Global existence, boundedness and stabilization in a high-dimensional chemotaxis system with consumption

This paper deals with the homogeneous Neumann boundary-value problem for the chemotaxis-consumption system \begin{eqnarray*} \begin{array}{llc} u_t=\Delta u-\chi\nabla\cdot (u\nabla v)+\kappa u-\mu u^2,\\ v_t=\Delta v-uv, \end{array} \end{eqnarray*} in $N$-dimensional bounded smooth domains for suitably regular positive initial data. We shall establish the existence of a global bounded classical solution for suitably large $\mu$ and prove that for any $\mu>0$ there exists a weak solution. Moreover, in the case of $\kappa>0$ convergence to the constant equilibrium $(\frac{\kappa}{\mu},0)$ is shown.


Introduction
Chemotaxis is the adaption of the direction of movement to an external chemical signal. This signal can be a substance produced by the biological agents (cells, bacteria) themselves, as is the case in the celebrated Keller-Segel model ( [5], [4]) or -in the case of even simpler organisms -by a nutrient that is consumed. A prototypical model taking into account random and chemotactically directed movement of bacteria alongside death effects at points with high population densities and population growth together with diffusion and consumption of the nutrient is given by u t = ∆u − χ∇ · (u∇v) + κu − µu 2 (1.1) which we have given in the form with logarithmic source terms paralleling that in (1.1), at first glance, (1.1) seems much more amenable to the global existence (und boundedness) of solutions -after all, the second equation by comparison arguments immediately provides an L ∞ -bound for v. However, such a bound is not sufficient for dealing with the chemotaxis term, and accordingly global existence and boundedness of solutions to (1.1) with κ = µ = 0 is only known under the smallness condition on the initial data ( [15]) or in a two-dimensional setting ( [22], [25] and also [9]). Their rate of convergence has been treated in [26]. In three-dimensional domains, weak solutions have been constructed that eventually become smooth [16].
For (1.2), the presence of logarithmic terms has been shown to exclude otherwise possible finite-time blow-up phenomena (cf. [23], [11]) -at least as long as µ is sufficiently large if compared to the strenght of the chemotactic effects ( [21]) or if the dimension is 2 ( [12]). If the quotient µ χ is sufficiently large, solutions to (1.2) uniformly converge to the constant equilibrium ( [24]); convergence rates have been considered in [3]. Explicit largeness conditions on µ χ that ensure convergence, also for slightly more general source terms, can be found in [10], see also [19]. For small µ > 0, at least global weak solutions are known to exist ( [7]), and in 3-dimensional domains and for small κ, their large-time behaviour has been investigated ( [7]).
Also the chemotaxis-consumption model (1.1) has already been considered with nontrivial source terms in [18]. There it was proved that classical solutions exist globally and are bounded as long as (1.3) holds -which is the same condition as for κ = µ = 0, thus shedding no light on any possible interplay between chemotaxis and the population kinetics.
In a three-dimensional setting and in the presence of a Navier-Stokes fluid, in [8] it was recently possible to construct global weak solutions for any positive µ, which moreover eventually become classical and uniformly converge to the constant equilibrium in the large-time limit.
It is the aim of the present article to prove the existence of global classical solutions if only µ is suitably large and to show their large-time behaviour. For the case of small µ > 0, we will prove the existence of global weak solutions (in the sense of Definition 6.1).
What largeness condition on µ might be sufficient for boundedness? For the Keller-Segel type model (1.2) the typical condition reads: 'If µ is large compared to χ, then the solution is global and bounded, independent of initial data.' In order to see why this condition would be far less natural for (1.1), let us suppose we are given suitably regular initial data u 0 , v 0 and a corresponding solution (u, v) of and let us define w := χv.
which is the same system, only with different chemotaxis coefficent and rescaled initial data for the second component. Consequently, in (1.1), large initial data equal high chemotactic strength. Hence, there cannot be any condition for global existence which includes µ and χ, but not v 0 L ∞ (Ω) . In light of this discussion, the requirement in Theorem 1.1 that µ be large with respect to χ v 0 L ∞ (Ω) seems natural. On the other hand, this observation does not preclude conditions that involve neither χ nor v 0 L ∞ (Ω) , and indeed µ > 0 is sufficient for the global existence of weak solutions. The first main result of the present article is global existence of classical solutions, provided that µ is sufficiently large as compared to χv 0 L ∞ (Ω) : Theorem 1.1. Let N ∈ N and let Ω ⊂ R N be a smooth, bounded domain. There are constants k 1 = k 1 (N) and k 2 = k 2 (N) such that the following holds: Whenever κ ∈ R, χ > 0, and µ > 0 and initial data has a unique global classical solution (u, v) which is uniformly bounded in the sense that there is some constant C > 0 such that The second outcome of our analysis is concerned with the large time behaviour of global solutions and reads as follows: Theorem 1.2. Let N ∈ N and let Ω ⊂ R N be a bounded smooth domain. Suppose that χ > 0, κ > 0 and µ > 0. Let (u, v) ∈ C 2,1 (Ω × (0, ∞)) ∩ C 0 (Ω × [0, ∞)) be any global bounded solution to (1.5) (in the sense that (1.6) is fulfilled) which obeys (1.4). Then and v(·, t) L ∞ (Ω) → 0 (1.8) as t → ∞. These solutions, too, stabilize toward ( κ µ , 0) as t → ∞, even though in a weaker sense than guaranteed by Theorem 1.2 for classical solutions:

