GLOBAL EXISTENCE AND BOUNDEDNESS IN A CHEMOREPULSION SYSTEM WITH SUPERLINEAR DIFFUSION

. In a bounded domain Ω ⊂ R n , where n ≥ 3, we consider the quasilinear parabolic-parabolic Keller-Segel system (cid:40) u t = ∇ · ( D ( u ) ∇ u + u ∇ v ) in Ω × (0 , ∞ ) v t = ∆ v − v + u in Ω × (0 , ∞ ) with homogeneous Neumann boundary conditions. We will ﬁnd that the con- dition D ( u ) ≥ Cu m − 1 suﬃces to prove the uniqueness and global existence of solutions along with their boundedness if D (0) > 0 and m > 1 + ( n − 2)( n − 1) n 2 which is a very diﬀerent result from what we know about the same system with nonnegative sensitivity. In the case of degenerate diﬀusion ( D (0) = 0) and for the same parameters, locally bounded global weak solutions will be established.

1. Introduction. The system considered in this work is, in its most general form, represented by u t = ∇ · (D(u)∇u) − ∇ · (S(u)∇v) in Ω × (0, ∞), v t = ∆v − v + u in Ω × (0, ∞), for a bounded domain Ω ⊂ R n , n ≥ 3, with smooth boundary, homogeneous Neumann boundary conditions and prescribed initial data. The biological phenomenon this class of systems of differential equations is used to describe is called chemotaxis, the movement of cells which is controlled by a chemical substance produced by the same cells. The most basic setting, proposed by Keller and Segel in 1970 ([16]), considers D ≡ 1 and S(u) = u and even there the detection of blow-up, the question whether it arises in finite or infinite time or, on the other hand, verifying the global existence and boundedness of solutions are nontrivial tasks. The only exception is the case n = 1 where no blow-up occurs at all ( [26]). The behaviour in two dimensions is more intricate: If the initial mass Ω u 0 is less than 4π, then every solution is bounded according to [10] and [24]. In higher dimensions a smallness condition on u 0 L same. On the other hand, if the initial data is large, solutions can be found which explode either in finite or infinite time ( [14]). In order to gain additional knowledge one can specify Ω to be a ball with radially symmetric functions. Then for n = 2 and Ω u 0 < 8π solutions are global and bounded ( [24]) while for initial data above this threshold [11] and [23] have detected blow-up in finite time. According to [34], for n ≥ 3 no such threshold for the initial mass exists and blow-up solutions can be constructed whenever Ω u 0 > 0. There are many potential choices for D and S, for a vast overview [1] is recommended. One approach is to link both functions via some Q ∈ C 2 ([0, ∞)) which describes the probability of a cell at (x, t) to find space nearby and the relations D(u) = Q(u) − uQ (u) and S(u) = uQ (u). [12] considers a decreasing function with decay at large densities as the best fit while in [37] an overview of hydrodynamic approaches or those involving cellular Potts models is given. To incorporate additional properties of the biological reality, one or both of the functions are often equipped with a signal-dependency: [33], [13] and [20] write S(u, v) to model saturation effects or activation thresholds for the cross-diffusion. Similar ideas for D are discussed in [8], [21], [29] and [30]. We now want to consider a more specific system, namely where D(u) ≥ C D u m−1 with some C D > 0 and m ≥ 1 and we point out the crucial difference the changed sign in the first equation will make for n ≥ 3. With D as here and the additional condition D(0) > 0 as well as S(u) = u, m > 1 + n − 2 n (m 0 ) has been identified as a crucial relation in (S 0 ): if it holds, then [32] shows the global existence and boundedness of solutions while for m below this threshold (see [35]) the following holds true: If Ω = B R (0) ⊂ R n , where n ≥ 2, then for any M > 0 we find T ∈ (0, ∞] and a radially symmetric solution (u, v) in Ω × (0, T ) such that Ω u ≡ M but u is unbounded in Ω × (0, T ). Such results can be extended (see [32], [17] and [4] as well as [27] and [15]) to more general choices of D and S as S(s) D(s) ≥ Cs α for some C > 0, α > 2 n and all s ≥ 1. For D(u) = (u + 1) m−1 and S(u) = (u + 1) κ−1 , [5] and [6] have determined the point of time at which blow-up occurs: If both m ≥ 1 and κ ≥ 1, then blow-up happens in finite time. On the other hand, if m < κ + n−2 n and κ < m 2 − n−2 2n , then solutions exist globally but we have lim sup t→∞ u(·, t) L ∞ (Ω) = ∞. The positive sensitivity in (S), resulting in repulsion instead of attraction, promotes global existence and boundedness of