Boundedness Through Nonlocal Dampening Effects in a Fully Parabolic Chemotaxis Model with Sub and Superquadratic Growth

This work deals with a chemotaxis model where an external source involving a sub and superquadratic growth effect contrasted by nonlocal dampening reaction influences the motion of a cell density attracted by a chemical signal. We study the mechanism of the two densities once their initial configurations are fixed in bounded impenetrable regions; in the specific, we establish that no gathering effect for the cells can appear in time provided that the dampening effect is strong enough. Mathematically, we are concerned with this problem ut=Δu-χ∇·(u∇v)+auα-buα∫ΩuβinΩ×(0,Tmax),τvt=Δv-v+uinΩ×(0,Tmax),uν=vν=0on∂Ω×(0,Tmax),u(x,0)=u0(x)≥0,v(x,0)=v0(x)≥0,x∈Ω¯,◊\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} u_t=\Delta u-\chi \nabla \cdot (u\nabla v)+au^\alpha -bu^\alpha \int _\Omega u^\beta &{}\textrm{in}\ \Omega \times (0, T_{max}),\\ \tau v_t=\Delta v-v+u &{}\textrm{in}\ \Omega \times (0, T_{max}),\\ u_\nu =v_\nu =0 &{}\textrm{on}\ \partial \Omega \times (0, T_{max}),\\ u(x, 0)=u_0(x)\ge 0, v(x,0)=v_0(x)\ge 0, &{}x \in {\bar{\Omega }}, \end{array}\right. } \quad {\Diamond } \end{aligned}$$\end{document}for τ=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau =1$$\end{document}, n∈N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\in {\mathbb {N}}$$\end{document}, χ,a,b>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi ,a,b>0$$\end{document} and α,β≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha , \beta \ge 1$$\end{document}. Herein u stands for the population density, v for the chemical signal and Tmax\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{max}$$\end{document} for the maximal time of existence of any nonnegative classical solution (u, v) to system (◊\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Diamond $$\end{document}). We prove that despite any large-mass initial data u0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_0$$\end{document}, whenever (The subquadratic case) 1≤α<2andβ>n+42-α,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le \alpha <2 \quad \text {and} \quad \beta >\frac{n+4}{2}-\alpha ,$$\end{document} (The superquadratic case) β>n2and2≤α<1+2βn,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta >\frac{n}{2} \quad \text {and} \quad 2\le \alpha < 1+ \frac{2\beta }{n},$$\end{document} actually Tmax=∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{max}=\infty $$\end{document} and u and v are uniformly bounded. This paper is in line with the result in Bian et al. (Nonlinear Anal 176:178–191, 2018), where the same conclusion is established for the simplified parabolic-elliptic version of model (◊\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Diamond $$\end{document}), corresponding to τ=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau =0$$\end{document}; more exactly, this work extends the study to the fully parabolic case Bian et al. (Nonlinear Anal 176:178–191, 2018). (The subquadratic case) 1≤α<2andβ>n+42-α,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le \alpha <2 \quad \text {and} \quad \beta >\frac{n+4}{2}-\alpha ,$$\end{document} (The superquadratic case) β>n2and2≤α<1+2βn,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta >\frac{n}{2} \quad \text {and} \quad 2\le \alpha < 1+ \frac{2\beta }{n},$$\end{document}

