Single-point blow-up in the Cauchy problem for the higher-dimensional Keller–Segel system

The Cauchy problem in R n for the Keller–Segel system u t = Δ u − ∇ ⋅ ( u ∇ v ) , v t = Δ v − v + u , is considered for n ⩾ 3. Using a basic theory of local existence and maximal extensibility of classical and spatially integrable solutions as a starting point, the study provides a result on the occurrence of finite-time blow-up within considerably large sets of radially symmetric initial data, and moreover verifies that any such explosion exclusively occurs at the spatial origin. The detection of blow-up is accomplished by analyzing a relative of the well-known Keller–Segel energy inequality, involving a modification of the corresponding energy functional which, unlike the latter, can be seen to be favourably controlled from below by the corresponding dissipation rate through a certain functional inequality along trajectories.


Introduction
The Keller-Segel system, in its most prototypical version coupling two parabolic equations according to Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
plays an important role in the biomathematical literature, and its essential ingredients form the respective core in a growing number of increasingly complex macroscopic models for migration processes at virtually all conceivable length scales. With applications ranging from paradigmatic cell aggregation phenomena such as in populations of Dictyostelium discoideum or E. coli ([16]), over models for tumor cell invasion ( [3,20]) for virus hotspot formation ( [29]) or for socially interacting animal populations ( [30]), up to the description of large-scale evolution in spatial ecology ( [7,33]), its relevance seems closely connected with its ability to describe spontaneous emergence of spatial structures. In fact, already shortly after its proposal in the 1970s the model (1.1) was conjectured to support even the formation of singular structures in the mathematically extreme sense of nite-time blow-up for some solutions ( [25]; see also the historical remarks in [12]); however, rigorous analytical detections of such explosions were accomplished only in the 1990s, and throughout a signi cantly long further period remained limited to either certain parabolic-elliptic simpli cations of (1.1) ( [1,15,22,23]), to the construction of particular and possibly non-generic initial data enforcing blow-up ( [10]), or to statements on mere unboundedness without option to determine whether such phenomena indeed occur in nite or only in in nite time ( [13]). This seems to rather well re ect the circumstance that in contrast to typical objects of parabolic blow-up analysis such as quite thoroughly understood scalar reaction-diffusion equations with zero-order or rst-order superlinear sources ( [26]), directional effects of the driving cross-diffusive nonlinearity in (1.1) follow substantially more complex mechanisms and hence require accordingly subtle analysis.
Correspondingly, only in the recent few years some additional methodological developments fostered further progress in this eld. Here a rst branch of novel activities concentrates on a ne analysis of the dynamics near explicit singular steady states of the twodimensional version of (1.1) ( [28]), and hence on the one hand remains somewhat local with respect to the choice of initial data, but on the other hand can be considered quite constructive by namely providing considerable qualitative information on the asymptotic behaviour of the obtained solutions near their blow-up time. An independent second recent development, though more destructive in the sense of simply con rming the occurrence of explosions without signi cant further qualitative description, has been found capable of identifying large sets of initial data which lead to nite-time blow-up in Neumann problems for (1.1) in ndimensional balls Ω, in both cases n 3 ( [32]) and n = 2 ( [21]). In such situations, namely, the quantities along reasonably regular solution curves of (1.1), could be shown to satisfy certain inequalities of the form F 0 −C · (D θ 0 + 1) with some C > 0 and θ ∈ (0, 1) (1.5) throughout suitably large sets of radially symmetric functions (u, v) = (u(r), v(r)) over Ω which, in particular, include radial trajectories of (1.1) ( [21,32]).
