Electronic Journal of Qualitative Theory of Differential Equations

This paper is devoted to studying the following quasilinear parabolic-ellipticelliptic chemotaxis system  ut = ∇ · (φ(u)∇u− ψ(u)∇v) + au− bu, x ∈ Ω, t > 0, 0 = ∆v− v + w1 , x ∈ Ω, t > 0, 0 = ∆w− w + u2 , x ∈ Ω, t > 0, with homogeneous Neumann boundary conditions in a bounded and smooth domain Ω ⊂ Rn(n ≥ 1), where a, b, γ2 > 0, γ1 ≥ 1, γ > 1 and the functions φ, ψ ∈ C2([0, ∞) satisfy φ(s) ≥ a0(s + 1)α and |ψ(s)| ≤ b0s(1 + s)β−1 for all s ≥ 0 with a0, b0 > 0 and α, β ∈ R. It is proved that if γ − β ≥ γ1γ2, the classical solution of system would be globally bounded. Furthermore, a specific model for γ1 = 1, γ2 = κ and γ = κ + 1 with κ > 0 is considered. If β ≤ 1 and b > 0 is large enough, there exist Cκ , μ1, μ2 > 0 such that the solution(u, v, w) satisfies ∥∥∥∥u(·, t)− ( b a ) 1 κ ∥∥∥∥ L∞(Ω) + ∥∥∥∥v(·, t)− ba ∥∥∥∥ L∞(Ω) + ∥∥∥∥w(·, t)− ba ∥∥∥∥ L∞(Ω) ≤ { Cκe1, if κ ∈ (0, 1], Cκe2, if κ ∈ (1, ∞), for all t ≥ 0. The above results generalize some existing results.

x ∈ Ω, t > 0, 0 = ∆w − w + u γ 2 , x ∈ Ω, t > 0, with homogeneous Neumann boundary conditions in a bounded and smooth domain Ω ⊂ R n (n ≥ 1), where a, b, γ 2 > 0, γ 1 ≥ 1, γ > 1 and the functions ϕ, ψ ∈ C 2 ([0, ∞) satisfy ϕ(s) ≥ a 0 (s + 1) α and |ψ(s)| ≤ b 0 s(1 + s) β−1 for all s ≥ 0 with a 0 , b 0 > 0 and α, β ∈ R. It is proved that if γ − β ≥ γ 1 γ 2 , the classical solution of system would be globally bounded.Furthermore, a specific model for γ 1 = 1, γ 2 = κ and γ = κ + 1 with κ > 0 is considered.If β ≤ 1 and b > 0 is large enough, there exist C κ , µ 1 , µ 2 > 0 such that the solution(u, v, w) satisfies 1 Introduction Chemotaxis is one of the basic physiological reactions of cells or organisms, which refers to the directional movement of biological cells or organisms along the concentration gradient of stimulants under the stimulation of some chemicals in the environment.The establishment of chemotactic mathematical model can be traced back to the pioneering work proposed by Keller and Segel [16] to describe the aggregation of cellular slime molds, which is given by where Ω ⊂ R n , τ ∈ {0, 1}, ν denotes the outward unit normal vector on ∂Ω, u(x, t) denotes the cell density and v(x, t) represents the concentration of the chemical signal.Here, f (u) describes cell proliferation and death, ∇ • (ϕ(u)∇u) and −∇ • (ψ(u)∇v) represent self-diffusion and cross-diffusion, respectively.It is well known that chemotaxis research has many important applications in both biology and medicine so that it has been one of the hottest research focuses in applied mathematics nowadays.In the past few decades, a large number of valuable theoretical results have been established.Among them, one of the main issues related to (1.1) is to study whether there is a global in-time bounded solution or when blow-up occurs.
