How far does logistic dampening influence the global solvability of a high-dimensional chemotaxis system?

This paper deals with the homogeneous Neumann boundary value problem for chemotaxis system {ut=Δu−∇⋅(u∇v)+κu−μuα,x∈Ω,t>0,vt=Δv−uv,x∈Ω,t>0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \textstyle\begin{cases} u_{t} = \Delta u - \nabla \cdot (u\nabla v)+\kappa u-\mu u^{\alpha }, & x\in \Omega, t>0, \\ v_{t} = \Delta v - uv, & x\in \Omega, t>0, \end{cases}\displaystyle \end{aligned}$$ \end{document} in a smooth bounded domain Ω⊂RN(N≥2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Omega \subset \mathbb{R}^{N}(N\geq 2)$\end{document}, where α>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha >1$\end{document} and κ∈R,μ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\kappa \in \mathbb{R},\mu >0$\end{document} for suitably regular positive initial data. When α≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha \ge 2$\end{document}, it has been proved in the existing literature that, for any μ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mu >0$\end{document}, there exists a weak solution to this system. We shall concentrate on the weaker degradation case: α<2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha <2$\end{document}. It will be shown that when N<6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$N<6$\end{document}, any sublinear degradation is enough to guarantee the global existence of weak solutions. In the case of N≥6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$N\geq 6$\end{document}, global solvability can be proved whenever α>43\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha >\frac{4}{3}$\end{document}. It is interesting to see that once the space dimension N≥6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$N\ge 6$\end{document}, the qualified value of α no longer changes with the increase of N.


