Properties of given and detected unbounded solutions to a class of chemotaxis models

This paper deals with unbounded solutions to a class of chemotaxis systems. In particular, for a rather general attraction–repulsion model, with nonlinear productions, diffusion, sensitivities, and logistic term, we detect Lebesgue spaces where given unbounded solutions also blow up in the corresponding norms of those spaces; subsequently, estimates for the blow‐up time are established. Finally, for a simplified version of the model, some blow‐up criteria are proved.

1. Introduction, motivations and state of the art 1.1. The continuity equation; the initial-boundary value problem. The well known continuity equation (1) u t = ∇ · F + h describes the transport of some quantity u = u(x, t), at the position x and at the time t > 0. In this equation the flux F models the motion of such a quantity, whereas h is an additional source idealizing some external action by means of which u itself may be created or destroyed throughout the time.
In this paper we are interested in the analysis of equation (1) in the context of self organization mechanisms for biological populations, i.e. phenomena for which organisms or entities direct their trajectory in response to one or more chemical stimuli. More precisely, we want to deal with the motion of a certain cell density u = u(x, t) whose flux has a smooth diffusive part and another contrasting this spread. In the specific, this counterpart accounts of an attractive and a repulsive effect, associated to two chemical signals and indicated respectively with v = v(x, t) and w = w(x, t); v (the chemoattractant ) tends to gather the cells, w (the chemorepellent ) to scatter them. Additionally an external source with an increasing and decreasing effect on the cell density is also included.
In this way, the diffusion is smoother and smoother for higher and higher values of m 1 , the aggregation/repulsion effects −χu(u + 1) m2−1 ∇v/ξu(u + 1) m3−1 ∇w increase for larger sizes of χ and ξ and m 2 and m 3 , and the cell density may increment with rate λu and may attenuate with rate −µu k . Naturally, since the flux is influenced by the two signals v and w, we will have to consider two more equations (which for the time being we indicate with P (v) = 0 and Q(w) = 0, but that will be specified later), one for the chemoattractant v and another for the chemorepellent w, to be coupled with the continuity equation. Furthermore, the analysis is studied in impenetrable domains (so homogeneous Neumann or zero flux boundary conditions are imposed) and some initial configurations for the cell and chemical densities are assigned: essentially, with position (2) in mind, we are concerned with this initial boundary value problem: (3) in Ω × (0, T max ), u 0 (x) = u(x, 0) ≥ 0; v 0 (x) = v(x, 0) ≥ 0; w 0 (x) = w(x, 0) ≥ 0 x ∈Ω, u ν = v ν = w ν = 0 on ∂Ω × (0, T max ).
The problem is formulated in a bounded and smooth domain Ω of R n , with n ≥ 1, u ν (and similarly for v ν and w ν ) indicates the outward normal derivative of u on ∂Ω. Moreover, T max ∈ (0, ∞] identifies the maximum time up to which solutions to the system can be extended. 1.2. A view on the state of the art: the attractive and the repulsive models and the attraction-repulsion model. The aforementioned discussion finds, of course, its roots in the well-known Keller-Segel models idealizing chemotaxis phenomena (see the celebrated papers [KS70,KS71a,KS71b]), that since the last 50 years have been attracting the interest of the mathematical community.
In particular, if we refer to chemotaxis models with single proliferation signal, taking in mind (2), problem (3) is a (more general) combination of this aggregative signal-production mechanism (4) u t = ∇ · (G 1 + H 1 ) = ∆u − χ∇ · (u∇v) and P (v) = P τ 1 (v) = τ v t − ∆v + v − u = 0, in Ω × (0, T max ), and this repulsive signal-production one (5) u t = ∇ · (G 1 + I 1 ) = ∆u + ξ∇ · (u∇w) and Q(w) = Q τ 1 (w) = τ w t − ∆w + w − u = 0, in Ω × (0, T max ). (Here τ ∈ {0, 1} and it distinguishes between a stationary and evolutive equation for the chemical.) The above models present linear diffusion and linear production rates; specifically, v and w are linearly produced by the cells themselves, and their mechanism is opposite when in P τ 1 (v) the term v − u (or in Q 1 (w) the term w − u) is replaced by uv (or uw); in this case the particle density consumes the chemical. (We will spend only few words on models with absorption.) As far as problem (4) is concerned, since the attractive signal v increases with u, the natural spreading process of the cells' density could interrupt and very high and spatially concentrated spikes formations (chemotactic collapse or blow-up at finite time) may appear; this is, generally, due to the size of the chemosensitvity χ, the initial mass of the particle distribution, i.e., m = Ω u 0 (x)dx, and the space dimension n. In this direction, the reader interested in learning more can find in [HV97,JL92,Nag01,Win10] analyses dealing with existence and properties of global, uniformly bounded, or blow-up (local) solutions to models connected to (4).