For τ = 1, ϕ(u) = 1, ψ(u) = χu and f (u) = 0 with χ > 0, it has been shown that the system (1.1) has globally bounded classical solution when n = 1 [24] or n = 2 and Ω u 0 dx < 4π χ [5,23], whereas the system (1.1) has finite time blow-up solution in the case of n = 2 and Ω u 0 dx > 4π χ [9,26] or in the case of n ≥ 3 [36,39].Inter alia, when f (u) = u − µu 2 with µ > 0, under the restrictions that τ = 1 and Ω is convex, Winkler [40] proved that if the ratio µ χ is sufficiently large, then the unique nontrivial spatially homogeneous equilibrium given by u = v ≡ 1 µ is globally asymptotically stable.Later on, Cao [2] used an approach based on maximal Sobolev regularity and improved Winkler's results without the restrictions τ = 1 and the convexity of Ω.When the chemical substance diffuses much faster than the diffusion of cells, the system (1.1) can be reduced to the simplified parabolic-elliptic model, i.e. τ = 0.Such model was first studied for ϕ(u) = 1, ψ(u) = χu and f (u) = 0 in [14].Recently, when f (u) = Au − bu α with α > 1, A ≥ 0 and b > 0, in [35], a concept of very weak solutions was introduced, and global existence of such solutions for any nonnegative initial data u 0 ∈ L 1 (Ω) was proved under the assumption that α > 2 − 1 n , moreover, boundedness properties of the constructed solutions were studied by Winkler.Thereafter various variants of (1.1) have been considered by many other scholars [6,11,31,34].In general, diffusion functions ϕ(u) and ψ(u) may not be linear forms, such as diffusion in porous media and volume filling effect.When ϕ(u), ψ(u) are nonlinear and f (u) = 0 or f (u) = 0, a lot of scholars have studied the finite time blow-up of solution and the existence of globally bounded classical solution to system (1.1).We refer the readers to [8,12,13,37,38] for more details.
With regard to the system (1.1), the term of chemotaxis signal production v is produced directly by the cell density u.However, the mechanism of signal production might be very complex in realistic biological processes.On the one hand, the signal generation usually undergoes intermediate stages, i.e. signal v is not produced directly by cells u, but is governed by some other signal substances w.The related models can be described as where u, v, w represent the density of cells, the density of chemical substances and the concentration of indirect signal, respectively.Such problem has been widely studied in recent years.For τ = 1, ϕ(u) = 1, ψ(u) = u and f (u) = µ(u − u γ ), the authors in [46] proved that if γ > n 4 + 1 2 , then the system possesses globally bounded classical solution.Moreover, if µ is large enough, the solution (u, v, w) converges to (1, 1, 1) in L ∞ -norm as t → ∞.When ϕ and ψ satisfy some nonlinear conditions and smoothness conditions, it also has been showed that the solution to system (1.2) is globally bounded in [30].Recently, the authors in [18] have studied the system (1.2) for τ = 0, where ϕ(s) ≥ a 0 (s + 1) α and |ψ(s)| ≤ b 0 s(1 + s) β−1 for all s ≥ 0 with a 0 , b 0 > 0 and α, β ∈ R.They have proved that the nonnegative classical solution to (1.2) is global in time and bounded.In addition, if µ satisfy some suitable conditions, the solution (u, v, w) converges to (1, 1, 1) in L ∞ -norm as t → ∞.More relevant results on the system with indirect signal production can refer to [10,19].
