Boundedness for a Fully Parabolic Keller–Segel Model with Sublinear Segregation and Superlinear Aggregation

<jats:p>This work deals with a fully parabolic chemotaxis model with nonlinear production and chemoattractant. The problem is formulated on a bounded domain and, depending on a specific interplay between the coefficients associated to such production and chemoattractant, we establish that the related initial-boundary value problem has a unique classical solution which is uniformly bounded in time. To be precise, we study this zero-flux problem 
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                  <mml:mi>u</mml:mi>
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                  <mml:mi>v</mml:mi>
                </mml:math></jats:alternatives></jats:inline-formula> are actually uniformly bounded in time.</jats:p><jats:p>The paper is in line with the contribution by Horstmann and Winkler (J. Differ. Equ. 215(1):52–107, 2005) and, moreover, extends the result by Liu and Tao (Appl. Math. J. Chin. Univ. Ser. B 31(4):379–388, 2016). Indeed, in the first work it is proved that for <jats:inline-formula><jats:alternatives><jats:tex-math>$g(u)=u$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
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                  <mml:mi>u</mml:mi>
                </mml:math></jats:alternatives></jats:inline-formula> the value <jats:inline-formula><jats:alternatives><jats:tex-math>$\alpha =\frac{2}{n}$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:mi>α</mml:mi>
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                </mml:math></jats:alternatives></jats:inline-formula> represents the critical blow-up exponent to the model, whereas in the second, for <jats:inline-formula><jats:alternatives><jats:tex-math>$f(u)=u$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:mi>f</mml:mi>
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                  <mml:mi>α</mml:mi>
                  <mml:mo>=</mml:mo>
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                </mml:math></jats:alternatives></jats:inline-formula>, boundedness of solutions is shown under the assumption <jats:inline-formula><jats:alternatives><jats:tex-math>$0< l<\frac{2}{n}$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:mn>0</mml:mn>
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                </mml:math></jats:alternatives></jats:inline-formula>.</jats:p>


