The Keller–Segel system on bounded convex domains in critical spaces

Consider the classical Keller–Segel system on a bounded convex domain Ω⊂R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varOmega \subset {\mathbb {R}}^3$$\end{document}. In contrast to previous works it is not assumed that the boundary of Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varOmega $$\end{document} is smooth. It is shown that this system admits a local, strong solution for initial data in critical spaces which extends to a global one provided the data are small enough in this critical norm. Furthermore, it is shown that this system admits for given T-periodic and sufficiently small forcing functions a unique, strong T-time periodic solution.

x 2 X; t [ 0; ou om ¼ ov om ¼ 0; x 2 oX; t [ 0; uðx; 0Þ ¼ u 0 ðxÞ; vðx; 0Þ ¼ v 0 ðxÞ; x 2 X; 8 > > > > < > > > > : ð1:1Þ has been investigated during the past decades by very many authors. Here u and v represent the density of a cell population and the concentration of a chemoattractant, respectively. In the situation of domains with smooth boundaries, the problem of local and global existence as well as blow-up of solutions are rather well understood, see e.g., the survey articles [9,10,16] and the references therein. In particular, the existence of global, strong solutions in L p -settings has been investigated by Hideo Kozono and coworkers in [13][14][15]. For further results in this context, see [17,18]. For recent results on the existence of unique, strong periodic solutions, see [8]. All these results deal with the setting of the whole space or domains with smooth boundaries. The situation is very different when one considers the Keller-Segel system in domains with nonsmooth boundaries, as e.g. Lipschitz domains. It was recently shown by Horstmann, Meinlschmidt and Rehberg [11] that under suitable conditions on the initial values and the geometry of the domain, one nevertheless obtains again the existence of a unique, strong, local solution to the Keller-Segel system.
In this article we study the Keller-Segel system in bounded convex domains and do not assume that the boundary of X is smooth. The situation of bounded convex domains with smooth boundaries was considered before e.g. in [22,23], as in this setting, due to the Neumann boundary condition, a classical solution to (1.1) satisfies ojrvj 2 om 0 on oX Â ð0; 1Þ. The latter property is helpful to establish a priori estimates for the solution to (1.1) which are used to prove global existence results. Here we use the convexity of the domain to study the Keller-Segel system, similarly to [11], in domains with rough boundaries.
The situation of bounded convex domains is, of course, less general than the framework of Lipschitz domains as considered in [11], howewer, in the situation of bounded convex domains, several main ingredients of the approach for smooth domains, such as knowledge of the domain of the Neumann Laplacian, its maximal L p -regularity as well as the mixed derivative theorem carry over to the given situation. This allows allows us in particular to treat the case of initial data lying in critical Besov spaces as stated precisely in Theorem 1.
Given a bounded convex domain X & R n , we are secondly interested in the question whether, given periodic functions f and g, there exist time periodic strong solutions to the classical Keller-Segel system. More precisely, for time periodic functions f 1 and f 2 , we consider the classical inhomogeneous Keller-Segel system We prove the existence of a T-periodic solution to (1.3), which is unique in the associated maximal regularity space, provided f ¼ ðf 1 ; f 2 Þ is T-periodic and sufficiently small in an appropriate norm. We will perform our analysis for f 2 L p ð0; T; L q ðXÞÞ, which corresponds to the weak setting. Here p; q 2 ð1; 1Þ need to satisfy certain conditions described in detail in Section 2. These conditions are essentially due to the mixed derivative theorem (see e.g. [19,Corollary 4.5.10]), allowing to transfer time into space regularity in a very precise sense, and to Sobolev emdeddings. The strategy of our approach may be described as follows: we first rewrite the Keller-Segel systems (1.1) and (1.3) as semilinear evolution equations, respectively, within the L p -setting. Maximal time weighted L p -regularity estimates as well as the existence of a bounded H 1 -calculus for the differential operators involved imply then the existence of a unique, local solution to the Keller-Segel system for initial data in critical spaces.
Secondly, the above maximal L p -regularity estimates for the linearization of the semilinear equation allow first for maximal periodic L p -solutions for the linear problem by the Arendt-Bu theorem, see [4]. The existence theory for maximal periodic L p -solutions for the linear evolution equations was generalized in [8] to the semi-and quasilinear setting. More precisely, for Banach spaces X we consider time-periodic quasilinear problems of the form u 0 ðtÞ þ AðuðtÞÞuðtÞ ¼ Fðt; uðtÞÞ; t 2 ð0; 2pÞ; uð0Þ ¼ uð2pÞ: Assuming natural Lipschitz conditions on A and F as well as that A(0) admits maximal periodic L p -regularity, the existence of a unique strong periodic solution to (1.4) is established provided kFðÁ; 0Þk F is small enough, where F :¼ L p ð0; 2p; XÞ. This article is organized as follows. In Sect. 2 we present the main results concerning the initial value problem as well as strong T-periodic solutions to the classical Keller-Segel system. Section 3 presents results on the Laplacian in the strong and weak setting for bounded convex domains. Sections 4 and 5 present the proofs of our main results.

