Boundedness in logistic Keller--Segel models with nonlinear diffusion and sensitivity functions

We consider the following fully parabolic Keller-Segel system \begin{document}$\left\{\begin{array}{ll}u_t=\nabla · (D(u) \nabla u-S(u) \nabla v)+u(1-u^γ),x'>over a multi-dimensional bounded domain \begin{document}$Ω \subset \mathbb R^N$\end{document} , \begin{document}$N≥2$\end{document} . Here \begin{document}$D(u)$\end{document} and \begin{document}$S(u)$\end{document} are smooth functions satisfying: \begin{document}$D(0)>0$\end{document} , \begin{document}$D(u)≥ K_1u^{m_1}$\end{document} and \begin{document}$S(u)≤ K_2u^{m_2}$\end{document} , \begin{document}$\forall u≥0$\end{document} , for some constants \begin{document}$K_i∈\mathbb R^+$\end{document} , \begin{document}$m_i∈\mathbb R$\end{document} , \begin{document}$i=1, 2$\end{document} . It is proved that, when the parameter pair \begin{document}$(m_1, m_2)$\end{document} lies in some specific regions, the system admits global classical solutions and they are uniformly bounded in time. We cover and extend [ 22 , 28 ], in particular when \begin{document}$N≥3$\end{document} and \begin{document}$γ≥1$\end{document} , and [ 3 , 29 ] when \begin{document}$m_1>γ-\frac{2}{N}$\end{document} if \begin{document}$γ∈(0, 1)$\end{document} or \begin{document}$m_1>γ-\frac{4}{N+2}$\end{document} if \begin{document}$γ∈[1, ∞)$\end{document} . Moreover, according to our results, the index \begin{document}$\frac{2}{N}$\end{document} is, in contrast to the model without cellular growth, no longer critical to the global existence or collapse of this system.