Introduction and motivations
Most of this article is dedicated to the following Cauchy boundary problem in Ω × (0, T max ), v t = ∆v − v + g (u) in Ω × (0, T max ), defined in a bounded and smooth domain Ω of R n , with n ≥ 2, and formulated through some functions f = f (s) and g = g(s), sufficiently regular in their argument s ≥ 0, and further regular initial data u 0 (x) ≥ 0 and v 0 (x) ≥ 0. Additionally, the subscription (·) ν indicates the outward normal derivative on ∂Ω whereas T max the maximum time up to which solutions to the system are defined. ∼ = u α for large values of u indicates that if α ∈ (0, 2 n ) any (u 0 , v 0 ) produces uniformly bounded classical solutions to problem (1) (see [19]), whilst for α > 2 n blow-up solutions either in finite of infinite time can be constructed, even for arbitrarily small initial data (see [24]). Moreover, the insight about the quantitative role of the diffusion of the cells on the evolution of the model reads as follows: for D(u) ∼ = u m−1 and f (u) ∼ = u α , m, α ∈ R, it is established in [1,2] (for the fully parabolic case) and in [26] (for the simplified parabolic-elliptic one) that α < m + 2 n − 1 is condition sufficient and necessary in order to ensure global existence and boundedness of solutions. (For completeness, we also refer to [13], where an estimate for the blow-up time of unbounded solutions to the simplified model is derived.) Unlike the case where D(u) = 1 and f (u) = u where the critical mass m c is n-independent, the above criterion implies that the size of the initial mass may have no crucial role on the existence of global or local-in-time solutions to nonlinear diffusion chemotaxis-systems. Conversely, the key factor is given by some specific interplay between the coefficients m, α and the dimension n; in particular, as mentioned, this is especially observed at high dimensions, for which a magnification of the diffusion parameter is required to compensate instability effects. A similar consideration, appropriately reinterpreted in that context, will be given below when the exponent associated to the nonlinear signal production is introduced.
Complementary, as far as nonlinear segregation chemotaxis models are concerned, when in problem (1) the case f (u) = u is considered, uniformly boundedness of all its solutions is proved in [12] for g(u) ∼ = u l , with 0 < l < 2 n . Moreover, by resorting to a simplified parabolic-elliptic version in spatially radial contexts, for f (u) = u and the second equation reduced to 0 = ∆v − µ(t) + g(u), with g(u) ∼ = u l and µ(t) = 1 |Ω| Ω g(u(·, t)), it is known (see [25]) that the same conclusion on the boundedness continues valid for any n ≥ 1 and 0 < l < 2 n , whereas for l > 2 n blow-up phenomena may appear. 2. Presentation of the main result and comparison with a simplified model. Plan of the paper 2.1. Claim of the main result. In accordance to what discussed above, we are interested in addressing situations concerning system (1) that, to our knowledge, are not yet discussed in the literature. To this aim, from now these assumptions, respectively identifying the actions associated to the chemoattractant and to the segregation of the chemical signal, are fixed: (3) f ∈ C 2 ([0, ∞)), f (0) = 0 and f (s) ≤ Ks α , for some K, α > 0 and all s ≥ 0, and (4) g ∈ C 1 ([0, ∞)) and 0 ≤ g(s) ≤ K 0 s l , for some K 0 , l > 0 and all s ≥ 0.
In particular, with a specific view to what mentioned for [6], linear productions of the chemical may be sufficient to emanate blow-up solutions when the impact from the chemoattractat, favoring gatherings in the motion of the species, is superquadratic, in any dimension, superlinear and subquadratic, in low dimensions, and sublinear in higher. Thus seems meaningful the following question: • May a sublinear signal segregation of the chemical enforce globability of solutions for superlinear chemosensitivitiy even in high dimensions? Our result positively addresses this issue in the sense that independently of the initial data, by weakening in an inversely proportional way to the dimension the impact associated to the production rate of the chemical, the uniform-in-time boundedness of solutions to model (1) is ensured, even for superlinear thrusts from the chemoattractant.
What said is formally claimed in this Let Ω be a bounded and smooth domain of R n , with n ≥ 2. Moreover, let f and g fulfill (3) and (4), respectively, with l ∈ (0, 2 n ) and α satisfying Then, for any nontrivial Remark 1. We make this considerations: • For α = 1 assumption (5) is simplified into l ∈ (0, 2 n ). In particular, our analysis is an extension of that developed in [12], in the sense that Theorem 2.1 recovers [12, Theorem 1.1] when f (u) = u in problem (1).
• From l ∈ (0, 2 n ), the comparison with the limit linear signal production model for system (1) makes sense only in two-dimensional settings; for l = 1 the upper bound in assumption (5) reads α < 1, and Theorem 2.1 is consistent with [6, Theorem 4.1].
• Since [6, Theorem 4.1] is applicable for any n ≥ 2 whenever α < 2 n and l = 1, a fortiori it holds true for l ∈ (0, 2 n ); this is the sole reason why we consider in our analysis α ≥ 2 n . • Considering that for linear production and nonlinear diffusion (with parameter m) models we discussed that the condition for boundedness reads α < m + 2 n − 1, from assumption (5) one can observe that the parameter l associated to the nonlinear segregation plays an opposite role with respect m: given α, for high values of n, smaller (larger) values of l (m) are needed to ensure globability and boundedness.

2.2.
A view to the parabolic-elliptic case. When the parabolic-parabolic problem (1) is simplified into parabolic-elliptic, with equation for the chemical replaced by 0 = ∆v − v + g(u), assumption (5) becomes sharper; precisely 2 n ≤ α < 1 + 2 n − l, which requires for compatibility only the restriction l ∈ (0, 1). We highlights this aspect in Figure 1, where we overlap the regions defined by the interplay between α and l in both models; in particular, we also distinguish the zones with superlinear chemoattractant (α > 1) and sublinear chemoattractant (α < 1). We understand that the observed gap between the range of parameters is not only justified by some technical reasons (see Remark 3 at the end of the paper, where few mathematical indications are given) but also by biological ones. Indeed, the fact that in the simplified version the values of v(·, t) only depend on the values of u(·, t) at the same time, is a strong modeling assumption. It corresponds to the situation where the signal responses to the concentration of the particles much faster than the organisms do to the signal; in particular, such difference in the relative adjustment of the bacteria and the chemoattractant makes that this one reaches its equilibrium instantaneously.

2.3.
Organization of the paper. The rest of the paper is structured in this way. Section 3 is concerned with the local existence question of classical solutions to (1) and some of their properties. Some general inequalities are included in Section 4. They are mainly devoted to establish how to ensure globability and boundedness of local solutions using their boundedness in some proper Sobolev spaces; a key cornerstone in this direction is the procedure to fix the corresponding exponents of theses spaces ( §4.2). Finally, the mentioned bound is derived in Section 5, which also includes the proof of Theorem 2.1.

Existence of local-in-time solutions and main properties
Let us dedicate to the existence of classical solutions to system (1). It is show that such solutions are at least local and, additionally, satisfy some crucial estimates.

