Radial solutions to a chemotaxis-consumption model involving prescribed signal concentrations on the boundary

The chemotaxis system \begin{align*} u_t&= \Delta u - \nabla \cdot (u\nabla v), \\ v_t&= \Delta v - uv, \end{align*} is considered under the boundary conditions $\frac{\partial u}{\partial\nu}- u\frac{\partial v}{\partial\nu}=0$ and $v=v_\star$ on $\partial\Omega$, where $\Omega\subset\mathbb{R}^n$ is a ball and $v_\star$ is a given positive constant. In the setting of radially symmetric and suitably regular initial data, a result on global existence of bounded classical solutions is derived in the case $n=2$, while global weak solutions are constructed when $n\in \{3,4,5\}$. This is achieved by analyzing an energy-type inequality reminiscent of global structures previously observed in related homogeneous Neumann problems. Ill-signed boundary integrals newly appearing therein are controlled by means of spatially localized smoothing arguments revealing higher order regularity features outside the spatial origin. Additionally, unique classical solvability in the corresponding stationary problem is asserted, even in nonradial frameworks.


Introduction
Chemotaxis systems, if posed in bounded domains, are usually studied with homogeneous Neumann boundary conditions. Especially where the chemotactic agents partially direct their motion toward higher concentrations of a signal which they consume instead of produce, however, other boundary conditions may become relevant.
In this article, we consider the chemotaxis consumption system posing Dirichlet boundary conditions for the signal concentration v and no-flux conditions for the bacterial population density u.
Arising from a line of investigations concerned with pattern formation in colonies of B. subtilis in a fluid environment [29], such chemotaxis systems with signal consumption, additionally coupled to a Stokes-or Navier-Stokes fluid have been studied extensively over the last decade (for pointers to the literature see e.g. Section 4.1 of the survey [2] or the introduction of [8]; for the model in the context of coral spawing, see e.g. [41]). While most works prescribed no-flux, homogeneous Neumann and homogeneous Dirichlet conditions for bacterial population density, signal concentration and fluid velocity, it turned out that these failed to adequately capture the colourful dynamics observed in the form of the patterns previously alluded to.
In particular, a common result for the long-term behaviour was convergence to a constant state (i.e. a state without any patterns), as e.g. in [26], [40], [34], [37], in [18], [19] and [36] in a system additionally including population growth, or also [11] in a related system with nonlinear diffusion.
Not least because of this, it has been suggested to use different, more realistic, inhomogeneous boundary conditions for the chemical signal (see [5] and [6], but also [29]), namely either Robin type boundary conditions where the rate of oxygen influx is controlled by the local oxygen concentration at the boundary or nonzero Dirichlet conditions directly prescribing the latter. (It has been confirmed [6,Prop. 5.1] that the latter kind of conditions arises as a limit case of the former.) In general, altering boundary conditions can have a profound impact on the solution behaviour in chemotaxis systems, see the appearance of a second critical mass in the Keller-Segel type system with Diricihlet conditions for v studied in [13] if compared to the same system with Neumann conditions; in both cases homogeneous. While -in these particular settings related to (1.1) -more realistic from a modelling perspective, the change of boundary conditions to inhomogeneous conditions brings about additional mathematical challenges.
Accordingly, only few results for chemotaxis-consumption models with boundary conditions different from homogeneous Neumann conditions are available: Those concerning the related system with slightly different chemotaxis and energy consumption studied in [16,17] with inhomogeneous Neumann and Dirichlet conditions are restricted to spatially one-dimensional domains. For Robin-type conditions of the form introduced in [5], also in higher dimensions, the stationary problem of (1.1) has been shown to be uniquely solvable (for any prescribed total mass Ω u of the first component, see [6]), and (under a moderate smallness condition) features as the limit of a parabolic-elliptic simplification of (1.1) (cf. [12]). Also in fluid-coupled systems solutions have been found in the presence of logistic source terms ( [5]), or superlinear diffusion ( [39,28]), or without both ( [7]).
