Blow-up profiles in quasilinear fully parabolic Keller–Segel systems

We examine finite-time blow-up solutions to in a ball , , where D and S generalize the functions with . We show that if as well as and is a nonnegative, radially symmetric classical solution to () blowing up at , then there exists a so-called blow-up profile satisfying Moreover, for all with we can find such that for all .


Introduction
The possibility of (finite-time) blow-up constitutes one of the most striking features of the quasilinear system in Ω × (0, T), proposed by Keller and Segel [17] to model chemotaxis, that is, the directed movement of bacteria or cells towards a chemical signal, and attracting interest of mathematicians for nearly half a century (see for instance [2] for a recent survey).
Therein Ω ⊂ R n , n ∈ N, is a smooth, bounded domain, T ∈ (0 For these selections, namely, solutions blowing up in finite time have been constructed in two- [11] and higher- [37] dimensional balls. On the other hand, if n = 1 [29], if n = 2 and Ω u 0 < 4π (or Ω u 0 < 8π in the radially symmetric setting) [28] or if n 3 and u 0 L n 2 (Ω) + v 0 W 1,n (Ω) is sufficiently small [3], all solutions are global in time and remain bounded. We should also note that if one replaces the second equation in (1.1) by a suitable elliptic counterpart, finite-time blow-up results have been achieved already in the 1990s [10,15,26].
Motivated inter alia by the desire to model volume-filling effects, it has been suggested to consider certain nonlinear functions D ≡ D(u) and S ≡ S(u) instead [12,30,41] and, in order to account for immotility in absence of bacteria [9,21] or receptor-binding and saturation effects [12,16], one might also (need to) choose functions D and S explicitly depending on v.
For the sake of exposition, we will for now confine ourselves with the choices D(u, v) = (u + 1) m−1 and S(u, v) = u(u + 1) q−1 for certain m, q ∈ R, but remark that all the works cited below allow for more general functions D and S as well. From a mathematical point of view, these are the most prototypical choices, as they generalize D ≡ 1, S(u, v) = u, which are obtained upon setting m = q = 1, and since estimates of the form D u m−1 , |S| u q , u 1, come in handy at several places (see for instance the proofs of the present article). Moreover, even these prototypical functions directly appear in biologically motivated models; by choosing m > 1 and q = 1 we arrive at (a nondegenerate version of) system (M5) in [12] while the choices m = 1 and q = 0 lead to model (M3b) in [12].
Regarding the question of global-in-time boundedness, the number n−2 n is critical: if Ω ⊂ R n , n ∈ N, is a smooth, bounded domain and m − q > n−2 n , then all solutions to (1.1) are global in time and bounded [13,14,34]. Conversely, if Ω ⊂ R n , n 2, is a ball and m − q < n−2 n , there exist initial data such that the corresponding solution blows up in either finite or infinite time [13,35].
If in addition to m − q < n−2 n one assumes n 3 as well as either m 1 (and hence q > 2 n > 0) or m ∈ R and q 1, finite-time blow-up is possible [5][6][7], while for q 0 solutions are always global in time [38]. Whether solutions may blow up in finite time given m − q < n−2 2 and q > 0 but q < 1 or m < 1 is, to the best of our knowledge, still an open question.
The picture is more complete if one replaces the second equation in (1.1) with a suitable elliptic equation. Again solutions are global and bounded provided that m − q > n−2 2 and in the radial symmetric setting there exist unbounded solutions if m − q < n−2 2 . Additionally, it is known for which parameters finite-time blow-up may occur: if q 0, these solutions are always global, while for q > 0 finite-time blow-up is possible [19,40]. An obvious conjecture, stated for instance in [38], is that the same holds true for the fully parabolic system (1.1).
A natural next step is to examine the qualitative behavior of (finite-or infinite-time) blowup solutions in more detail. While far from exhaustive, some results in this regard have been obtained for the classical Keller-Segel system, that is, for D ≡ 1 and S(u, v) = u.
In the two-dimensional settings some blow-up solutions collapse to a Dirac-type singularity (see [11,27] or also [32] for similar results for the parabolic-elliptic case). Additionally, for all n 2, temporal blow-up rates (even for S(u, v) = u q , q ∈ (0, 2)) have been established [24] and it is known that {u n 2 (·, t) : t ∈ (0, T max )} cannot be equi-integrable, where T max denotes the blow-up time [4].
Quite recently, the questions whether spatial blow-up profiles exist, that is, whether U := lim t Tmax u(·, t), T max again denoting the blow-up time, is meaningful in some sense, and, if this is indeed the case, properties of U have been studied.
Choosing Ω to be a ball in two or more dimensions, D ≡ 1 and S(u, v) = u, it has been shown in [39] that for all nonnegative, radially symmetric solutions blowing up at T max < ∞ there exists a blow-up profile U in the sense that u(·, t) → U in C 2 loc (Ω \ {0}) as t T max . Moreover, an upper estimate is available for U: for any η > 0 one can find C > 0 with If one simplifies (1.1) by not only setting D ≡ 1 and S(u, v) = u but also replacing the second equation therein with 0 = ∆v − 1 |Ω| Ω u 0 + u, more detailed information is available. In [33], the authors consider Ω := B R (0) ⊂ R n , R > 0, n 3, and construct a large class of initial data for which the corresponding solutions (u, v) blow up in finite time. The blow-up profile U := lim t Tmax u(·, t) exists pointwise and U(x) C|x| −2 for all x ∈ Ω holds for some C > 0, wherein the exponent 2 is optimal. Furthermore, the same paper also provides certain lower bounds for U.
Up to now, however, in the case of nonlinear diffusion there seems to be nearly no information available regarding behavior of finite-time blow-up solutions to (1.1) at their blow-up time. The present paper aims to be a first step towards closing this gap.