Preliminaries and main results
Consider the Keller-Segel system (1.1), where X & R 3 denotes a bounded convex domain, m the outward unit normal on oX and f a given T-periodic function. We study the initial value problem (1.1) for critical spaces as well as the existence of time periodic solutions to the Keller-Segel system (1.3).

ð2:1Þ
Here D N and D w N denote the Neumann Laplacian on L q ðXÞ and W À1;q ðXÞ ¼ W 1;q 0 ðXÞ 0 equipped with the domains DðD N Þ :¼ fu 2 W 2;q ðXÞ : o m u ¼ 0 on oXg and W 1;q ðXÞ, respectively. For details, see Section 3. Setting X 1 :¼ DðD w N Þ Â DðD N Þ, we see that X 1 ,!X 0 is densely embedded and that A : X 1 ! X 0 is bounded.
The Keller-Segel system (1.1) corresponds then to the equation Here u 0 denotes the time derivative of u in the distributional sense. We aim for solutions in the maximal regularity space The associated real interpolation space X lÀ1=p for the initial data is given by X lÀ1=p ¼ ðX 0 ; X 1 Þ lÀ1=p;p . Since in the given situation of bounded convex domains, the domains of D w N and D N are explicitly known, it follows as in Theorem 5.2 of [2] that q;p ðXÞ; s 2 ð0; 1=2 þ 1=2qÞ: ( We show in Section 4 that the critical value l c of l for which we obtain local wellposedness for (1.1) is given by l c ¼ 3=2q þ 1=p. It is thus natural to call this value of l the critical weight and it is hence meaningful to name X l c À1=p;p the critical space for (1.1). Our first result on the Keller-Segel system on bounded convex domains reads as follows.
Theorem 1 Let X & R 3 be a bounded and convex domain and p; q 2 ð1; 1Þ such that q 2 ð3=2; 2 and 3=2q þ 1=p 1. Then, for all ðu 0 ; v 0 Þ 2 B 3=qÀ1 q;p ðXÞ Â N B 3=q q;p ðXÞ the chemotaxis system (1.1) admits a unique solution w ¼ ðu; vÞ T with Remark 1 It is interesting to compare Theorem 1 with the results obtained in [11] dealing with a much more complex geometrical situation. It shows that the method of time-weights combined with the special situation of bounded convex domains allows to improve the regularity index for the initial data by more than 1, meaning from B s q;r ðXÞ for s [ 3=q þ 1 and r [ 2ð1 À 3=qÞ À1 in [11] to B 3=q q;p ðXÞ.
In order to extend Theorem 1 to a global existence result for small data, we consider for 1\r\1 the space L r 0 ðXÞ ¼ fu 2 L r ðXÞ : R X u ¼ 0g consisting of all L r -functions having mean zero. We then set X 0 :¼ À W 1;q 0 ðXÞ \ L q 0 0 ðXÞ Á 0 Â L q ðXÞ and consider the weak Corollary 1 Let X & R 3 be a bounded and convex domain and p; q 2 ð1; 1Þ such that q 2 ð3=2; 2 and 3=2q þ 1=p 1. Then there exists r 0 [ 0 such that the local solution w given in Theorem 1 exists globally and converges exponentially to zero in ðX 0 ; X 1 Þ 1À1=p;p provided kw 0 k ðX0;X1Þ 3=2q;p r 0 .
Our second main result concerns the existence of strong T-periodic solutions to the Keller-Segel model in bounded convex domains. For recent results on periodic solutions in the situation of bounded domains with smooth boundaries we refer to [8]. Similarly as above, we rewrite the Eq. (1.3) as an evolution equation on To this end, we recall A 0 and define for w ¼ ðu; vÞ T the mapping H as Here D N and D w N;0 denote the Neumann Laplacian on L q ðXÞ and À W 1;q 0 ðXÞ \ L q 0 0 ðXÞ Á 0 equipped with the domains DðD N Þ :¼ fu 2 W 2;q ðXÞ : o m u ¼ 0 on oXg and W 1;q ðXÞ \ L q 0 ðXÞ, respectively. For details, see Section 3. Setting X 1 :¼ DðD w N;0 Þ Â DðD N Þ, we see that X 1 ,!X 0 is densely embedded and that A 0 : X 1 ! X 0 is bounded.
The Keller-Segel system in the periodic setting corresponds then to the equation w 0 ðtÞ À A 0 wðtÞ ¼ Hðt; wðtÞÞ; t 2 ð0; TÞ; wð0Þ ¼ wðTÞ: Being interested again in strong solutions, we define ð2:5Þ as well as Our result on periodic solutions to (1.3) reads as follows.