Lemma 3.1 (Local existence).
Let Ω be a bounded and smooth domain of R n , with n ≥ 2. Moreover, let f and g fulfill (3) and (4), respectively, with l ∈ (0, 2 n ) and α satisfying (5). Then, for any nontrivial such that this dichotomy criterion holds true: In addition, the u-component obeys the mass conservation property, i.e.
whilst for some c 0 > 0 the v-component is such that Proof. We just mention that the conclusions concerning the local-in-time well-posedness as well as the dichotomy criterion (6) can be established by straightforward adaptations of widely used methods involving an appropriate fixed point framework and standard parabolic regularity theory; in particular we cite [6, Theorem 3.1] for the case g(u) = u and [21, Lemma 3.1] for g as in our hypotheses. Moreover, comparison arguments apply to yield both On the other hand, the mass conservation property easily comes by integrating over Ω the first equation of (1), in conjunction with the boundary and initial conditions. Finally, the last claim is derived as follows. From the assumption 0 < l < 2 n , we can first of all fix n 2 < γ < n complying with γ ≤ 1 l . In this way, thorough the Hölder inequality, taking in mind (4) and the mass conservation property (7), we have Henceforth, we can also pick 1 we invoke properties regarding the Neumann heat semigroup (e t∆ ) t≥0 (see Section 2 of [6] and Lemma 1.3 of [23]), so to obtain for some As a consequence, the introduction of the Gamma function Γ infers which combined with bounds (9) and (10) conclude the proof.
In the sequel of the paper with (u, v) we will refer to the gained local classical solution to problem (1); in particular we might tacitly avoid to mention that such solution is produced by the initial data (u 0 , v 0 ).

Preliminaries: inequalities and parameters
With the local solution (u, v) to problem (1) at disposal, its uniform boundedness on (0, T max ) is achieved when uniformin-time bound for u in some L p -norm and for |∇v| 2 in some L q -norm, with proper p and q, is derived. This will be obtained by constructing an absorption inequality satisfied by the functional (11) y(t) : In particular, the entire procedures requires, first, to adequately manipulate inequalities resulting by differentiating y(t) with respect to the time and, secondly, to figure out how to choose the parameters p and q. In view of its decisive role, this second step will be discussed with some details after this subsection.
4.1. Some algebraic and functional inequalities. This three coming lemmas will be used in the next logical steps. We start by considering a suitable version of the Gagliardo-Nirenberg interpolation inequality, commonly used to treat nonlinearities appearing in the diffusion and/or chemosensitivity terms (see [18,20,22]), and successively by recalling a particular boundary integral used to deal with terms defined in non-convex domains.
Let Ω be a bounded and smooth domain of R n , with n ≥ 1, and 0 < q ≤ p ≤ ∞ Let Ω be a bounded and smooth domain of R n , with n ≥ 1, and q ∈ [1, ∞). Then for any η > 0 there is C η > 0 such that for any w ∈ C 2 (Ω) satisfying ∂w ∂ν = 0 on ∂Ω this inequality holds: Proof. A proof can be found in [ Thanks to the next results (variants of Young's inequality), products of powers will be estimated by suitable sums involving their bases and powers of sums controlled by sums of powers.
Moreover, for further d 3 , d 4 > 0, it is possible to find positive d 5 and d such that Proof. We show the first inequality, being the proof of the second similar. By applying Young's inequality with conjugate exponents 1 d1 and 1 1−d1 , we obtain for any ǫ 1 > 0 and some c 1 (ǫ 1 ) > 0 that Moreover, due to d2 1−d1 < 1, a further application of the same inequality to the latter power provides for every positive ǫ 2 and proper c 2 (ǫ 2 ) > 0 this relation: . By putting together the two inequalities and choosing ǫ 1 = ǫ 2 , the first part of the lemma is concluded. All the details of the second inequality can be found in [14,Lemma 3.3].

