Chemotaxis can prevent thresholds on population density

We define and (for $q>n$) prove uniqueness and an extensibility property of $W^{1,q}$-solutions to $u_t =-\nabla\cdot(u\nabla v)+\kappa u-\mu u^2$ $ 0 =\Delta v-v+u$ $\partial_\nu v|_{\partial\Omega} = \partial_\nu u|_{\partial\Omega}=0,$ $ u(0,\cdot)=u_0 $ in balls in $\mathbb{R}^n$, which we then use to obtain a criterion guaranteeing some kind of structure formation in a corresponding chemotaxis system - thereby extending recent results of Winkler to the higher dimensional (radially symmetric) case. Keywords: chemotaxis, logistic source, blow-up, hyperbolic-elliptic system


Introduction
The Keller-Segel model of chemotaxis has been introduced by Keller and Segel in [11] to model the aggregation of bacteria (for instance, of the species Dictyostelium discoideum, with density denoted by u) in the presence of a signalling substance (cAMP, with density v) they emit in case of food scarceness. Their movement is governed by random diffusion and chemotactically directed motion towards higher concentrations of cAMP. The Keller-Segel model or variants thereof, as for example where all functions appearing in (KS) have a simple form and diffusion of the signalling substance is assumed to occur fast (instantaneously if τ = 0), have been widely used and incorporated in more complicated models in the mathematical depiction of biological phenomena, ranging from pattern formation in E. coli colonies [2] to angiogenesis in early stages of cancer [19] or HIV-infections [17]. For a survey of the extensive mathematical literature on the subject see the survey articles [7] or [8,9]. Often the occurence of the desired structure formation is identified with the blow-up of solutions to the model in finite time, i.e. the existence of some finite time T such that lim sup tրT u(·, t) L ∞ = ∞ -and the model -both for τ = 0 and τ > 0 -is known to possess such solutions for every sufficiently large initial mass or in space dimensions larger than two, whereas in dimension 2 for small initial mass all solutions exist globally in time and are bounded [10,6,14,15]. Moreover, blow-up of solutions with large enough initial mass has been shown to be a generic phenomenon of the equation in some sense even for the parabolic-parabolic version of the system [13,24].
Another point of view is that blow-up is "too much" and biologically inadequate, at least in some situations. Then, for example, terms preventing blow-up are added, e.g. some logistic growth term (cf. for example the tumor models in [1] or [18]), so that the model reads For this problem it is known [20,16,22] that classical solutions exist globally in time and are bounded if n ≤ 2 or µ is large (where for τ = 0, an explicit condition sufficient for this is µ > n−2 n ). For higher dimensions and small µ the existence or non-existence of exploding solutions is unknown. As [23] seems to indicate, superlinear absorption does not necessarily imply global existence.
The important question is: To what extent does the logistic term render the chemotaxis-term innocuous? Does there still emerge some structure? Recently this question has been answered affirmatively by Winkler [25] in the one-dimensional case: If the death rate µ is small enough (0 < µ < 1), then there is some criterion on (the L p -norm with p > 1 1−µ of) the initial data that ensures the existence of some time up to which any threshold of the population density will be surpassed -as long as the bacteria do not diffuse to fast. Of course, the biologically relevant situation is not that of only one space-dimension. With the present paper we give an answer to the question whether this phenomenon is restricted to this case or if it also occurs in higher dimensions.
We shall confine ourselves to the prototypical radially symmetric setting and in the end obtain Theorem 1.
For this purpose we set out to find estimates finally leading to the crucial extensibility criterion (25) for solutions of the "ε = 0-limit" model The extensibility criterion is analogous to (1.6) of [25], that in turn is built upon estimates, some of which heavily rely on one-dimensionality of the problem. Cornerstone of our analysis therefore will be Section 3.5, where we craft the inequality which also allows for higherdimensional and therefore more realistic scenarios. We will introduce our concept of solutions of (2) in Definition 21 and show their uniqueness -if u 0 ∈ W 1,q (Ω) for some q > n -in Theorem 23 and the existence of radially symmetric solutions that can be approximated by solutions of (1) in Theorem 27.
In contrast to the one-dimensional case, we are confronted with the challenge that we cannot, in general, rely on the existence of global classical bounded solutions to the approximate problem. Hence we prepare these results by finding a common existence time of such solutions -regardless of the value of ε (Theorem 19).
After collecting some additional boundedness property in Lemma 20, we can by a limiting procedure (Lemma 25) turn to solutions to (2).
If then µ is large enough, a global, in some cases even bounded solution is guaranteed to exist [Prop. 30]. However, if this is not the case, any radial solution to (2) with somehow (L p -)large enough initial mass blows up in finite time (Theorem 33). In combination with the fact that solutions to (2) can be obtained as limits of solutions to (1), this yields the announced theorem about nonexistence of thresholds to population density: If µ < 1 and u 0 L p (for p > 1 1−µ ) is large enough, before some time T any threshold on the population density will be exceeded at least at one point by any population that diffuses slowly enough.
After the following short section which recalls a few basic properties of solutions to the second equation in (1) and equation (1) with ε > 0, in Section 3 we focus our attention on existence of solutions to (1) and estimates yielding a common existence time (Theorem 19) as well as preparing for compactness arguments (by the estimates of Lemma 20 and Corollary 17). Section 4 will be devoted to definition, uniqueness, existence, estimates and a blow-up result for solutions to (2), followed in Section 5 by, finally, the proof of the "no threshhold" theorem 1.
Throughout the article, we assume that Ω ⊂ R n is a bounded domain with smooth boundary. Often we will speak of radially symmetric (or, for short, radial) functions. In this case Ω = B R (0) is to be understood to be a ball centered in the origin and we will interchange u(x, t), x ∈ Ω, and u(r, t), r = |x| ∈ [0, R).
Occasionally, we will abbreviate Ω × (0, T ) =: Q T for T > 0. We will often identify u(t) = u(·, t) and for the sake of brevity write u instead of u(x, t) or u(x, s).