This paper is in line with the result in [4], where the same conclusion is established for the simplified parabolic-elliptic version of model (✸), corresponding to τ = 0; more exactly, this work extends the study to the fully parabolic case [4].
If properly interpreted, this model idealizes a chemotaxis phenomenon, a mechanism from mathematical biology describing the directed migration of a cell in response to a chemical signal; more exactly, the movement of an organism or entity (such as somatic cells, bacteria, and other single-cell or multicellular organisms) is strongly influenced by the presence of a stimulus, and precisely the motion follows the direction of the gradient of the stimulus itself.
It is well known that the land marking event of chemotaxis was first introduced by Keller and Segel in 1970s ( [16], [15]).More expressly, by indicating with u = u(x, t) a certain cell density at the position x and at the time t, and with v = v(x, t) the stimulus at the same position and time, the pioneering study reads as (1) for the specific case a = b = 0.The partial differential equation modeling the motion of u, i.e. (2) essentially describes how a chemotactical impact of the (chemo)sensitivity (χ) provided by the chemical signal v may break the natural diffusion (associated to the Laplacian operator, ∆u) of the cells.Indeed, the term −∇ • (uχ∇v) models the transport of u in the direction χ∇v, the negative sign indicating the attractive effect that v has on the cells (higher for χ larger and for an increasing amount of v).As a consequence, when v is produced by the same cells, and in such a scenario v obeys the attractive impact may be so efficient as to lead the cell density to its chemotactic collapse (blow-up at finite time with appearance of δ-formations in the region).
1.2.An overview on the Keller-Segel system.Mathematically, it was proved that solutions to the initial-boundary value problem associated to equations ( 2) and (3), may be globally bounded in time or may blow up at finite time; this depends on the mass (i.e., Ω u 0 (x)dx) of the initial data, its specific configuration, and the value of the sensitivity χ.More precisely, in one-dimensional settings, all solutions are uniformly bounded in time, whereas for n ≥ 3 given any arbitrarily small mass m = Ω u 0 (x)dx > 0, it is possible to construct solutions blowing-up at finite time.On the other hand, when n = 2, the value 4π separates the case where diffusion overcomes self-attraction (if χm < 4π) from the opposite scenario where self-attraction dominates (if χm > 4π); respectively, all solutions are global in time, and initial data producing assembling processes at finite time can be detected.A detailed discussion on such analyses can be found in [11,22,21,32], which are undoubtedly classical results in this context.