1.
Introduction. This paper investigates the following fully parabolic Keller-Segel system for (u, v) = (u(x, t), v(x, t))        u t = ∇ · (D(u)∇u − S(u)∇v) + u(1 − u γ ), x ∈ Ω, t > 0, v t = ∆v − v + u, x ∈ Ω, t > 0, ∂u ∂ν = ∂v ∂ν = 0, x ∈ ∂Ω, t > 0, u(x, 0) = u 0 (x) ≥ 0, v(x, 0) = v 0 (x) ≥ 0, x ∈ Ω, (1.1) where Ω is a bounded domain in R N , N ≥ 2, with smooth boundary ∂Ω. (1.1) is a Keller-Segel model of chemotaxis, which describes the oriented movement of cellular organisms in response to heterogeneous spatial distribution of a chemical in the environment. Here u(x, t) denotes cell population density at space-time location (x, t), v(x, t) the chemical concentration. D(u) is the density-dependent motility of cells and it measures the ability of cells to move randomly in the environment, and S(u) reflects the variation of cellular sensitivity with respect to the levels of cell 5022 QI WANG, JINGYUE YANG AND FENG YU population density. The standard choice of logistic growth for the cell populations is made with intrinsic growth rate scaled to the unit; moreover, the attractive chemical is released by the cells and consumed by an enzyme in the environment meanwhile, both at a constant rate scaled to the unit. γ is a positive constant that measures the cellular competition intensity when the population is crowded. ν is the unit outer normal to the boundary ∂Ω and the non-flux boundary conditions interpret the assumption that the region is enclosed and inhibits both cellular and chemical flux across the boundary. The initial cell distribution u 0 ∈ C 0 (Ω) and chemical concentration v 0 ∈ C 1 (Ω) are assumed to be nonnegative but not identically zero. The goal of this paper is to establish the global existence and uniform boundedness for (1.1). Throughout this paper, we shall assume that D(u) and S(u) are C 2 -smooth functions of u and there exist constants K i ∈ R + , m i ∈ R, i = 1, 2, such that 0 < D(0), D(u) ≥ K 1 u m1 , ∀u ≥ 0, (1.2) and 0 ≤ S(u) ≤ K 2 u m2 , ∀u ≥ 0. (1.3) From the viewpoint of mathematical modeling, it is realistic to consider chemotaxis in 2D or 3D regions. Thus, we shall restrict our attention to the case of multidimensional domains with N ≥ 2, while the analysis carries over to 1D (See Remark 3 below). Our first main result is the following theorem. Then for any nonnegative initial data (u 0 , v 0 ) ∈ C 0 (Ω)×C 1 (Ω), there exists a unique couple (u, v) of nonnegative bounded functions in C 0 (Ω × [0, ∞)) ∩ C 2,1 (Ω × (0, ∞)) which solves (1.1) classically. Moreover, there exists a positive constant C such that the solution is uniformly bounded in time in the sense u(·, t) L ∞ (Ω) + v(·, t) L ∞ (Ω) < C, ∀t ∈ (0, ∞).
The logistic-type growth in first equation of (1.1) demonstrates the prevention of population overcrowding due to the competition between cellular organisms, in particular when the resource is limited. It is quite natural to expect that such damping effect tends to prevent blowup in (1.1) and a larger γ has a stronger damping effect, however, whether or not it is sufficient remains unknown to this date, even for the classical Keller-Segel models with m 1 = 0, m 2 = 1. See [15,27] for instance. Theorem 1.1 fails to reveal this fact due to technical reasons. To tackle this issue, we investigate the effect of γ and present the second main result of this paper in the following theorem. (1.5) Then (1.1) has a global and bounded classical solution and all the conclusions in Theorem 1.1 hold.
1.1. Keller-Segel models with nonlinear diffusion. Theoretical and mathematical modeling of chemotaxis dates back to the works of Patlak [18] in the 1950s and Keller-Segel [12,13,14] in the 1970s. The chemotaxis PDE systems and their variants, now often called Keller-Segel models, have been extensively studied by many authors over the past few decades. We refer to the review papers [2,6,7,8] for detailed descriptions of the models and their developments.
For example, in order to investigate the density-dependent chemotactic movement, considerable effort has been devoted to Keller-Segel models with nonlinear diffusion and sensitivity functions. For example, Painter and Hillen [17] proposed and studied the following volume-filling model   where Q(u) denotes the density-dependent probability of the cell finding space at its neighboring location. It is assumed in [17] that Q becomes identically zero for u large enough. In particular, the authors considered a specific example with Q(u) = max{ū − u, 0} to describe a situation where the intensity of cellular movement is lessened as the cell density increases, while cells become immobile where the density surpasses the critical levelū. This choice is made based on the so-called volumefilling effect: the movement of cells is inhibited near points where the cells are already packed. Global existence and boundedness of (1.6) are obtained in [5,17].
In [25], Winkler relaxed the decay assumption on Q(u) above and proposed the following model with general choices of smooth diffusion and sensitivity functions It is proved in [25] that, when Ω ⊂ R N , N ≥ 2, is a ball and S(u) D(u) ≥ Cu 2 N + , ∀u ≥ 1, for some C > 0 and > 0, (1.7) always possesses blow-up solutions regardless of the size of total cell population; under further technical conditions on D(u) and S(u), blowup always occurs for any bounded domain Ω ⊂ R N , N ≥ 3. The fact that the blowup is finite in time is revealed in [4]. On the other hand, Tao and Winkler [21] proved that, when the bounded domain Ω is convex, the solutions to (1.7) are global and bounded in time if S(u) D(u) ≤ Cu 2 N − , while the convexity condition was recently removed by Ishida et al. in [9]. Moreover, Ishida and Yokota [10] showed that (1.7) has global existence when S(u) D(u) ≥ Cu 2 N if the initial data are small; however the solutions may blow up in finite or infinite time if the initial data are large. Denote u α for u large, then generally speaking, the index 2 N is critical to (1.7) in the sense that its solutions exist globally and remain bounded in time if α < 2 N , while there may exist solutions which blow up in finite time if α ≥ 2 N . We would like to mention that the same criticality of S(u) [11,19] and the references therein. 3 ) is 1 2 . Note that in the lower right plot, , for better illustration.