On the other hand, for nonlinear segregation chemotaxis models like those we are interested in, when in problem (4) one has that P (v) = P 1 α (v) = v t − ∆v + v − u α = 0, with 0 < α < 2 n (n ≥ 1), uniform boundedness of all its solutions is proved in [LT16].
Concerning the literature about problem (5), it seems rather poor and general (see, for instance, [Moc74,Moc75] for analyses on similar contexts). In particular, no result on the blow-up scenario is available; this is meaningful due to the repulsive nature of the phenomenon.
Contrarily, the level of understanding for attraction-repulsion chemotaxis problems involving both (4) and (5) is sensitively rich; more specifically, if we refer to the linear diffusion and sensitivities version of model (3), for which F = F 1,1,1 and P τ α (v) = τ v t − ∆v + bv − au α and Q τ β (w) = τ w t − ∆w + dw − cu β , a, b, c, d, α, β > 0, equipped with regular initial data u 0 (x), τ v 0 (x), τ w 0 (x) ≥ 0, we can recollect the following outcomes. In the absence of logistics (h k ≡ 0), when linear growths of the chemoattractant and the chemorepellent are taken into consideration, and for elliptic equations for the chemicals (i.e. when P 0 1 (v) = Q 0 1 (w) = 0), the value Θ := χa − ξc measures the difference between the attraction and repulsion impacts, and it is such that whenever Θ < 0 (repulsion-dominated regime), in any dimension all solutions to the model are globally bounded, whereas for Θ > 0 (attraction-dominated regime) and n = 2 unbounded solutions can be detected (see [GJZ18,LL16,TW13,Vig19,YGZ17] for some details on the issue). Indeed, for more general expressions of the proliferation laws, modelled by the equations P 0 α (v) = Q 0 β (w) = 0, to the best of our knowledge, [Vig21] is the most recent result in this direction; herein some interplay between α and β and some technical conditions on ξ and u 0 are established so to ensure globality and boundedness of classical solutions. (See also [CY22] for blow-up results in the frame of nonlinear attractionrepulsion models with logistics as those formulated in (3) with F = F m1,m2,m3 and h = h k , and with linear segregation for the stimuli, i.e. with equations for v, w reading as P 0 1 (v) = Q 0 1 (w) = 0.) Putting our attention on evolutive equations for chemoattractant and chemorepellent, P 1 1 (v) = Q 1 1 (w) = 0, in [TW13] it is proved that in two-dimensional domains sufficiently smooth initial data emanate global-in-time bounded solutions whenever (As to blow-up results we are only aware of [Lan21], where unbounded solutions in three-dimensional domains are constructed.) When h = h k ≡ 0, for both linear and nonlinear productions scenarios, and stationary or evolutive equations (formally, P τ α (v) = Q τ β (w) = 0), criteria toward boundedness, long time behaviors and blow-up issues for related solutions are studied in [HTZ20, ZLZ22, CMTY21, RL22].