As several subsequent studies have revealed, functional inequalities of the avour in (1.5) can be derived for signi cantly larger classes of expressions generalizing those in (1.2) and (1.3), and can thus be applied to corresponding Neumann problems for several generalizations of (1.1) ( [4,18,19]). In problems posed in the entire space R n , however, the use of inequalities in the form of (1.4) for blow-up detection so far seems limited to situations in which the corresponding energy functional is constituted by sums of integrals among which each can favourably be bounded in its negative part by the associated dissipation rate in the style of e.g. (1.5). In particular, in the Cauchy problem for (1.1) the quantities F 0 and D 0 from (1.2) and (1.3) in this sense apparently become inappropriate, because in domains with in nite measure the expression Ω u ln u need no longer be bounded from below along trajectories when the only a priori information available for the rst component thereof seems to reduce to an L 1 bound obtained through mass conservation. Due to a corresponding lack of suitable energy-based arguments, thus somewhat contrasting with the development of a small-data solution theory in the special case Ω = R 2 in which (1.4) can indeed be accompanied by moment control techniques to establish results on global solvability for sucritical-mass data ( [2]), already the problem of verifying the mere existence of some non-global solutions to (1.1) on Ω = R n accordingly seems open up to now.
As a further complication encountered when passing from bounded to unbounded domains, we note that beyond such a basic nding on global nonexistence, the detection of blow-up in the spirit of a genuine application-relevant aggregation moreover should most favourably be accompanied by some statement on appropriate explosion localization, at least excluding the possibility that the corresponding blow-up set be empty. As impressive caveats in this regard, we recall some classical precedents which report on the appearance of so-called 'blow-up at space in nity' phenomena already in some scalar parabolic problems ( [9,17,27]).
Main results. Henceforth concerned with (1.1) in Ω = R n , the present work accordingly addresses two objectives: a rst goal consists in making this problem accessible to virialtype methods of blow-up detection, such as those introduced in [4,32] for bounded domains, by analyzing the evolution of a relative of the functional in (1.2) which is no longer genuinely nonincreasing along trajectories, but the possible growth of which can adequately be controlled, and which moreover enjoys favourable lower bounds, thus inter alia allowing for functional inequalities of the form in (1.5). Hence set in the position to verify the occurrence of nite-time blow-up throughout considerably large sets of initial data, as a second purpose we will pursue the problem of determining the corresponding blow-up sets, and thereby not only exclude the possibility of blow-up at space in nity, but actually even make sure that blow-up exclusively occurs at the spatial origin at least in frameworks of radially symmetric solutions.
To be more precise, for n 3 we shall subsequently consider under the assumptions that with some q > n, where as usual, BUC(R n ) denotes the Banach space of all bounded and uniformly continuous functions on R n . In most places we will moreover require that u 0 and v 0 are radially symmetric with respect to x = 0. (1.8) Then an indispensable prerequisite not only for our analysis of the functional F below, but also for our basic qualitative description of blow-up given in (1.14) and (1.15), is constituted by the following result on local existence and uniqueness of smooth solutions enjoying appropriate spatial decay features. As we predominantly intend to make use of this in the context of non-global solutions, besides the uniqueness feature that will ensure radial symmetry whenever (1.8) holds, we particularly stress the practically quite convenient extensibility criterion (1.10) here. Thanks to the choice of a xed point setting somewhat different from precedent approaches both to two-and to higher-dimensional versions of (1.6) ( [1,2,5,6]), this criterion will, up to an additional minor argument on regularity implied by L ∞ bounds on u (lemma 2.7), actually result as a fairly straightforward by-product from our construction of local solutions (lemma 2.2); in view of its accordingly signi cant importance for theorem 1.2 below, for reasons of full rigorousness we include an essentially complete demonstration of proposition 1.1 in section 2, although neither its outcome nor its derivation bear any considerable surprise. We note that in its main part, it does not require the symmetry assumption (1.8), and we may note that it actually extends quite immediately to the case n 2 not further pursued in the sequel.
which solve (1.6) in the classical sense in R n × (0, T max ), and which are such that if T max < ∞, then lim sup Moreover, u > 0 and v > 0 in R n × (0, T max ), and we have Finally, if in addition (1.8) holds, then u(·, t) and v(·, t) are radially symmetric with respect to x = 0 for all t ∈ (0, T max ).