1.
3. An overview on the Keller-Segel system with logistics.If the evolution of u in equation ( 2) is also influenced by the presence of logistic terms behaving as au − bu β , for β > 1, mathematical intuition suggests that superlinear damping effects should benefit the boundedness of solutions (this, for instance, occurs for ordinary differential equations of the type u ′ = au − bu β ).Actually, the prevention of δ-formations in the sense of finite-time blow-up for (4) when coupled with some equation implying the segregation of v with u (for instance (3)), has been established only for large values of b (if β = 2, see [31], [33]), whereas for some value of β near 1 a blow-up scenario was detected, first for dimension 5 or higher [34], (see also [9] for an improvement of [34]), but later also for n ≥ 3, in [35].
1.4.An overview on the Keller-Segel system with nonlocal sources.As anticipated, in this research we are interested in understanding how the introduction of external growth factors of logistic type defined in terms of the total mass of the some power of the population, and hence idealized by nonlocal external sources, may avoid blow-up mechanisms, exactly as logistics.To be precise, likewise to classical logistic effects, impacts behaving as ( 5) model a competition between a birth contribution, favoring instabilities of the species (especially for large values of a), and a death one opportunely contrasting this instability (especially for large values of b).
In this context, some questions naturally arise.Q: Can one expect that in a biological mechanism governed by the equation ( 6) the external dampening source suffices to enforce boundedness of solutions, even for any large initial distribution u 0 , arbitrarily small b > 0 and in any large dimension n? Are, conversely, some restrictions on n and/or a, b, α, β, u 0 required?To our knowledge, most of the analyses connected to the aforementioned questions can be found in the literature when the equation for v expressed as (or similarly to) (6) is of elliptic type, i.e. for some γ ≥ 1 As a matter of fact, when the equations for the cells and the stimulus are both evolutive, we are only aware of [24], where the authors consider, for τ = 1 = m, σ > 2, γ ≥ 1 and h = h(x, t) ≡ 0, the initial-boundary value problem associated to this model (7) Herein, the nonlocal term is where α ≥ 1, a 0 , a 1 > 0 and a 2 ∈ R; in particular, it is worthwhile mentioning that even though problem (1) is the limit case of (7) for m = 1 = γ and σ = 2 (and h = 0), these models are not directly comparable.
In fact, conversely to the mechanism we are dealing with (see again model (1)), in [24] the attractive driftsensitivity is nonlinear (i.e., σ > 2 in −χu(u + 1) σ−2 ∇v) and, more importantly, the nonlocal term of the reaction in (8) has both an increasing (a 2 > 0) and decreasing (a 2 < 0) effect on the cell density, whereas the dampening counterpart is of polynomial type; this contrasts with (5), where the nonlocal term is purely absorbing and the local one productive.
For model ( 7) the global-in-time existence of classical solutions and the convergence to the steady state are established in the same [24], under suitable regularity assumptions on the initial data and whenever the coefficients of the system satisfy (9) α ) Additionally, the suppression of some of the conditions in (9), might provide (at least from the numerical point of view) some blow-up solution.
As we said above, when the equation for the chemical v is elliptic (biologically this idealizes the situations where chemicals diffuse much faster than cells), some more results are available in the literature.In particular, in [23] the authors analyze, inter alia, problem (7) in the framework of what follows: τ = 0, σ = 2, m = γ = α = 1 and h = h(x, t) is a uniformly bounded function with suitable properties.Similar conclusions as those of the fully parabolic case are derived.
On the other hand, when the reaction term is taken exactly as in ( 5), these further results dealing with uniform-in-time boundedness of classical solutions emanating from sufficiently regular initial data have been obtained for problem (7), with τ = 0 and h ≡ 0: • for the special case where m = γ = a = b = 1 and σ = 2 in [4], whenever these assumptions (with [29]).For completeness, we add that another indication showing how rich is effectively the study in the framework of models with stationary equations for the stimulus, is given in these papers [3,7,6,20], where nonlocal problems alike those in (7) are studied in the whole space R n .(In this context, the equation for v is the classical Poisson's equation.) 1.5.Connection with the Fisher-KPP equation.In mathematics (10) is known (in its original one spatial dimensional version) as the Fisher-KPP equation, and it describes a reaction-diffusion phenomenon used to model population growth and wave propagation.(See [8,17].)In its more common form F , interpretable according to what said above as the rate of growth/death of the population, has this expression (a, b ≥ 0): Apart from the law of the corresponding sources, it appears interesting to discuss the parallelism between equations (10) and ( 4): essentially, in the latter the extra transport effect −∇ • (uχ∇v) appears.
In the specific, for χ = 0 no convection on the particle density u influences the mechanism, and pure Reaction/F (u)-Diffusion/∆u models (RDm) are obtained (see (10)).Oppositely, for χ > 0 the population is transported in the habitat toward the direction of ∇v; in this case, equation ( 4) is an example of Taxis/∇ • (uχ∇v)-Diffusion-Reaction models (TDRm).As a consequence, and at least intuitively, the sources being equal, TDRm are more inclined to present some instabilities with respect to RDm.Confining our attention to reactions F (u) of nonlocal type, for a general study on initial-boundary value problems (the majority of them with a homogeneous Dirichlet boundary condition, i.e. u = 0 on ∂Ω) associated to (10), we refer to [26,28] and references therein.Conversely, for results on more similar contexts to that considered in our analysis, we mention [2], where the authors study, among other things, globality and long-time behavior of solutions to a zero-flux nonlocal Fisher-KPP type problem.
2. Presentation of the main result and organization of the paper 2.1.Claim of the main result.In this research we intend to improve the degree of knowledge on chemotactic models described by two coupled partial differential equations, and with non-local logistic sources, when both are of parabolic-type.In particular, our overall analysis gives an answer to questions Q, in the sense that we establish that despite any fixed small value of the dampening parameter b and arbitrarily large growth parameter, any initial data (u 0 , v 0 ) (even arbitrarily large) produce uniform-in-time boundedness of solutions to model (1) for both subquadratic and superquadratic growth rate α, by properly magnifying the impact associated to the death rate β.
Additionally, for every 1 < q < ∞, let 0 ≤ u 0 , v 0 ∈ W 2,q (Ω) be given such that Structure of the paper.The rest of the paper is structured as follows.First, in §3, we collect some necessary and preparatory materials.Then, in §4, we give some hints on the local-well-posedness to model (1), so obtaining properties of related local solutions (u, v) on Ω × (0, T max ); additionally, through the extensibility criterion we establish how to ensure globability (i.e., T max = ∞) and boundedness (i.e., u(•, t) L ∞ (Ω) finite on (0, ∞)) by using their uniform-in-time L k (Ω)-boundedness, for k > 1.Such a bound is derived in §5, and successively used in §6 to prove Theorem 2.1.
Remark 1 (On the difficulties of the fully parabolic analysis).As we will see below, conversely to the parabolic-elliptic case analyzed in [4, (2.21)], in the fully parabolic case it is no longer possible to use the equation for v, so replacing ∆v appearing in the testing procedures with v−u.This complexity is circumvented by relying on Maximal Sobolev Regularity applied to the equation v t = ∆v − v + u.