One the other hand, the signal generation may be in a nonlinear form, which is given by where Ω ⊂ R n (n ≥ 2) is a bounded, smooth domain.When τ = 0, ϕ(u) = 1, ψ(u) = χu, f (u) = au − bu θ and g(u) = u κ with χ, b, κ > 0, a ∈ R and θ > 1, Xiang [44] obtained the global existence and boundedness of solution for (1.3) under either κ + 1 < max{θ, 1 kn χ.Besides, they studied the dynamical behavior of the solution on the interactions among nonlinear cross-diffusion, generalized logistic source and signal production.In addition, When τ = 1, ϕ(u) = 1, ψ(u) = χu, f (u) = 0 and g(u) ∈ C 1 ([0, ∞)) satisfying 0 ≤ g(u) ≤ Ku α with some constants K, α > 0, Liu and Tao [21] proved that the classical solution of the system (1.3) is globally bounded if 0 < α < 2 n .When the second equation degenerates into an elliptic equation (i.e.τ = 0), |Ω| Ω g(u), g(u) ≥ ku k for all u ≥ 0 with some k > 0, Winkler [43] derived a blow-up critical exponent k = 2 n , which asserted that the radially symmetric solution blows up in finite time if the parameter k satisfies k > 2 n .Conversely, when k < 2 n , they proved that there exists suitable initial value such that the system has globally bounded classical solution.Later on, the authors in [45] considered the case f (u) = λu − µu α with λ, µ > 0 and α > 1, and they generalized the blow-up results developed in [43] with k + 1 > α 2 n + 1 .Intuitively, the existing literatures show that the logistics source (i.e.f (u) = λu − µu α with λ, µ > 0 and α > 1) and its possibly damping behavior have important influences on the behavior of the solution.For instance, the strong logistic damping (i.e.µ is suitably large) may ensure the system has globally bounded classical solution, especially in higher-dimensional case.More precisely, when α = 2, Tello and Winkler [29] proved that for all suitably regular initial data, the system had a unique globally bounded classical solution if µ > max{0, n−2 n χ}.Afterwards, Cao and Zheng [3] proved that such global solution to a quasilinear system (1.3) is also known to exist for all nonnegative and smooth initial data if µ is suitably large.However, "logistic source" does not always prevent chemotactic collapse.When α = 2, such assertion was verified in [41] for one-dimensional case by Winkler, and also could be found in [15] for higher-dimensional setting.Recently, Winkler [42] obtained a condition on initial data to ensure the occurrence of finite-time blow-up to system (1.3) for Some boundedness or blow-up results to variants of system (1.3) can also be found in [20,22,25,32,33,47].Among the existing literatures, it is not difficult to find that there are very few papers to study the chemotaxis system, where chemical signal production is not only indirect but also nonlinear.Based on the complexity of biological process, such signal production mechanism could be more in line with the actual situation.Inspired by the above works, in this paper, we are concerned with the following system where Ω ⊂ R n (n ≥ 1) is a bounded domain with smooth boundary, ν denotes the outward unit normal vector on ∂Ω, the parameters satisfy a, b, γ 2 > 0, γ 1 ≥ 1 and γ > 1, and ϕ(u), ψ(u) are self-diffusion and cross-diffusion functions, respectively.Since from a physical point of view, the equation modeling the migration of cells should rather be regarded as nonlinear diffusion [27].Thus, here we assume that the diffusion functions ϕ, ψ for all s ≥ 0 with a 0 , b 0 > 0 and α, β ∈ R.
The main purpose of the present paper is to explore the interplay of nonlinear diffusion functions ϕ, ψ and logistic source term au − bu γ as well as nonlinear indirect signal production mechanism for system (1.4).To the best of our knowledge, studying the fully parabolic chemotaxis system need to use the method of variation-of-constants formula and heat semigroup, which can not be applied to the system (1.4).In this paper, we shall use a different method to reveal the influence of nonlinear diffusion functions ϕ, ψ and logistic source term au − bu γ as well as nonlinear indirect signal production mechanism on the dynamical behavior of the solution to system (1.4).
Firstly, we state our boundedness result to system (1.4) as follows.
Theorem 1.1.Let Ω ⊂ R n (n ≥ 1) be a bounded and smooth domain.Assume that a, b, for all t > 0.
In contrast to the boundedness criterion obtained in [18], the boundedness condition in Theorem 1.1 is more generalized involving nonlinear diffusion and logistic source term as well as nonlinear indirect signal production mechanism.
