Relaxed parameter conditions for chemotactic collapse in logistic-type parabolic–elliptic Keller–Segel systems

We study the finite-time blow-up in two variants of the parabolic–elliptic Keller–Segel system with nonlinear diffusion and logistic source. In n-dimensional balls, we consider JLut=∇·((u+1)m-1∇u-u∇v)+λu-μu1+κ,0=Δv-1|Ω|∫Ωu+u\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} u_t = \nabla \cdot ((u+1)^{m-1}\nabla u -u \nabla v) + \lambda u - \mu u^{1+\kappa }, \\ 0 = \Delta v - \frac{1}{|\Omega |} \int \limits _\Omega u + u \end{array}\right. } \end{aligned}$$\end{document}and PEut=∇·((u+1)m-1∇u-u∇v)+λu-μu1+κ,0=Δv-v+u,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} u_t = \nabla \cdot ((u+1)^{m-1}\nabla u -u \nabla v) + \lambda u - \mu u^{1+\kappa }, \\ 0 = \Delta v - v + u, \end{array}\right. } \end{aligned}$$\end{document}where λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} and μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} are given spatially radial nonnegative functions and m,κ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m, \kappa > 0$$\end{document} are given parameters subject to further conditions. In a unified treatment, we establish a bridge between previously employed methods on blow-up detection and relatively new results on pointwise upper estimates of solutions in both of the systems above and then, making use of this newly found connection, provide extended parameter ranges for m,κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m,\kappa $$\end{document} leading to the existence of finite-time blow-up solutions in space dimensions three and above. In particular, for constant λ,μ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda , \mu > 0$$\end{document}, we find that there are initial data which lead to blow-up in (JL) if 0≤κ<min12,n-2n-(m-1)+ifm∈2n,2n-2nor0≤κ<min12,n-1n-m2ifm∈0,2n,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} 0 \le \kappa&< \min \left\{ \frac{1}{2}, \frac{n - 2}{n} - (m-1)_+ \right\}&\quad \text {if } m\in \left[ \frac{2}{n},\frac{2n-2}{n}\right) \\ \text { or }\quad 0 \le \kappa&<\min \left\{ \frac{1}{2},\frac{n-1}{n}-\frac{m}{2}\right\}&\quad \text {if } m\in \left( 0,\frac{2}{n}\right) , \end{aligned}$$\end{document}and in (PE) if m∈[1,2n-2n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m \in [1, \frac{2n-2}{n})$$\end{document} and 0≤κ<min(m-1)n+12(n-1),n-2-(m-1)nn(n-1).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} 0 \le \kappa < \min \left\{ \frac{(m-1) n + 1}{2(n-1)}, \frac{n - 2 - (m-1) n}{n(n-1)} \right\} . \end{aligned}$$\end{document}

where λ and μ are given spatially radial nonnegative functions and m, κ > 0 are given parameters subject to further conditions. In a unified treatment, we establish a bridge between previously employed methods on blow-up detection and relatively new results on pointwise upper estimates of solutions in both of the systems above and then, making use of this newly found connection, provide extended parameter ranges for m, κ leading to the existence of finite-time blow-up solutions in space dimensions three and above. In particular, for constant λ, μ > 0, we find that there are initial data which lead to blow-up in (JL) if and in (PE) if m ∈ [1, 2n−2 n ) and

Introduction
How strong does a degrading term need to be in order to rule out chemotactic collapse in a Keller-Segel system? Or, phrased differently, when is chemotactic aggregation stronger than even superlinear dampening? On the one hand, in the absence of any degrading terms the minimal Keller-Segel model, proposed in the 1970s [20] to model chemotaxis, that is, the directed movement of, for instance, cells or bacteria toward a chemical signal, and given by (1.1) admits solutions blowing up in finite time. While all solutions are global and bounded in one dimension [31], blow-up does occur in two- [12,14,32] and higher-dimensional [46] domains. For a broader introduction to (1.1) and similar systems, we refer to the surveys [1,26].
