Boundedness in a two-dimensional chemotaxis system with signal-dependent motility and logistic source

In this paper, we study the following chemotaxis system with a signal-dependent motility and logistic source: {ut=Δ(γ(v)u)+μu(1−uα),x∈Ω,t>0,0=Δv−v+ur,x∈Ω,t>0,u(x,0)=u0(x),x∈Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \textstyle\begin{cases} u_{t}=\Delta {\bigl(\gamma (v)u\bigr)}+\mu u\bigl(1-u^{\alpha}\bigr), &x \in \Omega , t > 0, \\ 0=\Delta v-\ v+u^{r} , &x\in \Omega , t > 0, \\ u(x, 0) = u_{0}(x), &x\in \Omega \end{cases} $$\end{document} under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Omega \subset \mathbb{R}^{2}$\end{document}, where the motility function γ(v)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\gamma (v)$\end{document} satisfies γ(v)∈C3([0,∞))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\gamma (v)\in C^{3}([0,\infty ))$\end{document} with γ(v)>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\gamma (v)>0$\end{document}, and |γ′(v)|2γ(v)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\frac{|\gamma '(v)|^{2}}{\gamma (v)}$\end{document} is bounded for all v>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$v > 0$\end{document}. The purpose of this paper is to prove that the model possesses globally bounded solutions. In addition, we show that all solutions (u,v)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(u, v)$\end{document} of the model will exponentially converge to the unique constant steady state (1,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(1, 1)$\end{document} as t→+∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$t\rightarrow +\infty $\end{document} when μ≥K41+r\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mu \geq \frac{K}{4^{1+r}}$\end{document} with K=max0


Introduction
Experimental observations show that colonies of bacteria and simple eukaryotes can generate complex shapes and patterns.In order to understand the mechanism of pattern formations, extensive mathematical models were derived, including the Keller-Segel system in [10] modeling the pattern formations driven by chemotactic bacteria x ∈ , t > 0, (1.1) where the cell diffusion rate μ and the chemotactic sensitivity χ are assumed to depend only on the signal concentration.In (1.1), the cell diffusion rate μ and chemotactic sensi-tivity χ are linked via χ(v) = (σ -1)μ (v).
Here the parameter σ denotes the ratio of effective body length to step size, and γ (v) < 0 (resp.> 0) if the diffusive motility decreases (resp.increases) with respect to the chemical concentration.
One notices that σ = 0 implies that χ = -μ , the motion of cells is biased by the local concentration chemotactic signal and is prescribed by the motility γ (v) of cells, so that the system (1.1) can be written as x ∈ , t > 0, with homogeneous Neumann boundary conditions.Recently, the mathematical research on the system (1.2) has attracted a lot of interest.Tao and Winkler [22] considered the chemotaxis model with τ = 1 under the condition that the motility function γ (v) satisfies γ ∈ C 3 ([0, ∞)), C 1 < γ (v) < C 2 , and |γ (v)| ≤ C 3 for all v > 0, and they showed that the chemotaxis model has globally bounded solutions.Yoon and Kim [23] proved that there exists a globally bounded solution for the system (1.2) with τ = 1 and γ (v) = C 0 v k (k > 0) for small constant C 0 > 0 in any dimension.Lately, the smallness assumption of C 0 > 0 was removed for the parabolic-elliptic case of (1.2) with τ = 0 when n ≤ 2, k > 0 or n ≥ 3 with k < 2 n-2 ; Ahn and Yoon in [1] proved that the model (1.2) possesses a globally bounded and classical solution.
In [7], Jiang and Laurençot proved the existence of a global and bounded classical solution to the model (1.2) with τ = 0 when γ ∈ C 3 ((0, ∞)), γ > 0, K v = sup s∈[v,∞] {γ (s)} < ∞, v > 0 and if n ≥ 2, u 0 satisfying u 0 (x) ∈ C 0 ( ), u 0 (x) ≥ 0, u 0 (x) ≡ 0, x ∈ . (1.3) In addition, these solutions were shown to be uniformly bounded with respect to time when γ ∈ C 3 ((0, ∞)), γ ≤ 0, and there are k ≥ l > 0 such that lim v→∞ inf v k γ (v) > 0 and lim v→∞ sup v l γ (v) < ∞ for any k < n n-2 and kl < 2 n-2 if n ≥ 3 and u 0 satisfies (1.3).Considering the presence of cell generation and death in a biological realistic setting, this can be expressed through logical sources.Researchers usually considered the following density-suppressed motility chemotaxis model with logical sources: x ∈ , t > 0. (1.4) Essentially, (1.4) with μ > 0 has been used in [4] to justify that the bacterial motion with density-suppressed motility (i.e., γ (v) < 0) can produce the stripe pattern formation observed in the experiment of [11].Fujie and Jiang [5] proved the global existence of a classical solution for (1.4) in the two-dimensional setting when u 0 was assumed to be as in (1.3) and and γ (v) also satisfies lim v→+∞ v k γ (v) < +∞, for k > 0, the global solution to (1.4) in the two-dimensional case is proven to be bounded uniformly in time.Recently, Lyu and Wang [12] explored how strong the logistic damping can warrant the global boundedness of solutions to (1.4) under the minimal conditions for the density-suppressed motility function γ (v), and further established the asymptotic behavior of solutions under some conditions.In order to address the dependence of dynamical behaviors of solutions on the interactions between nonlinear cross-diffusion, generalized logistic source, and nonlinear signal production, Tao and Fang [17] considered the density-suppressed motility mode They proved that if β < 2 N+2 α, then the system has a globally bounded classical solution, and further established the asymptotic behavior of solutions for sufficiently large μ.
Inspired by the above works, in this paper we consider the initial-Neumann boundary value problem of the following parabolic-elliptic system: in a smooth bounded domain ⊂ R 2 .Here μ, α, r > 0 are any given constants.We denote the cell density by u and the chemical concentration by v.In order to study the large-time behavior of the system (1.5), we assume that the motility function γ (v) satisfies: In this paper we shall develop some results on the global boundedness and large-time behavior of solutions to the system (1.5) with general motility γ (v).The main result of this paper reads as follows.