From the viewpoint of biological evolution, it has profound theoretical and practical significance to study the long time behavior of chemotaxis system.Based on [7,18,44], we have also studied the long time behavior of solution to a special case (see system (3.1) in Section 3) of system 1.1 (i.e.γ 1 = 1, γ 2 = κ and γ = κ + 1 with κ > 0).Here, it should be pointed out that from the above Theorem 1.1 if β ≤ 1, the corresponding system has globally bounded classical solution for this case.Thus, from Theorem 1.1, there exists R > 0 independent of a, b, α, β, a 0 , b 0 and κ such that holds on Ω × [0, ∞).Moreover, we can also find λ > 0 independent of a, b, a 0 , b 0 and κ such that Therefore, the second conclusion of this paper can be stated as then there exists C κ > 0 large enough such that the classical solution (u, v, w) to system for all t ≥ 0, where and with R > 0 and λ > 0 defined in (1.8) and (1.9), respectively.
The results in Theorem 1.2 are similar to those in [44, Theorem 5.1(i)], but more general, since self-diffusion, cross-diffusion and indirect secretion mechanism are involved.We need to modify the method in [44] to overcome the difficulties from these terms (see (3.10) and (3.25) in the proof of Lemma 3.2).In addition, our conclusion in Theorem 1.2 can also be seen as an extension of [7] or [18].Comparing with [7], in Theorem 1.2, we calculate the exponential convergence rate explicitly in terms of the model parameters with diffusion functions, generalized logistic source and nonlinear indirect secretion.But in [7], the convergence rate estimates were derived but not stated explicitly (see [7,Theorem 1]) for special logistic source and linear secretion.Comparing with [18], since our model is nonlinear indirect production, we have to divide the range of κ into (0, 1] and (1, +∞) to construct different functionals A(t) and H(t) (see Lemma 3.2) to prove Theorem 1.2.
Remark 1.3.It is relevant to point out that by the limitation of the method, we also have no idea the long time behavior of solution to system (1.4) for generalized parameters γ 1 , γ 2 and γ satisfying the condition in Theorem 1.1.
The outline of this paper is as follows.In Section 2, the global existence and boundedness of classical solution to (1.4) is proved.In Section 3, by applying the method of energy functional, we obtain that the solution to system (3.1)exponentially converges to the point

Global existence and boundedness
In this section, we will obtain the existence and boundedness of globally classical solution to system (1.4).At the beginning, we give a statement on the local existence of classical solutions.The proof depends on the Schauder fixed theorem.We omit it for brevity and refer the readers to [30] for more details.
Proof.Multiplying the first equation of system (1.4) by (u + 1) p−1 and integrating by parts over Ω, we derive for all t ∈ (0, T max ).Since ϕ satisfies (1.5), we can estimate the first term on the right-hand side of (2.18) as and for all t ∈ (0, T max ).From (2.20) and (2.21), we can get (p − 1) for all t ∈ (0, T max ).By the basic inequality (u + 1) γ < 2 γ (u γ + 1) with γ > 1, we have for all t ∈ (0, T max ), where we have made use of the second identity 0 = ∆v − v + w γ 1 in system (1.4).In the sequel, we estimate (2.24) in two different cases.Case 1 (γ − β > γ 1 γ 2 ).In this case, using Young's inequality, we can derive with applications of Young's inequality, we get from Lemma 2.2 with θ = p+γ−1 where for all t ∈ (0, T max ), which means that (2.31) Thus we can get the conclusion immediately by the ODE comparison principle.