Preliminaries: The elliptic equation
In the proofs we will mainly be concerned with u, therefore it would be desirable to estimate various terms involving the solution v of or its derivatives in terms of u. The following lemmata will be the tools to make this possible: Proof. As in [25, Lemma 2.1], for p ∈ (1, ∞), testing the equation by v(v 2 + η) p 2 −1 as 0 < η → 0 yields this estimate, which can then be extended to p ∈ {1, ∞} by limiting procedures.
Applying these two inequalities to v p+1 2 and using Lemma 2 for p = 1, we arrive at We also recall useful facts on maximal regularity for elliptic PDEs: Proof. Proof. This is a consequence of the elliptic maximum principle.

Existence
We prepare the following two lemmata with this estimate from [21, Lemma 1.3 iv)] about the (Neumann) heat semigroup: Then there exists C > 0 such that for all t > 0 and for all w ∈ (L q (Ω)) n Proof. [21, Lemma 1.3 iv)]. Although the lemma in that article is stated only for q < ∞, the proof actually already covers the case q = ∞, because C ∞ (Ω) is dense in L 1 (Ω).
One of the first steps in dealing with solutions of (1) is to show that they exist, at least locally. Let us briefly give the corresponding fixed point arguments.
Proof. For u ∈ C(Ω) denote by v u the solution of Let R > 2 u 0 W 1,q (Ω) be given. Fix constants C 1 as in Lemma 6, C 2 and the function C : and note that C is monotone and continuous with C(0) = 0. Choose T ∈ (0, 1) such that Then Φ : C(Q T ) → C(Q T ) is well-defined and, in fact, even Φ(u) ∈ C ∞ (Q T ). In addition, Φ is a contraction in M := f ∈ C(Q T ); f L ∞ (QT ) ≤ R , as can be seen as follows: Furthermore Φ maps M to M as well: With the aid of Banach's fixed point theorem, this procedure yields a solution on (0, T ). Successively employing the same reasoning on later time intervals (then with different u 0 and possibly larger R) the existence of a solution on a maximal time interval (0, T max ) is obtained where either T max = ∞ or lim sup tրTmax u(·, t) L ∞ (Ω) = ∞.

