Approaching optimality in blow-up results for Keller-Segel systems with logistic-type dampening

Nonnegative solutions of the Neumann initial-boundary value problem for the chemotaxis system \begin{align}\label{prob:star}\tag{$\star$} \begin{cases} u_t = \Delta u - \nabla \cdot (u \nabla v) + \lambda u - \mu u^\kappa, \\\\ 0 = \Delta v - \overline m(t) + u, \quad \overline m(t) = \frac1{|\Omega|} \int_\Omega u(\cdot, t) \end{cases} \end{align} in smooth bounded domains $\Omega \subset \mathbb R^n$, $n \ge 1$, are known to be global-in-time if $\lambda \geq 0$, $\mu>0$ and $\kappa>2$. In the present work, we show that the exponent $\kappa = 2$ is actually critical in the four- and higher dimensional setting. More precisely, if \begin{alignat*}{3} \qquad n&\geq 4,&&\quad \kappa \in (1, 2) \quad&&\text{and} \quad \mu>0 \\\\ \text{or}\qquad n&\geq 5,&&\quad \kappa = 2 \quad&&\text{and} \quad \mu \in \left(0, \frac{n-4}{n}\right), \end{alignat*} for balls $\Omega \subset \mathbb R^n$ and parameters $\lambda \geq 0$, $m_0>0$, we construct a nonnegative initial datum $u_0 \in C^0(\overline \Omega)$ with $\int_\Omega u_0 = m_0$ for which the corresponding solution $(u, v)$ of \eqref{prob:star} blows up in finite time. Moreover, in 3D, we obtain finite-time blow-up for $\kappa \in (1, \frac32)$ (and $\lambda \geq 0$, $\mu>0$). As the corner stone of our analysis, for certain initial data, we prove that the mass accumulation function $w(s, t) = \int_0^{\sqrt[n]{s}} \rho^{n-1} u(\rho, t) \,\mathrm d\rho$ fulfills the estimate $w_s \le \frac{w}{s}$. Using this information, we then obtain finite-time blow-up of $u$ by showing that for suitably chosen initial data, $s_0$ and $\gamma$, the function $\phi(t) = \int_0^{s_0} s^{-\gamma} (s_0 - s) w(s, t)$ cannot exist globally.


Introduction
A considerable amount of the literature on chemotaxis systems deals with detecting critical parameters distinguishing between global existence and finite-time blow-up. Such a dichotomy is already present in the minimal Keller-Segel system proposed by Keller and Segel to model chemotactic behavior of bacteria attracted by a chemical substance they produce themselves [10]. Considered in two-dimensional balls, the mass of u 0 is critical: If the initial datum u 0 is sufficiently regular, radially symmetric and satisfies Ω u 0 < 8π, then the corresponding solutions are global-in-time and bounded [18] while for any m 0 > 8π, there exists u 0 ∈ C 0 (Ω) with Ω u 0 = m 0 leading to finite-time blow-up [5,15]. (See also [17] for corresponding results in a parabolic-elliptic simplification of (1.1).) Let us note that this specific critical mass phenomenon is limited to the two-dimensional setting: While solutions to (1.1) are always global-in-time an bounded if considered in one-dimensional domains [20], in the spatially higher dimensional cases, finite-time blow-up has been detected even for arbitrary positive initial masses [29].
Other dichotomies between boundedness and blow-up include critical exponents both for nonlinear diffusion as well as nonlinear sensitivity [7] and nonlinear signal production [32]. Instead of presenting them in detail here, we refer to the surveys [1] and [13] for a broader overview of chemotaxis systems and related results.
Aiming to further enhance our understanding of the exact strength of the destabilising taxis term, in this article, we present another critical parameter distinguishing between global existence and finite-time blowup, namely the exponent κ = 2 in Keller-Segel systems with logistic-type degradation.
Before stating our main result, let us introduce systems featuring such dampening terms and recall some of the corresponding results. That is, we will first consider the Keller-Segel system with logistic source in smooth, bounded domains Ω ⊂ R n , n ≥ 2, and given parameters λ ∈ R, µ > 0 and τ ≥ 0. (We note that in view of the global existence result for λ = µ = 0 in one-dimensional domains mentioned above, at least for the question whether finite-time blow-up occurs, we may confine ourselves to the assumption n ≥ 2.) The system (1.2) and variations thereof describe several biological processes such as population dynamics [6,23], pattern formation [37] or embryogenesis [21] (see also [6] for an overview).