1.3. The nonlocal case. In this work we are mainly interested to the so-called nonlocal models tied to (3), for which, specifically, P (v) = P α (v) := ∆v − 1 |Ω| Ω u α + u α = 0, and Q(w) = Q β (w) := ∆w − 1 |Ω| Ω u β + u β . In particular, recalling the position in (2), we herein mention the most recent researches we are aware about and inspiring our study; to this purpose we refer to the only attraction version and the attraction-repulsion one For the linear diffusion and sensitivity version of model (6), i.e. with flux G 1 + H 1 , if h k ≡ 0, it is known that boundedness of solutions is achieved for any n ≥ 1 and 0 < α < 2 n , whereas for α > 2 n blow-up phenomena may be observed (see [Win18]). When dampening logistic terms take part in the mechanism, for the linear flux G 1 + H 1 appearances of δ-formations at finite time have been detected for some sub-quadratic growth of h k , and precisely for h k with 1 < k < n(α+1) n+2 < 2 (see [YMXD21]). But there is more; for the limit linear production scenario, corresponding to P 1 (v) = 0, some unbounded solutions have been constructed in [Fue21] even for quadratic sources h = h 2 , whenever n ≥ 5 and µ ∈ 0, n−4 n .
In nonlinear models without dampening logistic effects (i.e., general flux G m1 + H m2 and h k ≡ 0) and linear production (i.e. P 1 (v) = 0), in [WD10] it is shown, among other things, that for m 1 ≤ 1, m 2 > 0, m 2 > m 1 + 2 n − 1 situations with unbounded solutions at some finite time T max can be found. (See also [MNV20] for questions connected to estimates of T max .) Some results have been also extended in [Tan22] when h k ≡ 0 and for nonlinear segregation contexts, P α (v) = 0; in particular, inter alia, for m 1 ∈ R, m 2 > 0, blow-up phenomena are seen to appear if m 2 + α > max m 1 + 2 n k, k , whenever m 1 ≥ 0 or m 2 + α > max 2 n k, k , provided m 1 < 0. On the other hand, for the attraction-repulsion models, in [LL21] it is proved, together with other results, that if P α (v) = Q β (w) = 0, α > 2 n and α > β ensure the existence of unbounded solutions to (7) for the linear flux F = F 1,1,1 , without logistic (h k ≡ 0). Conversely, detecting gathering mechanisms for the nonlinear situation is more complex and we are only aware of [WZZ23]; indeed this issue is therein addressed only for F = F m1,1,1 , with m 1 ∈ R, but even in presence of dampening logistics. Since in our research we will show the existence of blow-up solutions for a larger class of fluxes, precisely for F = F m1,m2,m2 with m 1 ∈ R and any m 2 = m 3 > 0, we will spend more words to analyze details of [WZZ23] below, precisely in §3.2.
Remark 1. Let us clarify that in nonlocal models, and henceforth in this work, v stands for the deviation of the chemoattractant; the deviation is the difference between the signal concentration and its mean, and that it changes sign in contrast to what happens with the cell and signal densities (which are nonnegative). In particular, it follows from the definition of v itself that its mean is zero (as specified in the last positions of the problem (8)), which in turn ensures the uniqueness of the solution of the Poisson equation under homogeneous Neumann boundary conditions. The same comments apply for the chemorepellent w. (We did not introduce different symbols to indicate the chemicals and their deviations since it is clear from the context.)

2.2.
Presentation of the Theorems. Overall aims of the paper. Our project finds its motivations in the observation that there is no automatic connection between the occurrence of blow-up for solutions to model (8) in the L ∞ (Ω)-norm and that in L p (Ω)-norm (p > 1). Indeed, for a bounded domain Ω, it is seen that so that unboundedness in L p (Ω)-norm implies that in L ∞ (Ω)-norm, but oppositely Ω u p might even remain bounded in a neighborhood of T max when max Ω u uncontrollably increases at some finite time T max . In light of this, in order to bridge the gap between the analysis of the blow-up time T max in the two different mentioned norms, we aim at (i) detecting suitable L p (Ω) spaces, for certain p depending on n, m 1 , m 2 , m 3 , α and β, such that given unbounded solutions also blow up in the associated L p (Ω)-norms; (ii) providing lower bounds for the blow-up time of the aforementioned solutions in these L p (Ω)-norms.
Additionally, another objective of our work is (iii) giving sufficient conditions on the data of the model such that related solutions are actually unbounded at finite time.
