A family of mass‐critical Keller–Segel systems

The no‐flux initial‐boundary value problem for the quasilinear Keller–Segel system * ut=∇·(D(u)∇u)−∇·(S(u)∇v),vt=Δv−v+u,\begin{equation} \hspace*{6pc}{\left\lbrace \def\eqcellsep{&}\begin{array}{l}u_t=\nabla \cdot (D(u)\nabla u) - \nabla \cdot (S(u)\nabla v), \hspace*{-6pc}\\[3pt] v_t=\Delta v-v+u, \end{array} \right.} \end{equation}is considered in smoothly bounded domains Ω⊂Rn$\Omega \subset \mathbb {R}^n$ , n⩾3$n\geqslant 3$ , where D∈C2([0,∞))$D\in C^2([0,\infty ))$ and S∈C2([0,∞))$S\in C^2([0,\infty ))$ are such that D>0$D>0$ on [0,∞)$[0,\infty )$ and that S(0)=00$s>0$ . A particular focus is on cases in which there exist κ>0,CSD>0$\kappa >0, C_{SD}>0$ and f∈L1((1,∞))$f\in L^1((1,\infty ))$ such that ** −f(s)⩽D(s)S(s)−κs2/n⩽CSDsforalls⩾1.\begin{equation} \hspace*{6pc}- f(s) \leqslant \frac{D(s)}{S(s)} - \frac{\kappa }{s^{2/n}} \leqslant \frac{C_{SD}}{s} \quad \mbox{for all } s\geqslant 1.\hspace*{-6pc} \end{equation}It is first shown that then there exists m0>0$m_0>0$ such that whenever u0$u_0$ and v0$v_0$ are reasonably regular and nonnegative with ∫Ωu0