In the Dirichlet setting this article is concerned with we mention [20] where an asymptotic analysis of the vanishing diffusivity limit for v in the stationary system seems to confirm the potential of (1.1) to capture pattern dynamics. In the time-dependent problem (including fluid flow), solutions in R 2 ×[0, 1] were constructed in [22] if the signal consumption was strong, at least quadratic with respect to u, and in Ω ⊂ R 2 in [32], in both cases under a smallness condition on the initial data. Without smallness conditions, solutions to (1.1) coupled to a fluid flow governed by the Stokes equations were constructed in [30] (Ω ⊂ R N with linear diffusion for N = 2 and porous medium type diffusion in higher dimensions) and in [31] (Ω ⊂ R 3 ). Nevertheless, the solution concepts pursued in these works are rather weak and do not yield comparable regularity as [33] and [25] for solutions to the system with homogeneous Neumann conditions.
As to the above-mentioned difficulties concerning (1.2), different strategies have been employed: Exploiting the Robin condition in their systems, the works [5] and subsequently [39] and [28] rely on a Lions-Magenes type transformation converting v to a function with homogeneus Neumann boundary conditions. The energy functional is enhanced by additional "boundary energy" terms in [7]. In [22] and [12], a trace theorem is used to control the boundary integrals by integrals over the domain involving higher derivatives, which are available either due to the simpler elliptic form of the second equation (in [12]) or due to a smallness condition ( [22], also in the result on long-term limit in [12]). In [31], a localized modification of the energy functional was investigated, the localization being detrimental to the regularity information near the boundary. Leaving (1.2) behind, the approaches in [30] and [32] used different energy functionals (or small-data energy functionals), giving rise to less potent a priori estimates, as reflected in the generalized sense of solvability obtained.
Regularity control on the boundary for radial solutions. Main results. In this article, we plan to use radial symmetry as a mean to unravel difficulties related to possible effects that the change from Neumann to Dirichlet boundary conditions for the signal may have on boundary regularity of solutions. Specifically, in a ball Ω = B R (0) ⊂ R n with R > 0 and n ≥ 2, and with a given positive constant v ⋆ , we shall consider the initial-boundary value problem assuming that where, as throughout the sequel, radial symmetry of a function ϕ on Ω is to be understood as referring to the spatial origin.
To make appropriate use of these symmetry assumptions, at a first stage of our analysis we shall rely on the essentially one-dimensional framework thereby generated in order to step by step turn the basic properties of mass conservation in the first component, and uniform L ∞ boundedness in the second, into knowledge on higher order regularity features locally outside the spatial origin (see Section 3 and especially Corollary 3.7). This will particularly enable us to appropriately control boundary integrals which due to the presence of possibly nonzero normal derivatives arise in a spatially global energy analysis related to that in (1.2) (Section 4).
In the spatially planar case, this will be found to entail a priori bounds actually in L ∞ × W 1,∞ , and to thus imply the following statement on global classical solvability and boundedness in (1.3): Theorem 1.1. Let R > 0 and Ω = B R (0) ⊂ R 2 , and suppose that v ⋆ ≥ 0, and that u (0) and v (0) satisfy (1.4). Then there exist unique functions which are such that u > 0 and v > 0 in Ω × [0, ∞), that (u(·, t), v(·, t)) is radially symmetric for all t > 0, and that (u, v) solves (1.3) in the classical sense in Ω × (0, ∞). Moreover, there exists C > 0 such that But also in some higher-dimensional situations, the regularity information gained from our energy analysis can be used to establish a result on global solvability, albeit in a slightly weaker framework: such that u(·, t) and v(·, t) are radially symmetric for a.e. t > 0, and that (u, v) forms a global weak solution of (1.3) in the sense of Definition 6.1 below. Furthermore, there exists C > 0 such that Ω u(·, t) ln u(·, t) ≤ C for almost all t > 0 (1.7) and Ω |∇v(·, t)| 2 ≤ C for almost all t > 0 (1.8) We have to leave open here the question how far information on the large time behaviour of the above solutions that goes beyond the boundedness features in (1.5) and in (1.7)-(1.9) can be derived, especially in the presence of large initial data. After all, a steady state analysis guided by the approach developed in [6] provides the following result which may be viewed as an indication for nontrivial dynamics involving structured states in (1.3): Let Ω ⊂ R n be a bounded domain with smooth boundary, and suppose that v ⋆ ∈ β∈(0,1) C 2+β (Ω) is nonnegative. Then for every m ≥ 0, the stationary problem (7.19) has a unique solution (u, v) ∈ (C 2 (Ω)) 2 satisfying Ω u = m. If Ω = B R (0) and v ⋆ is constant, then this solution is radially symmetric, and both u and v are convex.