4.2.
The right procedure in fixing the parameters p and q. In this sequence of lemmas, we will verify that the mutual relation between the parameters l in (3) and α and l in (4), i.e. relation (5), is such that the mentioned parameters p and q may be chosen in the appropriate way to make sure our general machinery work.
Lemma 4.4. For any n ∈ N, with n ≥ 2, let l ∈ (0, 2 n ) and α comply with assumption (5). Then there exist 1 < θ < n n−2 and µ > n 2 such that Proof. From now on, we first precise that for n = 2, 1 < θ < n n−2 indicates that θ might also be fixed large as we want; despite that, this is not the case, and we will take θ always sufficiently close to 1.
Precisely, for n ≥ 2, 1 < θ < n n−2 implies that 2nθ + n 2 − n 2 θ > 0; in particular, we can consider θ > 1 small enough so to have n(θ + 1 − 2αθ) + 2θ > 0. Hence the function is the difference of two positive terms. Now, from our assumptions and the claim is proved by means of continuity arguments.
Lemma 4.5. Let the hypotheses of Lemma 4.4 be satisfied, and 1 < θ < n n−2 and µ > n 2 be therein fixed. Then there is q r ∈ [1, ∞) such that for all q > q r one has this compatibility relation: Proof. Some easy computations show that for any n ≥ 2 the claim follows once it is established that for 1 < θ < n n−2 and µ > n 2 as in the hypotheses, and for there is a q ≥ 1 entailing As we justify and explain it in Figure 2, the above occurs whenever A − C < 0.  (14) we intend to find q r such that for some θ and µ we have that k(θ, µ; q) < 0 for all q > q r . Let θ close to 1 and µ sufficiently large be taken from Lemma 4.4. As a consequence, inequality (12) leads to A − C < 0, whereas B − D ∈ R. In particular, considering that ∂k(·,·;q) ∂q and lim q→+∞ k(·, ·; q) = A − C, the illustration shows the qualitative behavior of the function k(θ, µ; q) for these values of θ and µ, assuming the nontrivial situation B − D < 0. (If, indeed, θ and µ infer B − D ≥ 0, k is negative for all q.) Then, by indicating with q r = B−D A−C the root of k, any q ∈ (q r , ∞) satisfies relation (14). (In order to clarify the role of µ and θ, we observe that for n = 2 the chain of inequality in (13) is more manageable; in fact, it reads lq < q(2θ − 2αθ + 1), directly coming from l < 2θ − 2αθ + 1, corresponding to (12) when n = 2, and it is µ-independent and true for some θ approaching 1, once α < 3−l 2 from (5) is considered.) belong to the interval (0, 1) and, additionally, imply that these other relations hold true: (15) β 1 + γ 1 = p − 2 + 2α p a 1 + 1 q a 2 ∈ (0, 1) and β 2 + γ 2 = 2l p a 3 + q − 1 q a 4 ∈ (0, 1).

Remark 2.
Let us spend some words on how to treat the introduced parameters p and q in accordance with our overall purposes. This technical detail makes the analysis of the present work different and in some sense more thorough with respect those presented in many references above mentioned; therein, indeed, no undesired smallness assumption on p, generally, appears.
(i) Taking the "lower extremes" for q in (q r , ∞) and for p in (f 1 (q), f 2 (q)), as specified in Lemma 4.6, might not be appropriate when dealing with other computations where they are involved. In particular, as we will perform in the last step toward the proof of Theorem 2.1, it could be necessary to enlarge each one of this values in order to ensure the validity of certain inequalities/inclusions. Despite that, we understand that some care is needed when this procedure has to be adopted; indeed, p cannot be taken large as we want independently by q, but this is possible when the order f 1 (q) < p < f 2 (q) related to relation (13) is preserved. (This was already imposed in the same Lemma 4.6.) (ii) In support to the previous item, we point out that even though asymptotically we have µ ′ (n−2(q+1)) ր 1 increasing with q, this is not sufficient to ensure that there exists a couple (p, q) for which both p−2+2α p a 1 + 1 q a 2 < 1 and 2l p a 3 + q−1 q a 4 < 1 are satisfied. Surely each one of this inequality holds true for two different couples, let's say (p 0 , q 0 ) and (p 1 , q 1 ), but the identification of a single (p, q) producing simultaneously those inequalities requires the extra condition p ∈ (f 1 (q), f 2 (q)), intimately linked to the main assumption (5).