Theorem 1.2 Under the same assumptions as in Theorem
, then the globally bounded solution of (1.5) satisfies Moreover, the convergence rate is exponential in the sense that there exist constants λ, C such that

Preliminaries
We first recall the local existence of classical solutions to equations (1.5 to (1.5).Here, T max ∈ (0, ∞] denotes the maximal existence time. At the end of this section, we state some a priori estimates on u, v, which shall be used in the sequel.Lemma 2.2 Let (u, v) be the solution of the system (1.5).Then it holds that u(•, t) dx ≤ max u 0 L 1 ( ) , | | := m * , for all t ∈ (0, T max ), (2.2) and v(•, t) dx ≤ C, for all t ∈ (0, T max ). (2.3) Proof Integrating the first equation of (1.5) over , we obtain Due to the Hölder inequality, we conclude that , for all t ∈ (0, T max ), (2.5) and hence (2.2).Now (2.3) results from a time integration of to the second equation of (1.5).As the parameters satisfy 1+2α 2(1+α) ≤ r ≤ 1 ≤ α, one has and hence (2.3) follows.

Boundedness of solutions
In this section, we shall prove Theorem 1.1.First, we show the global existence of uniformly-in-time bounded solutions.
Lemma 3.1 Let the same assumptions as in Theorem 1.1 hold.Then there exists a constant C > 0 independent of t such that the solution of (1.5) satisfies Proof Multiplying the first equation of (1.5) by ln u, and integrating the result by parts, one has for all t ∈ (0, T max ).From the assumptions in (H), we can find a constant K > 0 such that We now estimate the first integrals on the right of this inequality (3.2).Using the Hölder and Young inequalities, we have for all t ∈ (0, T max ), and substituting into (3.2) gives Applying the Agmon-Douglis-Nirenberg L p estimates (cf.[1,2]) to the second equation of (3.5) with homogeneous Neumann boundary conditions, we know that for all p > 1, there exists a constant The Sobolev embedding theorem yields ∇v in two dimensions (i.e., n = 2) which, together with (3.6), implies 1+2α ( ) On the other hand, using the L p -interpolation inequality and the fact u(•, t) L 1 ( ) ≤ m * (see Lemma 2.2), with positive parameters satisfying 1+2α 2(1+α) ≤ r ≤ 1 ≤ α, we have the last inequality holding due to the Young inequality.We substitute (3.7) and (3.8) into (3.5) to obtain where we have used the following fact (see [21,Lemma 3.1]): Let μ > 0, α ≥ 1, and b ≥ 0, then there exists a constant l := l(μ, b, α) > 0 such that Hence from (3.9), we obtain which gives u ln u dxu dx ≤ C 9 and then Since u ln u ≥ -1 e , we derive which yields (3.1).
Next, we will show that there exists some p > 1 close to 1 such that u p dx is uniformly bounded in time.The following basic statement can be found in [6,Chap. 4 Proof The assertion easily follows by simply maximizing x ∈ (0, ∞) → xy-1 β x ln x for fixed β > 0 and y > 0.
We next provide a specific estimate for the solution of a linear elliptic equation with a source term in L 1 .Lemma 3.4 Let ⊂ R 2 be a bounded domain with a smooth boundary.Then for all M > 0 there exist β > 0 and C > 0 such that and the solution v of Therefore, invoking Jensen inequality, (3.12), and Fubini theorem, we find that for each β > 0, dy.
If we now fix β > 0 to be small enough fulfilling βLM < 2, then C 1 := sup y∈ |x -y| -βLM dx is finite due to the boundedness of , and therefore holds for any such β > 0.
