EMERGENCE OF LARGE POPULATION DENSITIES DESPITE LOGISTIC GROWTH RESTRICTIONS IN FULLY PARABOLIC CHEMOTAXIS SYSTEMS

. We consider the no-ﬂux initial-boundary value problem for Keller-Segel-type chemotaxis growth systems of the form in a ball Ω ⊂ R n , n ≥ 3, with parameters χ > 0 ,ρ ≥ 0 and µ > 0. By means of an argument based on a conditional quasi-energy inequality, it is ﬁrstly shown that if χ = 1 is ﬁxed, then for any given K > 0 and T > 0 one can ﬁnd radially symmetric initial data, possibly depending on K and T , such that for arbitrary µ ∈ (0 , 1) the corresponding local-in-time classical solution ( u,v ) satisﬁes with some x ∈ Ω and t ∈ (0 ,T ); in fact, this growth phenomenon is actually identiﬁed as being generic in the sense that the set of all initial data having this property is dense in the set of all suitably regular radial initial data in a certain topology. Secondly, turning a focus on possible eﬀects of large chemotactic sensitivi- ties, on the basis of the above it is shown that when ρ ≥ 0 and µ > 0 are ﬁxed, then for all L > 0 ,T > 0 and χ > µ one can ﬁx radial initial data ( u 0 ,χ ,v 0 ,χ ) which decay in L ∞ (Ω) × W 1 , ∞ (Ω) as χ → ∞ , and which are such that for the respective solution ( u χ ,v χ ) there exist x ∈ Ω and t ∈ (0 ,T ) fulﬁlling L.


1.
Introduction. Including logistic proliferation terms may substantially influence the dynamics in chemotaxis systems. This firstly concerns the ability of the respective system to spontaneously generate singularities, as known to constitute one of the most striking features of the classical Keller-Segel system which has widely been accepted as the simplest reasonable macroscopic model for the collective behavior in cell populations, quantified through their density u = u(x, t), in chemotactic response to a signal produced by themselves and represented by its concentration v = v(x, t) ( [10], [7]). Indeed, whereas the nonlinear crossdiffusion process in (1) is known to enforce finite-time blow-up of some solutions with respect to the norm in L ∞ of its first component in two-or higher-dimensional cases ( [6], [26]; cf. also the surveys [8], [3]), in the correspondingly modified variant thereof given by the additional dissipative effect of the quadratic zero-order death term is known to rule out any such collapse when either n = 2 and µ > 0 is arbitrary ( [17]), or n ≥ 3 and µ is sufficiently large ( [25]); if in the latter case n ≥ 3 the number µ > 0 is arbitrary, then at least certain global weak solutions can be constructed, and if moreover n = 3 and ρ is suitably small, then these solutions eventually become smooth and classical ( [13]). In line with this, systems of type (2) appear as subsystems at the core of numerous more complex models for chemotactic cell migration at large time scales, especially in situations when infinite densities turn out to be unrealistic, and thus seem of particular relevance in the modeling of tumor invasion processes ( [4], [22], [20]), also in the context of multiscale approaches ( [14], [21]). However, effects of logistic source terms in fact may go significantly beyond such aspects of global existence and boundedness theory, and thus the interplay of Fishertype cell kinetics with diffusion and chemotactic cross-diffusion is considerably more colorful than with merely diffusion. This is, inter alia, indicated by numerical evidence revealing quite a multifaceted and possibly even chaotic solution behavior already in spatially one-dimensional versions of (2) ( [19]), as well as rich structures of associated steady-state sets in two-dimensional cases, including the occurrence of hexagonal patterns ( [11]).
Apparently, however, up to now only few aspects of the solution behavior in (2) have been captured by rigorous analysis. For instance, it is known that if µ > µ 0 with some µ 0 = µ 0 (d, χ, ρ, Ω), then the corresponding nontrivial spatially homogeneous equilibrium of (2) is globally asymptotically stable (see [27] for a proof in the prototypical case d = χ = ρ = 1), where even an explicit bound for µ 0 can be obtained in an associated parabolic-elliptic simplification of (2) in which the signal evolution is governed by the elliptic equation 0 = ∆v − v + u ( [23]). In presence of small values of µ when no such proliferation-dominated behavior can be expected, only little seems known beyond results on existence and dimension of exponential attractors in two-dimensional frameworks ( [17], [16], [2]); after all, large-time extinction phemonena, as numerically observed to occur in large spatial regions ( [19]) and initially discussed in [1] from a rigorous perspective, have recently been shown to necessarily be of local nature in the sense that for each global solution, the associated total mass of cells always persists throughout evolution ( [24]).