L p -bounds and global existence
Bounds on L p -norms are of great utility, not only for the deduction of global existence. A standard testing procedure (see also [25]) yields Lemma 8. Let κ ≥ 0, µ > 0, u 0 ∈ C(Ω) nonnegative. Let (u, v) solve (1) classically in Ω × (0, T ) for some T > 0, ε > 0. Then for p ≥ 1 and on the whole time interval (0, T ), we have Proof. Multiplication of the first equation of (1) by u p−1 and integration by parts yield Another integration by parts in combination with the second equation of (1) and Lemma 5 show which gives formula (4).
This estimate directly leads to the following bound on L p -norms of u.
Proof. An application of Hölder's inequality gives and transforms (4) into the differential inequality for y = Ω u p . An ODE-comparison then yields the result.
If even µ ≥ 1, bounds can be given in a more explicit form and independent of ε.
Proof. (Cf. [25,Lemma 4.6].) This can be obtained by comparison with the solution y of

Radial solutions
In the following sections we will restrict ourselves to the prototypical radially symmetric situation. In this case, equations (1) can be rewritten in the form We begin by preparing an inequality for the derivative of v. Gained by the radial symmetry, it will be one of the most important tools for the calculations preparing the estimation of ∇u L q (Ω) in terms of u L ∞ (Ω) .
Proof. Fix t > 0. Equation (6) can also be written in the form 1 which leads to (7).

Compatibility
We say that a function u 0 satisfies the compatibility criterion (or, for short, that u 0 is compatible) if u 0 ∈ C 1 (Ω) and ∂ ν u 0 | ∂Ω = 0. If functions with this property are used as initial condition in parabolic problems, the solutions they yield have bounded first [spatial] derivatives on a time interval containing 0 ( [12]). This will be important in the derivation of the crucial estimate of Ω |∇u| q for solutions u of (1) (Lemma 17) in terms of ∇u 0 L q instead of only ∇u(·, τ ) L q for arbitrary small τ > 0. At first we show that any function u 0 ∈ W 1,q (Ω) can be approximated by compatible functions preserving all kind of 'nice properties': Lemma 13. Let q > n, u 0 ∈ W 1,q (Ω) be radially symmetric and nonnegative, let ε > 0. There is (Ω) < ε and also u 0 is radial and nonnegative.

The most important estimate
In this section we are going to derive an inequality which shows that we can control the · W 1,q (Ω) -norm of solutions to (1) by their L ∞ (Ω)-norm. For the following computation we define, for η > 0, In preparation for later calculations we also note that for a, s ∈ R For any radial classical solution u of (1) in Ω×(0, T ) with radial initial data u 0 ∈ W 1,q (Ω), and arbitrary τ ∈ (0, T ), t ∈ (τ, T ), we have (with K as C from Lemma 4) Proof.
Denote Ω δ = Ω \ B δ (0) and let 0 < τ < t < T . Note that on Ω δ ×(τ, t) all derivatives of u appearing in the following calculation are smooth and bounded, and we can change the order of integration and differentiation to start with Here we use equation (5) for u t : Now we integrate by parts twice in the first term where we also used (9), and once in the second integral Upon addition, some of these summands vanish and estimating I 3 ≤ 0 by (8), we obtain Also the next term can be rewritten by integration by parts and using v r (R, t) = 0 for t ∈ (0, T ).
Inserting (6) to express v rrr in I 6 differently, we obtain (among others) terms to cancel out I 7 and I 9 : Together with the trivial observation that I 11 ≤ 0 by (8), these estimates and reformulations give Passing to the limit δ ց 0 by boundedness of u r , u rr , v r on (τ, t) and the dominated convergence theorem we arrive at and with the help of (10), the first of these integrals can be rewritten as Treating the second term similarly and inserting (10) and (6) gives where also (u − v)u 2 r ≤ u u 2 r + η . For the sum of these terms we are thereby led to where we can use Lemma 12 to infer Furthermore adding the other terms and making use of (8) in I D , Here an application of Young's inequality gives Merging first and third term, with Lemma 4 (and K as provided by that lemma) and Lemma 2 we have In total, these estimates show the claim.