Now the core of our results restricts to the radial setting and, indeed, asserts occurrence of nite-time blow-up, localized at the spatial origin, within sets of initial data enjoying a certain density property. This will be achieved on the basis of an appropriate and rigorously veri able relative of the identity formally ful lled by smooth solutions of (1.6) which decay suitably fast in space, where According to a functional inequality favourably controlling R n uv in terms of the dissipated quantities in (1.12) along radial trajectories (section 4), thanks to the trivial fact that u ln(u + 1) is nonnegative it can be shown that F , along with some meaningful replacement of D 0 , satis es a lower estimate of the form in (1.5), and hence ensures nite-time blow-up for all radial initial data with suitably large negative energy (sections 5 and 6). Finally, a bootstrap-like regularity reasoning will reveal boundedness of any such non-global radial solution outside arbitrary neighbourhoods of the spatial origin (section 7). In summary, we will obtain the following statement on blow-up in which, as throughout the sequel, for R > 0 we abbreviate B R := B R (0) ⊂ R n . Theorem 1.2. Suppose that n 3 and that with some q > n the functions u 0 and v 0 satisfy (1.7) and (1.8) and are positive on R n . Then for any choice of p ∈ (1, 2n n+2 ) one can nd such that u 0j and v 0j are radially symmetric and positive for all j ∈ N, that (1.13) and that for each j ∈ N the associated classical solution (u j , v j ) of (1.6) from proposition 1.1 blows up in nite time at the spatial origin, in the sense that the corresponding maximal existence time T max,j > 0 actually satis es T max,j 1, that lim sup To prepare our construction of local-in-time solutions, for ϕ ∈ p∈ [1,∞] with G(z, t) := (4πt) − n 2 e − |z| 2 4t for z ∈ R n and t > 0. Then the following lemma collects some essentially well-known facts that can readily be derived using basic integrability and regularity properties of G, and a proof of which is thus omitted here.
(ii) Whenever 1 p q ∞ and ω ∈ N n 0 , one can nd C(p, q, ω) > 0 with the property that given any ϕ ∈ L p (R n ) we have and for all i ∈ {1, . . . , n}, Then the following statement on local existence of smooth solutions, along with a rst though not yet very convenient extensibility criterion, can be established by application of a contraction mapping argument. In view of our eventual goal to achieve even (1.10), the function space setting chosen here will differ from those underlying apparently all precedent relatives (see [1,5,6] and also [14], for instance).

Lemma 2.2.
Suppose that (1.7) holds with some q > n. Then there exist T max ∈ (0, ∞] and at least one classical solution (u, v) of (1.6) in R n × (0, T max ) ful lling (1.9) which is such that and v(·, t) = e t(∆−1) v 0 + t 0 e (t−s)(∆−1) u(·, s) ds in R n for all t ∈ (0, T max ), (2.2) and that if T max < ∞, then lim sup Proof. According to lemma 2.1, let us pick c 1 > 0, c 2 > 0 and c 3 > 0 such that and as well as where Then in the Banach space equipped with the norm · X de ned by letting we consider the closed set Since 0 t → e t∆ u 0 can easily be seen to belong to C 0 ([0, ∞); BUC(R n )) due to (1.7), two applications of lemma 2.1 (iii) then readily entail that Φ maps S into X. Moreover, recalling the well-known fact that e t∆ is nonexpansive on L p (R n ) for all p ∈ [1, ∞] and t > 0, for (u, v) ∈ S we can use the rst restriction contained in (2.7) to estimate because (2.4) clearly warrants that ∇ · e t∆ ϕ L ∞ (R n ) nc 1 t − 1 2 − n 2q ϕ L q (R n ) for all t > 0 and each ϕ ∈ L q (R n ; R n ).