On the other hand, adding logistic terms to (1.1), for instance, in order to model population dynamics [13,33] or tumor invasion processes [3], leads to the system where λ, μ > 0 and κ = 1 are given parameters. Here, all solutions are global and bounded in two [30] and, provided that μ ≥ μ 0 for some μ 0 > 0 depending on the space dimension, also in higher dimensions [44]. (See also [38] for the corresponding parabolic-elliptic system.) Moreover, without any restriction on μ > 0, in all space dimensions global weak solutions have been constructed, which in three-dimensional convex domains additionally become eventually smooth if λ is small enough [24].
In order to better understand the relative strengths of the possibly explosion-enhancing cross-diffusive chemotaxis effect and the damping force of the logistic terms, also systems with weaker damping (e.g., κ < 1) have been investigated with respect to the existence of classical, weak or generalized solutions (see, e.g., [29,40,41,53,54]).
For a more complete answer, however, it seems indispensable to also search for the opposite case, that of blow-up: What happens for κ ∈ (0, 1) (or for κ = 1 and small μ > 0)? For which values of κ can solutions blowing up in finite time be constructed?
Even beyond solutions that grow on smaller timescales in case of slow diffusion [23,47], some partial results in this direction are available: The first blow-up result for a chemotaxis system with superlinear degradation apparently goes back to [45]. Following a simplification introduced in [19] by Jäger and Luckhaus, there it was shown that for the system in a ball in R n , n ≥ 5, finite-time blow-up is possible provided that λ, μ > 0 and κ < 1 2 + 1 2n−2 . Moreover, chemotactic collapse may even happen in the physically (most) relevant space dimension three. In [49], it was shown that the parabolic-elliptic system in a ball in R n , n ≥ 3, admits solutions blowing up in finite time provided λ, μ > 0 and In two-dimensional domains, however, known results seem to be limited to the case of space-dependent functions μ. That is, if one replaces the first equation in (1.2) with u t = Δu−∇·(u∇v)+λu−μ 1 |x| α u 1+κ , then solutions blowing up in finite-time have been constructed if again the domain is a ball, λ, μ 1 , α > 0 and κ < α 2 . Phrased differently, given any κ > 0, there exist blow-up solutions even in 2D-provided α is large enough [10].
There is another effect that can hinder blow-up and is often included in the model, be it for reasons of biological modelling, for example, of tumor cells, cf. [35] or [11,21], or from a purely mathematical motivation: nonlinear, porous-medium type diffusion (i.e., the replacement of Δu by, e.g., ∇·((u+1) m−1 ∇u)). If sufficiently strong, it can prevent blow-up even on its own (see [15,17,34,37] for boundedness results in case of m > 2 − 2 n ) or at least in combination with logistic sources, [27,42,[55][56][57]. However, for the regime of slightly weaker diffusion, the occurrence of blow-up may still be possible. And indeed, in the absence of logistics (κ = 0, μ = λ), m < 2n−2 n for Ω being a ball in R n allows for some unbounded solutions (i.e., blow-up after either finite or infinite time, see [15,43]), with finite-time blow-up having been detected in [6]. As to blow-up for different combinations of diffusion and sensitivity terms, refer to [4,5,50] or to [16,18] for the case of degenerate diffusion.
Parabolic-elliptic analogues were investigated in [7,25,52], revealing blow-up after finite or infinite time for different parameter ranges.
If logistics and nonlinear diffusion both are incorporated into the model, at least in space dimensions n ≥ 5 sometimes blow-up is possible: In [28], it was shown that the techniques of [45] can be applied for diffusion rates m ∈ [1, 2n−4 n ) and dampening exponents κ ∈ (0, mn 2(n−1) ), resulting in finite-time blow-up for some radial solutions of the system in which the signal's equation is simplified analogously to the system from the famous work [19] by Jäger and Luckhaus. In [28,Remark 1.2], it was conjectured that blow-up should occur for m < 2n−2 n but the answer was left open to further research. For an earlier extension of the same methods from [45] to systems with nonlinear diffusion and logistics, combined with superlinear sensitivity functions, see [59].
The recent advances of [49] in the linear-diffusion case with logistic raise some hope that also in a nonlinear setting, the discovery of blow-up is also possible in the slightly less simplified parabolic-elliptic system and, more importantly, even in the physically more relevant case of n = 3, for instance.