From this we gain
which implies the assertion.
Corollary 17. In addition to the hypotheses of Lemma 14, let u 0 be compatible. Then Proof. Since from the dominated convergence theorem we know that Ω |∇u(·, τ )| q → Ω |∇u 0 | q as τ ց 0 due to the boundedness of ∇u for solutions of (1) with compatible initial data, this is a direct consequence of Lemma 16.

Epsilon-independent time of existence
We begin this section with some Gronwall-type lemma which we will need during the next proof: Lemma 18. Let f : [0, ∞) → R nondecreasing and locally Lipschitz continuous, let y 0 ∈ R. Denote by y the solution of y(0) = y 0 , y ′ (t) = f (y(t)) on some interval (0, T ) and assume that the continuous function z : [0, T ) → R satisfies Then z(t) ≤ y(t) for all t ∈ (0, T ).
which is contradictory.
The next lemma prepares the ground for the approximation procedure to be carried out in Theorem 27. It guarantees that solutions to (1) exist "long enough". Its proof is an adaption of that of [25,Lemma 4.5], where an assertion similar to our Lemma 27 is shown.
Proof. For any ε > 0, the classical solution (u ε , v ε ) of (1) exists on some interval (0, T ε max ) and satisfies lim sup tրT ε max u ε L ∞ (Ω) = ∞, unless T ε max = ∞. It is therefore sufficient to show boundedness of u ε L ∞ (Ω) on (0, T (D)) for some ε-independent T (D) > 0. Fix constants c 1 , c 2 such that for all ψ ∈ W 1,q (Ω) where we use q > n, as well as K as in Lemma 14 and c 3 = c 3 (D) such that so that by Corollary 9 applied to p = 1, Let furthermore denote y D the solution to and denote by T (D) > 0 a number, such that y D (t) ≤ ( √ 2D) q + 2 for all t ∈ (0, T (D)). Now, let u 0 ∈ W 1,q (Ω) be as specified in the lemma, especially with u 0 W 1,q (Ω) ≤ D. For ε > 0 denote by u ε the solution of the corresponding equation (1).

Preparations for convergence: boundedness of u t in an appropriate space
In order to use the Aubin-Lions-type estimate of Lemma 24, we will need at least some regularity of the time derivative of bounded solutions.
Here taking the supremum over ψ with ψ 4 Hyperbolic-elliptic case

What is a solution?
We want to name a function "solution" if it is a solution in a sense similar to that in [25,Def. 4.1]: holds true for all ϕ ∈ L 1 ((0, T ); W 1,1 (Ω)) that have compact support in Ω × [0, T ) and satisfy ϕ t ∈ L 1 (Ω × (0, T )). If additionally T = ∞, we call the solution global.

Remark 22.
Due to density arguments, it is of course possible to formulate Definition 21 for ϕ ∈ C ∞ 0 (Ω × [0, T )) and obtain the same solutions.