Main results
At first, we will deal with (a slight generalization of) the first sub-problem in (1.1) and derive pointwise estimates for its solutions.
Then for any we can find C > 0 with the following property: and is a classical solution of in Ω, for all ρ > 0 and as well as (1.11) as integrating the PDI in (1.5) over Ω and integrating by parts (all boundary terms are nonpositive because of the second condition in (1.5)) assert Ω u(·, t) Ω u 0 for all t ∈ (0, T max ). As a second step, we then apply this result to radially symmetric solutions to (1.1) and obtain Theorem 1.3. Let n 2, R > 0 and Ω := B R (0) as well as (1.14) For any and any β > n − 1, there exists C > 0 with the following property: let T ∈ (0, ∞]. Any nonnegative and radially symmetric classical  (i) Let us briefly discuss the conditions in (1.14). On the one hand, observe that m − q − 1 n implies α ∞. On the other hand, [34] proves that all solutions to (1.1) for a large class of functions D, S are global in time and bounded, provided m, q ∈ R satisfy m − q > n−2 n . In both cases a statement of the form (1.12) would not be very interesting. (However, for m − q > n−2 n the statement still holds if one sets α := n because if (1.9) is fulfilled for some q ∈ R then also for all larger q.) The second condition in (1.14), however, is purely needed for technical reasons and we conjecture that theorem 1.3 holds even without this restriction, albeit the constant C may then depend on T as well. (ii) In [8, corollary 2.3], it has been shown that (1.12) cannot hold for any As m − q < n−2 n implies α > n > α, we do not know whether (1.15) is in general optimal. However, in the case of m − q = n−2 n (and m > n−2 n ) we have α = n = α, hence at least in this extremal case the condition α > α is, up to equality, optimal.
The third and final step will then consist of proving that lim t Tmax u(·, t) and lim t Tmax v(·, t) exist in an appropriate sense provided the diffusion mechanism in the first equation in (1.1) is nondegenerate. Theorem 1.5. Let n 2, R > 0, Ω := B R (0) and suppose that the parameters in (1.13) and the functions in (1.16) comply with (1.11), (1.14) and (1.17)- (1.20). Furthermore, suppose also that there is (1.21) Then for any nonnegative and radially symmetric classical solution (u, v) blowing up in finite time in the sense that there is T max < ∞ such that Moreover, for any α > α (with α as in (1.15)) and any β > n − 1 we can find C > 0 with the property that (1.23) Remark 1.6. Obviously, theorem 1.5 is only of interest if, given S and D, there are indeed initial data leading to finite-time blow-up. Therefore, we stress that, for instance, the choices D(ρ, σ) := (ρ + 1) m−1 and S(ρ, σ) := ρ(ρ + 1) q−1 for ρ, σ 0 and m ∈ R, q 0 satisfying (1.14) as well as q 1 or m 1 not only comply with (1.16)-(1.19) and (1.21) for certain parameters but also allow for finite-time blow-up [5,7]. That is, there exist initial data Moreover, let us emphasize that our results can indeed be applied to models stemming from a biological motivation, for instance to (a nondegenerate version of) the system (M5) in [12], that is, to (1.1) with m > 1 and q = 0. Furthermore, even the degenerate version thereof is covered by theorem 1.3. Remark 1.7. As a final remark, let us point out that theorem 1.5 includes the result in [39, corollary 1.4], as in the case of m = 1 and q = 1 we have α = n(n − 1).