Some preliminaries and auxiliary tools
We will make use of this functional relation, obtainable by manipulating the well known Gagliardo-Nirenberg inequality.We underline that for the case Ω = R n the proof is given in [1, Lemma 2]; we did not find a reference covering bounded domains and henceforth herein we dedicate ourselves to this issue.Lemma 3.1.Let Ω be a bounded and smooth domain of R n , with n ∈ N and let, for n ≥ 3, (11) p := 2n n − 2 .
We will also need this comparison argument for Ordinary Differential Equations.
By contradiction, if there were some t 0 ∈ S y1 then by the continuity of y and y 0 < y 1 we could find I = (t, t) (with possibly t 0 = t) such that y 1 < y(t) < y( t), y 1 < y(t) on I; henceforth, by hypothesis, φ(t, y) ≤ 0 for all t ∈ I.At this stage, the Lagrange theorem would provide a proper ξ ∈ I leading to this inconsistency:
Proof.The first part of the proof can be obtained by adapting to the fully parabolic case the reasoning in [4, Proposition 4] developed for the simplified parabolic-elliptic scenario.
As to the boundedness of the mass, we integrate over Ω the first equation of problem (1) so that by Hölder's inequality, and γ(t) := Ω u α ≥ 0 on (0, T max ),

Now we apply Lemma 3.3 with
, so concluding with m 0 = max{y 0 , y 1 }.
Once the classical local well posedness to model (1) provided by Lemma 4.1 is ensured (in particular from now on with (u, v) we refer to the local solution defined on Ω × (0, T max )), a suitable uniform-in-time boundedness criterion is required.In the specific, the next result based on an iterative method connected to the Moser-Alikakos technique addresses the issue.Lemma 4.2.Whenever for every k > 1 there exists C > 0 such that Ω u k ≤ C for all t ∈ (0, T max ), actually u is uniformly bounded, in the sense that u ∈ L ∞ ((0, ∞); L ∞ (Ω)).Automatically, v is also uniformly bounded.

A priori estimates
Since the uniform-in-time boundedness of u is implied whenever u ∈ L ∞ ((0, T max ); L k (Ω)) for some k > 1, here under we dedicate to the derivation of some a priori integral estimates.
Proof.The Young inequality directly provides the claim.
Let us now distinguish the analysis of the subquadratic case from the superquadratic one, exactly starting from this last situation.