Local solvability, approximation and basic properties
In order to simultaneously address, throughout large parts of our analysis, the case n = 2 in which classical solvability is strived for, and the case n ∈ {3, 4, 5} in which we intend to construct a solution via approximation, for ε ∈ [0, 1) let us consider the variants of (1.3) given by for all ξ ≥ 0 and ε ∈ [0, 1); indeed, these choices ensure that (2.1) coincides with (1.3) when ε = 0.
These solutions clearly preserve mass in their first component and are bounded in the second.
for all ε ∈ (0, 1) as well as v| t=0 ≥ v (0) and v| ∂Ω ≥ v ⋆ , the latter due to the fact that Also for the gradient of the second solution component some first a priori estimates are available.

Local estimates outside the origin
In line with common abuse of notation, we occasionally write u ε (r, t) and v ε ( and ε ∈ [0, 1), and in order to prepare our derivation of local estimates outside the origin, we observe that whenever As the above basic estimates imply L 1 bounds for b ε , a straightforward argument based on parabolic smoothing in the one-dimensional heat equation (3.1) yields the following information on regularity of the taxis gradient outside the origin.
, and that χ ≡ 1 in [δ, R], and let (e −tA ) t≥0 denote the one-dimensional heat semigroup generated by the operator A := −(·) rr under homogeneous Dirichlet boundary conditions on ( δ 2 , R). Then known regularization features of the latter ( [10], [23]) ensure that if we fix q ∈ (1, ∞), then we can find (3.5) Apart from that, a combination of Lemma 2.3 with (2.5) and (2.6) shows that since supp χ ⊂ [ δ 2 , R], we can pick c 3 = c 3 (δ) > 0 in such a way that for any ε ∈ [0, 1), On the basis of (3.1) and the fact that , we can therefore utilize (3.4) and (3.5) to estimate This has consequences for bounds on certain powers of u ε and their derivative outside a neighbourhood of the spatial origin.
and that Relying on the positivity of u ε in Ω × (0, T max,ε ) for ε ∈ [0, 1), from (2.1) we then obtain that due to Young's inequality, Here by the Hölder inequality, (0) we may apply Lemma 3.1 to q := 2 1−p to see that thanks to (2.5), with some c 1 = c 1 (p, δ) > 0 we have Therefore, an integration in (3.9) shows that again due to the Hölder inequality and (2.5), because τ ε ≤ 1. This implies (3.7), whereupon (3.8) readily results from (3.7) according to the fact that the Gagliardo-Nirenberg inequality provides for all t ∈ (0, T max,ε ) and ε ∈ [0, 1), and because for all t ∈ (0, T max,ε ) and ε ∈ [0, 1), In contrast to settings with homogeneous boundary conditions, in the present situation it will become necessary to deal with non-vanishing boundary terms. While this section will culminate in corresponding estimates, a key to these becomes visible in the following corollary already.
Proof. By means of the Gagliardo-Nirenberg inequality, we can pick c 1 > 0 with the property that Combining (2.5) with an application of Lemma 3.2 to p := 1 2 thus shows that with some c 2 > 0 we have from which (3.10) follows upon employing the Hölder inequality.