Already in 2007, Tello and Winkler showed that for τ = 0, any λ ∈ R, µ > n−2 n and any reasonably smooth initial data, the system (1.2) possesses global, bounded classical solutions [25]. Moreover, for n ≥ 3 and µ = n−2 n (and again τ = 0 and at least λ ≥ 0) solutions to (1.2) are global-in-time [9], but to the best of our knowledge it is unknown whether these are also always bounded. For the parabolic-parabolic case, that is, for τ > 0, the situation is similar: In the two-dimensional setting, assuming merely µ > 0 suffices to guarantee global existence of classical solutions [19], even for dampening terms growing slightly slower then quadratically [40]. Moreover, for higher dimensional convex domains, global classical solutions have been constructed for µ > µ 0 for some µ 0 > 0 in [27], where explicit upper bounds of µ 0 then have been derived in [16,39] and the convexity assumption has been removed in [38] at the cost of worsening the condition on µ. In all these settings, however, the known upper bounds for µ 0 are larger than n−2 n . However, if one resorts to more general solution concepts, further existence results are available. Under rather mild conditions, global weak solutions have been constructed in [25] and [12] for the cases τ = 0 and τ = 1, respectively. Moreover, if the degradation term −µu 2 in (1.2) is replaced by a weaker but still sufficiently strong superlinear dampening term, global generalized solutions have been obtained, again both for the parabolic-elliptic [26] and the fully parabolic case [34,35,41].
On the other hand, it has been observed that despite the presence of quadratic dampening terms, structures may form on intermediate time scales which even surpass so-called population thresholds to an arbitrary high extent (cf. [9,11,30] for the parabolic-elliptic and [31] for the parabolic-parabolic case).
While these findings already show that the aggregating effect of the chemotaxis term is strongly countered although not completely nullified by quadratic degradation terms, the question arises whether the most drastic form of spatial aggregation-finite-time blow-up-still occurs in Keller-Segel systems with superlinear degradation terms. A first partial (and affirmative) answer has been given in [28]: There, the compared to (1.2) with τ = 0 slightly simplified system is considered in balls Ω ⊂ R n , n ≥ 5 and, for any λ ≥ 0 and κ ∈ (1, 3 2 + 1 2(n−1) ), initial data leading to finite-time blow-up are constructed. The second important finding in this direction transfers this result to physically meaningful space dimensions. More concretely, [33] detects finite-time blow-up even in the system (1.2) with τ = 0 (inter alia) for balls Ω ∈ R n , n ∈ {3, 4}, λ ≥ 0 and κ ∈ (1, 7 6 ). Recently, the regime of exponents allowing for finite-time blow-up in (1.3) has been further widened to κ > 4 3 and κ > 3 2 in the three-and four-dimensional settings, respectively [2]. Moreover, in planar domains, chemotactic collapse can be obtained if one replaces the term −u κ in (1.3) with certain heterogeneous dampening terms such as −|x| 2 u 2 [4]. (Let us additionally note that similar finite-time blow-up results are also available for systems with nonlinear diffusion [2,14] or sublinear taxis sensitivity [24].) Main results. At least in the four-and higher dimensional settings, the journey of detecting finite-time blow-up in (1.3) for ever increasing values of κ comes to an end with the present article; we obtain the corresponding result up to (and for n ≥ 5 even including) the optimal exponent κ = 2.
More precisely, our main result reads is positive, radially symmetric as well as radially decreasing (1.5) and fulfills Ω u 0 = m 0 as well as the following holds: There exist T max < ∞ and a classical solution in Ω, (1.8) which blows up at T max in the sense that lim tրTmax u(0, t) = ∞.
Main ideas. Following Jäger and Luckhaus [8], we rely on the mass accumulation function given by w(s, t) := n √ s 0 ρ n−1 u(ρ, t) dρ, which transforms (1.8) to a scalar equation, see Lemma 3.1. The predecessors [2] and [33] of this article, which deal with (variations of) the system (1.3), then proceed to show that the function φ defined by cannot, at least not for certain initial data, s 0 ∈ (0, R n ) and γ ∈ (0, 1), exist globally in time, implying that u must blow up in finite time. One of the most challenging terms to estimate arises from the degradation term; one essentially has to control the integral s0 0 w κ s (s, t) ds. At this point, pointwise estimates for w s come in handy, which due to the identity w s (s, t) = u(s 1 n , t) are available once pointwise estimates for u are known. These in turn can for instance be obtained by analyzing general parabolic equations in divergence form [3,36] or by arguments similar to Lemma 3.3 below. In fact, one of the main points in [2] is to discuss how pointwise upper estimates for u of the form u(x) ≤ C|x| −p influence the possibility to detect finite-time blow-up.