solutions, especially for m = 1 we already have relevant results: If n = 2, global solutions and their boundedness have been established while for n ∈ {3, 4} locally bounded global weak solutions have been found (both in [7]). Additionally we also have a result for nonlinear sensitivity: [31] has found uniform-in-time bounds for classical solutions and convergence to the average of the initial mass. In this work we want to consider the case of superlinear diffusion and to achieve a condition similar to (m 0 ). To this end we begin by proving Theorem 1.1. We assume that we have some positive constants C D and C D as well as some M ≥ m > 1 + (n−2)(n−1) n 2 and a function D ∈ C 1 ([0, ∞)) with for all s ≥ 0. Then for any u 0 ∈ C 0 (Ω) and v 0 ∈ C 1 (Ω) the system (S) has a unique classical solution. This solution is global and bounded in the sense that there is some C > 0 with Moreover, for the case of degenerate diffusion, this result can be used to detect global weak solutions that are locally bounded using the fact that D vanishing at zero influences the construction of solutions but not the size of the bounds derived in this work: Theorem 1.2. We assume that we have some positive constants C D and C D as well as some M ≥ m > 1 + (n−2)(n−1) for all s ≥ 0. Then for any u 0 ∈ L 1 (Ω) and v 0 ∈ L 1 (Ω) we find a locally bounded global weak solution to(S 0 ) in the sense of definition 4.1.
Note that theorems 1.1 and 1.2 could be modified to incorporate the case n = 2 with the expected condition M ≥ m > 1. However, this is the same bound as the one for the chemoattraction case discussed in [32] and one can easily check that the proof therein covers the two-dimensional version of our system as well -only in higher dimensions do we see the effect of the different sign in the first equation of (S). Alternatively, very simple changes to lemma 3.4 can be made to include n = 2, but since nothing substantial could be gained this way, for the sake of clarity and readability we are only considering n ≥ 3.
2. Helpful lemmata. Before we present our findings, let us collect some known results that have been established before and also verify general estimates that will help with other upcoming proofs. Of central importance to us are the well-known Gagliardo-Nirenberg inequality (see [25] for example) and a variant for fractional Sobolev spaces (see Section III in [2]).
(n−p)+ for p ≤ n and r ∈ (0, p). Then there are some C > 0 and a constant a ∈ (0, 1) with for all w ∈ C 1 (Ω) and a is given by Lemma 2.2 (Gagliardo-Nirenberg inequality for fractional Sobolev spaces). For fixed r ∈ 0, 1 2 there are C > 0 and a ∈ (0, 1) as in lemma 2.1 such that w holds for any w ∈ C 1 Ω .
Next and in preparation for the discussion of the case of degenerate diffusion we cite a version of the Aubin-Lions lemma which can be found as a corollary in chapter 8 of [28].
Furthermore, there is a simple consequence of Young's inequality that will help us in utilising Hölder's inequality. It can in fact be proven by consecutively employing Young's inequality multiple times.
Finally, we want to use integration by parts to tackle certain integrals involving the second component of a solution to (S), but for nonconvex domains ∂w ∂ν ∂Ω = 0 is not sufficient to deduce nonpositivity of ∂|∇w| 2 ∂ν on the boundary of Ω. Hence, the following lemma will be introduced.
3. Uniform boundedness of functions solving (S). In this part we will prove that any classical solution (u, v) to our system (S) is uniformly bounded during its entire existence time. Furthermore, the bounds we find do not depend on D(0) which enables us to utilise these results in an upcoming approximation process. Under the overarching condition which is less strict than (m 0 ), we shall assume that we have been given some time T ∈ (0, ∞] and a pair of classical solutions to (S) in Ω × (0, T ). Without stating so in every lemma, none of the arising constants will depend on T . The main result of this section accordingly reads as follows: When dicussing chemoattraction systems, initial steps for the regularity of any solution often consist of proving L 1 -boundedness and then using L p -L q -estimates to show v ∈ W 1,q (Ω) for any q ∈ (1, n n−1 ). Here however, the small difference in the first equation enables us to go even further as was first shown by [7].
Proof. While the first part can be seen upon computing d dt Ω u = 0, the second half follows from x ln x ≥ − 1 e for x > 0 and d dt which holds for all t ∈ (0, T ).