The following elementary observation, a proof of which can be found in [38,Lemma 3.4], will be referred to in Lemma 3.5, Lemma 4.3 and Lemma 5.1.
Whereas the previous estimates for u ε were concerned with temporally integrated quantities, the following lemma provides a temporally uniform bound.
Proof. We again take a function ζ ∈ C ∞ (Ω) fulfilling 0 ≤ ζ ≤ 1 and ζ| B δ 2 (0) ≡ 0 as well as ζ| Ω\B δ (0) ≡ 1, and once more rely on (2.1) to see by means of Young's inequality and (2.3) that Here, taking any p 0 = p 0 (p) > p such that p 0 < 3, we may again draw on Young's inequality to estimate and so that invoking Lemma 3.1 we find c 1 = c 1 (p, δ) > 0 fulfilling From (3.12) we therefore obtain that 1 2 T max,ε } is finite according to Lemma 3.2 and the fact that p 0 < 3, by using an ODE comparison argument along with Lemma 3.4 we infer that for all t ∈ (0, T max,ε ) and ε ∈ [0, 1), and hence conclude as intended, because With these bounds at hand, we can even estimate the derivative of the second component uniformly, again outside a neighbourhood of the origin.
Proof. Once more explicitly relying on radial symmetry, we may use that according to Lemma 2.1 the second equation in (2.1) holds up to ∂Ω throughout (0, T max,ε ), which namely ensures that on ∂Ω we have the one-sided inequality Again thanks to (2.3), this implies that and that, similarly, for all t ∈ (0, T max,ε ) and ε ∈ [0, 1), the claim results from Lemma 3.6 when combined with Corollary 3.3.

Energy analysis
Our approach toward deriving a suitable relative of (1.2) is now launched by the following observation.
Proof. According to the no-flux boundary condition accompanying the first equation in (2.1), while on the basis of the second equation in (2.1) we first compute Here two integrations by parts show that and that so that since ∇|∇v ε | 2 = 2D 2 v ε · ∇v ε , we obtain that Therefore, (4.3) is equivalent to (4.1).
5 The two-dimensional case. Proof of Theorem 1.1 In this section we concentrate on the two-dimensional setting of Theorem 1.1. Since the solutions there will already turn out to be bounded and classical, it is not necessary to resort to an approximation by means of (2.1) for ε > 0. Throughout this section, we will therefore directly address the solutions (u, v) := (u 0 , v 0 ) of (2.1) obtained for ε = 0.
Based on the information provided by Lemma 4.3, we can combine the outcomes of two further testing procedures applied to (2.1) in quite a standard manner, and thereby achieve the following key toward higher order bounds: Then there exists C > 0 such that the solution (u, v) ≡ (u 0 , v 0 ) of (2.1), as corresponding to the choice ε = 0, satisfies where T max := T max,0 is as accordingly provided by Lemma 2.1.
For the proof of Theorem 1.1, we are merely lacking a transfer of the boundedness properties we have just obtained to the spaces that actually occur in the extensibility criterion (2.4): Proof of Theorem 1.1. Based on the outcome of Lemma 5.2, we may again utilize known smoothing properties of the Dirichlet heat semigroup on Ω, and additionally employ a standard result on gradient Hölder regularity in scalar parabolic equations ( [21]), to find c 1 > 0, c 2 > 0 and θ 1 ∈ (0, 1) such that v(·, t) W 1,∞ (Ω) ≤ c 1 for all t ∈ (0, T max ) (5.10) and ∇v where again τ 0 = min{1, 1 2 T max }. According to (5.10), we may thereafter rely on the latter token once again to infer from Lemma 5.2 and the first sub-problem in (2.1) that with some c 3 > 0 and θ 2 ∈ (0, 1) we have which combined with (5.10) and (2.4) shows that Lemma 2.1 indeed asserts that T max = ∞, whereupon (1.5) becomes a consequence of Lemma 5.2 and (5.10). 20 6 The case n ≥ 3. Proof of Theorem 1.2 The solution concept to be pursued in higher-dimensional cases appears to be quite natural.