1.2.
Literature review and illustration of main results. Concerning the question to what extent the logistic growth of u allows or prevents chemotactic collapse, (1.1) has been investigated by various authors. First of all, according to our discussions above, it is easy to see that the arguments for the global well-posedness of (1.7) in [21] naturally carry over to (1.1) thanks to the presence of logistic damping. This observation was confirmed by Wang et al. in [22], and they proved that, if 0 < m 2 − m 1 < 2 N and γ > 0, (1.1) possesses a unique global classical solution which is nonnegative and uniformly bounded in time. Note that the index 3N +2 is larger than 2 N for N ≥ 3 (both equal 1 when N = 2) and Theorem 1.1 covers and extends [22] when N ≥ 3 and γ ≥ 1. Recently, Zhang and Li [28] when γ ≥ 1, (1.1) possesses a unique global classical solution which is uniformly bounded. Moreover, we note that Theorem 1.2 covers and extends [28]. In Figure 1, we present a schematic illustration of our results in Theorem 1.1 and Theorem 1.2. See Remarks 1 and 2. For each pair of (m 1 , m 2 ) in the shaded region of each plot as γ varies, (1.1) admits global bounded classical solutions. In particular, when γ ≥ 1, Theorem 1.1 extends one borderline L 1 to L * 1 . Here the borderlines have equations . On the other hand, it seems necessary to point out that, under conditions (1.2) and (1.3), Cao [3] proved that if m 2 < 1 and γ = 1, (1.1) admits a solution which is classical and uniformly bounded for any m 1 ∈ R. Therefore, the damping effect of logistic growth alone is sufficient to guarantee the global existence and boundedness for (1.1) as long as the sensitivity function is a sublinear function of cell density, regardless of the size of m 1 . Combining the results from [22], Zheng [29] generalized [3] Figure 2, we give a schematic illustration on these results and provide a relatively complete summary on the global existence for (1.1) with nonlinear diffusion and sensitivity functions. In this paper, indicate that, in contrast to (1.7) without cellular growth, the index 2 N is no longer critical to (1.1) in the presence of logistic cellular growth. Our results overlap with the results in [3,22,28,29] when N = 2 and extends theirs when N ≥ 3, in particular when m 1 is large.
2. Local existence and preliminary results. We first study the local existence of classical solutions of (1.1) following the fundamental theory developed by Amann [1].
and the following dichotomy holds: We now collect some basic properties of the local solutions.
(2.6) 3. A priori estimates. According to Proposition 1, in order to prove Theorem 1.1 and Theorem 1.2, it is sufficient to show that u(·, t) L ∞ (Ω) + v(·, t) L ∞ (Ω) is bounded for all t ∈ (0, T max ), then T max = ∞ and the solution is global in time. Indeed, we shall show that the solution is uniformly bounded for t ∈ (0, ∞). To this end, it suffices to prove that u(·, t) L p (Ω) is bounded for some p large thanks to Lemma 2.2 as we shall see later on. The main vehicle of our approach is the combined estimate on Ω u p + Ω |∇v| 2q for both p and q large, based on an idea recently developed in [21,24] etc.
3.1. Combined a priori estimates. For any p ≥ 2, we multiply the u-equation in (1.1) by u p−1 and then integrate it over Ω by parts Here and in the sequel we skip dx in the integrals for the succinctness. To estimate (3.1), we have from D(u) ≥ K 1 u m1 in (1.2) that where the identity holds due to the fact u p+m1−2 |∇u| 2 = 4 (p+m1) 2 |∇u p+m 1 2 | 2 . Moreover, we can apply Young's inequality to obtain where C 31 is a positive constant dependent on p. Thanks to (3.2)-(3.4), (3.1) implies On the other hand, for any q ≥ 2, we have from the v-equation in (1.1) In light of the pointwise identity ∇v · ∇∆v = 1 2 ∆|∇v| 2 − |D 2 v| 2 , we first estimate I 1 in (3.6) through the integration by parts To estimate I 11 , we recall inequality (2.4) in [9] which states that: there exists a positive constant C Ω depending only on the curvatures of ∂Ω such that ∂|∇v| 2 ∂ν ≤ C Ω |∇v| 2 , ∀x ∈ ∂Ω, then we obtain . (3.9) Choose k = r + 1 2 and s = 2 in Lemma 2.4, and then we have that α 1 ∈ (0, 1) satisfies or equivalently hence we can apply (2.6) to f = |∇v| q to obtain where the second inequality is due to the boundedness of ∇v L 2 in (2.2). In view of (3.9) and (3.10), we apply Young's inequality to (3.8) and have where C 35 is a positive constant dependent on q. On the other hand, through the pointwise identity |∇v| 2q−4 ∇|∇v| 2 2 = 4 q 2 ∇|∇v| q 2 , we can rewrite I 12 as Substituting (3.11) and (3.12) into (3.7) gives us To estimate I 2 in (3.6), we first have from the integration by parts Then we can apply Young's inequality to derive 15) where the second inequality follows from the pointwise inequality |∆v| 2 ≤ N |D 2 v| 2 , and Collecting (3.15) and (3.16), we infer from (3.14) Finally, by collecting (3.5) and (3.18) we conclude that d dt I32 Ω u 2 |∇v| 2q−2 +C 36 , (3.19) where C 36 = C 31 +C 35 . We are now ready to present the following a priori estimates.
First of all, we see that (3.28) is equivalent as which, in terms of (3.21), (3.22) and respectively. Then it is easy to check that all the inequalities hold if p and q are sufficiently large hence (3.28) is verified.

3.2.
Alternative estimates. In this subsection, we shall provide estimates for (3.19) by a different approach in the following lemma. Our proof is based on and slightly extends [28].

4.
Global existence and boundedness. Finally, we present the proofs of our main results.
Proof of Theorem 1.1. The proof is rather standard and we shall only sketch the main steps. First of all, by taking p > N , we have from Lemma 2.2 and Lemma 3.1 that v(·, t) W 1,∞ is bounded for all t > 0. Then one can apply the classical Moser-Alikakos L p iteration arguments of Lemma A.1 in [21] to show that u(·, t) L ∞ is uniformly bounded for t ∈ (0, T max ). Therefore T max = ∞ and the solution is global in time according to Proposition 1.
Proof of Theorem 1.2. The proof is the same as that of Theorem 1.1 by using Lemma 3.2.