It is our goal to prove uniform boundedness of both components of any solution (u, v) to (S) and the next step on this way is concerned with a higher regularity for u. To prepare for this we prove holds for all t ∈ (0, T ).
Proof. Using our estimate for D, integration by parts and Young's inequality as well as lemma 3.2, we find positive constants C 1 and and therefore, together with some C 3 > 0 given by lemma 2.5, holds for all t ∈ (0, T ). Utilising |∆v| 2 ≤ n|D 2 v| 2 , integration by parts and Young's inequality we see for all t ∈ (0, T ). Adding this to the previous result completes the proof.
In the next lemma, which is also the source of our restrictions for m and n, we will fix several parameters which combined with the previous result will show that u belongs to L p (Ω) for any finite p.
In the converse case we choose q 1 ≥q such that nq n(q − 1) + 2 < n n − 2 holds for all q ≥ q 1 and conclude as before with some θ ∈ nq1 n(q1−1)+2 , n n−2 . Setting p 2 := max {p 1 , n − 1 − m} we fix µ ∈ n 2 , n n−2 m+p2−1 2 and accordingly we have (µ 1 ) as well as (µ 2 ) for any choices of p ≥ p 2 and q ≥ q 2 := max{q 1 , 2}. Following the definition p 3 := max {p 1 , n + 1}, for p ≥ p 3 we consider and, possibly after another adjustment, we identify p 4 ≥ p 3 such thatq(p) > q 2 holds for every p ≥ p 4 . Now, for p ≥ p 4 and q ∈ (q 2 ,q(p)) we consider For fixed p we quickly see shows (σ 2 ) for p ≥ p 4 and q ∈ (q 2 ,q(p)). On the other hand, for p ≥ p 4 we see which tends to 2 n as p → ∞. The derivative of the latter term with respect to p is and due to our additional condition θ > n−2 n 1 2m+ 2 n −3 which we introduced at the beginning of the proof as well as a rough estimate for µ > n 2 it is positive. Therefore picking any p ≥ p 4 and some large q ∈ (q 2 ,q(p)) ensures that both (σ 1 ) and (σ 2 ) hold while the other four properties remain intact.
As announced, we now use these parameters together with lemma 3.3 in order to obtain a useful regularity result for u. and Ω u 2µ and Ω |∇v| 2(q−1)µ for all t ∈ (0, T ). Firstly we have some where we have used (θ 1 ) and where a is given by which leads to a = n(m + p − 1) 2(−m + p + 1) Since (µ 1 ), we also have . With respect to the integrals containing v we use which utilises (µ 2 ). In a next step we use the boundedness of u(·, t) L 1 (Ω) and ∇v(·, t) L 2 (Ω) as provided by lemma 3.2, so that lemma 2.4 gives us some positive constant C 2 with d dt for all t ∈ (0, T ). Similarly to before, the Gagliardo-Nirenberg inequality allows for the comparison of the occurent terms. We see for all t ∈ (0, T ) and this gives us some positive constants κ and C 4 and the ordinary differential inequalityẏ for y(t) := Ω u p (·, t) + Ω |∇v(·, t)| 2q . A comparison argument shows for all t ∈ (0, T ) which completes the proof.
From this and L p -L q -estimates we can deduce uniform boundedness of v and its gradient similarly to results in section 3 in [9]: Lemma 3.6. There is a positive constant C with v(·, t) W 1,∞ (Ω) ≤ C for all t ∈ (0, T ).
Together with the representation above we therefore see dτ for every t ∈ (0, T ) and the integral on the right-hand side is finite. We can estimate similarly for the norm of the derivative of v: dτ holds for all t ∈ (0, T ) and here, too, the right-hand side is a positive number.
With these regularity results for u and v achieved, we are now able to prove even u ∈ L ∞ (Ω × (0, T )) using an estimate that basically removes the dependence on p in the L p -norms of u.
Proof of lemma 3.1. We combine the two previous lemmata 3.7 and 3.6 to find the asserted estimate.

Existence of solutions and conclusions.
The crucial question concerning the existence and uniqueness of solutions is whether D vanishes at u = 0 or not. In the case of nondegenerate diffusion we will detect the existence of classical solutions and the previous section proves their boundedness. Using an approximation process, this will also result in us finding weak solutions for systems where D(0) = 0 and for this it is crucial that the bounds from before do not depend on the precise value of D at u = 0. 4.1. Definition of weak solutions and proofs of the theorems. After the results of the previous section and together with some technical assumptions, the global existence of classical solutions in the nondegenerate case can be proven relatively quickly: Proof of theorem 1.1. Using standard arguments (namely from [18]) we find a local solution to (S) in Ω × (0, T max ) for some T max ∈ (0, ∞]. We also see that either Lemma 3.1 shows that the second alternative cannot occur.