In order to deal with issues (i), (ii) and (iii), we fix the following relations, determining some precise interplay involving constants defining system (8): Assumptions 2.1. Let n ∈ N, m 1 , m 2 , m 3 ∈ R and α, β, χ, ξ, λ, µ > 0 and k > 1 be such that With the support of the above position, and as far as the analysis of (i) is concerned, in the spirit of [Fre18, Theorem 2.2] we will prove this first result, dealing with properties of given unbounded solutions to model (8). Specifically, the proof is based on the analysis of the functional ϕ(t) = 1 p Ω (u + 1) p , defined for local solutions to problem (8) on (0, T max ). We will show that if for some p sufficiently large ϕ(t) is uniformly bounded in time, then it also is for any arbitrarily large p > 1, so contrasting with the unboundedness of u itself.
Theorem 2.2. Let Ω be a bounded and smooth domain of R n , and condition (H 1 ) as well as one of (H 2 ),(H 3 ),(H 4 ), (H 5 ) in Assumptions 2.1 hold true. Moreover, for f i and u 0 complying with (10) and (11), let (u, v, w), with be a solution to problem (8) which blows-up at some finite time T max , in the sense that Then, for any p > n 2 (m 2 − m 1 + α), we also have lim sup t→Tmax u(·, t) L p (Ω) = +∞.
Successively aim (ii) is achieved by establishing for the same functional ϕ(t) a first order differential inequality (ODI) of the type ϕ ′ (t) ≤ Ψ(ϕ(t)) on (0, T max ). In particular, for any τ 0 > 0 the function Ψ(τ ) obeys the Osgood criterion ([Osg98]), so that an integration on (0, T max ) of the ODI implies, whenever lim sup t→Tmax ϕ(t) = ∞, the following lower bound for T max and thereby a safe interval of existence [0, T ) for solutions to the model itself.
Theorem 2.3. Let the hypotheses of Theorem 2.2 be satisfied. Then there existp > 1 and positive constants A, B, C, D, as well as γ > δ > 1, such that the blow-up time T max complies with both the implicit estimate and the explicit one Finally, the last theorem (connected to item (iii)) establishes (at least in a particular case) the existence of unbounded solutions to system (8), so as to make the two previous statements meaningful. The basic idea consists on the analysis of the temporal evolution of the functional φ(t) : , being U (s, t) the so-called mass accumulation function of u, obeying a superlinear ODI.

Miscellaneous and general comments
3.1. On the parameters p andp. We herein want to spend some words on the role of the parameters p andp appearing in Theorems 2.2 and 2.3. In particular, as we will observe in Lemma 4.2 below, p andp depend on n, m 1 , m 2 , m 3 , α and β, and Figure 1 collects some of their values on the p axis. In the specific, for a blowing-up solution to (8), we will have that Ω u p is bounded on (0, T max ) for p = 1, while it blows up for p ≥ p; in general, the behaviour of Ω u p on (0, T max ) for p ∈ (1, p) is unknown. On the other hand, an estimate for T max is given in terms of 1 p Ω (u 0 + 1) p , for p ≥p, but not when p ∈ (p,p). 3.2. Improving Theorem 1.1 in [WZZ23] and addressing an open question in Remark 1.2 of [WZZ23]. Herein we want to compare Theorem 2.4 and [WZZ23, Theorem 1.1], both dealing with blow-up solutions to attraction-repulsion models with logistics. First, in order to have consistency between these results, we have to fix m 2 = 1 in (H 6 ). Additionally, for the ease of the reader, let us also rephrase the related blow-up assumptions: (17) Blow-up conditions in Theorem 2.4: We easily note that if m 1 ≥ 1 the two conditions (17) and (18) coincide. Now, we analyze the cases m 1 ≤ 0 and 0 < m 1 < 1, separately.

Local existence and necessary parameters
By an adaption of standard reasoning in the frame of the fixed-point theorem, we can show the following result on local existence and extensibility of classical solutions to (8).
Proof. The proof can be achieved by following well established results: for instance, we refer the interested reader to [CW08,Nag95,Wan16,WD10]. In particular, an integration of the first equation in (8) and an application of the Hölder inequality give bound (19).
