Global boundedness in a two-dimensional chemotaxis system with nonlinear di ﬀ usion and singular sensitivity

: In this study, we investigate the two-dimensional chemotaxis system with nonlinear di ﬀ usion and singular sensitivity


Introduction and sketch of the main results
In 1970, Keller and Segel proposed a mathematical model concerning about cell's life cycle, especially an aggregation process.In this model, "chemotaxis" plays an essential role to induce the aggregation process of the cell.When cells are starving, cells move toward increasing concentrations of the signal substance that is produced by cells [12]: where n denotes the cell density and w is the chemical concentration.The simplified model of Model (1.1) was proposed by Nanjundiah [24]: The mathematical analysis of (1.2) and the variants thereof mainly concentrates on the boundedness and blowup of the solutions (refer to, e.g., [3,11,22,23,25,32,36,38] and references therein).When = − + τw w w n Δ t is replaced by = w w Δ t , Li et al. [22] proved that a multidimensional chemotaxis system is ill-posedness in ( ( ) ) and = τ 0, Lyu and Wang [23] explored how strong the logistic damping can warrant the global boundedness of solutions and further established the asymptotic behavior of solutions on top of the conditions.Keller and Segel [13] introduced a phenomenological model of wave-like solution behavior without any type of cell kinetics, a prototypical version of which is given by: where u represents the density of bacteria and v denotes the concentration of the nutrient.The second equation models the consumption of the signal.In the first equation, the chemotactic sensitivity is determined according to the Weber-Fechner law, which says that the chemotactic sensitivity is proportional to the reciprocal of signal density.Winkler [39] proved that if initial data satisfies appropriate regularity assumptions, System (1.3) possesses at least one global generalized solution in two-dimensional bounded domains.
Moreover, he took asymptotic behavior of solutions to System (1.3) into account and proved that ⋅ ⇀ v t , * 0 ( ) in , where m and M are positive constants.When uv is replaced by g u v ( ) , ∈ g C 1 ( ) and ≤ ≤ g u u 0 β ( ) , ∈ β 0, 1 ( ), ∈ χ 0, 1 ( ) and any suffi- ciently regular initial data, Lankeit and Viglialoro [20] showed that System (1.3) has a global classical solution.Moreover, if additionally = m u L 0 Ω 1 ‖ ‖ ( ) is sufficiently small, then also their boundedness is achieved.When System (1.3) has a logistic source f u ( ), Lankeit and Lankeit [17] showed that System (1.3) possesses a global generalized solution for any ≥ χ 0, ≥ r 0, and and v 1 is replaced by as → ∞ v .Zhao and Zheng [50] proved that System (1.3) possesses a unique positive global classical solution provided > k 1 with = N 1 or > + k 1 N 2 with ≥ N 2. For more recent outcomes, one can see [16,19,33,34,40,45].When v does not stand for a nutrient consumed but for a signaling substance produced by the bacteria themselves, which is given by: and when v 1 is replaced by χ v ( ), ) with some > χ 0 0 , > α 0, and > k 1, Winkler [35] proved that for any choice of appropriate initial data, System (1.4) possesses a unique global classical solution that is bounded in × ∞ Ω 0, ( )for ≥ N 1.Stinner and Winkler [31] showed that for any ∈ λ 0, min 1, χ ) is a global weak power-λ solution of (1.4) for ≥ N 2. Winkler [37] proved that < < χ 0 N 2 and then for any such data, there exists a global-in-time classical solution.Moreover, global existence of weak solutions is established whenever 4 .The boundedness of solution is left as an open problem.Fujie [7] solved the open problem of uniform-in-time boundedness of solutions for < < χ 0 N 2 , which was conjectured by Winkler [37].Recently, Winkler and Yokota [44] proved that System (1.4) possesses a uniquely determined global classical solution if ), > δ 0 are the constants.Furthermore, the solution of (1.4) con- verges to the homogeneous steady state u u ¯, 0 0 ( )at an exponential rate with respect to the norm in ∞ L Ω 2 ( ( )) as

