Improvements and generalizations of results concerning attraction-repulsion chemotaxis models

We enter the details of two recent articles concerning as many chemotaxis models, one nonlinear and the other linear, and both with produced chemoattractant and saturated chemorepellent. These works, when properly analyzed, leave open room for some improvement of their results. We generalize the outcomes of the mentioned articles, establish other statements and put all the claims together; in particular, we select the sharpest ones and schematize them. Moreover, we complement our research also when logistic sources are considered in the overall study.


Preamble
For details and discussions on the meaning of the forthcoming model, especially in the frame of chemotaxis phenomena and related variants, as well as for mathematical motivations and connected state of the art, we refer to [1,2]. These articles will be often cited throughout this work.
Then we prove these two theorems.
When the logistic term h does not take part in the model, problem (1) has been already analyzed in [1] for the nonlinear diffusion and sensitivities case, and in [2] for the linear scenario; nevertheless, in these papers only small values of α are considered. Precisely, for α belonging to (0, 1 2 + 1 n ), boundedness is ensured: • in [1, Theorem 2.1] for m 1 , m 2 , m 3 ∈ R and l ≥ 1, under the assumption or in [2, Theorem 2.2] for m 1 = m 2 = m 3 = 1 and any l > 1. In light of Theorems 2.1 and 2.2, herein we develop an analysis dealing also with values of α larger than 1 2 + 1 n . Additionally, for α belonging to some sub-intervals of (0,  3. Local well posedness, boundedness criterion, main estimates and analysis of parameters For Ω, χ, ξ, δ, m 1 , m 2 , m 3 and f, g, h as above, from now on with u, v, w ≥ 0 we refer to functions of (x, t) ∈ Ω × [0, T max ), for some finite T max , classically solving problem (1) when nonnegative initial data (u 0 , v 0 ) ∈ (W 1,∞ (Ω)) 2 are provided. In particular, u satisfies Further, globality and boundedness of (u, v, w) (in the sense of (2)) are ensured whenever (boundedness criterion) the u-component belongs to L ∞ ((0, T max ); L p (Ω)), with p > 1 arbitrarily large, and uniformly with respect t ∈ (0, T max ). These basic statements can be proved by standard reasoning; in particular, when h ≡ 0 they verbatim follow from [1, Lemmas 4.1 and 4.2] and relation (5) is the well-known mass conservation property. Conversely, in the presence of the logistic terms h as in (4), some straightforward adjustments have to be considered and the L 1 -bound of u is consequence of an integration of the first equation in (1) and an application of the Hölder inequality: precisely for and we can conclude by invoking an ODI-comparison argument.
In our computations, beyond the above positions, some uniform bounds of v(·, t) W 1,s (Ω) are required. In this sense, the following lemma gets the most out from L p -L q (parabolic) maximal regularity; this is a cornerstone and for some small values of α the succeeding W 1,s -estimates are sharper than the W 1,2 -estimates derived in [1,2], and therein employed.

Remark 1.
In view of its importance in the computations, we have to point out that from the above lemma s can be chosen arbitrarily large only when α ∈ 0, 1 n . In particular, as we will see, in such an interval the terms Ω (u + 1) p+2m2−m1−1 |∇v| 2 and Ω (u + 1) 2α |∇v| 2(q−1) , appearing in our reasoning, can be treated in two alternative ways: either invoking the Young inequality or the Gagliardo-Nirenberg one.

4.
A priori estimates and proof of the Theorems 4.1. The non-logistic case. Recalling the globality criterion mentioned in §3, let us define the functional y(t) := Ω (u + 1) p + Ω |∇v| 2q , with p, q > 1 properly large (and, when required, with p = q), and let us dedicate to derive the desired uniform bound of Ω u p .
In the spirit of Remark 1, let us start by analyzing the evolution in time of the functional y(t) by relying on the Young inequality.  Proof. Let p = q > 1 sufficiently large; moreover, in view of Remark 1, from now on, when necessary we will tacitly enlarge these parameters.
Finally, ODE comparison principles imply u ∈ L ∞ ((0, T max ); L p (Ω)), and the conclusion is a consequence of the boundedness criterion in §3.
Remark 2 (On the validity of the theorems in [1] and [2] for α ≥ 1 2 + 1 n ). In the proofs of [1, Theorem 2.1] and [2, Theorems 2.1 and 2.2], it is seen that the L 2 uniform estimate of ∇v is used to control some integral on ∂Ω (and this allows us to avoid to restrict our study to convex domains), as well as to deal with the term Ω |∇v| 2p with the Gagliardo-Nirenberg inequality; for instance we are referring to [2, (28) and (39)], respectively. Such finiteness of Ω |∇v| 2 is related to the values of α in these articles: α ∈ 0, 1 2 + 1 n (see [2,Lemma 4.1]). Apparently only ∇v ∈ L ∞ ((0, T max ); L 1 (Ω)) suffices to address these issues. Indeed, as far as the topological property of Ω is concerned, we can invoke [3, (3.10) of Proposition 8] with s = 1; on the other hand, for the question tied to the employment of the Gagliardo-Nirenberg inequality, we may operate as done in (35). As a consequence, in view of Lemma 3.1, we have that ∇v ∈ L ∞ ((0, T max ); L 1 (Ω)), so that [1, Theorem 2.1] and [2, Theorems 2.1 and 2.2] hold true for any α ∈ (0, 1).