Remark 2. For reasons which will be exploited later on, and precisely in Lemma 5.1, it appears important to point out that assumptions (H 3 ), (H 4 ) and (H 5 ) imply that the ratios (20f), (20h) and (20m) in Lemma 4.2 can be also taken equal to 1.

A priori estimates and proof of Theorems 2.2 and 2.3
In this section we will use (H 1 )-(H 5 ). Moreover, without explicitly computing their values, we underline that the constants c i appearing below and throughout the paper depend inter alia on p, are positive and their subscripts i start anew in each new proof.
Lemma 5.1. Under the hypotheses of Lemma 4.2, let p =p and p be any of the constants therein defined. If (u, v, w) is a classical solution to problem (8), u ∈ L ∞ ((0, T max ); L p (Ω)) and ϕ(t) is the energy function then there exist c 1 , c 2 such that Proof. Let us differentiate the functional ϕ(t) = 1 p Ω (u + 1) p . Using the first equation of (8) and the divergence theorem we have for every t ∈ (0, T max ) By considering the definition of F j (u) above, again the divergence theorem, the second and third equation of (8), we have for every t ∈ (0, T max ), Let us now specify how each of the constrains in Assumptions 2.1 takes part in our computation. First, from (H 1 ) we have p > 1; this makes meaningful our assumption u ∈ L ∞ ((0, T max ); L p (Ω)). (Recall that u ∈ L ∞ ((0, T max ); L 1 (Ω)) is always met by (19).) Now, by exploiting (H 2 ) and (22), we can see that from (23), if we neglect the nonpositive terms we get on (0, T max ) As to the third term, by using twice Hölder's inequality (recall (20f) and (20g)), we obtain, for every t ∈ (0, T max ), Moreover, since for any ε > 0 there is d(ε) > 0 such that this inequality (see [FV21,Lemma 4 is true, by virtue of (20h), we have that (26) c 9 Ω (u + 1) p+m3−1 Ω (u + 1) β ≤ Ω (u + 1) p+m2+α−1 + c 11 for all t ∈ (0, T max ).
Proof. With a view to inequality (21), as already done in (30), a further application of the Gagliardo-Nirenberg inequality, supported by (20d), leads also thanks to (29) to Ω (u + 1) p ≤ c 1 ∇(u + 1) or, equivalently Finally, by using relations (21) and (34), we arrive at this initial problem for every t ∈ (0, T max ), for all t < T max .
By taking advantage from the previous lemma, let us show the uniform-in-time boundedness of u.
Since all the hypotheses of [TW12, Lemma A.1.] are fulfilled, we have the claim.
Now we are in a position to prove our first results.
Proof of Theorem 2.2: Let (u, v, w) be a given blow-up solution at some finite time T max to problem (8). If u was not unbounded in some L p (Ω)-norm, Lemma 5.3 would imply the uniform-in-time boundedness of u, contradicting hypothesis (13).
As to the derivation of the explicit expression for the lower bound T , let us reduce (38) as follows: from |Ω| ≤ Ω (u + 1) p = pϕ(t), we can estimate C in relation (38) as so that (38) can be rewritten in this form: Now, since ϕ blows up at finite time T max , there exists a time t 1 ∈ [0, T max ) such that From γ > δ > 1 (recall (20j)), we can estimate the second and third term on the rhs of (39) by means of ϕ γ : By plugging expressions (40) into (39) we obtain for so that an integration of (41) on (t 1 , T max ) yields this explicit lower bound for T max : Remark 3. For completeness, we observe that it is also possible to avoid estimate (36); indeed, in relation (35) the term Ω (u + 1) p is directly pϕ(t), so that (38) would read ϕ ′ (t) ≤Ãϕ γ (t) +Bϕ δ (t) +Cϕ(t) +D on (0, T max ).