Main results
The present article is dedicated to this question. Aiming for blow-up, we study (JL) and (PE) in a ball Ω ⊂ R n , n ≥ 3, for sufficiently smooth nonnegative functions λ, μ and a parameter κ ≥ 0. We refer to the introduction of [10] for a motivation for logistic source terms with spatial dependence. We extend the methods of [49] to nonlinear diffusion and show that they are applicable in (JL) as well as in (PE). At the same time, we would like to offer a different perspective on these, seeing them as a bridge connecting pointwise upper estimates of solutions to the occurrence of explosions. We therefore give our main result in the following form: , α ≥ 0, μ 1 > 0, p ≥ n, T > 0 as well as K > 0 and suppose that λ, μ are such that and comply with Assume moreover that κ ≥ 0 and m > 0 satisfy Then, we can find r 1 ∈ (0, R) with the following property: If C β (Ω) being nonnegative, radially symmetric and radially decreasing (1.8) and as well as If upper estimates as in (1.10) are known, this theorem shows that finite-time blow-up is possible in (PE) and (JL); that is, there are initial data such that (1.11) holds with some T max < ∞. This results in the following: Assume moreover that λ, μ satisfy (1.4) and (1.5). Moreover, for m = 1, the condition (1.12)-(1.13) is equivalent to Furthermore, for m = 1, the condition (1.14) reads Before we provide a more detailed comparison to the conditions on the existence of solutions blowing up in finite-time established in previous works, let us note the following.
for all q < n p and certain C(q) > 0, assuming that (1.10) holds for some p < n and a large class of initial data, these initial data would automatically be uniformly bounded in L n+p 2p (Ω), say, by C . However, as can be seen by applying Hölder's inequality, their mass on B r1 (0) would then be bounded by C |B r1 (0)| n−p n+p , which converges to 0 for r 1 0. Thus, it would not be clear if one of these initial data could still fulfill (1.9) for the value of r 1 given by Theorem 1.1. (iii) To the best of our knowledge, Theorem 1.2 provides the first detection of finite-time blow-up for Keller-Segel systems with nonlinear diffusion and superlinear damping terms in space dimensions 3 and 4. For (PE) and m = 1, it is furthermore the first such result in higher dimensions. (iv) The finite-time blow-up result for (PE) also constitutes a partial answer to the second part of Open Problem (i) in [58]. Now, let us take a more in-depth look at the new ranges for the parameter κ in some different spatial dimensions under the assumption of α = 0 for some special values of m. In this setting, earlier works have established a certain κ * (provided in Table 1) for which blow-up has been proven for κ < κ * .
Evidently, the findings of [45] and [28] (and also of the related [59]) only cover higher dimensions. For n ≥ 5 and m ∈ [1, 2n−4 n ), however, these results still provide better ranges than the one we could attain with our method. To the best of our knowledge, for larger values of m or for small space dimensions, however, our results provide the first proof of finite-time blow-up in (JL).
Regarding (PE), for the linear diffusion case we are able to match the range previously established in [49], while also providing first results for the nonlinear diffusion setting in higher dimensions.
When comparing the parameter ranges across the two different systems for m = 1 and n ∈ {2, 3}, we see that our results for (JL) yield a wider regime for κ than the corresponding results obtained in [49] for (PE). Indeed, 1 3 > 1 6 and 1 2 > 1 6 . In general, known results for (JL) are stronger than for (PE). However, lacking global existence results for κ < 2, it is yet unclear whether blow-up is actually more prominent in (JL) or just easier to detect.
Here, means that for the prescribed values of n and m, Theorem 1.2 asserts the existence of solutions blowing up in finite time for certain κ > 0, for whose precise values we refer to Theorem 1.2

Main ideas
As is meanwhile well established in the context of finite-time blow-up proofs for chemotaxis systems and has first been proposed by Jäger and Luckhaus in [19], we consider the mass accumulation function which transforms (JL) into the scalar equation (and (PE) at least into a system that is easier to handle than (PE) itself). The main difficulty for detecting finite-time blow-up lies in the fact that the term +nww s , stemming from the cross-diffusion in (JL), has to counter the quite different terms n 2 s 2− 2 n w ss and −n s 0 μ(σ 1 n )w 2 s (σ, t) dσ originating from the diffusion and logistic terms, respectively.