However, a natural limitation of this approach is the exponent p = n since for fixed C > 0 and p < n, nonnegative functions u 0 ∈ C 0 (Ω) with u 0 (x) ≤ C|x| −p cannot have their mass concentrated arbitrary close to the origin; that is, depending on the value of C and p, none of these functions may fulfill (1.6). However, as seen in [2], even the choice p = n 'only' yields finite-time blow-up for the system (1.3) for certain κ < 3 2 . Thus, in the present article, where we handle exponents κ ≤ 2, we choose a slightly different path. At the basis of our analysis stands Lemma 3.3: There, we derive the key estimate w s ≤ w s , which due to w(0, ·) ≡ 0 actually improves on w s ≤ C s . Its proof is surprisingly simple: As already observed in similar contexts (cf. [2,4,32]), for radially decreasing initial data, w s (·, t) is decreasing for all times t, see Lemma 3.2. The desired estimate is then just a consequence of the mean value theorem.
Another major difference of our methods compared to [2] and [33] is that we do not limit our analysis of (1.9) to γ ∈ (0, 1) but also allow for parameters γ being larger than 1. In the five-and higher dimensional settings, this will then allow us to obtain finite-time blow-up even for κ = 2. (In 3D and 4D, the term stemming from the diffusion forces γ to be smaller than 1 and hence we cannot employ the same method as in higher dimensions.) We also note that the realization of the idea of taking γ > 1 is made possible by the new crucial estimate w s ≤ w s . The rest of the article is organized as follows: After stating some preliminary results in Section 2, in Section 3 we derive w s ≤ w s in Lemma 3.3. Section 4 then starts with the definition of the function φ and a calculation of its derivative, see Lemma 4.1, Next, in the Lemma 4.2, we suitably estimate the term originating in the logistic source, before dealing with the remaining terms and the initial datum of φ in the subsequent lemmata. In Lemma 4.6, we then finally prove finiteness of the maximal existence time T max .
Proof. This immediately follows from integrating the first equation in (1.8) and using that µ > 0.
As used multiple times in the sequel, let us also state the following elementary

The mass accumulation function w
Given u 0 as in (1.5) (and thus (u, v) as in Lemma 2.1), we denote the mass accumulation function by which has been introduced in the context of chemotaxis systems in [8]. In this section, we prove some of its properties, most notably the crucial estimate w s ≤ w s in Lemma 3.3. Lemma 3.1. For every u 0 satisfying (1.5), the function w given by (3.1) as well as Proof. This can be seen by a direct calculation. In fact, the asserted regularity is a consequence of Lemma 2.1, the identity (3.2) follows from the chain rule, and [28, equation (1.4)] asserts that (3.3) holds.
Next, as a major step towards proving w s ≤ w s , we show that for initial data fulfilling (1.5), the first solution component is radially decreasing throughout evolution.
As can be rapidly seen by writing the second equation As already advertised multiple times, this lemma now allows us to rapidly obtain the important estimate w s ≤ w s . Lemma 3.3. Assume that u 0 satisfies (1.5). For all s ∈ [0, R n ] and t ∈ [0, T max ),

holds. In particular, for all
Proof. For fixed t ∈ [0, T max ) and s ∈ [0, R n ], the mean value theorem provides us with ξ ∈ (0, s) such that w(s, t) = sw s (ξ, t), which already proves (3.4) since w s is decreasing by Lemma 3.

A supersolution to a superlinear ODE: finite-time blow-up
We will construct initial data leading to finite-time blow-up and hence prove Theorem 1.1 in this section. As already mentioned in the introduction, our argument is based on constructing a function φ which cannot exist globally, implying that the solution of (1.8) also can only exist on a finite time interval. In fact, we define φ as in [2] or [33]; that is, for given u 0 as in (1.5) and γ ∈ (0, 2), we set However, as the parameter γ herein may be larger than 1 (unlike as in [2] or [33]), some care is needed for calculating the time derivative of φ. This is done in the following Lemma 4.1. Suppose that u 0 complies with (1.5). Let γ ∈ (0, 2) and φ be as in (4.1). For every s 0 ∈ (0, R n ), φ(s 0 , ·) belongs to C 0 ([0, T max )) ∩ C 1 ((0, T max )) and fulfills Proof. We first fix s 0 ∈ (0, R n ) and note that φ(s 0 , ·) ∈ C 0 ([0, T max )) because of (3.5) and 1 − γ > −1.