More subtle qualitative facets of chemotaxis-growth interaction could up to now be rigorously detected only in simplified parabolic-elliptic settings and under the essential additional assumption that cell diffusion is suitably weak: In the hyperbolicelliptic limit case d = 0 of such sytsems, namely, it can be observed that some solutions blow up in finite time with respect to the spatial L ∞ norm of the component u, even in spatially one-dimensional intervals ( [28]), but also in radial higherdimensional situations ( [12]). Based on a suitable perturbation analysis, it can be shown that in either of these cases, under an appropriate assumption on the initial data it is possible to find T > 0 with the property that for each M > 0 there exists d 0 > 0 such that whenever d ∈ (0, d 0 ), one can find a point x d in the spatial domain Ω and t d ∈ (0, T ) for which the solution (u d , v d ) of an associated Neumann initial-boundary value problem in Ω × (0, T ) satisfies In particular, this means that even in situations when solutions are known to be global and bounded, the influence of chemotactic cross-diffusion may force some solutions to exceed any given threshold dynamically, at least on intermediate time scales, which is in sharp contrast to the solution behavior e.g. in the diffusive Fisher-KPP problem corresponding to the choice χ = 0 in (2), where such phenomena are ruled out by the availability of a maximum principle.
Main results.
To the best of our knowledge, however, no rigorous results on solution behavior far from equilibrium are available for the fully parabolic system (2), nor for any chemotaxis-growth system involving possibly large diffusion rates. The purpose of the present work consists in developing an approach which enables us to accomplish some first steps in this direction, and especially to show that the dynamical emergence of structures, extreme in the sense that arbitrarily large population densities are involved, need not necessarily be a small-diffusion phenomenon.
For this purpose, firstly focusing only on the parameters relevant to cell proliferation we will consider the initial-boundary value problem in a ball Ω ⊂ R n , n ≥ 3, where the numbers ρ ≥ 0 and ε > 0 as well as the initial data u 0 and v 0 are given. As for this problem, the first of our main results reveals an unboundeness phenomenon, possibly transient in time, which can even be viewed generic with respect to the choice of initial data within an appropriate topology, and which can moreover be quantified in terms of the parameter ε in (3). Theorem 1.1. Let n ≥ 3 and Ω = B R (0) ⊂ R n with some R > 0, let ρ ≥ 0, and suppose that u 0 ∈ C 0 (Ω) and v 0 ∈ W 1,∞ (Ω) are radially symmetric and positive in Ω. Then for all K > 0 and each T ∈ (0, 1) there exist sequences (u 0k ) k∈N ⊂ C 0 (Ω) and (v 0k ) k∈N ⊂ W 1,∞ (Ω) of radially symmetric positive functions u 0k and v 0k on Ω such that that u 0k → u 0 in L p (Ω) for all p ∈ 1, 2n n + 2 and v 0k → v 0 in W 1,2 (Ω) as k → ∞, and that for all k ∈ N and ε ∈ (0, 1) one can find t ε,k ∈ (0, T ) with the property that (3) possesses a classical solution In particular, this implies the following quantitative result on dynamical growth in (3) for a fixed pair of initial data. Corollary 1. Let n ≥ 3 and Ω = B R (0) ⊂ R n with some R > 0, and let ρ ≥ 0. Then for all K > 0 and any T > 0 there exist radially symmetric positive functions u 0 ∈ C 0 (Ω) and v 0 ∈ W 1,∞ (Ω) with the property that for each ε ∈ (0, 1) one can find t ε ∈ (0, T ) with the property that (3) possesses a classical solution As a second by-product of Theorem 1.1, the particular quantitative information (6) contained therein will enable us to study possible effects of large chemotactic sensitivities in presence of a fixed logistic source. Specifically, for the version of (2) given by we shall obtain the following.