Using an approximation process, this allows us to obtain a result for degenerate diffusion as well. Let us begin by defining an appropriate solution concept: To prove the existence of such a solution we will use ε ∈ (0, 1) and the function D ε := D(· + ε) to approximate D. Clearly, upon an appropriate discussion of the initial data, this choice allows for the employment of theorem 1.1 since holds for all s ≥ 0. While this may seem to couple the estimates to ε ∈ (0, 1), which is now a necessary part of such a lower bound for the diffusion, the proofs only rely on D ε (s) ≥ C D s m−1 which is valid independently of ε ∈ (0, 1).
As a basis for all following steps we want to fix the used approximations and their properties. for all s ≥ 0. Furthermore, let nonnegative u 0 ∈ L 1 (Ω) as well as v 0 ∈ L 1 (Ω) and for ε ∈ (0, 1) define D ε := D(· + ε). Then we have for all s ≥ 0. Additionally there are K > 0 and two sequences of functions, (u 0ε ) ε∈(0,1) ⊂ C 0 (Ω) and (v 0ε ) ε∈(0,1) ⊂ C 1 (Ω), such that for every ε ∈ (0, 1) as well as Proof. The estimates for D ε are an immediate consequence of the properties given to D and the rest is a matter of choosing a helpful approximation.
Having fixed these quantities, we now consider a slightly different system than before. Aside from the adapted initial data ensuring sufficient regularity we also change the first equation in such a way that the diffusion is no longer degenerate. The resulting system is the following: in Ω.

(S ε )
This system meets all the requirements we have previously seen in deriving classical solutions with the helpful properties of uniqueness and global existence. Additionally, since all D ε share the quality D ε (s) ≥ C D s m−1 for every s ≥ 0, we find a common upper bound for the family of approximating solutions: Lemma 4.3. For the quantities from lemma 4.2 and every ε ∈ (0, 1) the system (S ε ) has a unique classical solution (u ε , v ε ) that is global and there is C > 0 with for every ε ∈ (0, 1) and every t ∈ (0, ∞).
Proof. Firstly, theorem 1.1 gives us unique classical solutions along with their global existence. The parameter-independent estimate D ε (s) ≥ C D s m−1 for all s ≥ 0 and every ε ∈ (0, 1) then guarantees the uniform boundedness together with theorem 3.1 and thereby finishes the proof.
These solutions to the approximate problems (S ε ) will be shown to converge to solutions of the actual system (S) in a suitable fashion. In preparation of the discussion of this convergence we will now find and fix several bounds that will enable us to start a process where we repeatedly choose subsequences along which u and v converge in a certain sense. Here and in the subsequent proof we follow ideas from [19].

GLOBAL EXISTENCE IN CHEMOREPULSION 5957
Setting X := L 1 ((0, T ); (W 1,n+1 0 (Ω)) * ) we see X * = L ∞ ((0, T ); W 1,n+1 0 (Ω)) and thus any ϕ ∈ X * with ϕ X * ≤ 1 gives us for all ε ∈ (0, 1). Firstly we have Moreover, with Young's inequality and for some C 3 > 0, we see and for the final integral we have Now we are only left with the task of proving the fourth and fifth claim. If m = 2, then we can use Young's inequality to see Together with an elementary bound for x → x log x, this shows that the for this case relevant quantities proves the equivalent of the two estimates above. Therefore, for any admissable m all four claims hold true.
We can now prove the existence of a weak solution (u, v) to (S) by taking a zero sequence (ε k ) k∈N ⊂ (0, 1) and solutions (u ε k , v ε k ) to the approximating problems (S ε ) for ε = ε k and letting k → ∞.
Proof of theorem 1.2. For ε ∈ (0, 1) we again use D ε (s) := s 0 D(σ) dσ and by lemma 4.3 we have a uniquely determined global classical solution to (S ε ) so that there can be no confusion concerning the functions we are dealing with in the upcoming proof.
We start with an arbitrary monotonous zero sequence (ε 0,l ) l∈N ⊂ (0, 1) and so we can assume that for some k ∈ N we have a sequence (ε k−1,l ) l∈N with the desired properties. Thanks to lemma 4.4 we find C 1 (k) > 0 with ≤ C 1 (k) ∀l ∈ N.