Plan of the paper
The reasoning from [39], where estimates on blow-up profiles to solutions to (1.1) with D ≡ 1 and S(u, v) = u have been derived, is to consider w := ζ α u with ζ(x) ≈ |x| and to make use of semi-group arguments as well as L p -L q estimates in order to derive an L ∞ bound for w which in turn implies the desired estimate of the form (1.12) for u. However, through their mere nature, these methods are evidently inadequate to handle equations with nonlinear diffusion.
The present paper is built upon the belief that, generally, an iterative testing procedure should be as strong as semi-group arguments. While the latter method may be quite elegant, the former has the distinct advantage of being applicable not only to equations with linear diffusion but also to (1.5).
Indeed, iteratively testing with w pj−1 for certain 1 p j ∞ allows us to obtain an L ∞ bound for w at the end of section 2-provided the critical assumption (1.3) is fulfilled.
Applying theorem 1.1 to solutions of (1.1) mainly consists of adequately estimating f := −∇v. To that end we may basically rely on the results in [39]. It probably should also be noted that this is the only part where we explicitly make use of the radially symmetric setting.
Finally, the existence of blow-up profiles is shown in section 4 by considering global solutions (u ε , v ε ), ε ∈ (0, 1), to suitably approximative problems which converge (along a subsequence) on all compact sets in We then prove that these functions coincide which u and v on Ω × [0, T max ) such that we may set U := u(·, T max ) as well as V := v(·, T max ) and make use of regularity of u and v.
In order to identify ( u, v) with (u, v) we crucially need uniqueness of solutions to (1.1) which we show in lemma A.1-provided that the first equation is nondegenerate. As this might potentially be of independent interest, we choose to prove uniqueness for a class of systems slightly generalizing (1.1).

Pointwise estimates for subsolutions to parabolic equations in divergence form
Unless otherwise stated, we assume throughout this section that Ω ⊂ R n , n 2, is a smooth, bounded domain with 0 ∈ Ω, set R := sup x∈Ω |x| and suppose that the parameters (all henceforth fixed) in (1.1) as well as α comply with (1.2) and (1.3). Moreover, we may also assume since whenever (1.10) is fulfilled for some β > 0, then also for all β > β (provided one replaces K f by max{R, 1}β −β K f ). In order to simplify the notation, we also fix T ∈ (0, ∞] and functions in (1.6) satisfying (1.4) and (1.7)-(1.11) as well as a nonnegative classical solution u ∈ C 0 (Ω × [0, T)) ∩ C 2,1 (Ω × (0, T)) of (1.5), but emphasize that all constants below only depend on the parameters in (1.1) as well as on α.
Our goal, which will be achieved in lemma 2.10 below, is to prove an L ∞ bound for the function which in turn directly implies the desired estimate (1.12).
To this end, we will rely on a testing procedure to obtain L p bounds for all p ∈ (1, ∞). Due to an iteration technique, this will then be improved to an L ∞ bound-hence the constants in the following proofs need also to be independent of p .
In order to prepare said testing procedure, we will need for all ρ 0 and some K g > 0.

Another important ingredient will be
Proof. This is an immediate consequence of (2.2) and (1.4). □ As further preparation, we state a quantitative Ehrling-type lemma. Since this will be also used in the proof of the quite general lemma A.1 below we neither require n 2 nor 0 ∈ Ω.
Hence we may invoke the Gagliardo-Nirenberg inequality (which holds even for r, s ∈ (0, 1), see for instance [22, lemma 2.3]) to obtain c 1 > 0 with the property that Therein we have by Young's inequality (with exponents 1 a , 1 1−a ) for all ε ∈ (0, 1) and all ϕ ∈ W 1,2 (Ω) This already implies the statement for C := c 1 (1 + c 2 ). □ In order to be able to apply lemma 2.5, we first rewrite the dissipative term in (2.5).