Lemma 3.5 Let ⊂ R 2 be a bounded domain with a smooth boundary.There exists C > 0 such that the solution of (3.13) satisfies Proof Now in order to prove (3.16), we use v as a test function in (3.13) to obtain We find β > 0 and C 1 , C 2 > 0 such that e βv dx ≤ C 1 , u ln u dx ≤ C 2 .We employ Lemma 3.3 to estimate which, together with (3.17), establishes (3.16).
Next, we will show that there exists some p > 1 close to 1 such that u p dx is uniformly bounded in time.
Lemma 3.6 Suppose the conditions in Theorem 1.1 hold.Then there exists p > 1 close to 1 such that u(•, t) L p ( ) ≤ C, for all t ∈ (0, T max ), (3.18)where C > 0 is a constant independent of t.
Proof We multiply the first equation of (1.5) by u p-1 to obtain The Cauchy-Schwarz inequality and (3.3) allow us to have (3.20) Using the Hölder and Gagliardo-Nirenberg inequalities, as well as (3.16) and (3.6), one has Next, we will show v(•, t) L ∞ ( ) is uniformly bounded in time, which rules out the possibility of degeneracy.Lemma 3.7 Suppose the conditions in Theorem 1.1 hold.Then there exist constants C, γ 1 , γ 2 > 0 such that v(•, t) L ∞ ( ) ≤ C, for all t ∈ (0, T max ). (3.26) (3.27) Proof From Lemma 3.6, we can find a constant C 1 > 0 such that u(•, t) L p ( ) ≤ C 1 for some p > 1.Then applying the elliptic regularity estimate to the second equation of (1.5), one has which, combined with (3.27), gives We differentiate the second equation of the system (1.5) and multiply the result by 2∇v to obtain where we have used the identity |∇v| 2 = 2∇v • ∇ v + 2|D 2 v| 2 .Then multiplying (3.32) by |∇v| 2 and integrating the results, we have With the inequality ∂|∇v| 2 ∂ν ≤ 2λ|∇v| 2 on ∂ (see [13,Lemma 4.2]) and the following trace inequality (see [16, Remark 52.9]) for any ε > 0: By the Gagliardo-Nirenberg inequality and the fact |∇v| 2 L 1 ( ) = ∇v 2 L 2 ( ) ≤ C 2 (see Lemma 3.5), we have (3.39) With the boundedness of u L 1 ( ) and u ln u L 1 ( ) and the inequality in [15, Lemma 3.5], we can choose ε small enough to obtain (3.40) On the other hand, using the Gagliardo-Nirenberg inequality, we can derive that This gives (3.29) with the help of the ODE comparison principle.
Next, we shall show the boundedness of u(•, t) L ∞ ( ) .To this end, we first improve the regularity of v.More precisely, we have the following results.Lemma 3.9 Suppose the conditions in Theorem 1.5 hold.Then we have where C > 0 is a constant independent of t.
Proof Using (3.6) and (3.29), we can derive that v(•, t which, by the Gagliardo-Nirenberg inequality, gives Then multiplying the first equation of (1.5) by u 2 and integrating over by parts, one obtains 1 3 which, subject to the facts (3.3) and (3.43), gives rise to 1 3 Using the Gagliardo-Nirenberg inequality with the fact u On the other hand, using the Hölder and Young inequalities, one has By the ODE comparison principle, we have Using the elliptic regularity (3.6) and Sobolev embedding theorem again, from (3.47) we derive This finishes the proof.
Proof of Theorem 1.1 With the aid of Lemma 3.9 and a Moser-type iteration (cf.Lemma 3.6 in [9] or Lemma A.1 in [18]), we obtain that u is bounded in (0, T max ).Thus, we can find a positive constant C independent of t such that which, together with Lemma 2.1, shows that T max = ∞.Therefore, (u, v) is a global bounded classical solution to the system (1.1) and the proof of Theorem 1.1 is completed.