Then for any choice of L > 0, T ∈ (0, 1) and χ > µ one can find radially symmetric positive functions w 0χ ∈ C 0 (Ω) and z 0χ ∈ W 1,∞ (Ω) such that and that for any χ > µ there exists t χ ∈ (0, T ) such that (8) The main idea: Exploiting a conditional quasi-energy inequality. Our approach is rooted in a contradictory argument based on an analysis of the quantity which is well-known to play the role of a genuine Lyapunov functional for the unforced normalized Keller-Segel system obtained on letting holds along the respective trajectories, with the nonnegative dissipation rate given by ( [15]). Whereas this subtle structure is apparently destroyed in presence of the kinetic terms in (2), it will turn out that at least a certain quasi-energy inequality can be derived under an appropriately mild boundedness hypothesis on the solution component u. Relying on a functional inequality relating F(u, v) to the associated dissipation rate, as obtained in [26] by making essential use of the fact that n ≥ 3 (Lemma 3.2), under the assumption that within a suitably small time interval the solution of (2) satisfies u ≤ K ε with some K > 0, this will enable us to establish an autonomous ordinary differential inequality for F (u, v) (Lemma 3.1, Lemma 3.6 and Lemma 3.10) which cannot hold throughout this time interval (Lemma 3.11). Exploiting this will yield the statements from Theorem 1.1 and Corollary 1 in Section 3, whereupon a stratightforward variable transformation will lead to a proof of Theorem 1.2 in Section 4.

2.
Preliminaries. For definiteness in our subsequent arguments, let us first recall from [26] that any given pair of suitably regular positive radial functions on Ω can conveniently be approximated by low-energy data.
such that for all j ∈ N, u 0j and v 0j are radially symmetric and positive in Ω with as j → ∞, and that with F as in (11) we have Proof. This immediately results from the statement in [26, Lemma 6.1].
When employed as initial data in (3), all these functions give rise to corresponding local-in-time classical solutions. Lemma 2.2. For all ε ∈ (0, 1) and j ∈ N, there exists T ε,j ∈ (0, ∞] such that the problem (3) with u 0 := u 0j and v 0 := v 0j possesses a positive classical solution Proof. It is well-known ( [25]) that the problem in question is solvable in the indicated class, with some T ε,j ∈ (0, ∞] which is such that To see that actually (14) holds, assuming on the contrary that T ε,j be finite, but that u ε,j be bounded in Ω×(0, T ε,j ), by applying standard arguments from parabolic regularity theory to the second equation in (3) ([9]) we could find c 1 > 0 such that v ε,j (·, t) W 1,∞ (Ω) ≤ c 1 for all t ∈ 1 2 T ε,j , T ε,j .
This contradicts (15) and thereby verifies (14). (3). The following generalization of the energy identity (12) to the chemotaxis-growth system (3) is straightforward but fundamental to our approach.
In order to draw appropriate conclusions from (16), we recall from [26] that in the case n ≥ 3 considered here, the expression Ω uv can essentially be controlled by a sublinear power of the dissipation rate D(u, v) from (13) in the sense of the following functional inequality that is actually valid for a large class of radially symmetric functions on Ω.  for all t ∈ 0, min{1, T ε,j } .
Secondly, based on Lemma 3.3 and features of parabolic regularization, also the second solution component can be seen to comply with the requirements contained in Lemma 3.2.
Proof. Thanks to Lemma 3.3 and the assumed radial symmetry, this can be seen by straightforward modification of the reasoning in [26, Section 3]; for completeness, let us briefly outline a proof: Without loss of generality assuming that κ ≤ n − 1 and then writing p := n κ+1 > 1, we have p < n n−1 , so that a standard result on regularization in the inhomogeneous linear heat equation v t = ∆v − v + u ( [9]) applies so as to provide c 1 > 0 such that for all t ∈ (0, T ε,j ), whence by Lemma 3.3 and the boundedness of (v 0j ) j∈N in W 1,2 (Ω), as asserted by Lemma 2.1, we can find c 2 > 0 such that where T := min{1, T ε,j }. For each fixed t ∈ (0, T ), we can therefore find r 0 (t) ∈ for all r ∈ (0, R). As can be verified by explicit evaluation, herein we have r r0(t) whence on using (23) we can readily derive (22) from (24). Therefore, Lemma 3.2 indeed becomes applicable for the solutions from Lemma 2.2 at least for suitably small times: Lemma 3.5. There exist θ ∈ (0, 1) and C 0 > 0 with the property that for all ε ∈ (0, 1) and any j ∈ N, the solution gained in Lemma 2.2 satisfies where D is taken from (13).