→ ∞ t
, where . Lankeit and Winkler [21] introduced an apparently novel type of generalized solution and proved that under the hypothesis: all initial data satisfying suitable assumptions on regularity and positivity, an associated no-flux initialboundary value problem admits a globally defined generalized solution.This solution inter alia has the Winkler [43] proved that if ≥ N 3 and > − χ N N 2 , a statement on spontaneous emergence of arbitrarily large values of u for appropriately small ε is derived.When the second equation degenerates into an elliptic equation, and Senba [8] showed that System (1.4) admits a global classical positive solution that is uniformly bounded.
Black [4] showed that System (1.4) admits at least one global generalized solution if < < − χ 0 N N 2 .Zhigun [53] showed that System (1.4) admits a generalized supersolution with appropriate condition and assumed that u v , ( ) satisfies additional conditions and that the generalized supersolution is a classical solution.When System , Fujie et al. [9] proved that System (1.4) possesses global bounded ( ) , Kurt and Shen [14] showed that in any space dimensional setting, logistic kinetics is sufficient to enforce the global existence of classical solutions and hence prevents the occurrence of finite-time blow-up even for arbitrarily large χ.In addition, the solutions are shown to be uniformly bounded under the conditions > r α χ inf 4 ( ).For the fully parabolic system, Zhao and Zheng [49]
This article is concerned with the following chemotaxis system with singular sensitivity and signal production: where ⊂ Ω 2 is a bounded convex domain with smooth boundary ∂Ω and ∂ ∂ν denotes the derivative with respect to the outer normal of ∂Ω, u represents the cell density, and v denotes the concentration of the chemical signal.> χ 0, > θ 1 is a given parameter and initial data u 0 and v 0 are the known functions satisfying Ω with 0 and 0 in Ω, Ω with 0 in Ω.
Chemotaxis system with nonlinear diffusion and singular sensitivity  3 The additional external production of the signal chemical Recently, Ren and Ma [29] have proved that if > θ 2, System (1.5) possesses at least one global weak solution, which is locally bounded in the sense that , for all 0 and 2.
When the first equation in (1.6) is replaced by Furthermore, suppose g satisfies (1.7) and (1.10) as well as Then, System (1.5) possesses at least one global weak solution in the sense of Definition 2.1, which is globally bounded in the sense that In this article, we use symbols C i and c i ( = i 1, 2,… ) as some generic positive constants that may vary in the context.For simplicity, u x t , The contents of this article are as follows.In Section 2, we first introduce the concept of weak solutions and then give the global existence result for System (1.5).In Section 3, we give some fundamental estimates for the solution to System (1.5) and prove Theorem 1.1.