Likewise, relying on (2.5) and the second requirement entailed by (2.7) we see that while the third implication of (2.7) guarantees that Since furthermore from (2.6) we know that due to the Hölder inequality and (2.7) we have and since, again by (2.5), the condition (2.7) furthermore warrants that from the de nition of · X it follows that In quite a similar manner, given (u, v) ∈ S and (u, v) ∈ S we can use (2.4) and the rst requirement in (2.8) to estimate whereas (2.5) together with the second restriction in (2.8) ensures that Apart from that, the three rightmost conditions contained in (2.8) imply that for (u, v) ∈ S and (u, v) ∈ S we have and, by (2.6), as well as according to (2.5). In summary, which combined with (2.10) enables us to invoke the Banach xed point theorem to nd an A standard argument (cf e.g. [11, lemma 3.3] for a detailed demonstration in a closely related setting) thereafter shows that actually (u, v) belongs to (C 2,1 (R n × (0, T)) 2 and, since ∇· and e t∆ commute on C 1 (R n ; R n ) ∩ L 1 (R n ; R n ), solves (1.6) classically in R n × (0, T), and from our de nition of T it becomes clear through another standard reasoning that (u, v) can be extended up to a maximal T max ∈ (0, ∞] ful lling (2.3), and that (2.1) and (2.2) actually hold throughout the entire interval (0, T max ).
As it directly refers to the integral identity (2.2) and to lemma 2.1, let us include the following basic integrability property of ∇v already here, although it will only be used in the course of our blow-up argument in lemma 4.1, and in the part identifying x = 0 as blow-up point (lemma 7.1).
Proof. To the integral identity (2.2), we only need to once again apply lemma 2.1 along with fact that e t∆ has Lipschitz constant 1 in L 1 (R n ). In view of (1.11), namely, this shows that with some c 1 > 0 we have from which (2.11) follows by niteness of ∞ 0 σ − 1 2 e −σ dσ.

Uniqueness
To prepare our subsequent localization arguments not only in this but also during the next sections, we x a nonincreasing cut-off function ξ ∈ C ∞ (R) ful lling ξ ≡ 1 in (−∞, 0] and ξ ≡ 0 in [1, ∞), and for R > 0 we let Then ζ R is radially symmetric about the origin, with ζ R ≡ 1 in B R and supp ζ R ⊂ B R+1 as well as supp ∇ζ R ⊂ B R+1 \B R , and moreover we have 0 ζ R 1 in R n .
A rst use of the family (ζ R ) R>0 enables us to conclude uniqueness of classical solutions within spaces of functions satisfying spatial decay conditions in the avour of those from lemma 2.2. (2.13) Proof. If (u, v) and (u, v) are two classical solutions in R n × (0, T) ful lling the above regularity assumptions, then w := u − u and z : so that with (ζ R ) R>0 as de ned in (2.12), for R > 0 we have according to the pointwise identity ∇z · ∇∆z = 1 2 ∆|∇z| 2 − |D 2 z| 2 and Young's inequality. Now in view of (2.13), the numbers c 1 := sup t∈(0,T) ∇v(·, t) L q (R n ) and c 2 := sup t∈(0,T) u(·, t) L ∞ (R n ) are nite, so that on the right-hand side of (2.14), using the Hölder inequality and the fact that 0 ζ R 1 we can further estimate where thanks to our assumption that q > n and hence 2q q−2 < 2n n−2 , we may employ the Gagliardo-Nirenberg inequality and Young's inequality to nd c 3 > 0 and c 4 > 0 such that Next, further applications of the Hölder inequality show that abbreviating On combining (2.14) with (2.15) and with (2.16)-(2.18), we hence infer that Since y R (0) = 0, an integration thereof yields ds for all t ∈ (0, T), and that thus But by de nition of (y R ) R>0 , (2.19) means that w(·, t) ≡ 0 and ∇z(·, t) ≡ 0 in R n for all t ∈ (0, T), and that hence (u, v) ≡ (u, v) in R n × (0, T).
we once more utilize a localization argument involving the functions from (2.12) to derive the following statement on positivity.