Uniqueness
These solutions are unique as can be proven very similar to the one-dimensional case.
Proof. (Cf. [25,Lemma 4.2]). Let q > n and u 0 ∈ W 1,q (Ω) with u 0 ≥ 0. Let (u, v), ( u, v) be strong W 1,q -solutions of (2) and note that (w, z) : for all ϕ ∈ L 1 ((0, T ); W 1,1 (Ω)) which have compact support in Ω × [0, T ) and satisfy ϕ t ∈ L 1 (Q T ) and Let T 0 ∈ (0, T ). Then by Definition 21 (and by Lemma 4), we can define constants such that and let c 4 denote the constant from Lemma 4. We set By Lemma 4, (15) implies z W 2,q (Ω) ≤ C w L q (Ω) and hence, as for q > n W 1,q (Ω) ֒→ L ∞ (Ω), In (14) we use some function ϕ we construct as follows: For t 0 ∈ (0, T 0 ) define χ δ ∈ W 1,∞ (R) by for δ ∈ (0, T0−t0 2 ) and let Then for δ ∈ (0, T0−t0 2 ), h ∈ (0, T0−t0 2 ), 1 > η > 0, ϕ is a valid test function in (14) and yields: Taking the limit δ ց 0, which is possible for the first term because (x, t) → w(x, t) 1 h t+h t w(x, s)(w 2 (x, s)+ η) q 2 −1 ds is continuous and on the right hand side by Lebesgue's theorem, since ∇z, ∆z, u, ∇v, ∆v, w, u, are bounded and ∇w is uniformly bounded in L q (Ω) up to time t 0 + h (according to Definition 21), we obtain With the abbreviations w = w(x, t), w h = w(x, t + h) we observe that where we have used that s ≤ (s 2 +η) 1 2 and Young's inequality. Converting the time shift in the arguments to a change of integration limits, we obtain where in the limit η ց 0 the last line vanishes. Furthermore, by the continuity of w and because |w(w 2 + η) q 2 −1 | can be bounded by the integrable function (w 2 + 1) as well as in L ∞ (Ω × (0, t 0 )) as η ց 0. Therefore we can conclude from (17) that Here we will estimate the integral on the right hand side to obtain an expression that allows to conclude w = 0 by means of Gronwall's lemma. We will consider the summands seperately: Also for the next term, the second equation of (1) and nonnegativity of v are helpful: For the last we make use of the nonnegativity of both u and u: Boundedness of u in W 1,q (Ω) and Lemma 4 play the main role in the following estimate.
And finally, once more employing the second equation, Gathering all these estimates together we gain the constant C from (16) such that for all t 0 ∈ (0, T 0 ) hence by Gronwall's lemma w = 0 (and therefore also z = 0), which proves uniqueness of solutions.

Local existence, approximation
We will prove existence of solutions to (2) by means of a compactness argument whose key lies in: Lemma 24. Let X, Y, Z be Banach spaces such that X ֒→ Y ֒→ Z, where the embedding X ֒→ Y is compact. Then for any T > 0, p ∈ (1, ∞], the space Proof. The proof uses the Arzelá-Ascoli theorem and Ehrling's lemma and can be found in [25,Lemma 4.4].
We directly take this tool to its use and employ it with a slightly different choice of spaces than in the one-dimensional case to obtain a similar result.
With the choice of X = W 1,q (Ω), Y = C α (Ω), Z = (W 1, q q−1 (Ω)) * , Lemma 24 allows to conclude relative compactness of (u εj ) j in C([0, T ], C α (Ω)). Due to this and (4.3), given any subsequence of (ε j ) j , we can pick a further subsequence thereof such that as i → ∞ and also by the propagation of the Cauchy-property from ( The limit (u, v) is a strong W 1,q -solution of (2), as can be seen by testing (1) by an arbitrary ϕ ∈ C ∞ 0 (Ω × [0, T )) and taking ε = ε ji → 0 in each of the integrals seperately, as possible by (19) to (21): Hence the limit of all these subsequences u εj i of subsequences is the same, namely the unique (Lemma 23) solution of (2) and therefore the whole sequence converges (in the spaces indicated in equations (19) to (21)) to the solution (u, v) of (2).
In Lemma 25 we assumed uniform boundedness of the approximating solutions. Fortunately, on small time scales we are entitled to do so and can prove the following: Lemma 26. Let κ ≥ 0, µ > 0, q > n. Then for D > 0 there is some T (D) > 0 such that for any radial symmetric nonnegative u 0 ∈ W 1,q (Ω) fulfilling u 0 W 1,q (Ω) < D there is a unique W 1,q (Ω)-solution (u, v) of (2) in Ω × (0, T (D)). Furthermore, if u 0ε are compatible functions satisfying u 0ε − u 0 W 1,q (Ω) < ε, this solution (u, v) can be approximated by solutions (u ε , v ε ) of (1) (with initial condition u 0ε ) in the following sense: Moreover, with K as in Lemma 14 this solution satisfies for a.e. t ∈ (0, T (D)).
Proof. For ε ∈ (0, 1) let u 0ε be compatible and u 0ε − u 0 W 1,q (Ω) < ε. Apply Theorem 19 with D + 1 to obtain T (D) such that the solutions u ε to (1) with initial data u 0 exist on (0, T (D)) and are bounded by M (D) on that interval. Here Lemma 25 applies to provide a strong W 1,q -solution with the claimed approximation properties. The inequality results from Corollary 17 as follows: According to Corollary 17, for all t ∈ (0, T (D)), Let t ≥ 0 be such that u(·, t) ∈ W 1,q (Ω). Convergence of the right hand side is obvious because of the uniform convergence u ε → u and u 0ε → u 0 in W 1,q (Ω). This implies boundedness of (∇u εj ) j in L q , hence L q -weak convergence along a subsequence and -due to the weak lower semicontinuity of the norm -

