 0 and χ >χ 1 , we conclude from (4.17) and the properties of λ + that Hence, in either case, we obtain (4.20) Now, if (σ k ) is an odd number, then by (4.13) and (4.20) the topological structures of L ± k and hence of H ± k change when χ crossesχ k . Indeed, by the well-known bifurcation from "eigenvalues" of odd multiplicity (cf. [17,18]), it follows thatχ k is a bifurcation value. Consequently, there exists a bifurcation branch C k containing (ũ,ṽ, χ k ) such that either C k is not compact in X × X × R or C k contains (ũ,ṽ, σ j ) with σ j = σ k .
Case 1: If, for some k, the bifurcation branch C k is not compact in X ×X ×R, then C k extends to infinity in χ due to the elliptic regularity that any closed and bounded subset of the solution triple (u, v, χ) of our chemotaxis system (4.1) in X × X × R is compact; this can be easily shown by the Sobolev embeddings and results from [14,Chapter 3], see similar discussions in [29,Proposition 4.1]. Clearly, in this case, we can find a sequence {χ k (ũ)} ∞ k=1 fulfilling the statement of the theorem. Case 2: If, for any k, the branch C k contains (ũ,ṽ,χ j ) withχ j =χ k , then we define , the system (4.1) has at least one non-constant solution. From this and the fact that σ k → ∞ andχ k = (λ + ) −1 (σ k ) → ∞ as k → ∞, a sequence {χ k (ũ)} ∞ k=1 satisfying the description of the theorem can be readily constructed. Finally, the theorem follows by unifying allũ ∈ Z.
For the constant steady state (ũ,ṽ), the length of the associated interval (χ 2k−1 , χ 2k ) of existence of nonconstant solutions is positive: This, joined with χ k → ∞, illustrates that the set P χ specified in the theorem is unbounded. However, it is yet unknown whether or not (4.1) has a nonconstant solution for χ in the complement of the unbounded set P χ .
Based on Theorem 4.3, we naturally wish to explore the asymptotic behavior of the nontrivial solutions (u, v) of (4.1) as χ → ∞. By using the a priori estimates in Lemma 4.1, we obtain the following result on their asymptotic behavior as χ → ∞.
and let (u χ , v χ ) be any positive solution of (4.1). Then there is a subsequence for some nonnegative constant M , where if θ κ ≥ n. are uniformly bounded with respect to χ. Hence, the reflexivity and Sobolev embedding allow us to find a subsequence {χ j } with lim j→∞ The last convergence in (4.23) follows from the compact Sobolev embedding W 2, θ κ (Ω) → C 0 (Ω) since θ/κ > n/2. One can easily infer from (4.2) and (4.23) that On the other hand, multiplying the first equation in (4.1) by w ∈ W 2, θ θ−1
Remark 4. Based on the merely weak convergence of {u j } in L θ (Ω), we are unfortunately unable to determine the precise values of M . The natural candidate for M is 0 or (a/b) 1 θ−1 because of (4.28). Indeed, Kuto et al [13] claimed either M = 0 or a/b for the specific choices θ = 2 and κ = 1. We underline that their claim is in general incorrect as to be discussed below. Indeed, they claimed from (4.28) that {u j } contains a subsequence, still denoted by {u j }, which converges to u ∞ almost everywhere in Ω as j → ∞. However, the equality (4.28) does not exclude oscillating functions (a priori, we do not know whether or not the the solution u j will behave like this), and hence the claim is not guaranteed in general. For example, if we take u j (x) = 1 + sin(jx), then it follows that u j 1 weakly in L 2 (0, 2π), which contradicts (4.29). Therefore, u j has no subsequence that converges a.e. to 1 in (0, 2π). The other gap of their proof lies in the application of Lebesgue dominated convergence theorem without finding the dominating function for u j . Typically, there is no dominating function for u j , since, on the one hand, the cells will aggregate when chemotactic effect is strong, and, on the other hand, we would get a stronger convergence if a dominating function was found. However, a stronger convergence than that of Theorem 4.4 seems unavailable, since boundedness results in Lemma 4.2 are not uniform with respect to χ, even in L p -topology.