Finite-time blow-up to a simplified version of problem (8)
This section is dedicated to prove finite time blow-up for solutions to problem (8) in a more specific case; in our computations we will be inspired by [Tan22,WZZ23], where respectively blow-up is established in a model with only attraction, nonlinear diffusion and sensitivity and logistic term, and in an attraction-repulsion one, with nonlinear diffusion but linear sensitivities and logistics. 6.1. Detecting unbounded solutions to problem (8) for m 2 = m 3 > 0. Let us fix m 2 = m 3 > 0 in model (8), and in turn let us set z = χv − ξw, m(t) = χm 1 (t) − ξm 2 (t) and f (u) = χf 1 (u) − ξf 2 (u), being m 1 (t) and m 2 (t) defined in (9); in this way, problem (8) itself is reduced into in Ω × (0, T max ), In particular, if we confine our study to radially symmetric cases, by setting r := |x| and by considering Ω = B R (0) ⊂ R n , n ≥ 1 and some R > 0, the radially symmetric local solution (u, z) = (u(r, t), z(r, t)) to model (42) solves the following scalar problem In the same spirit of [JL92], we introduce the mass accumulation function f (nU s (σ, t)) dσ for s ∈ (0, R n ) and t ∈ (0, T max ), In addition, given s 0 ∈ (0, R n ), γ ∈ (−∞, 1) and U as in (44), we introduce the moment-type functional which is well defined and belongs to C 0 ([0, T max )) ∩ C 1 ((0, T max )). Moreover, we define where M is the bound of the L 1 (Ω)-norm of u established in (19) and ω n = n|B 1 (0)|. With these preparations in our hands, let us give a series of necessary lemmas, some of which are not new. We start with a result dealing with the concavity of U and some estimate for m(t).
Proof. As to the concavity property, the proof is based on minor adjustments of [ (We point out that c 0 will be used in some other places below.) Let us now start with the analysis of the temporal evolution of φ.
Lemma 6.2. Under the same assumptions of Lemma 6.1, let s 0 ∈ 0, R n 6 . Then Proof. By the definition of φ (recall (48)) and exploiting (47), we get this estimate Finally, by applying (51), we obtain the thesis.
The next results provide some lower bounds of I 1 , I 2 , I 3 and I 4 in respect of ψ(t) defined in (49).
As to the estimate of I 2 + I 3 we need to rearrange some computations; this is exactly where we have to go beyond the analysis in [Tan22] and [WZZ23].
Lemma 6.4. Under the same assumptions of Lemma 6.1, let moreover m 2 , α > 0 comply with m 2 + α > 1. Then for some c 1 , c 2 and c 3 we have for any s 0 ∈ 0, R n 6 and for all t ∈ S φ .
Proof. Since α > β, by applying the Young inequality and (12), we get From the concavity of U in Lemma 6.1, it is seen that U s is nonincreasing, namely U s (σ, t) ≥ U s (s, t) for any σ ∈ (0, s).
In order to obtain the desired superlinear ODI for φ, we have to rely on some relations involving U and ψ and φ.
Then there exist c 1 , c 2 such that for any s 0 ∈ 0,  Proof. The proof of (61) and (62)  The following is precisely the lemma relying on assumption (H 6 ).
Proof of Theorem 2.3: We focus only on the situation where m 1 > 0, the cases m 1 = 0 and m 1 < 0 being similar. Since (H 6 ) holds, we can apply Lemma 6.6 and find γ ∈ (−∞, 1), c 7 , c 8 such that for each u 0 fulfilling the related restrictions in (10) and (12), and s 0 ≤ R n 6 , one deduces (63) φ ′ (t) ≥ c 7 s into the simplified version (42) is not longer employable, by reasoning as in Lemma 6.2, the corresponding inequality (52) would read φ ′ (t) ≥ n 2 s0 0 s 2− 2 n −γ (s 0 − s)(nU s + 1) m1−1 U ss ds − χf 1γ valid for all t ∈ S φ and with f 1γ = f 1 8n 2 γ (3−γ)ωn . In particular, the extra terms involving the repulsion coefficient ξ make the analysis more complex. This is also connected to the signs of such terms (exactly opposite than those of the integrals associated to the attraction coefficient χ), which do not allow to use the right inequalities tied to the hypotheses on f i , and in turn on m i .