Proof. In view of the strong maximum principle and (1.7), it is suf cient to make sure that both u and v are nonnegative in R n × (0, T) for each T ∈ (0, T max ). To verify this, for R > 0 we take ζ R from (2.12) and use the continuous differentiability of R ∋ s → s 2 − , with s − := max{−s, 0} for s ∈ R, to see that thanks to Young's inequality, Here we may proceed similarly to the proof of lemma 2.4 in employing the Hölder inequality and the Gagliardo-Nirenberg inequality to see that since ∇v belongs to L ∞ ((0, T); L q (R n )), with some c 1 = c 1 (T) > 0 and c 2 = c 2 (T) > 0 and for all R > 0 we have As, furthermore, using (2.12) we can nd c 3 > 0 and c 4 = c 4 (T) > 0 such that for all R > 0, . When integrated over time, this entails that since y R (0) = 0, ds for all t ∈ (0, T) and each R > 0, whence observing that for all p ∈ (1, ∞) we have Having thereby asserted nonnegativity of u in R n × (0, T), we can make use this to see that once more due to Young's inequality, and hence completes the proof. Two further but now quite simple testing procedures involving (2.12) next allow for controlling the mass functionals of both components in quite an expected manner. Lemma 2.6. Assume (1.7) with some q > n. Then the solution (u, v) of (1.6) from lemma 2.2 enjoys the mass conservation property (1.11), and moreover we have for all t ∈ (0, T max ). (2.21) Proof. Again taking (ζ R ) R>0 from (2.12), we use (1.6) to see that for any T ∈ (0, T max ), and that thus (1.11) holds, for T ∈ (0, T max ) was arbitrary. Likewise, for xed T ∈ (0, T max ) the second equation in (1.6) implies that due to (1.11), by an ODE comparison argument. Noting that lemma 2.2 ensures that on taking R → ∞ we readily obtain (2.21) from this. With these preparations at hand, we can return to the mild formulation (2.2) to conclude that the extensibility criterion (2.3) can be re ned so as to actually reduce to (1.10).

Lemma 2.7.
Under the assumption that (1.7) is satis ed with some q > n, the solution of (1.6) from lemma 2.2 has the property that (1.10) holds.
Proof. Let us assume on the contrary that T max < ∞, but that there exists c 1 > 0 such that u(·, t) L ∞ (R n ) c 1 for all t ∈ (0, T max ). (2.22) Then since from lemma 2.6 and the nonnegativity of u we know that by using the Hölder inequality we see that for all t ∈ (0, T max ).
As lemma 2.2 ensures validity of (2.2), on applying lemma 2.1 (ii) to the latter identity we thus infer the existence of c 4 > 0 such that while lemma 2.1 (ii) in conjunction with (2.23) shows that with some c 5 > 0 we have Proof of proposition 1.1. Existence, uniqueness and validity of (2.1) and (2.2) have been found lemmas 2.2 and 2.4, whereas positivity of u and v have been asserted by lemma 2.5. Due to lemma 2.7, this solution satis es the re ned extensibility criterion (1.10), and the mass conservation identity (1.11) is part of the statement from lemma 2.6. Based on the uniqueness property, a standard argument thereupon reveals the claimed radial symmetry feature under the additional hypothesis (1.8).

A quasi-energy functional bounded from below by − R n uv
Next addressing the problem of detecting blow-up, motivated by (1.12) we shall perform one more localized testing procedure using (ζ R ) R>0 from (2.12) to achieve the following counterpart of (1.12) in which time differentiation is avoided due to possibly lacking regularity features, and in which unfavourable contributions have already been estimated in a convenient manner.