5.
Large time behavior for the K-S model. In this section, we shall study the large time behavior for a specific chemotaxis-growth model with nonlinear production in the chemical equation as follows: x ∈ Ω, t > 0, x ∈ Ω, t > 0, ∂u ∂ν = ∂v ∂ν = 0, x ∈ ∂Ω, t > 0, u(x, 0) = u 0 (x), x ∈ Ω, where a ∈ R, b > 0, χ > 0, κ > 0 and Ω ⊂ R n is a bounded smooth domain with n ≥ 1. For κ = 1 and b = a, under the assumption b > 2χ, Tello and Winkler in [21] used comparison arguments to show that the solution of (5.1) converges in L ∞topology to its constant steady state (1,1). Recently, such methods were extended for a model with nonlinear chemosensitivity and secretion [5]. On the other hand, for κ = 1, He and Zheng [6] modified the energy functional method from [1] to obtain the stabilities of the constant equilibria (0, 0) and (a/b, a/b) for κ = 1 with convergence rate estimates. Here, we extend the energy functional method to undergo a comprehensive analysis for the global stabilities with explicit convergence rates of the constant steady sates ((a/b) 1 κ , a/b) and (0, 0). Our precise long time behaviors for (5.1) as t tends to infinity go as follows.
Then the global bounded solution (u, v) of (5.1) converges exponentially: for all t ≥ 0 and some large constant C κ independent of t. Here (ii) In the case of a = 0, the global solution (u, v) of (5.1) converges algebraically: for all t ≥ 0 and some large constant C κ independent of t. (iii) In the case of a < 0, the global solution (u, v) of (5.1) converges exponentially: for all t ≥ 0 and some large constant C κ independent of t.
Corollary 3. In the case of (i), the equilibrium ((a/b) 1 κ , a/b) is globally asymptotically stable; in the case of (ii) or (iii), (0, 0) is globally asymptotically stable. Thus, under the conditions of the theorem, the chemotaxis system (5.1) has no nonconstant steady state. Remark 5. Theorem 5.1 gives explicit convergence rates for (u, v), which were not explicitly stated in [6, Theorems 1 and 2] for κ = 1; besides, it extends their linear secretion case (κ = 1) to nonlinear secretion case (κ = 1). As can be easily seen from the proof below, the condition b ≥ κn−2 κn χ is merely used to ensure uniform boundedness and hence global existence. While, if we have only b > 0, then we can adapt the arguments in [21,26] to infer that the chemotaxis system (5.1) has a global weak solution which will become eventual smooth and bounded. Therefore, the decay estimates (5.4), (5.5) and (5.6) will continue to hold for t ≥ T 0 with some T 0 > 0 .
The key of the proof of Theorem 5.1 relies on finding so-called Lyapunov functionals, which are inspired from [1,6]. Here, we will present all the necessary details for the clarity of obtaining the explicit convergence rates.
Lemma 5.2. In the case of (i) of Theorem 5.1, the solution (u, v) of (5.1) Using the first equation in (5.1), we deduce from Cauchy-Schwarz inequality that (5.17) Testing the second equation in (5.1) by (u κ − a b ), we have A substitution of (5.18) into (5.17) gives rise to ).

(5.19)
Multiplying the second equation in (5.1) by (v − a b ), we get

(5.21)
A simple calculation from the second assumption in (5.2) shows that > 0 and then an integration of the above inequality from any fixed t 0 ≥ 0 to t entails and thus the nonnegativity of H yields

TIAN XIANG
Again, the global boundedness and uniform continuity of A simple use of Hölder inequality to (5.20) immediately shows and so The fact κ > 1 leads to Finally, the L 2 -convergence in (5.7) follows from (5.23) and (5.22).
Proof of (i) of Theorem 5.1. We conclude from the Gagliardo-Nirenberg inequality This together with (5.24) allows one to find t 1 ≥ 0 such that and so 1 4κc This in conjunction with (5.24) gives the existence of t 2 ≥ 0 such that and so Remark 6. In the absence of chemotaxis, i.e. χ = 0, we get from (5.17) that Consequently, the estimates (5.31) and (5.32) imply the exponential convergence: This holds true for all a > 0. While, in the presence of chemotaxis, especially, with super-linear secretion, i.e., κ > 1, we need further restrict a to satisfy a > (1 − 1 κ ) 2 as stated in (5.2) in order to have such exponential convergence. Hence, there is a gap left as to whether or not the exponential stabilization of solution still occurs when 0 < a ≤ (1 − 1 κ ) 2 ? Proof of (ii) and (iii) of Theorem 5.1. It is straightforward to check from the proofs in sections 2-4 that the sign of a does not play any role in the boundedness and global existence. Thus, (u, v) is still a global bounded classical solution under the condition of Theorem 5.1. In the case of a = 0, we integrate the first equation in shows L ∞ (Ω) Ω u, if κ > 1. This combined with (5.33) and (5.34) yields if 0 < κ ≤ 1, Then we conclude from (5.34) with u replaced by v that, t > 0, v(·, t) L ∞ (Ω) ≤ , if κ > 1.