INTRODUCTION
Critical mass thresholds in the context of blow-up phenomena belong to the apparently most striking subtleties going along with the interplay of diffusion and self-enhanced cross-diffusion in frameworks of Keller-Segel type chemotaxis systems [29,42,45]. On the one hand, such a dependence of the possibility to enforce singularity formation on total population masses seems to reflect fairly well the presence of spatially global quorum-sensing type mechanisms of apparent relevance for taxis-driven aggregation in various experimental settings [21,29,43]. Beyond this, however, a considerable motivational aspect of essentially mathematical nature rests on the circumstance that effects of this flavor apparently presuppose quite a precise balance of those system ingredients that incorporate the strength of diffusion and of cross-diffusive interaction, respectively.
Constituting the undoubtedly most prominent example in this regard, the spatially twodimensional version of the classical Keller-Segel system [32] { = Δ − ∇ ⋅ ( ∇ ), = Δ − + , (1.1) is known to exhibit critical mass phenomena with respect to the occurrence of unbounded solutions both in its fully parabolic variant with = 1, and in its parabolic-elliptic simplification with = 0. In fact, when posed along with homogeneous Neumann boundary conditions in bounded planar domains Ω, the initial value problem for (1.1) with = 1 possesses global bounded solutions for all reasonably regular initial data whenever ∫ Ω (⋅, 0) < 4 [40], whereas for all values of ∈ (4 , ∞) ⧵ {4 | ∈ ℕ}, one can find unbounded solutions fulfilling ∫ Ω (⋅, 0) = , provided that Ω is simply connected [30]; in the simpler case when = 0, the knowledge in this regard is even much more complete, inter alia asserting the existence of solutions genuinely blowing up within finite-time, without any restriction on the prescribed size > 4 of their initial mass, and on the shape of the domain ( [39]; cf. also [23] and [10] for results on mass-dependent bubbling in the associated fully stationary problem). Some further simplifications of the second equation allow for yet more comprehensive results on corresponding Cauchy problems posed in the whole plane (see, for example, [5,6,8,9,12,42] and the references therein).
Beyond these particular scenarios in close connection to (1.1), however, only few rigorous detections of mass criticality in specific chemotaxis systems significantly different from (1.1) can be found in the literature. This may be viewed as partially reflecting the substantial mathematical challenges linked to proving occurrence of blow-up in systems less simple than (1.1), especially in fully parabolic cases in which, besides the equation for the population density , also that describing the evolution of the signal concentration is parabolic. Accordingly, the essentially only further result in this direction that addresses such a parabolic-parabolic Keller-Segel system asserts finite-time blow-up of some large-mass solutions in three-and four-dimensional variants of (1.1) involving certain critical nonlinear cell diffusion mechanisms for which the resulting diffusion operator is exactly of porous medium type [37]. Apart from that, rigorous proofs for the occurrence of critical mass phenomena seem available only for some relatives of (1.1) with certain particular structures, inter alia presupposing that the signal evolution is governed by elliptic equations (cf., for instance, [7,14,2,13,36,47] and [3]).
Even in the quasilinear generalization of (1.1), in the framework of a no-flux initial-boundary value problem in a smoothly bounded domain Ω ⊂ ℝ given by and arising for different choices of the parameter functions and in refined models for chemotactic migration in various contexts [28], despite a certain structural proximity to (1.1), to be described in more detail below, available results seem to essentially concentrate on identifying conditions on and that ensure either global existence and boundedness of widely arbitrary solutions on the one hand, or the occurrence of unboundedness phenomena at arbitrarily small mass levels on the other. Quite well understood in this regard seems the subclass of (1.2) determined by the prototypical choices ( ) = ( + 1) −1 and ( ) = ( + 1) −1 , ⩾ 0, (1 .3) for which, namely, it is known that whenever ∈ ℝ and ∈ ℝ are such that > + 1 − 2 , given any suitably regular nonnegative initial data one can find a global classical solution for which is bounded throughout Ω × (0, ∞) [31,33,44,46], while if conversely < + 1 − 2 , and if moreover ⩾ 2 and Ω is a ball, then for arbitrary > 0, it is possible to find classical solutions for which cannot remain uniformly bounded [50]. Further studies even have provided further information on subcases of the latter in which the respective blow-up phenomenon either must occur within finite time [16][17][18], or only arises in the sense of an infinite-time growup [16,18,54]; cf. also [35,15,22,24,25] for related results concerned with parabolic-elliptic analogues, one-dimensional versions, and small-data solutions in some supercritical parameter settings). Especially in the context of modeling procedures based on the inclusion of volume-filling effects in the style of the seminal work [41] in this regard, however, diffusion and cross-diffusion rates with quite rapid decay at large population densities arise in a natural manner when saturation effects in cell motility at large population densities are accounted for; in fact, the particular approach in [41] (cf. also [55]) suggests to link and through the relationships ( ) = ( ) − ′ ( ) and ( ) = ( ), ⩾ 0, (1.4) where ( ) represents the probability for a cell, when positioned at a place with current population density , to find space in some neighboring site. Accordingly, functions and exhibiting exponential or even faster decay appear whenever is chosen so as to reflect strong saturation in decreasing faster than algebraically. For such more general classes of diffusion rates and chemotactic sensitivity functions in (1.2), the knowledge seems much sparser; while the outcome of [50] actually extends so as to provide unboundedness in the sense described above, also of small-mass solutions, under quite mild conditions on and which essentially reduce to the assumption that for all ⩾ 1 and some > 0 and > 0, ( 1 . 