Letting 0 < t 0 < t 1 < T max , we then make use of Lemma 3.
Aiming to derive that φ is a supersolution to a superlinear ODE, we now estimate the terms I 1 , . . . , I 4 in (4.2) and begin with I 4 , the term stemming from the logistic source. In the following proof, we will crucially make use of the estimate (3.4) to improve on corresponding results obtained by the predecessors [2] and [33]. (i) If κ = 2, γ > 1 and u 0 fulfills (1.5), then . There exists C 4 > 0 such that whenever u 0 fulfills (1.5), then for all s 0 ∈ (0, min{1, R n }) and t ∈ (0, T max ). Proof. We let γ ∈ (0, ∞) \ {1} and also fix u 0 as in (1.5) but will make sure that C 4 can be taken independently of u 0 . By Fubini's theorem, we first observe that for all s 0 ∈ (0, R n ) and t ∈ (0, T max ).
In the case of γ > 1 and κ = 2, we drop a positive term and employ (3.4) in calculating for all s 0 ∈ (0, R n ) and t ∈ (0, T max ), which already implies (4.3).
If on the other hand γ ∈ (0, 1) and κ ∈ (0, 1), going back to (4.5) and making use of use of (3.4), we see that for all s 0 ∈ (0, R n ) and t ∈ (0, T max ). By Hölder's inequality (with exponents 2 2−κ , 2 κ ), we have therein for all s 0 ∈ (0, R n ) and t ∈ (0, T max ). We assume now moreover that γ > 2(κ−1) κ and hence γ − 1 > κ−2 κ as well as a := (γ−1)κ 2−κ > −1, so that applying Lemma 2.3 (with B as in that lemma) gives The remaining integrals in (4.2) can be estimated as in [33] or [2]. However, at least for the statement concerning I 1 , we would like to give a full proof here in order to show the basis of the restriction on κ in Theorem 1.1. Indeed, while in Lemma 4.2 above, γ has to be taken sufficiently large, for estimating I 1 , we need γ to be suitably small. We will obtain finite-time blow-up precisely in the cases where the set of admissible γ for both these lemmata is nonempty. Moreover, compared to [33], the proof below makes use of the estimate (3.4) and is hence somewhat shorter.
Proof. For convenience, we fix u 0 as in (1.5), albeit we emphasize that the constants below do not depend on u 0 . An integration by parts gives Herein, the second term on the right hand side is positive and the last one is zero because of γ < 2− 4 n < 2− 2 n . Setting c 1 := 2 − 2 n − γ > 0, we hence infer from (3.4) and Hölder's inequality that Next, for estimating the integrals I 2 and I 3 in (4.2), we basically recall the corresponding results from [33].
Proof. Arguing as in [33,Lemma 4.4], we obtain for all s 0 ∈ (0, R n ) and t ∈ (0, T max ) (4.11) for some c 1 > 0 independent of u 0 . Taking both the left and the right hand side therein to the power 2 already yields (4.9). Moreover, as by Lemma 2.2 and (3.4), another consequence of (4.11) is (4.10).
As a final preparation, we note that, under certain circumstances, φ(s 0 , 0) can be shown to be sufficiently large. A combination of the results obtained above now reveals that for initial data whose mass is sufficiently concentrated near the origin, the corresponding solution cannot exist globally in time. Again, the argument is not too different from [28] or [33], but we choose to give it nonetheless in order to show that s 0 and u 0 can be chosen in such a way that φ would blow up in finite time if (u, v) were a global solution.
Dividing by the right hand side therein yields upon an integration in time for all t ∈ (0, min{1, T max }), implying T max ≤ 1 2 .
Finally, we conclude that Theorem 1.1 is now merely a direct consequence of the lemmata above.
Proof of Theorem 1.1. Lemma 4.6 asserts that there is r 1 ∈ (0, R) such that under the conditions of Theorem 1.1, the maximal existence time T max is finite. By Lemma 3.2 and Lemma 2.1, this then implies u(0, t) = u(·, t) L ∞ (Ω) → ∞ as t ր T max .