Following [49], our approach consists of showing that for certain initial data, γ ∈ (0, 1) and s 0 ∈ (0, R n ), the function cannot exist globally in time, which due to the blow-up criterion asserted in Lemma 2.1 implies the desired finite-time blow-up result (1.11). That is, in Sect. 3 we show that φ is a supersolution to the ODI φ = aφ 2 − b for certain a, b > 0 and in Sect. 4, we conclude the existence of initial data leading to finite-time blow-up of φ and hence u.
Let us briefly discuss how we deal with the two most problematic terms stemming from the degradation and diffusion terms, respectively. As we will see in Lemma 3.5, in order to handle the former, we essentially need to control At this point, the assumption (1.10) comes into play, which due to w s (s, this estimate to w κ s in (1.15) and then integrate by parts. Moreover, by (3.11), the term arising from the diffusion can be estimated against (some positive multiple of) For m ≥ 1, we can proceed as above; that is, we apply the pointwise upper bound to w m−1 s and then integrate by parts, while for m < 1 we can follow at least two different paths: For m ∈ (0, 2 p ), we apply this bound to w m and do not integrate by parts and for m ≥ 2 p , we estimate (nw s + 1) m ≤ nw s + 1 and integrate by parts without using the pointwise upper estimate for w s at all. The fact that depending on the value of m we employ two different methods here is the reason for the different conditions in (1.6) and (1.7).
At last, we show that pointwise upper estimates of the form (1.10) are indeed available both for (JL) and (PE). While for the former system we make use of the comparison principle applied to u r in Lemma 5.1, for the latter we resort to the recent study on blow-up profiles [9] to obtain the desired bounds in Lemma 5.2.
By (u, v), we will refer to a solution to either of the systems (JL) or (PE), and we also set and, in case of (JL), Ω v(·, t) = 0 for all t ∈ (0, T max ).
Proof. Local existence can be proved by a standard fixed point argument, which is explained in more detail in [7] or [38], for instance, while nonnegativity of u follows by the maximum principle and preservation of radial symmetry is a consequence of uniqueness.
As a first basic observation, we note that, at least locally in time, the mass of u can be controlled by the parameters we fixed above-and thus, independently of the precise choice of u 0 . Proof. Due to λ ≤ λ 1 and nonnegativity of μ, integrating the first equation over Ω gives d dt so that the statement follows by an ODE comparison argument.

Proving finite-time blow-up
Following [2,19,49], we define and, given s 0 ∈ (0, R n ) and γ ∈ (0, 1), introduce the functions If, by the usual slight abuse of notation, we identify the radially symmetric function u ∈ C 0 (Ω×[0, T max )) with u ∈ C 0 ([0, R]×[0, T max )) and write u r for its radial derivative, we can compute the spatial derivatives of w: For φ, which we later want to show to blow up, the following differential inequality holds: for all t ∈ (0, T max ) in the case of (JL). For (PE), the same estimate holds with Proof. The regularity of φ follows from (3.2). Written in radial coordinates, the differential equations in (JL) read In the remaining part of this section, we further estimate the terms of the right-hand side of (3.5), aiming to show that φ(s 0 , ·) fulfills a certain superlinear ODE. These results will then be combined in Sect. 4; ultimately, the consolidation of the lemmata will show that at least for certain values of γ and s 0 , φ(s 0 , ·) cannot exist globally.
In order to streamline the arguments below, let us first state two elementary lemmata. Proof. This is an evident consequence of the properties of the beta function.
The estimate (3.4), which originates in the crucial assumption (1.10), will come into play at two different places. The first of these is the following lemma, where said upper estimate is the most important ingredient for controlling the term arising from the logistic source, namely I 4 in (3.5). Lemma 3.5. Let γ ∈ (0, 1) and p ≥ n satisfy pκ n − α n < γ 2 . Whenever (1.10) is fulfilled for some K > 0, T > 0 and s 0 ∈ (0, R n ), then I 4 from Lemma 3.2 satisfies Proof. This can be proved analogously to [49,Lemma 4.5]: Firstly, Fubini's theorem asserts for all t ∈ (0, min{T, T max }). Finally, we note that according to Lemma 3.3 since α n − pκ n + γ 2 > 0 by assumption. The statement follows by combining the estimates above.