Introduction
Chemotaxis is a characteristic of organisms that move toward an environment conducive to their own growth. A pioneering mathematical model for chemotaxis was proposed by Keller and Segel in 1970s [1]. The so-called Keller-Segel minimal model consists of two equations of the form: where u and v denote the cell density and chemosignal concentration, respectively. It has been shown that all the solutions to the homogenous Neumann initial-boundary value problem associated with (1) in ⊂ R N (N ∈ N) are bounded when either N = 1 or N = 2 and total mass u 0 is small [2,3], while some finite/infinite-time blow-up may occur when N ≥ 3 or N = 2 and u 0 = m > 0 is large [4][5][6].
The classical Keller-Segel model only considers the diffusion and chemotaxis of cells. However, in the actual biological context, the reproduction and death of cells or population themselves need to be considered. A prototypical choice to achieve this is the logistic type source κuμu α with birth and death rates κ and μ, respectively. In the past decades, the homogeneous Neumann initial boundary value problem of the following Keller-Segel system with logistic source has been widely investigated: where ⊂ R N (N ≥ 1) is a bounded smooth domain, κ ≥ 0, μ > 0, α > 1. The presence of logistic source has been shown to have an effect of blow-up prevention. When α = 2, if the spatial dimension N ≤ 2, the system with nonnegative regular initial data only allows for global and uniformly bounded solutions even for arbitrarily small μ > 0 [2,7,8]. In the higher dimension setting, it has been proved in [9] that if μ is large enough, for all sufficiently smooth and nonnegative initial data, the problem possesses a unique bounded and global classical solution (we also refer to [10][11][12][13][14] and the references therein for more recent and specific results). Some recent studies show that the presence of weaker degradation terms may also exhibit significant relaxation effects in comparison with (1). Actually, by resorting to some weaker notions of solvability, global solutions have recently been constructed for systems merely containing certain subquadratic degradation terms, where it is required that α ≥ 2-1 N when N ≥ 2 [15,16] and even the wider ranges α > 2N+4 N+4 [17]. However, despite the presence of superlinear degradation, some unboundedness phenomena have been detected in the literature for problems of type (2) and certain parabolic-elliptic versions. Even in some situations in which solutions are known to remain bounded globally, such as e.g. in the quadratic case α = 2, certain results on spontaneous emergence of arbitrarily large densities have been reported when there is a diffusion coefficient before u in the first equation of (2) and the diffusion coefficient is small. See [18][19][20] for parabolic-elliptic case and [21] for parabolic-parabolic case, respectively. On the other hand, it has been proved in [22] that the solutions of parabolic-elliptic model, in which the second equation of (2) is replaced by 0 = vv + u, exhibits a finite-time blow-up phenomenon under the condition α < 7 6 when N ∈ {3, 4} or α < 1 + 1 2(N+1) when N ≥ 5. Some similar results were also derived for the simplified version by replacing the second [23,24]. Apart from (2), another typical chemotaxis system is the following consumption-type model: The reaction term -uv in the second equation indicates that the cells consume chemicals during the overall chemotaxis process. In contrast to the models with production mechanism (1), (2), the chemoattractant consumption mechanism in this system is more prone to the global existence of solutions due to the fact that the second equation immediately provides an L ∞ -bound for v. However, such a bound is not sufficient for dealing with the chemotaxis term. Actually, global existence and boundedness of solutions to (3) with κ = μ = 0 are only known under the smallness condition v(·, t) L ∞ ( ) ≤ 1 6(N+1) [25] or in a two-dimensional setting [26,27]. The three-dimensional version admits a global weak solution which after some waiting time eventually becomes classical [28].
When the logistic dampening is in consideration, in the high-dimensional setting, the global solvability seems still restricted to the quadratic degradation case (α = 2) [29]. When the logistic-type degradation is weaker, that is, α < 2, the global existence result obtained so far concentrates on the small-data solutions [30]. As for the global existence of arbitrarily large initial data solutions to ( 4) in the sub-qudratic case 1 < α < 2, to the best of our knowledge, it still remains unknown. In this short paper, we shall do some work and give a definite answer in this respect. We shall include the case of any high-dimensional domain.
Precisely, we will consider the problem where ν is the unit outer normal vector on the boundary. We are interested which α can guarantee the global solvability of this system even in the high-dimensional case.
To formulate our main results, we assume throughout that the initial data satisfy Our main result can be read as follows.
Our result shows that when the spatial dimension is smaller than 6, any superlinear degradation is enough to guarantee the global existence of solutions to system (4), and that if the domain dimension is no less than 6, the qualified value of α is bigger than 4 3 . We define the weak solution in the natural way as follows.
such that uv and u∇v ∈ L 1 loc ([0, ∞); L 1 ( )) and the following integral equalities hold: The rest of this paper is organized as follows. In Sect. 2, we establish the global existence of a family of approximate problems to system (4). Section 3 is devoted to the estimates of the time derivatives for the approximate problems. Finally, we obtain the global weak solution of our problem by an approximate procedure in Sect. 4.
In contrast to the situation without source terms, we cannot hope for mass conservation in the first component. Nevertheless, the following result still holds (see also other works involving logistic source e.g. [32][33][34]).
Proof By Hölder's inequality, We can obtain (12) by an ODE comparison argument. Estimate (13) is a consequence of the parabolic comparison principle.
Next we want to derive a (quasi-)energy inequality for the functional to get some essential estimates of (u ε , v ε ). The method used here is from [35].
Proof Fix p ∈ (1, 1 + 2 N ) and observe that An application of the Gagliardo-Nirenberg inequality yields ε-independent positive constant C 1 such that where θ := (p-1)N 2p ∈ (0, 1) and 2p · θ < 2 due to p ∈ (1, 1 + 2 N ). Thereupon, we can find some ε-independent positive constants C 2 , C 3 such that by making use of (12). On the other hand, making use of the boundedness of v ε L ∞ ( ) and the Young inequality, we know there exist ε-independent positive constants C 4 , C 5 fulfilling Substituting (25), (26) into (16), we conclude that there exist positive constants C 6 and C 7 such that, for any ε > 0, with K as given by Lemma 2.4. We can conclude the validity of (21). The result of (22) and (23) can be obtained by an integration of (16) and the fact u ε lnu ε ≥ -| | e . Furthermore, for (24), by Lemma 2.2, for any ε > 0, which is bounded due to (21).
By using interpolation inequalities, we can derive some further estimates from Lemma 2.5.
Proof From (22), we know that t+τ t |∇u 1 2 ε | 2 ≤ C 1 with some ε-independent constant C 1 > 0. Then with the aid of the Gagliardo-Nirenberg inequality, we obtain with some ε-independent positive constants C 2 , C 3 . Furthermore, we can make use of the Young inequality to obtain that with some ε-independent positive constants C i (i = 4, 5, 6).
We are now in the position to prove that the classical solution (u ε , v ε ) to the approximate systems (9) is global for each ε ∈ (0, 1).

Time regularity
In preparation of an Aubin-Lions type compactness argument, we shall supplement the estimates obtained in Sect. 2 with bounds on time-derivatives, since in Lemma 4.1 these will be used to warrant pointwise convergence.
Proof of Theorem 1.1 The statement is evidently implied by Lemma 4.2.