Large time behavior
In this section, we will study the large-time behavior of the solution for the system (1.5).Let and Then based on some ideas in [8,19], we shall show that the constant steady state (1, 1) is globally asymptotically stable by showing A(t) is a Lyapunov functional under the conditions μ ≥ K 4 1+r .We next introduce the following lemma, which is a useful tool in this section.
We construct an appropriate energy function to the system (1.5), which is prepared for the proof of the large-time behavior.Lemma 4.2 Suppose (u, v) is the solution of (1.5) obtained in Lemma 2.1.Let K and A(t) be defined by (4.1) and (4.2), respectively.Then we have the following results: (1) A(t) ≥ 0 for any t > 0.
Next, we shall show that the convergence rate is exponential.Lemma 4.4 Assume that μ ≥ K 4 1+r and (u, v) is the global classical solution of the system (1.5).Then there exist two positive constants ε > 0, C > 0 such that for all t > T 1 > 0,

.26)
Proof By using L'Hôpital's rule, we obtain According to (4.5) and (4.27), there exists ε > 0 such that By the Grönwall inequality, we readily conclude that Similarly, combining with (4.24), we can obtain which finishes the proof of Lemma 4.4.
Next, we shall show the boundedness of ∇u L 4 to obtain the convergence rate with the L ∞ ( )-norm.More precisely, we have the following result.Using the identity |∇u| 2 = 2∇u • ∇ u + 2|D 2 u| 2 , we can estimate the term J 1 as follows: Using the boundedness of u L ∞ ( ) and v W 1,∞ ( ) obtained in Theorem 1.1, assumptions (H), as well as the fact v = vu r , we have ) satisfies e βv dx ≤ C. (3.14) Proof Since (3.14) is trivial in the special case u ≡ 0, we may assume that u ≡ 0. Then with G denoting Green's function of -+1 in under homogeneous Neumann boundary conditions, v can be represented as v(x) = G(x, y)u r (y) dy, a.e.x ∈ , see [14, p. 43].Now Lemma 3.2 provides A > 0 and L > 0 such that v(x) ≤ L ln A |x -y| • u r (y) dy, a.e.x ∈ .
Lemma 3.2 Suppose that ⊂ R 2 is a bounded domain with a smooth boundary, and let G denote Green's function of -+ 1 in subject to Neumann boundary conditions.Then we have G(x, y) ≤ L ln A |x -y| , for all x, y ∈ with x = y.(3.10) 2 • ∇u∇u • ∇v dx (4.32) 2 |∇u| 2 |∇v| dx.