Preliminaries
Under the assumptions of u, the first equation of System (1.5) may be degenerate at = u 0. Therefore, System (1.5) does not allow for classical solvability in general as the well-known porous medium equations.We introduce the following definition of weak solutions.) and In order to construct weak solutions by an approximation procedure, we introduce the following regularized problems: for ∈ ε 0, 1 ( ).All of these problems (2.3) are, indeed, globally solvable in the classical sense.Lemma 2.2.Let ⊂ Ω 2 be a bounded domain with smooth boundary, > χ 0. Assume that g fulfills (1.7), and let > θ 1 and ∈ ε 0, 1 ( ).Then, there exist functions Proof.This can be seen by a straightforward adaptation of the reasoning in [42] on the basis of standard results on local existence and extensibility, as provided by the general theory in [2].This completes the proof.
where the nonnegative function ∈ g L loc 1 ( ) has the property that with some < < τ T 0 and > a 0.Then, ).We say that L fulfills (L) if L has the property that L T sup whenever 1.10 and 1.11 hold .
T 0 ( ) and as well as L T for all t T and ε , , 0, 0, 1 .
Proof.The essential idea has been illustrated in [ First, by the positivity of > v 0 0 in Ω and the maximum principle, we have , for all 0.
Next, the representation formula of v, the maximal principle, and (2.3) imply that ( ) { } for all ≥ t 0. We obtain (2.4) immediately. (2.5) and (2.6) follow almost immediately from integrating the first and second equations of (2.3), and we omit giving details here.The proof is complete.□ 3 Proof of Theorem 1.1 In this section, we will obtain the integral inequalities of u ε over Ω, whose core is to improve the regularity of v ε with regard to W Ω q 1, ( ) for any > q 2 and to obtain the ∞ L -boundedness of u ε by applying a standard Moser-type recursive argument.Lemma 3.1.(Lemma 3.1 in [29]) Assume that > θ 1, and let ∈ p 0, 1 ( ).Then, there exists ) such that Proof.The proof is divided into two cases.
Proof.The proof is divided into two cases.
. From Lemmas 2.5 and 3.1, there exist )such that (L) holds and that for all > T 1, and Using the Gagliardo-Nirenberg inequality, we obtain )and ∈ ε 0, 1 ( ), which yields the first inequality in (3.16).
Proof.The proof can be found in our recent work [29,Lemma 3.5]; for the reader's convenience, we present the detailed proof here.The following proof is divided into two cases.
. Let ≔ − p θ 2 1; owing to > θ 3 2 , it is easy to see that > p 2. By the continuous embedding as well as From Lemmas 2.5 and 3.4, there exist and that Applying the variation-of-constants formula of ⋅ v t , ε ( ) and (3.26), we have for all ∈ * and (1.7) guarantees that for all ∈ * t t t ˜, ( ].For the second last summand on the right of (3.30), as ≥ * t 2, we obtain where 2, in a way similar to (3.28), so we omit it.
Similarly, we can take ≔ + p θ 1 satisfying > p 2 in line with ≥ θ 2, and by following the same procedure, we can obtain the second inequality in (3.44).The proof is complete.
L T for all t T and ε , 0 , Proof.Multiplying the first equation in (2.3) by − u ε p 1 and using Young's inequality, we have From Lemmas 2.2 and 3.5, there exists a On the one hand, if , applying Hölder's inequality, we obtain ) .By the Gagliardo-Nirenberg inequality, there exists = > c c p 0 1 , for all 0, and 0, 1 .
The desired result is directly from ordinary differential equations comparison argument.
Proof.Based on Proof.The proof is similar to our recent work [27]; to avoid repetition, we omit giving details here.) , integrating by parts, and using Young's inequality, we , for all 0, and 0, 1 .
By Lemmas 2.5, 3.1, and 3.8, there exist and as well as Then, for all > T 0, there exists Proof.From Lemma 3.8, given > T 0, we can find 3 , and thus, Proof.Fixed > t 0 and ∈ ∞ ψ C Ω ( ), by the straightforward calculation, we have for all ∈ ε 0, 1 ( ).Given > T 0, from Lemmas 3.8, 2.2, and 3.5, and equation (1.7), there exist c T i ( ) Using Young's inequality, we have Integrating (3.62) over T 0, ( ) and using Lemmas 3.1 and 3.9, we obtain (3.54).The proof is complete.□ Lemma 3.12.Let > θ (3.63) ) for ∈ ε 0, 1 ( ); from Lemma 3.8 and (1.7), we know that , and (3.63) is directly from standard theory on the Hölder regularity in parabolic equations [26].The proof is complete.□ Now, we are preparing to extract a suitable sequence of number ε along with the respective solutions that approach a limit in appropriate topologies.□ Finally, we prove the main theorem.
Proof of Theorem 1.1.The global existence directly results from Lemma 3.13.Considering the global weak solution of (1.5) obtained in Lemma 3.13, it is straightforward that the boundedness of ) for all > q 2 can be derived due to Lemmas 3.7 and 3.8 together with Hypotheses (1.10) and (1.11) and property (L).Accordingly, the additional Property (1.12) can be obtained by equations (3.65) and (3.67).The proof is complete.□ lack of continuity of the solution for the Cauchy problem.When =

3 2 .
Then, for all > T 0, there exist = ∈ r, and μ are replaced by r x t continuous in ∈x Ω uniformly with respect to ∈ t , and there are positive constants r μ .27) Chemotaxis system with nonlinear diffusion and singular sensitivity  17 is a global weak solution of equation(1.5)in the sense of Definition 2.1.From Lemma 3.8 and the Vitali convergence theorem, we know that (3.65) and the first result in (3.64) hold.From Lemma 3.7 and Lemma 3.12, the Arzelá-Ascoli theorem, and the Banach-Alaoglu theorem, we know that (3.66), (3.67), and the second result in (3.64) hold.Depending on (3.65)-(3.67)when taking = ↘ ε ε 0 j in the corresponding weak formulation associated with (2.3), we readily obtain (2.1) and (2.2).The proof is complete.