Lemma 3.1. Assume (1.7) for some q > n, and let and
Having thus asserted continuity of F = F (1) + F (2) + F (3) + F (4) as well as its claimed initial behaviour, we are left with the veri cation of (3.3). To accomplish this, for R > 0 we once more take ζ R as de ned in (2.12) and note that is evidently continuous on [0, T max ) and continuously differentiable on (0, T max ), and satis es Here integrating by parts and using Young's inequality shows that due to the second equation in (1.6), whereas from the rst equation in (1.6) we obtain that for all t ∈ (0, T max ). Using the identity

Dissipation controls superlinear powers of R n uv: a functional inequality along radial trajectories
Now inspired by the strategy from [32], as a key step toward revealing blow-up we shall establish a link between the negative contribution − R n uv to F and the dissipation rate functional D from (3.2). This will be achieved in lemma 4.5 below, and is to be prepared by four lemmata, each of which has quite a close relative in [32], but in the derivation of each of which we need to adequately account for the unboundedness of the domain on the one hand, and for the differences between (F , D) and (F 0 , D 0 ) from (1.2) and (1.3) on the other. Here and below, whenever convenient we shall without further explicit mentioning switch to the standard notation for functions radially symmetric about the origin, thus writing e.g. u = u(r, t) for r = |x| 0. We begin with a pointwise estimate for v gained upon combining lemma 2.3 with (2.21) and making essential use of radial symmetry. where Proof. We rst recall that due to lemmas 2.3 and 2.6 there exists c 1 > 0 such that whenever (1.7) and (1.8) hold, with K 1 as accordingly de ned by (4.2) we have ∞ 0 r n−1 |v r (r, t)|dr c 1 K for all t ∈ (0, T max ) (4.3) where the latter especially implies that for xed (u 0 , v 0 ) and each t ∈ (0, T max ) we can pick r 0 (t) ∈ [1, 2] ful lling v(r 0 (t), t) r n−1 0 (t)v(r 0 (t), t) = Now xing any such (u 0 , v 0 ), for t ∈ (0, T max ) and r ∈ (0, r 0 (t)] we can use (4.4) together with whereas if t ∈ (0, T max ) and r r 0 (t), then similarly v(r, t) K + r r 0 (t) |v r (ρ, t)|dρ because r 0 (t) 1. In combination, (4.5) and (4.6) yield (4.1). By means of the previous lemma, we can next apply a standard testing procedure to the second equation in (1.6), involving one speci c among the cut-off functions from (2.12), to relate the integral under consideration to a Dirichlet integral of v, up to a sublinear, and hence favourably small, power of D.
Proof. We take ζ := ζ 1 with ζ 1 as de ned in (2.12), and then observe that since lemma 4.1 provides c 1 > 0 ful lling v(r, t) c 1 K for all r 1 and t ∈ (0, T max ), (4.8) in the decomposition we may estimate according to (1.11). To appropriately handle the rst integral on the right of (4.9), we write f := − ∆v + v − u and test this de ning identity by ζ 2 v to see that due to Young's inequality and the Hölder inequality, because 0 ζ 1. Since supp ∇ζ ⊂ B 2 \B 1 , we may once again rely on (4.8) in estimating with c 2 := c 2 1 R n |∇ζ| 2 , while invoking the Gagliardo-Nirenberg inequality and again Young's inequality as well as (4.11) we can nd positive constants c 3 , c 4 , c 5 and c 6 such that for all t ∈ (0, T max ) (4.13) and that, similarly, (4.14) as due to lemma 2.6 we have R n ζv R n v K for all t ∈ (0, T max ). It thus only remains to insert (4.12) and (4.13) into (4.11) and combine the latter with (4.10) to infer (4.7) from (4.9).