Remark 1.4.
Under the restriction N = 3, the existence of global weak solutions that eventually become smooth and uniformly converge to ( κ µ , 0) has been proven in [8], where a coupled chemotaxis-fluid model is treated.
Plan of the paper. In Section 2 we will prepare some general calculus inequalities. In the following for some a > 0 we will then consider For ε = 0, this system reduces to (1.5); for ε ∈ (0, 1) we will be able to derive global existence of solutions without any concern for the size of initial data and hence obtain a suitable stepping stone for the construction of weak solutions. Beginning the study of solutions to this system in Section 3 with a local existence result and elementary properties of the solutions, we will in Section 4 consider a functional of the type Ω u p + Ω |∇v| 2p and finally, aided by estimates for the heat semigroup, obtain globally bounded solutions, thus proving Theorem 1.1. In Section 5 where κ is assumed to be positive, we will let a := µ κ and employ the functional in order to derive the stabilization result in Theorem 1.2 and already prepare Theorem 1.4. Section 6, finally, will be devoted to the construction of weak solutions to (1.5), and to the proofs of Theorem 1.3 and Theorem 1.4.

Remark 1.5.
In (1.9), the additional term −εu 2 ε ln au ε could be replaced by −εΦ(u ε ) with some other continuous function Φ which satisfies: Φ(s) → 0 as s ց 0, Φ(s) s 2 → ∞ as s → ∞ and, for the stabilization results in Section 5, Φ < 0 on (0, κ µ ) as well as Φ > 0 on ( κ µ , ∞). We will always let and note that the choice for the case κ ≤ 0 was arbitrary and that in Sections 4 and 6, the precise value of a plays no important role.
Notation. For solutions of PDEs we will use T max to denote their maximal time of existence (cf. also Lemma 3.1). Throughout the article we fix N ∈ N and a bounded, smooth domain Ω ⊂ R N .

General preliminaries
In this section we provide some estimates that are valid for all suitably regular functions and not only for solutions of the PDE under consideration.

Lemma 2.1. a) For any c ∈ C 2 (Ω):
Proof. a) Straightforward calculations yield Let us now derive the following interpolation inequality on which we will rely in obtaining an estimate for holds, where D 2 c denotes the Hessian of c.