Continuation and existence on maximal time intervals
Solutions constructed up to now may only exist on very short time intervals. With the following theorem (which parallels [25,Thm. 1.2] in statement and proof) we ensure that they can be glued together to yield a solution on a maximal time interval -to all eternity or until blow-up.
Assume T max < ∞ and lim sup tրTmax u(·, t) L ∞ (Ω) < ∞. Then there exists M > 0 such that for all Let N ⊂ (0, T max ) be a set of measure zero, as provided by the definition of S, such that (26) holds for all t ∈ (0, T max ) \ N . Together with u ≤ M this would imply for some positive D 1 and for each t 0 ∈ (0, T max ) \ N . Lemma 26 would yield the existence of a strong W 1,q -solution of on Ω × (0, T (D 1 )), which would satisfy for almost every t ∈ (0, T (D 1 )). Upon the choice of t 0 ∈ (0, would define a strong W 1,q -solution of (2) in Ω × (0, t 0 + T (D 1 )) which clearly would satisfy (26) for a.e. t < t 0 . For t > t 0 on the other hand, a combination of (27) and (26) would give This would finally lead to a contratiction to the definition of T max as supremum, because then obviously ( u, v) would satisfy (26) for a.e. t ∈ (0, t 0 + T (D 1 )).

An estimate for strong solutions: boundedness in L 1
Having confirmed existence and uniqueness of solutions, we set out to explore some more of their properties. And as in [25,Lemma 4.1], one of the first facts that can be observed (and proven like in the 1-dimensional case) is their boundedness in L 1 .

Global existence for large µ
Bounds on the L ∞ (Ω)-norm are the only thing we need to guarantee existence of solutions for longer times. They arise as a corollary to Lemma 11, which directly implies the following.

Blow-up for small µ
The contrasting -and more interesting -case is that of small values of µ. Here we will show blow-up. We borrow the following technical tool from [25]: Lemma 31. Let a > 0, b ≥ 0, d > 0, κ > 1 be such that Then if for some T > 0 the function y ∈ C([0, T )) is nonnegative and satisfies y(t) ≥ a − bt + d To be of any use to us, this estimate must be accompanied by lower bounds for (some norm of) u. We prepare those by the following lemma Lemma 32. Let κ ≥ 0 and µ > 0. For all p > 1 and η > 0 there is B(η, p) > 0 such that for all q > 1 all W 1,q -solutions (u, v) of (2) with nonnegative u 0 in Ω × (0, T ) satisfy Proof. The same testing procedure as in [25,Lemma 4.8] leads to success. We repeat it (with the necessary adaptions) for the sake of completeness, because Lemma 32 is a main building block of the blow-up result. Let T 0 ∈ (0, T ), t 0 ∈ (0, T 0 ), δ ∈ (0, T − t 0 ), χ δ as in the proof of Lemma 23: For each ξ > 0, the function (u + ξ) p−1 belongs to L ∞ loc ([0, T ), W 1,q (Ω)) and for δ ∈ (0, T 0 − t 0 ), h ∈ (0, 1), ξ > 0, is a test function for (13), if we set u(·, t) = u 0 for t < 0. This yields 1 δ