To appropriately estimate the crucial contribution B 2 |∇v| 2 to the right-hand side of (4.7), we subdivide the ball appearing therein and rst concentrate on certain annuli with yet ex-ible radii. On multiplying the second equation in (1.6) by the positive but sublinear power v 1 2 of v, by means of another localization using (2.12) we can achieve the following estimate of the corresponding integral against small portions of our original target object, as well as two summands explicitly containing certain negative powers of the respective cutting radius.  7) and (1.8) hold with some q > n, then for any choice of r 0 ∈ (0, 2), the solution of (1.6) from lemma 2.2 satis es where again K is taken from (4.2).

Proof.
We once more abbreviate f := − ∆v + v − u, and we take ζ := ζ 2 with ζ 2 introduced in (2.12). Then multiplying the equation −∆v = u − v + f by ζ 2 v 1 2 , upon an integration by parts we obtain that thanks to Young's inequality and the nonnegativity of ζ 2 v 3 2 , for all t ∈ (0, T max ), where since supp ∇ζ ⊂ B 3 \B 2 , lemma 4.1 says that with some c 1 > 0 independent of (u 0 , v 0 ) we have for all t ∈ (0, T max ). for all x ∈ B 2 \B r 0 and t ∈ (0, T max ), and since thus for any choice of r 0 ∈ (0, 2) the inequality holds for all t ∈ (0, T max ), from (4.16) and (4.17) we infer that for all t ∈ (0, T max ) and each r 0 ∈ (0, 2), Here given ε > 0 we may again use Young's inequality to see that for all t ∈ (0, T max ) and r 0 ∈ (0, 2), (4.19) whereas employing the Cauchy-Schwarz inequality along with lemma 2.6 we nd that whenever r 0 ∈ (0, 2). We nally follow an idea from [32, lemma 4.4] in deriving an inequality for the associated inner Dirichlet integral in terms of, essentially, the product of D with a factor that contains a positive power of the dividing radius, and hence can be enforced to become conveniently small.

Blow-up of low-energy radial solutions
As a last ingredient for our analysis of the inequaliy (3.3), let us add a Gronwall-type statement on blow-up in an integral inequality that can be viewed as a counterpart of a superlinearly forced differential inequality.
We are thereby prepared to combine lemma 3.1 with lemma 4.5 in order to reveal a criterion on radial initial data as suf cient for nite-time blow-up: Lemma 5.2. There exist M > 0 and γ > 0 with the property that if for some q > n, u 0 and v 0 comply with (1.7) and (1.8) and are such that the corresponding solution of (1.6) satis es Proof. According to lemma 4.5, we can nd θ ∈ ( 1 2 , 1) and c 1 > 0 such that whenever (1.7) and (1.8) hold, taking D and K as de ned in (3.2) and (4.2) we have R n uv c 1 K 2 (D θ (t) + 1) for all t ∈ (0, T max ). (5.7) Using that θ < 1, we may employ Young's inequality here to nd c 2 > 0 such that for any such solution we moreover have and (5.11) Then assuming (1.7) and (1.8) to hold for some (u 0 , v 0 ) which is such that for the corresponding solution we have with K as accordingly de ned through (4.2), we claim that necessarily (5.6) must be valid. To verify this, we suppose for contradiction that T max > 1, and then rst observe that by the continuity property of F asserted by lemma 3.1, would be a well-de ned element of (0, 1], because since 2 1−θ > 2 and K 1, from (5.12) and (5.10) we especially know that F (0) < −(2c 1 + 4c 2 )K 2 1−θ < −2c 1 K 2 . To see that actually we note that the converse assumption t 0 < 1 implies that, again by continuity of F , F (t) < −2c 1 K 2 for all t ∈ (0, t 0 ) and F (t 0 ) = −2c 1 K 2 . (5.14) In view of lemma 3.1 and (5.8), this particularly entails that because t 0 1. Therefore, by (5.14), the nonnegativity of D, (5.12) and (5.10), which is absurd and hence con rms that indeed t 0 = 1. But since, on the other hand, by de nition (3.1) of F and (5.7) we have according to the fact that 1 −F (t) 2c 1 K 2 for all t ∈ (0, t 0 ), from (5.15) we infer that As, by (5.11), however, we may invoke lemma 5.1 to conclude from (5.16) that which is incompatible with (5.13) and thus shows that in fact our hypothesis that T max > 1 must have been wrong, and that hence the claimed implication holds with M := c 4 and γ := 2 1−θ .