5.
Deriving uniform-in-time L p × L q -bounds for (u, |∇v| 2 ). Proof of the main result The coming lemma provides a uniform-in-time bound on (0, T max ) for u in L p (Ω) and for |∇v| 2 in L q (Ω).
Proof. With θ, µ, p and q as in Lemma 4.6, the validity of all the computations along this lemma is justified. As announced, let us differentiate with respect to the time y(t) defined in (11) and split the resulting derivations in three main steps, altogether yielding the proof.
as test function for the first equation in (1), so that by integrating by parts we obtain, also in view of the no-flux boundary conditions, that Through an application of Young's inequality and (3), the latter term reads (23) and the second integral at the right-hand side is estimated by the Hölder inequality so to have Now we can apply Lemma 4.1 with p = 2(p−2+2α)θ p , q = 2 p and, once the following inequality (used in the sequel without mentioning) (x + y) s ≤ 2 s (x s + y s ) for any x, y ≥ 0 and s > 0 is also considered, we obtain for every t ∈ (0, T max ) ≤ c 1 ∇(u + 1) where c 1 > 0 depends on K, C GN , and with a 1 ∈ (0, 1) taken from Lemma 4.6 and belonging to (0, 1) as therein proved. As a consequence, by observing that the mass conservation property (7) implies the boundedness of (u + 1) (25) we have that for some c 2 > 0 and β 1 ∈ (0, 1) deduced from Lemma 4.6 (26) In a similar way, we can again invoke the Gagliardo-Nirenberg inequality, with an evident choice of p and q, to have for some c 3 > 0 and a 2 ∈ (0, 1) as in Lemma 4.6 for all t < T max .
With such a relation in mind, by using |∇v| 2q−2 as test function, a differentiation of the second equation of problem (1) implies that on (0, T max ) this estimate holds: Now, an application of Lemma 4.2 allows us to find C η > 0 such that for some suitable η > 0 (to be chosen later) we have By integrating by parts the latter integral above and using Young's inequality, we get in view of the pointwise inequality |∆v| 2 ≤ n|D 2 v| 2 . Henceforth, by exploiting (30) and recalling assumption (4), we can rephrase (29) as where c 6 is a positive constant depending also on K 0 . Let us now estimate the last integral in the previous bound. By employing the Hölder inequality, we first obtain the following estimate (32) whereas by relying on Lemma 4.1, we find a constant c 7 > 0, depending on C GN , such that for a 3 ∈ (0, 1) from Lemma 4.6 we arrive at On the other hand, by arguing as before, we infer for some c 8 > 0, β 2 ∈ (0, 1) taken by Lemma 4.6 and by the finiteness of (u + 1) (immediately coming from (7) in view of 0 < l < 2 n < 1) (Let us note that in (33) we have intentionally applied the Gagliardo-Nirenberg inequality with exponent q = 2l p only for exhibiting reasons; in particular, the expression of B in Lemma 4.5 appears more compact than the one that would be obtained by considering the optimal exponent q = 2 p . This does not preclude the sharpness of the assumption because, since µ is taken indefinitely large, the exponent p has the control on a 3 , and not q.) At this point, by making use again of the Lemma 4.1, positive constants c 9 and c 10 satisfy where, once more through Lemma 4.6, γ 2 ∈ (0, 1) and a 4 ∈ (0, 1). Finally, by plugging relations (32), (34) and (35) into bound (31), a further application of the δ-Young inequality with δ 3 , δ 4 > 0 and c 11 being a proper positive constant which depends on δ 3 and δ 4 .
With these gained bounds, we exploit a general boundedness result to quasilinear parabolic equations (see [19]) so to ensure uniform-in-time boundedness of the local solution (u, v) to system (1).
Proof of Theorem 2.1. Let (u, v) be the local classical solution to (1). Upon enlarging p and q accordingly to what said in item (i) of Remark 2, we can obtain that the term f (u)∇v ∈ L ∞ ((0, T max ); L q1 (Ω)), for some q 1 > n + 2. So we conclude thanks to Lemma 5.1, [19, Lemma A.1] and the dichotomy criterion (6). This estimate is essentially the same than that derived in [25, §4] so that, as therein, in order to take advantage from a combination of the Gagliardo-Nirenberg and Young's inequalities, one has to impose α − 1 + l < 2 n (coinciding, exactly as discussed in §2.2, with the parabolic-elliptic version of assumption (5)); consequently, the integral KK0 p+α−1 Ω (u + 1) p+α+l−1 can be suitably treated. Standard procedures, successively, provide that u ∈ L ∞ ((0, T max ); L p (Ω)) for arbitrarily large p > 1, and hence also g ∈ L ∞ ((0, T max ); L p (Ω)) for any l ∈ (0, 1). Finally, elliptic regularity theory applied to the second equation in problem (41) infers uniform bound of ∇v, on (0, T max ), so that u and v are uniformly bounded for all t > 0.
We note that the necessary regularity of ∇v is gained only by differentiating Ω (u + 1) p , by using the initial-boundary value problem (41) and, solely, the mass conservation property; neither an estimate like that in (8) is a priory needed nor the analysis of the term Ω |∇v| 2q , involving the extra parameter q, has to be developed.