Then (u, v) will be called a global weak solution of (1 . In order to construct such solutions, we now utilize solutions of the approximate versions of (2.1), that is, those corresponding to positive values of ε. As can easily be seen, the strength of the accordingly regularizing features F ε is sufficient to ensure that each of these solutions is global in time: Lemma 6.2. Let n ≥ 3 and ε ∈ (0, 1). Then T max,ε = ∞.
In passing to the limit ε ց 0, we will use the following consequences of Lemma 4.3 on further spatiotemporal bounds. Proof. From the Gagliardo-Nirenberg inequality and (2.5) we obtain c 1 > 0 and c 2 > 0 such that for all t > 0 and ε ∈ (0, 1), and an application of Young's inequality shows that for all t > 0 and ε ∈ (0, 1).
This essentially establishes our main result on global weak solvability in (1.3) already.

Stationary states
We finally consider the stationary problem associated with (1.3), that is, the boundary value problem and our arguments in this regard will be closely related to those in [6], where the second equation was instead supplemented by Robin-type boundary conditions. Dropping the requirement of radial symmetry here, we will assume that Ω ⊂ R n is a bounded bounded with smooth boundary, and that with some β ∈ (0, 1), v ⋆ belongs to C 2+β (Ω) and is nonnegative on ∂Ω.
A first essential observation is that we can eliminate u from the stationary system and subsequently deal with a single equation only. Especially regarding the question of uniqueness, the appearance of a constant parameter α in said equation could turn out to be unfortunate. However, we will later show that α is in one-to-one correspondence with Ω u. (An alternative would be to compute α = m Ω e v and work with the nonlocal equation for v, see [20], where this approach was used for the special case of constant boundary value.) If, on the other hand, (7.21) holds for some v ∈ C 2 (Ω) and α ∈ R, then (7.20). Furthermore, the signs of Ω u and α coincide.
Proof. While the second part of the statement directly follows from the chain rule and the last part is obvious after integration of (7.21), the first is identical to [6,Lemma 4.1].
We now take care of solvability and some a priori estimates for solutions of the second equation of (7.19) if we insert (7.21), firstly in a related linear problem.
With this, solving the second equation of (7.19) with (7.21) is possible: For every α ≥ 0, the boundary value problem has a solution v ∈ C 2 (Ω), and v satisfies (7.23) and (7.24).
As a particular consequence of Lemma 7.4, for every α ≥ 0 the solution to (7.25) is unique. From now on, we will denote it by v α . That, according to Lemma 7.4, v α is decreasing with respect to α is of little help with regard to the monotonicity of αe vα (or rather α Ω e vα ). For further information we study the derivative of v α w.r.t. α.
Lemma 7.6. For every α > 0, Proof. We abbreviate v ′ = v ′ α and v = v α . From Lemma 7.4, we obtain that 0 ≥ v ′ . We let x 0 ∈ Ω be such that v ′ (x 0 ) = min Ω v ′ . Then v ′ ≡ 0 (which would finish the proof) or x 0 ∈ Ω and due to (7.27) Consequences of Lemma 7.6 on the desired relation between α and m = Ω u are as follows: In the case when Ω is a ball and v ⋆ is constant, radial symmetry follows from the above uniqueness statement. Since u = α exp(v) by Lemma 7.1 and exp is monotone and convex, to complete the proof it is sufficient to show convexity of v, that is of the solution to (7.25). But when written in radial coordinates, (7.25) turns into with v r (0) = 0 due to radial symmetry and differentiability of v. Hence, Nonnegativity of v (cf. (7.23)) shows that hence v r ≥ 0; thus the rightmost expression in (7.31) is clearly increasing with respect to r, which shows monotonicity of v r and therefore convexity of v.