A density property of blow-up enforcing radial data
In order to complete our argument ensuring blow-up within large sets of initial data, we now only need to resort to a known and essentially explicit construction of explosion-enforcing initial data in arbitrarily small neighbourhoods of any prescribed pair of positive functions ful lling (1.7) and (1.8).

Localization of blow-up points: bounds for u outside the origin
Our nal objective consists in establishing the inequality (1.15) for a radial solution that is already known to blow up within nite time. Although our nal result in this direction will essentially parallel knowledge on behaviour in Neumann problems for (1.1) on planar disks ( [24]), its derivation here will need to considerably deviate from that in the corresponding precedent, inter alia due to unboundedness of the physical domain. To accomplish uniform bounds outside the origin through several steps on the basis of arguments from parabolic regularity theory, for localization procedures different from those previously performed let us choose a family (χ δ ) δ∈(0,1) of cut-off functions χ δ ∈ C ∞ 0 (R) ful lling 0 χ δ 1 on R as well as χ δ ≡ 1 on [0, 1] and supp χ δ ⊂ (− δ 2 , 2) for all δ ∈ (0, 1), and let for δ ∈ (0, 1) and R > δ. Then , R + 2 for all δ ∈ (0, 1) and R > δ, (7.2) and furthermore sup R>δ χ ′ δR L ∞ (R) + χ ′′ δR L ∞ (R) < ∞ for each xed δ ∈ (0, 1).
Lemma 7.1. Assume that (1.7) and (1.8) hold with some q > n, and that T max < ∞. Then for all p > 1 and each δ ∈ (0, 1) there exists C(p, δ) > 0 such that the solution of (1.6) from lemma 2.2 satis es R+1 R |v r (r, t)| p dr C(p, δ) for all t ∈ 1 2 T max , T max and R > δ. (7.9) Proof. We rst recall lemma 2.3 to x c 1 > 0 such that Given δ ∈ (0, 1), we thereby see that In view of (7.3) and the second relation in (7.2), we thus readily obtain c 4 (δ) > 0 such that the accordingly de ned function b from (7.8), extended by zero to all of R × (0, T max ) if necessary, satis es b(·, t) L 1 (J R ) c 4 (δ) for all ∈ (0, T max ) and R > δ, (7.11) where we have set J R := (R − 1 2 , R + 2). To make appropriate use of this, we note that by translation invariance of the one-dimensional heat equation and known smoothing properties of the Neumann heat semigroup (e t∆ J ) t 0 over open bounded intervals J ⊂ R ([31]), given p > 1 we can x c 5 (p, δ) > 0 such that for any choice of R ∈ R, ∂ r e t∆ J R ϕ L p (J R ) c 5 (p, δ)t −1+ 1 2p ϕ L 1 (J R ) for all t > 0 and any ϕ ∈ L 1 (J R ). (7.12) Then using that with χ = χ δR , and once more with a trivial extension if appropriate, we have (χv) r = 0 on ∂J R by (7.2), we may apply (7.12) to a variation-of-constants representation associated with (7.5) to see that again thanks to (7.10), max for all t ∈ ( 1 2 T max , T max ) and R > δ. As (χv) r ≡ v r in (R, R + 1) × (0, T max ) ⊂ J R × (0, T max ) by (7.2), this establishes (7.9).
By tracking suitably localized versions of u p 0 with some sublinear p 0 > 0, as a rst consequence of lemma 7.1 we can derive an integrability property of u involving arbitrary subcubic powers.