As a first important application of the latter, we can use (25) to adequately control the crucial ill-signed summand ε Ω u 2 v on the right of (16) whenever εu satisfies an upper estimate which we finally plan to disprove. We can thereby turn the identity (16) into an inequality exclusively containing F and D as follows.
Then with F and D as in (11) and (13), for all t ∈ (0, T ).
In order to relate the summands in (28) containing D to certain expressions only involving F, we once more apply Lemma 3.5 to achieve the following estimate on D from below in terms of F. Lemma 3.7. Let θ ∈ (0, 1) and C 0 > 0 be as in Lemma 3.5. Then for any choice of ε ∈ (0, 1) and j ∈ N, where D and F are as in (13) and (11).
As long as F attains suitably large negative numbers, this implies that up to a multiplicative constant, D even dominates a superlinear power of F itself.
We next intend to make sure that as long as εu is conveniently small and −F is suitably large, D also substantially exceeds the last three summands in (28), the first among which is considered in the following.
Proof. Due to (34), Corollary 2 may be applied so as to guarantee that with (u, v) := (u ε,j , v ε,j ). Therefore, using (37) and the fact that θ < 1 we can estimate which is equivalent to (38).
The last two summands in (28) can be dealt with similarly.
Lemma 3.9. Let K > 0 and T ∈ (0, 1), and assume that ε ∈ (0, 1) and j ∈ N have the properties that T ε,j ≥ T and that (34) is valid, and such that with θ ∈ (0, 1) and C 0 > 0 from Lemma 3.5 we have where F is as in (11). Then the functional D from (13) satisfies Proof. We again use that thanks to Corollary 2 our assumption that (34) holds ensures that for (u, v) := (u ε,j , v ε,j ) we have Therefore, namely, from (40) we immediately obtain that for all t ∈ (0, T ), as claimed.
In conclusion, if all of the above hypotheses are met, F will satisfy a superlinear autonomous ordinary differential inequality. Lemma 3.10. Let F be as in (11), let K > 0 and T < 1, and suppose that ε ∈ (0, 1) and j ∈ N are such that T ε,j ≥ T , and that (36), (34), (37) and (40) are valid with θ ∈ (0, 1) and C 0 > 0 taken from Lemma 3.5. Then Proof. According to the assumed inequality in (34), we particularly know that (u, v) := (u ε,j , v ε,j ) satisfies whereas the hypotheses that (36), (37) and (40) be valid guarantee that as well as for all t ∈ (0, T ) due to Lemma 3.8 and Lemma 3.9. Therefore, from Lemma 3.6 we obtain that for all t ∈ (0, T ), so that another application of Corollary 2 establishes (42).
The latter inequality, however, cannot hold throughout the considered time interval if the energy functional attains suitably large negative values initially. The contradiction thereby obtained leads to the following conlcusion.
Proof. Given K > 0 and T ∈ (0, 1), we abbreviate Then in order to verify that j 0 has the claimed property, assuming this to be false we could find j > j 0 and ε ∈ (0, 1) such that in view of (14) we would have and u ε,j (x, t) ≤ K ε for all x ∈ Ω and t ∈ (0, T ).
For these fixed values of ε and j, we would thus obtain that y(t) := −F u ε,j (·, t), v ε,j (·, t) , t ∈ [0, T ), is well-defined with its initial value satisfying y(0) > y 0 := max{c 1 , c 2 , c 3 , c 4 } according to (44). Therefore, by continuity of y, S := T ∈ (0, T ) y(t) > y 0 for all t ∈ [0, T ) would be nonempty and hence also T := sup S well-defined. To see that we actually must have T = T , we observe that (47) especially entails that y ≥ c 1 and y ≥ c 2 as well as y ≥ c 3 on (0, T ), which along with (45) and (46) asserts the hypotheses of Lemma 3.10. An application of the latter thus shows that so that, in particular, y ≥ 0 on (0, T ) and hence y ≥ y(0) > y 0 on (0, T ). This would clearly be incompatible with the assumption that T < T , meaning that indeed T = T and that hence the inequality in (48) is valid for all t ∈ (0, T ). On integration, however, this would entail that