5 ) a comparably exhaustive result on boundedness in the presence of correspondingly subcritical asymptotics of is yet lacking: Only under the crucial overall assumption that ( ) be bounded from below by a function decreasing at most algebraically fast as → ∞, the hypothesis that for all ⩾ 1 and some > 0 and > 0 (1.6) has been shown to imply comprehensive statements on global existence of bounded classical solutions so far [46]. The lack of farther-reaching knowledge in this regard appears to be inherently linked to missing information on suitably regularizing features of the associated diffusion operators in (1.2) especially in cases when ( ) decays at rates faster than algebraic as → ∞, in which accessibility to standard arguments, for example, based on Moser-type iteration procedures, seems limited; accordingly, even for simple choices of and exhibiting essentially exponential decay only partial results concerned with issues of boundedness and unboundedness in frameworks of classical solutions are available [19,20,53].
Main results. The purpose of the present work is to reveal that under quite mild assumptions on and not substantially exceeding the requirement that any three-or higher dimensional constellation in which holds with some > 0 and some decaying suitably fast as → ∞ enforces a certain mass criticality, and that hence a separation between the two mass-insensitive scenarios discussed above around (1.5) and (1.6) is achieved within a considerably large family of model ingredients, rather than merely nongeneric lines. In fact, such settings will be shown to go along with some critical mass phenomenon with respect to an boundedness property in a suitable context of generalized solvability (Theorem 1.3, Theorem 1.2 and Corollary 1.4), and for essentially powertype nonlinearities such as those in (1.3) this phenomenon will be seen to actually take place in contexts of classical solutions and their pointwise boundedness (Theorem 1.3 and Corollary 1.4).
In the case when additionally is assumed to be algebraically bounded from above and below, this basic information can be seen to be sufficient as a starting point for an iterative argument ultimately leading to ∞ estimates for , in Section 3 hence implying the following first of our main results.  then ( (⋅, ), (⋅, )) are radially symmetric with respect to = 0 for all > 0.
If the additional requirement (1.13) is dropped, however, then we shall see in Section 4 that the above basic regularity properties, together with (1.9), after all allow for the construction of a certain global generalized solution, the first component of which remains bounded in some reflexive space: We shall next complement the above by making sure that the requirements on ∕ and on smallness of ∫ Ω 0 both in Theorem 1.1 and in Theorem 1.2 actually cannot be substantially relaxed. To this end, in Section 5 we shall first see by means of an explicit construction that if an inequality opposite to that in (1.12) holds, then the energy functional  has a certain property of unboundedness from below, even when restricted to radial functions (Lemma 5.1). Intending to use such as initial data, in Lemma 5.10 we shall perform a Pohozaev-type testing procedure to derive, at each fixed level of the total population mass, an a priori bound from below for energies of suitably regular radially symmetric steady state solutions to (1.2) in balls, where in order to be able to include the critical relationship with = 0 in (1.5), in contrast to previous related approaches [50], we will rely on some decisive refinement which at its core rests on a simple consequence of standard elliptic regularity to be stated in Lemma 5.9. Combined with knowledge on natural connections between -limit sets of global solutions and steady states, to be carefully derived in the considered context of poor regularity information in Section 5.2, this will lead to our following main result on nonexistence of solutions enjoying boundedness features in the flavor of (1.18). Theorem 1.3. Suppose that Ω = (0) ⊂ ℝ with some ⩾ 3 and > 0, and that and satisfy (1.7) and, with some > 0 and > 0, as well as where is as defined in (1.10), and where Then for any choice of ( 0 , 0 ) fulfilling (1.14) and (1.17), one can find functions 0 and 0 , ∈ ℕ, such that ( 0 , 0 ) complies with (1.14) and (1.17) for all ∈ ℕ, that 0 → 0 in (Ω) and 0 → 0 in 1, (Ω) as → ∞ for all ∈ (0, 1) and ∈ [1, −1 ), Here a considerably large fund of concrete examples can be generated by observing that the structural condition (1.20) indeed is fulfilled in quite a large number of situations in which ∕ approaches some multiple of the critical function 0 < ↦ 2∕ suitably fast: Remark. Let us emphasize that according to the remarkable sizes of the funnel-type regions determined by (1.25), the latter can be interpreted as confirming that the occurrence of critical mass phenomena in chemotaxis systems is not limited to constellations involving ingredients of precise functional nature, such as those addressed in precedent literature. Indeed, by admitting not only fairly arbitrary behavior of and as long as merely ∕ remains asymptotically near ∕ 2∕ for some > 0, but by furthermore even including cases of noticeably strong oscillations in ∕ , Corollary 1.5 indicates a considerable stability of mass criticality in quasilinear Keller-Segel systems with respect to the model components.
When applied to the prototypical situation determined by the power-type laws in (1.3), Corollary 1.5 asserts mass criticality in the classical sense along the entire line = + 1 − 2 in the ( , ) plane whenever ⩾ 3, thus manifesting a remarkable contrast to the corresponding onedimensional analogue in which at least the particular point ( , ) = (0, 1) is known to belong to the regime of unconditional boundedness of solutions [4]: With regard to the particularly application-relevant version of (1.2) that accounts for volumefilling effects through the relations in (1.4), our results finally reveal mass criticality whenever the probability distribution function therein exhibits a certain type of exponentially fast decay: Corollary 1.7. Let ⩾ 3, > 0 and Ω = (0) ⊂ ℝ , and let and be defined through (1.4) with where > 0 and > 0. Then (1.2) exhibits a critical mass phenomenon in the sense that the number ( ) from Corollary 1.4 is finite and positive.