Local existence and basic properties of solutions
We first recall a result on local solvability of (1.9): , let κ ∈ R, µ > 0, χ > 0 and q > N. Then for any ε ∈ [0, 1) there exist T max ∈ (0, ∞] and unique classical solution (u ε , v ε ) of system (1.9) with a as in (1.10) in Ω × (0, T max ) such that Proof. Apart from minor adaptions necessary if ε > 0 (see also [19,  Even thought the total mass is not conserved, an upper bound for it can be obtained easily: Proof. Because s 2 ln(as) ≥ − 1 2a 2 e for all s > 0, integrating the first equation in (1.9) over Ω and applying Hölder's inequality shows that and the claim results from an ODI-comparison argument.
For the second component, even uniform boundedness can be deduced instantly: and is monotone decreasing.
Proof. This is a consequence of the maximum principle and the nonnegativity of the solution.
Also the gradient of v can be controlled in an L 2 (Ω)-sense: There exists a positive constant M such that for all ε ∈ [0, 1) the solution of (1.9) with a as in (1.10) satisfies Proof. Integration by parts and the Young inequality result in Adding (3.5) to (3.6) and taking into account that for any ε ∈ [0, 1) and s ≥ 0, we obtain that Since Lemma 3.2 shows that Ω u ε (x, t)dx ≤ m 1 for any ε ∈ [0, 1) and t ∈ (0, T max ), a comparison argument leads to holding true on (0, T max ), which in particular implies (3.4)

Existence of a bounded classical solution
We now turn to the analysis of the coupled functional of Ω u p and Ω |∇v| 2p . We first apply standard testing procedures to gain the time evolution of each quantity.
Proof. Testing the first equation in (1.5) against u p−1 ε and using Young's inequality, we can obtain on (0, T max ), which by using the fact that directly results in (4.1).
, any ε ∈ [0, 1), we have that the solution of (1.9) with a as in (1.10) satisfies Proof. We differentiate the second equation in (1.5) to compute Upon multiplication by (|∇v ε | 2 ) p−1 and integration, this leads to on (0, T max ). Then integrating by parts, we achieve throughout (0, T max ), were we have used Lemma 3.3. Next by Young's inequality and Lemma 2.1 a) we have that Thereupon, (4.4) implies that on (0, T max ).
Next we will show that if µ is suitably large, then all integrals on the right side in (4.1) and (4.3) can adequately be estimated in terms of the respective dissipated quantities on the left, in consequence implying the L p estimate of u and the boundedness estimate for |∇v|.
The previous lemma ensures boundedness of u in some L p -space for finite p only. Fortunately, this is already sufficient for the solution to be bounded -and global.
In fact, the assumption of Lemma 4.4 suffices for even higher regularity, as we will see in Lemma 5.1. For the moment we return to the proof of global existence of solutions.

Stabilization
In this section, we shall consider the large time asymptotic stabilization of any global classical bounded solution.
In a first step we derive uniform Hölder bounds that will facilitate convergence. After that, we have to ensure that solutions actually converge, and in particular must identify their limit. In the spirit of the persistence-of-mass result in [17], showing that v → 0 as t → ∞ would be possible by relying on a uniform lower bound for Ω u and finiteness of ∞ 0 Ω uv (see also [8, Lemmata 3.2 and 3.3]). We will instead focus on other information that can be obtained from the following functional of type already employed in [8] (after the example of [19]), namely This way, in Lemma 5.3 we will achieve a convergence result for v ε that will also be useful in the investigation of the large time behaviour of weak solutions in Section 6. Then there are α ∈ (0, 1) and C > 0 such that valid for any t > 0.
Proof of Theorem 1.2. The statement of Lemma 5.4 is even slightly stronger than that of Theorem 1.2.
Proof of Theorem 1.3. The assertion of Theorem 1.3 is part of Lemma 6.3.
Remark 6.1. If N ≤ 3, the uniform bound on t+1 t Ω |∇v| 4 contained in Lemma 6.1 proves to be sufficient for (6.25) even to hold for p = ∞, which can be used as starting point for derivation of eventual smoothness of solutions via a quasi-energy-inequality for Ω u p (η−v) θ with suitable numbers θ and η. This result is already contained in [8].
Proof of Theorem 1.4. Lemma 6.4 is identical with Theorem 1.4.