The Laplacian on bounded and convex domains
For bounded domains X & R n with smooth boundary oX it is well known that the Laplacian D with domain DðDÞ ¼ fu 2 W 2;q ðXÞ : o m u ¼ 0 on oXg generates an analytic semigroup of positive contractions on L q ðXÞ for all q 2 ð1; 1Þ and that it satisfies the maximal L p -regularity property, see [5].
The results by Wood [24] show that this is no longer the case for arbitrary Lipschitz domains. However, under suitable assumptions on the Lipschitz domain X and the exponent q, the above operator still generates a positive, analytic and contractive semigroup on L q ðXÞ.
In the following, we consider bounded, convex domains X & R n , n ! 2, and define the Neumann-Laplacian D N on L q ðXÞ for 1\q\1 by D N u :¼ Du; For the particular situation of n ¼ 3 and 1\q 2 it was proved by Wood [24] that D N generates an analytic semigroup of positive contractions on L q ðXÞ provided X is bounded and convex. He proved in addition that ÀD N admits the property of L p ½0; T-regularity on L q ðXÞ provided 1\q 2 and 0\T\1.
Since D N generates a positive semigroup on L q ðXÞ, it follows from the results in [6] (see also Section 10.7 of [12]) that ÀD N admits a bounded H 1 -calculus on L q ðXÞ as well as on L q 0 ðXÞ with / 1 ÀD N p=2 provided X is bounded, convex and n ¼ 3 and 1\q 2. Since the semigroup generated by D N À x is bounded analytic on L q ðXÞ for suitable x ! 0, we conclude by Theorem 10.7.13 of [12] that / 1 ÀD N þx \p=2. This reproves in particular the fact that ÀD N admits maximal L p -regularity on L q ðXÞ provided X is bounded, convex and 1\q 2.
We next introduce the complex interpolation-extrapolation scale ðX a ; A a Þ, a 2 R, generated by ðX 0 ; A 0 Þ ¼ ðL q ðXÞ; D N Þ. Consider the weak Neumann-Laplacian D w N on W À1;q ðXÞ defined by D w N :¼ A À1=2 : W 1;q ðXÞ ! W À1;q ðXÞ; which has the explicit representation for ðu; /Þ 2 W 1;q ðXÞ Â W 1;q 0 ðXÞ with 1=q þ 1=q 0 ¼ 1. Since kf ðÀD w N Þk LðW À1;q Þ ¼ kB À1 f ðÀD w N ÞðB À1 Þ À1 k LðW À1;q Þ Ckf ðÀD N Þk LðLqÞ for f 2 H 1 0 ðR u Þ it follows that ÀD w N has a bounded H 1 -calculus on W À1;q ðXÞ with the same angle as ÀD N . Here H 1 0 ðR u Þ and f ðÀD w N Þ are defined as in [5]. In particular, D w N admits maximal L p -regularity on W À1;q ðXÞ. Concerning bounded imaginary powers of D N note that the transference principle of Coifman-Weiss implies that kðÀD N Þ it k L q Mð1 þ t 2 Þe pjtj=2 for t 2 R. Since D N is selfadjoint on L 2 ðXÞ it follows by interpolation that ÀD N has bounded imaginary powers on L q ðXÞ of angle h, where h [ pj1=q À 1=2j. As before, here n ¼ 3 and q 2 ð1; 2.
We summarize our calculations in the following proposition.
(a) The operator ÀD N admits a bounded H 1 -calculus on L q ðXÞ and there exists x ! 0 such that ÀD N þ x admits a bounded H 1 -calculus on L q ðXÞ of angle / 1 ÀD N þx \p=2. (b) The operator ÀD w N admits a bounded H 1 -calculus on W À1;q ðXÞ and there exists x ! 0 such that ÀðD N þ xÞ w admits a bounded H 1 -calculus on W À1;q ðXÞ of angle / 1 ÀðDN þxÞ w \p=2. (c) The operators ÀD N and ÀD w N admit bounded imaginary powers on L q ðXÞ and W À1;q ðXÞ, respectively, of angle u [ pj1=q À 1=2j. (d) The operators ÀD w N and ÀD N admit maximal L p ð½0; TÞ-regularity on W À1;q ðXÞ and L q ðXÞ, respectively.
We also consider the Neumann Laplacian D N;0 on L q 0 ðXÞ as well as the weak Neumann Laplacian on ðW 1;q 0 ðXÞ \ L q 0 0 ðXÞÞ 0 , where as above L r 0 ðXÞ ¼ fu 2 L r ðXÞ : R X u ¼ 0g for 1\r\1 and set where X & R 3 is bounded and convex.