We now turn our attention to the integral involving the effects of nonlinear diffusion. This is the second place where (at least for certain m) we make use of the assumption (1.10). Lemma 3.6. Suppose that (1.10) holds for some p ≥ n, K > 0 and T > 0 and let I 1 be as in (3.5).
Then, there is C > 0 such that for any s 0 ∈ (0, R n ) and all t ∈ (0, min{T, T max }), we have Then, there is C > 0 such that for any s 0 ∈ (0, R n ) and all t ∈ (0, min{T, T max }), we have Proof. Direct calculation gives for every s 0 ∈ (0, R n ) The last two terms therein are positive, since (3.9) and (3.10) both entail 2 − 2 n − γ > 0, leading to (3.11) Now, let us start by considering the case that (3.9) holds. First, we find that w s ≥ 0 implies for m ≥ 1 that (nw s + 1) m ≤ 2 m−1 (n m w m s + 1) in (0, T max ). By (3.4), this entails that (nw On the other hand, for m ∈ (0, 1) we have (nw s + 1) m ≤ nw s + 1 in (0, R n ) × (0, min{T, T max }). Thus, letting c 1 := max{n, 2 m−1 , 2 m−1 nK (m−1)+ } we find that from combining these two estimates, we have and hence, from (3.11), in (0, min{T, T max }) and for every s 0 ∈ (0, R n ). An integration by parts therefore yields The third term on the right-hand side is nonnegative, and in the other terms, we use s 0 − s ≤ s 0 and s ≤ s 0 for all s ∈ (0, s 0 ) as well as the conditions γ < 2 − 2 n and γ > 1 − 2 n − p n (m − 1) + contained in (3.9) to see that in (0, min{T, T max }) and for s 0 ∈ (0, R n ). To estimate further, we make use of Lemma 3.4, the fact that (3.9) entails 0 < 1 − 2 n − p n (m − 1) + − γ 2 and Lemma 3.3 to obtain Collecting (3.12) and (3.13) proves the estimate of I 1 in the case that (3.9) holds.
To verify the asserted inequality in the case of (3.10), we return to (3.11) and note that due to w s ≥ 0 and m ∈ (0, 1), we have (nw s + 1) m ≤ n m w m s + 1 on (0, min{T, T max }). Here, we rely on (3.4) to conclude that (nw s + 1) m ≤ K m s − pm n + 1 in (0, min{T, T max }) and hence in (0, min{T, T max }) and for s 0 ∈ (0, R n ), where the conditions γ < 2 − 2 n − pm n and γ < 2 − 2 n contained in (3.10) together with s 0 − s ≤ s 0 entail The arguments for estimating the remaining integrals in (3.5) rely on the following relation between φ and ψ, which was also obtained in [49,Lemma 3.4]. Lemma 3.7. Let γ ∈ (0, 1). For every s 0 ∈ (0, R n ) and t ∈ (0, T max ), and the claim follows from Lemma 3.3.
Remark 3.10. For (JL), we can even obtain the stronger estimate in place of (3.14).
Combining these lemmata shows that φ(s 0 , ·) is indeed a supersolution to a superlinear ODE as long as s 0 is sufficiently small and γ can be chosen in a suitable way.

First conclusion: Proof of Theorem 1.1
While Lemmata 3.11 and 3.12 already show that φ(s 0 , ·) is (for certain s 0 and γ, at least) a supersolution to a superlinear ODE, we still need to show that φ(s 0 , 0) can be arranged to be suitably large. We take care of this last step in the following lemma; Theorem 1.1 will then be proven directly thereafter. We therefore can pick s 0 ∈ (0, s 1 ) so small that

Pointwise upper estimates for u: Proof of Theorem 1.2
The goal of this section is to prove Theorem 1.2. To that end, we first derive estimates of the form (1.10) both for (JL) (Lemma 5.1) and (PE) (Lemma 5.2) and then apply Theorem 1.1.