Proof of Theorem 1.1.Let a, b, γ 2 > 0, γ 1 ≥ 1, γ > 1 and (u, v, w) be a solution of system (1.4).From Lemma 2.3, for any p > max{1, 1 − β}, there exists C 13 > 0 such that u L p (Ω) ≤ C 13 for all t ∈ (0, T max ).By the elliptic L p -estimate applied to the second and third equations in system (1.4), we have for all t ∈ (0, T max ), with some C 14 > 0. Using the Sobolev imbedding theorem, we can get for all t ∈ (0, T max ), with some C 15 > 0. Thus by standard Alikakos-Moser iteration ( [28, Lemma A.1]), we can find a constant C 16 > 0 such that for all t ∈ (0, T max ), which together with Lemma 2.1 implies that T max = ∞.Hence, by standard elliptic regularity theory, we know that (u, v, w) is a globally bounded classical solution of system (1.4).The proof of Theorem 1.1 is completed.

Long time behavior of the solution for a specific model
In this section, we shall study the long time behavior of the solution for a specific model (i.e.
with nonlinear indirect signal production and logistic source as follows where Ω ⊂ R n (n ≥ 1) is a bounded and smooth domain, the parameters a, b, κ > 0 and functions ϕ, ψ ∈ C 2 [0, ∞) satisfy conditions (1.5) and (1.6), respectively.Based on Theorem 1.1, it is easy to check that if β ≤ 1, then the system (3.1)admits a unique globally bounded classical solution (u, v, w).Furthermore, such classical solution (u, v, w) may be strictly positive which can be ensured by choosing some suitable 0 ≤ u 0 ∈ C( Ω) from Theorem 1.1.Thus let us assume that the classical solution (u, v, w) to system (3.1) is strictly positive throughout the proof of Theorem 1.2.For the convenience, we repeat the description stated in (1.8) and (1.9), i.e. there exists R > 0 which does not depend on a, b, α, β, a 0 , b 0 and κ such that 0 holds on Ω × [0, ∞).Moreover, we can also find λ > 0 independent of a, b, a 0 , b 0 and κ such that In order to prove Theorem 1.2, we introduce a useful lemma.
Using Young's inequality and the fact (3.3), we deduce from the first equation of system (3.1) Multiplying the third equation in system (3.1) by w − c κ , we get Similarly, multiplying the second equation in system (3.1) by v − c κ , we derive Substituting (3.8) and (3.9) into (3.7), by Young's inequality we see For κ ∈ (0, 1], we have the following basic inequality Thus, from (3.10) and (3.11), we derive where δ = b − λb 2 0 c κ 16a 0 .For any t 0 ≥ 0, integrating both sides of (3.12) on [t 0 , t], one can obtain Since A(t) ≥ 0 and δ is nonnegative ensured by b > b 0 4 λa a 0 .Thus From Theorem 1.1, we know that (u, v, w) is a globally bounded classical solution.Hence, by standard parabolic regularity for parabolic equations [17], we can find σ ∈ (0, 1) and C > 0 such that This clearly implies that Ω (u − c)(u κ − c κ ) is globally bounded and uniformly continuous with respect to t.Using (3.11) once again, we can obtain from Lemma 3.1 On the other hand, using Young's inequality to (3.8), we get and so Similarly, and so for some ξ between R κ and c κ .Thus For κ ∈ (1, +∞), we define the following functional for u > 0. We can easily obtain the function h(s) = s − a b − a b ln( bs a ) has global minimum zero over (0, ∞) at s = a b .Thus where ϑ = Similarly, for the third equation, we get where = b − ϑ 4 .By the assumption (1.10) in Theorem 1.2, we know that > 0. Then for any t 0 ≥ 0, an integration of the inequality (3.28) from t 0 to t entails Thus the nonnegativity of H yields From Lemma 3.1, the global boundedness and uniform continuity of A simple use of Young's inequality to (3.27) immediately shows and so Similarly, we have L ∞ (Ω) L 2 → 0 as t → ∞. (3.38) For κ ∈ (0, 1], by the L'Hospital rule, we get Based on (3.38) and(3.39),we choose t 1 > 0 such that and so ≤ C κ (4κc 2κ−1 A(t 1 ))