The initial value problem on convex domains
We start this section by recalling from [21] and [20] (see also Chapter I of [7]) some results on semilinear evolution equations, on which we will base the proof of Theorem 1. Let X 0 ; X 1 be Banach spaces such that X 1 ,!X 0 is densely embedded, and let A : X 1 ! X 0 be bounded. For 0\T 1 consider the semi-linear problem Here u 0 denotes the time derivative of u in the distributional sense. We aim for solutions in the maximal regularity space E l ðJÞ :¼ H 1;p l ðJ; X 0 Þ \ L p l ðJ; X 1 Þ: As space for the initial data u 0 we introduce the real interpolation space u 0 2 X c;l ¼ ðX 0 ; X 1 Þ lÀ1=p;p and for f 2 F l ðJÞ :¼ L p l ðJ; X 0 Þ; where p 2 ð1; 1Þ. We define for b 2 ½0; 1 the space X b as the complex interpolation space The following existence and uniqueness results are based on the following assumptions: (H1) A has maximal L p -regularity for p 2 ð1; 1Þ.
For the definition and properties of UMD spaces see e.g. [12]. Furthermore, the solution u depends continuously on the data.
When investigating the question of a global solution, we consider t þ ðu 0 Þ :¼ supfT 0 [ 0 : equation ð4:1Þ admits a solution v 2 E l ð0; T 0 Þg: By the above Proposition 2, this set is non-empty, and we say that (4.1) has a global solution if for f 2 L q l ð0; T; X 0 Þ, 0\T\1, which can be extended trivially to ð0; 1Þ, one has t þ ðu 0 Þ ¼ 1. Global existence results can be derived from suitable a priori bounds following [19,Theorem 5.7.1].
In the particular case of bilinear nonlinearities, i.e. FðuÞ ¼ Gðu; uÞ with G : X b Â X b ! X 0 bilinear and bounded one obtains a global solution to (4.1) provided the data are small enough in the ðX 0 ; X 1 Þ 1À1=p;p -norm. More precisely, the following corollary to Proposition 2 holds.
The assertion of Corollary 1 follows from Corollary 3.

Strong time periodic solutions
We start by recalling from [4] that time periodic solutions to linear Cauchy problems within the maximal L p -regularity class are well understood. Indeed, let X 0 be a Banach space and A : X 1 ! X 0 be a linear operator. For p 2 ð1; 1Þ and setting For definitions of UMD-spaces and R-bounded families of operators as well as their properties, we refer to [5,12,19]. The relationship between maximal periodic L p -regularity and maximal L p -regularity for the Cauchy problem can be described, following again the results of Arendt and Bu [4], as follows. & An extension of the above result to the quasilinear and semilinear situation was given in [8]. To formulate this result, let X 0 ; X 1 be Banach spaces with X 1 densely embedded in X 0 and for 1\p\1 set X c ¼ ðX 0 ; X 1 Þ 1À1=p;p . We introduce the following Lipschitz condition on the nonlinearity F. To prove Theorem 2 we note first that ÀA 0 on X 0 defined as in (2.3) admits maximal periodic L p -regularity on X 0 . In fact, we have 0 2 qðD w N;0 Þ and 0 2 qðD N À 1Þ and hence 0 2 qðÀA 0 Þ. Due to Corollary 2, ÀD w N;0 on ðW 1;q 0 ðXÞ \ L q 0 0 ðXÞÞ 0 as well as ÀðD N À 1Þ on L q ðXÞ admit maximal L p -regularity. Hence, the triangular structure of A 0 implies that ÀA 0 admits maximal periodic L p -regularity on X 0 .
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Declarations
Conflict of interest The authors declare that they have no conflict of interest.
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