ENERGY-INDUCED BOUNDS FOR SMALL-MASS SOLUTIONS
The core of our existence analysis both in classical and in generalized frameworks can be found in the following lower estimate for the energy functional from (1.8) and (1.9) at suitably small mass levels of the first among its arguments.
for all ∈ 1,2 (Ω) and each ∈ 0 (Ω) fulfilling > 0 in Ω as well as Proof. We first estimate , and then rely on the continuity of the Then given > 0, we choose 0 = 0 ( ) > 0 small enough such that and assume that (1.7) and (1.12) hold, and that ∈ 1,2 (Ω) and ∈ 0 (Ω) are such that > 0 in Ω and that (2.2) is valid. Then according to (1.12), the function in (1.10) satisfies Now combining the Hölder inequality with (2.4) and Young's inequality, we see that here where again due to the Hölder inequality, because of (2.2) and (2.5). Therefore, (2.6) entails that Indeed, due to the energy identity (1.8), the latter implies some basic information on integrability properties of and ∇ .

7)
and Proof. By means of Lemma 2.1, we can choose 1 = 1 ( ) > 0 and 2 = 2 ( ) > 0 such that the functional from (1.9) has the property that for all ∈ 1,2 (Ω) and any positive ∈ 0 (Ω) fulfilling (2.2). According to (1.8), for the solution under consideration, we thus have Especially, through the bound in (2.8), the above lemma can be relied on to derive further regularity properties by means of the quasi-energy inequality in the following statement. This will be used in Lemma 3.2 to address classical solutions, and later on again in Lemma 4.5 in the course of our construction of very weak energy solutions. then for all ∈ (0, ), Proof. By (1.2) and the definition of Φ , due to (1.12) and Young's inequality we have Now once more due to Young's inequality, as well as we have and In view of (2.15), from (2.14) we thus infer that and conclude as intended. □

GLOBAL BOUNDED CLASSICAL SOLUTIONS. PROOF OF THEOREM 1.1
Now in order to suitably utilize the above in classical solution frameworks, let us recall from standard theory [1,34] that smooth solutions exist at least locally in time.  .7) and (1.14). Then there exist ∈ (0, ∞] and functions
Then under the additional hypothesis in (1.13), appropriate selection of the parameter in Lemma 2.3 enables us to derive bounds for the first component of these solutions, as well as for the gradient of the second, in Lebesgue spaces involving arbitrary finite integrability exponents, provided that the assumption on smallness of mass from Lemma 2.3 is satisfied. (1.12) and (1.13) with some > 0, ⩾ 0 and ⩾ 0, and suppose that (1.14) holds with ∫ Ω 0 < 0 , where 0 = 0 ( ) > 0 is as in Lemma 2.1. Then for all > 1 there exists ( ) > 0 such that and Proof. Given > 1, we fix = ( ) ⩾ 2 large such that and invoke Lemma 2.3 along with (3.2) and Lemma 2.2 to see that writing ∶= 2( −1) we can find We moreover use (1.13) to see that with some positive constants = ( ), ∈ {3, 4, 5, 6}, we have and hence We furthermore introduce and note that the first two restrictions in (3.5) ensure that ∈ (0, 1), and that thus the Gagliardo-Nirenberg inequality applies so as to show that once more due to (3.2), with some 7 = 7 ( ) > 0 and 8 = 8 ( ) > 0, we have Writing ∶= min{ ( + ) , 1}, by using Young's inequality we thus infer from (3.7) that and that hence, by comparison, In view of (3.7), this shows that and thereby establishes (3.3) and (3.4), because according to Young's inequality and the two rightmost conditions in (3.5). □ Once more due to (1.13), a standard iterative argument thereupon becomes applicable so as to assert even an ∞ bound for the first component of any such small-mass solution. (1.12) and (1.13) hold with some > 0, ⩾ 0 and ⩾ 0, and that (1.14) is satisfied with ∫ Ω 0 < 0 , where 0 = 0 ( ) > 0 is as in Lemma 2.1. Then there exists > 0 such that