Lemma 2.6.
There are c 1 , c 2 > 0 and p 0 1 such that for all p p 0 we have for all t ∈ (0, T) and
Herein we may now finally apply lemma 2.5 together with Young's inequality to obtain c 4 > 0 such that By combining (2.20), (2.22)-(2.24) and lemma 2.7 (with ε = c1 2 ) we may find c 5 > 0 such that The assumption s Proof. Let p 0 > 1 and s 0 > 0 be as in lemma 2.8. By Hölder's inequality we may without loss of generality assume that p M Fuest Nonlinearity 33 (2020) 2306 in (0, T) by Hölder's inequality and lemma 2.6, we may apply lemma 2.8 to obtain for all j ∈ N.
Suppose first that there is a strictly increasing sequence ( j k ) k∈N ⊂ N such that A j k c 2 for all k ∈ N. As then w(·, t) L ∞ (Ω) = lim k→∞ w(·, t) L p j k (Ω) c 2 for all t ∈ (0, T) since lim k→∞ p j k = ∞ by (2.31), this already implies (2.27) for C := c 2 .
Hence, suppose now that on the contrary there is j 0 ∈ N such that A j > c 2 for all j j 0 . Since then also A j 1 for all j j 0 and because of pj s > 1 for all j ∈ N 0 , we conclude from (2.33) that for all j > j 0 .

As (2.31) entails
pj−1 pjs where c 2 := 2c 1 , and hence by induction and (2.32) which in turn directly implies the statement. □ The main result of this section now follows immediately.

Pointwise estimates in quasilinear Keller-Segel systems
Suppose henceforth that n 2, R > 0 and Ω := B R (0). In order to apply theorem 1.1 to the system (1.1)-and hence prove theorem 1.3-we need some integrability information about ∇v. This is provided by

Then any classical, radially symmetric solution
Proof. See [39, lemma 3.4]. □ We are now indeed able to employ theorem 1.1 in order to obtain pointwise estimates for solutions to systems slightly more general than (1.1). (The generality is needed as the following lemma will be used not only to prove theorem 1.3 but also in in the proof of lemma 4.3 below.) Lemma 3.2. Suppose that the parameters in (1.13) comply with (1.14) and set K g > 0. Then for any α > α, with α as in (1.15), and any β > n − 1, there exists C > 0 with the following property: Given functions in (1.16) and g ∈ C 0 ([0, ∞)) complying with (1.11), (1.17)- (1.20) and any nonnegative and radially symmetric classical fulfills (1.12) and |∇v(x, t)| C|x| −β for x ∈ Ω and t ∈ (0, T).
Proof. We fix such a solution (u, v) and functions in (1.16) as well as g ∈ C 0 ([0, ∞)), but emphasize that all constants below only depend on the parameters in (1.13) as well as on K g , α and β.

Existence of blow-up profiles
Throughout this section we suppose n 2, R > 0, Ω := B R (0), and that (1.11) and (1.17)- (1.19) are fulfilled for certain parameters and functions in (1.13) and (1.16), respectively. In addition-and in contrast to the preceding sections-we will also assume (1.21), that is, that D η, for some η > 0. Furthermore, fix T max < ∞ and a solution (u, v) to (1.1) (with T max instead of T) with the property lim sup t Tmax u(·, t) L ∞ (Ω) = ∞.
Proof. Local existence and extensibility can be proved as in [20, lemma 2.1-2.4] which essentially relies on regularity theory for nondegenerate parabolic equations and Schauder's fixed point theorem-while nonnegativity follows by the maximum principle. □ For all ε ∈ (0, 1) fix henceforth u ε , v ε and T max,ε as given by lemma 4.1. By quite standard methods we see that the regularized solutions are global in time. Proof. Since G ε is bounded, L p -L q estimates (see [36,  Testing the first equation in (4.1) with u p−1 ε , p > 2, gives 1 Because of u, v ∈ C 0 ([0, T max ]; C 2 loc (Ω \ {0})) a consequence thereof is (1.22) if we set U := u(·, T max ) and V := v(·, T max ). Finally, (1.23) follows by theorem 1.3. □ in Ω × (0, T ), testing with u 1 − u 2 and integrating by parts gives =: I 1 + I 2 + I 3 + I 4 + I 5 in (0, T ). Therein we make first use of the nondegeneracy, that is, the crucial assumption that D η, to see that Also, by Young's inequality Thus, by Young's and Hölder's inequalities (with exponents p 2 , p p−2 ) L ∞ ((0,T );W 1,p (Ω)) . As our assumptions on p imply r := 2p p−2 < 2n (n−2)+ , we may invoke lemma 2.5 to find c 4 > 0 with the property that