The superquadratic growth
Then there exist k 0 ≥ 1, L 0 > 0 such that for all k > k 0 , Ω u k ≤ L 0 for all t ∈ (0, T max ).
Proof.Let us start fixing k 0 = 1, and when necessary we will enlarge this initial value.For all k > k 0 , we have from the first equation in (1) and integration by parts that Here, from bound (22) in Lemma 5.1 we have that ( 26) for all t ∈ (0, T max ).
A combination of relations ( 25) and (26) implies that for all t ∈ (0, T max ) We now estimate the second integral on the right-hand side of (27).From the identity , our aim is exploiting Lemma 3.1 with ϕ := u k 2 and proper q and r.In the specific, for n ≥ 3 (at the end of this proof we will discuss the cases n = 1 and n = 2) in order to make meaningful the forthcoming computations, let us take k 0 = max{β − α + 1, 1}.From the definition of k 0 and condition (24), for any k > k 0 it is possible to set (28) In this way, for q := 2(k + α − 1) k , r := 2k ′ k a number of calculations yield 1 ≤ r < q < p and q r < 2 r + 1 − 2 p .Therefore we infer from ( 12) that for all c > 0 for all t ∈ (0, T max ).
Here, the interpolation inequality (see [5, page 93]) yields for all t ∈ (0, T max ), where (32) b We note that recalling the expression of k ′ in (28) and the range of α in (24), some computations provide As a consequence, we can invoke Young's inequality so that relation (31) reads which in conjunction with (30) implies for all t ∈ (0, T max ), Now we focus on the second integral at the right-hand side: the Gagliardo-Nirenberg inequality and (21) produce for this bound on (0, T max ): In turn, we have from the Young inequality that for all ĉ > 0 ĉ Coming back to (27), in order to estimate the term c 1 Ω |∆v| k+α−1 α−1 , let us exploit Lemma 3.2 with q = k+α−1 α−1 : we have (35) ds for all t ∈ (0, T max ).
Since from the condition α ≥ 2 we have that k+α−1 α−1 ≤ k + α − 1, the Young inequality leads to (Naturally for the limit case α = 2, the constant c 9 can be taken equal to 0.) We now add to both sides of ( 27) the term Ω u k and then we multiply by e t .Since e t d dt Ω u k + e t Ω u k = d dt e t Ω u k , an integration over (0, t) provides for all t ∈ (0, T max ) By inserting estimate ( 35) into (37) and taking into account bounds (36), ( 33) and (34), we arrive at with L 0 := c 12 + Ω u k 0 , so the claim is proved.For n ∈ {1, 2} the arguments are similar once relation ( 29) is, respectively, replaced by

The subquadratic growth
Proof.Let us consider k 1 = 1; as done before, we will enlarge this initial value when necessary.By following the same argument of Lemma 5.2 for all k > k 1 , we arrive for all t ∈ (0, T max ) at Since α ≥ 1, an application of relation (23) of Lemma 5.1 to the second integral at the right-hand side of (39) gives ( 40) |∆v| k+1 for all t ∈ (0, T max ), whereas from the condition α < 2, the Young inequality leads to (41) ka Ω u k+α−1 ≤ Ω u k+1 + c 13 for all t ∈ (0, T max ).
Combining estimates (40) and (41) with bound (39), we have for all t ∈ (0, T max ), Now let us focus on the second integral on the right-hand side of (42).Since , we can apply Lemma 3.1 with ϕ := u k 2 and suitable q and r.In the specific, for any k > k 1 := max {1, 1 − α + β} , by posing it is possible to check that In this way, and for n ≥ 3, letting q := 2(k + 1) k , r := 2k ′ k we can establish that 1 ≤ r < q < p and q r < 2 r + 1 − 2 p .Consequently, we deduce from ( 12) that for all c > (44) for all t ∈ (0, T max ).
Now an application of the interpolation inequality yields for all t ∈ (0, T max ), (A comparison between the couple (a 2 , b 2 ) above and (a 1 , b 1 ) in (32) shows that a 1 = a 2 , whereas b i , i = 1, depends on q.) From straightforward calculations and the condition (38), we observe that This in conjunction with (44) implies that t ∈ (0, T max ), As to the term Ω |∆v| k+1 in expression (42), by exploiting Lemma 3.2 with q = k + 1, we obtain

Proof of Theorem 2.1
We apply Lemma 5.2 and Lemma 4.2, and Lemma 5.3 and Lemma 4.2 to give the proof for the subquadratic and superquadratic case, respectively.

Contents 1 . 1 . 1 .
Introduction and motivations 1.1.Basic description of the research 1.2.An overview on the Keller-Segel system 1.3.An overview on the Keller-Segel system with logistics 1.4.An overview on the Keller-Segel system with nonlocal sources 1.5.Connection with the Fisher-KPP equation 2. Presentation of the main result and organization of the paper 2.1.Claim of the main result Introduction and motivations 1.Basic description of the research.In this paper we consider (1)