The notion of global very weak energy solutions
In order to facilitate applicability of Lemmas 2.2 and 2.3 also in the presence of asymptotic behavior of that is too degenerate to be compatible with (1.13), we shall resort to a generalized solution concept which with regard to the fulfillment of the differential equations in (1.2) is very weak, especially through the requirement that the first sub-problem in (1.2) be rather satisfied in the sense of some weak supersolution property only, after all accompanied with an additional information on mass conservation. Thus orienting our approach along precedents of solution constructions in potentially quite irregular chemotaxis systems [49,52], we thereby particularly rely on a notion of solution that is even yet weaker than that of natural weak solvability, but, as will become important in our nonexistence arguments in the context of Theorem 1.3, despite this substantial weakening we will retain some crucial information on the energy structure expressed in (1.8) in a sense that will turn out to be suitable for our subsequent approximation procedure.
Our formulation thereof will be prepared by the following lemma which introduces an auxiliary function Σ, and which for later reference in Section 5.2 summarizes some evident basic properties thereof.

Constructing initial data at large negative energy levels
Our approach toward the statement on nonexistence in Theorem 1.3 will be based on the observation that contrary to the small-mass scenario addressed in Lemma 2.1, throughout favorably large sets of radial initial data, thus necessarily involving large-mass functions at least in cases of critical behavior of , the energy functional from (1.9) is unbounded from below. This will be seen by means of an essentially explicit construction which, especially in order to cover widely general and also critical nonlinearities, relies on an approximation process quite different from related precedents in the literature [18,30,35,51,54].

2)
and that where again  is taken from (1.9).

Lower bounds for energy levels of -limits
The core of our analysis can now be found in this section, the purpose of which is to establish a priori bounds from below for all conceivable energy levels of initial data which evolve into global very weak solutions enjoying the additional boundedness property in (1.23). According to the nonincrease of  , this essentially reduces to the derivation of corresponding bounds for respective -limits, and the main part of our analysis in this direction will be focused on the challenge to appropriately cope with the circumstance that all conclusions need to be based on the considerably poor regularity information provided by (1.23) and Definition 4.2) for such solutions. Indeed, significant efforts appear necessary already at the first stage of our considerations in this regard, aiming at a verification of the intuitive guess that the energy inequality (4.6) should enforce any such solution to possess a nonempty -limit set consisting of solutions ( ∞ , ∞ ) to a corresponding stationary problem, located at some energy level below that of the initial data.
Fortunately, in the presently considered radial framework, some additional regularity features are available outside the spatial origin: According to the second-order information contained in (5.15) and one-dimensional Sobolev embeddings, the component ∞ can easily be seen to actually possess the first-order properties listed in the following lemma which, besides being used in Lemma 5.6 addressing the above identity, will moreover be relied on in Lemmas 5.7 and 5.8. . Upon integration, this entails that being finite due to our restriction on . To derive the claim of the lemma, given a radial ∈ 2, (Ω) let us choose radial functions ∈ 2 (Ω), ∈ ℕ, such that → in 2, (Ω) as → ∞. Then an application of (5.32) to ∶= − ′ for ∈ ℕ and ′ ∈ ℕ shows that ( ) ∈ℕ is a Cauchy sequence in 1 (Ω ⧵ (0)) for all ∈ (0, ), thus implying that indeed ∈ 1 (Ω ⧵ {0}). Thereupon, taking ∶= , ∈ ℕ, and letting → ∞ in (5.32) and (5.33) yields (5.30) and (5.31). □ In Lemma 5.6, we shall furthermore draw on the following elementary observation.

A C K N O W L E D G E M E N T S
The author is grateful to the anonymous reviewer for a very thorough evaluation of this manuscript. He furthermore acknowledges support of the Deutsche Forschungsgemeinschaft in the context of the project Emergence of structures and advantages in cross-diffusion systems (Project No. 411007140, GZ: WI 3707/5-1).

J O U R N A L I N F O R M AT I O N
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