Sharp conditions on global existence and blow-up in a degenerate two-species and cross-attraction system

We consider a degenerate chemotaxis model with two-species and two-stimuli in dimension $d\geq 3$ and find two critical curves intersecting at one same point which separate the global existence and blow up of weak solutions to the problem. More precisely, above these curves (i.e. subcritical case), the problem admits a global weak solution obtained by the limits of strong solutions to an approximated system. Based on the second moment of solutions, initial data are constructed to make sure blow up occurs in finite time below these curves (i.e. critical and supercritical cases). In addition, the existence or non-existence of minimizers of free energy functional is discussed on the critical curves and the solutions exist globally in time if the size of initial data is small. We also investigate the crossing point between the critical lines in which a refined criteria in terms of the masses is given again to distinguish the dichotomy between global existence and blow up. We also show that the blow ups is simultaneous for both species.


INTRODUCTION
The interaction motion of two cell populations in breast cancer cell invasion models in R d (d ≥ 3) have been described by the following chemotaxis system with two chemicals and nonlinear diffusion (cf. [20,30]) where m 1 , m 2 > 1 are constants. Here, u(x, t) and w(x, t) denote the density of the macrophages and the tumor cells, v(x, t) and z(x, t) denote the concentration of the chemicals produced by w(x, t) and u(x, t), respectively. For simplicity, the initial data are assumed to satisfy Since the solutions to the Poisson equations can be written by the Newtonian potential such as 2 and c d is the surface area of the sphere S d−1 in R d , the original system (1.1) can be regarded as the interaction between two populations where it follows that the solutions obey the mass conservation The associated free energy functional F for (1.1) or (1.3) is given by which is non-increasing with respect to time since for smooth case it satisfies the following decreasing property Only one-single population and chemical signal consisting of chemotaxis system is the well-known Keller-Segel model by taking into account volume filling constraints (see [28,38,9]) reading as which has immensely investigated over the last decades. See [3,23,28,39,13] for the biological motivations and a complete overview of mathematical results for related more general aggregation-diffusion models. Here the diffusion exponent m 1 is taken to be supercritical 0 < m 1 < m c := 2 − 2/d, critical m 1 = m c and subcritical m 1 > m c if d ≥ 3. The critical number m c is chosen to produces a balance between diffusion and potential drift in mass invariant scaling. For the subcritical m 1 > m c in the sense that diffusion dominates, the solutions are globally solvable without any restriction on the size of the initial data [29,43,45]. However, in the supercritical case, the attraction is stronger leading to a coexistence of global existence of solutions and blow-up behavior. More precisely, finite-time blow up occurs for large initial data, see [11] for m 1 = 1, [17] for m 1 = 2d/(d + 2), [16] for 2d/(d + 2) < m 1 < m c , and [43] for 1 < m 1 < m c . But there also exists a global weak solution with decay properties under some smallness condition on the initial mass [4,17,18,45]. The critical case m 1 = m c is investigated in [6,44] showing the existence of a sharp mass constant M * allowing for a dichotomy: if u 1 = M 1 < M * the solutions exist for all time, whereas if M 1 ≥ M * there exists solution with non-positive free energy functional blowing up. In addition, such similar dichotomy was found in [8,19,24] earlier in dimension d = 2 and linear diffusion m 1 = 1 for (1.4) with K(x) = −1/(2π) log |x| , where M * was replaced by 8π. We also note that the results in [7] prove that solutions blow up as a delta Dirac at the center of mass as time increases in critical mass M 1 = 8π. Sufficient conditions for nonlinear diffusion m 1 > 1 to prevent blow up are derived in [9].
The variational viewpoint to analyse problems of the type (1.4) has also been an active field of research. For instance, there have been recent results about the properties of global minimizers of the corresponding free energy functional, including the existence, radial symmetry and uniqueness and so on, since they not only correspond to steady states of (1.4) in some particular cases, but also are candidates for the large time asymptotics of solutions to (1.4). Lion's concentration-compactness principle [36] (see also [2]) can be directly applied to the subcritical m 1 > m c if d ≥ 3 and allows the existence of minimizer which further satisfies some regularities properties (see [15]). The uniqueness of minimizer in this case is ensured in [33] and such unique minimizer is also an exponential attractor of solutions of (1.4) when the initial data is radially symmetric and compactly supported by using the mass comparison principle (see [29]). In the critical case m 1 = m c , the free energy functional doses not admit global minimizers except for the critical mass case M 1 = M * introduced above [10]. Such minimizers were used in [6] to describe the infinite time blow-up profile. For the nonlinear-diffusion in two dimension, the long time asymptotics of solutions is fully characterized in [14] based on the unique existence of radial minimizer of F [12]. We refer to [5] for a discussion on the existence of many stationary states for m 1 = 1 and d = 2 in the critical case M 1 = 8π and their basins of attraction.
Back to linear two-species system (1.1) in d = 2, similar to the role of the critical mass 8π in (1.4) ( [8,19]), the critical curve M 1 M 2 − 4π(M 1 + M 2 ) = 0 for two species is discovered in [22]: solutions exist globally if The key tool for the proof of the global existence part is using the Moser-Trudinger inequality as in [42] in two dimensions. One can use partial results in [42] to check that mimimizers indeed exist in the case M 1 M 2 − 4π(M 1 + M 2 ) = 0. We also mention that such nonlinear system (1.1) and the one-single population system (1.4) can be formally regarded as gradient flows of the free energy functional in the probability measure space with the Euclidean Wasserstein metric [1,25]. For general n-component multipopulations chemotaxis system, in [26,27] the authors have made considerable progress on these aspects and obtain the global arguments in subcritical and critical cases. The Neumann initial-boundary value problem is analysed in [34,35,47,48].
The aim of this paper is to give a thorough understanding of the well-posedness and asymptotic behavior for (1.1) and (1.3) in d ≥ 3 and to show the existence or non-existence of global minimizers in critical cases. We make use of bold faces m, A, B, I, M, · · · to denote two-dimensional vectors through the paper and assume that A = (a 1 , a 2 ) ≤ (≥)B = (b 1 , b 2 ) means that a 1 ≤ (≥)b 1 and b 1 ≤ (≥)b 2 , respectively. If (u, w) is a solution of (1.3), then for any λ > 0 the following scaling is also a solution, where the above scaling becomes mass invariant for both u and w if and only if m := (m 1 , m 2 ) = (m c , m c ). When m satisfy the mass conservation law only holds for w, whereas only u preserves L 1 -norm if The curves (1.5) and (1.6) can be shown to be the sharp conditions separating the global existence and blow up. Our main result in Theorem 1.3 shows the following dichotomy: above the two red curves in Figure 1, in the sense that m 1 m 2 + 2m 1 /d > m 1 + m 2 or m 1 m 2 + 2m 2 /d > m 1 + m 2 , weak solutions globally exists and blow up occurs below the red curves for certain initial data regardless of their initial masses (see Theorem 1.3). Several results are also obtained at the critical curves (see Theorem 1.4). In addition, both two lines will intersect at the point (m c , m c ). Therefore, we consider the (m 1 , m 2 ) ∈ (1, ∞) 2 parameter range divided by the following three critical cases (red curve in Figure 1): Parameter lines determining the critical regimes.
Based on the above discussion, we say that m = (m 1 , m 2 ) is subcritical if Notice that this corresponds to be above (subcritical) or below (supercritical) the red curves in Figure 1. We also define subsets of L 1 (R d ) as For a given weak solution, we also define: is called a free energy solution with some regular initial data (u 0 , w 0 ) on (0, T) if (u, w) is a weak solution and moreover satisfies (u (2m 1 −1)/2 , w (2m 2 −1)/2) ) ∈ (L 2 (0, T; H 1 (R d ))) 2 and for all t ∈ (0, T) with v = K * w and z = K * u.
Our first main result for (1.1) or (1.3) above or below lines L 1 and L 2 is:

2). Then i) If m is subcritical, there exists a global free energy solution.
ii) If m is supercritical, then one can construct large initial data ensuring blow up in finite time.
On the lines L 1 , L 2 and intersection point I, our second main result is as follows.
If m is on L 2 , there exists M 1c > 0 with the similar properties for M 1

and blow-up solution exists if
It is an open problem to determine the sharp relation between the masses leading to dichotomy in the intersection point I and the long time asymptotics on the red curves L 1 and L 2 in Figure 1.
The organization of the paper is as follows: we first construct an approximated system for (1.1) in Section 2, and provide an sufficient condition for global existence of smooth solution and then obtain global weak solution or free energy solution of (1.1) by passing limits upon a prior estimate. Section 3 deals with properties of free energy functional, including the lower and upper bounds, and the existence or non-existence of non-zero minimizers if m is critical. Finally, we prove that the solutions are globally solvable if m is subcritical or critical with small initial data in Section 4 and construct blow-up solutions if m is supercritical or critical with large masses in Section 5.

APPROXIMATED SYSTEM
As mentioned in the introduction, we first consider an approximated system with u ǫ 0 and w ǫ 0 being the convolution of u 0 and w 0 with a sequence of mollifiers and u ǫ 0 1 = u 1 = M 1 and w ǫ 0 1 = w 1 = M 2 . Then the uniform a priori estimate for solutions to (2.1) is given if m 1 and m 2 are suitably large, thus global weak solution or even free energy solution exists by letting ǫ tends to 0.
By virtue of the local existence of strong solution for only one-single population chemotaxis system (see [43, Now we recall the Hardy-Littlewood-Sobolev (HLS) inequality which we frequently use later (see [31] or [32,Chapter 4] Then for h ∈ L κ 1 (R d ) and for κ 1 , κ 2 > 1 with 1 An equivalent form of the HLS inequality can be stated that if Inspired by [46], the global solvability of (2.1) can be achieved based on assumptions on the boundedness for u ǫ m 1 and w ǫ m 2 with some large m 1 and m 2 .
Proof. We split the proof into three steps.
Step 1. The choices of p and q. There existp > 1,q > 1, r 1 > 1 and r 2 > 1 such that for some p >p and q >q one has as well as Let us first pick r 1 > 1 and r 2 > 1 fulfilling and and let In (2.15), p > m 1 + 1 implies q > m 2 + 1. The assertions in (2.6)-(2.7) and (2.9) easily hold by sufficiently large p ≥p with somep > 1 and q ≥q with someq > 1.
To see the possible choice of r 1 satisfying (2.7)-(2.8), we first observe that 1 − 1 (q+m 2 −1)d for any q > 1. Thus the asserted r 1 can be actually found. When m 2 < d 2 , one has 1 The first inequality is guaranteed by (2.13) and the second is due to , and the assertion is true.
Step 2. Inequalities for both u and w. For p > 1 and q > 1, we test (2.1) 1 by u p−1 ǫ and integrate to find that by Hölder's inequality with r 1 , We begin with estimating the right sides of (2.17)-(2.18) based on the choices of p, q, r 1 and r 2 in Step 1. The assumption (2.6) ensures and by (2.8). Then by a variant of the Gagliardo-Nirenberg inequality (see [45,Lemma 6]), Likewise, (2.7)-(2.8) warrants that which allows one to make use of the Gagliardo-Nirenberg inequality and the upper bound for w m 2 in (2.3) to estimate To estimate the right side of (2.18), we use (2.10) and (2.9) to obtain Then the Gagliardo-Nirenberg inequality implies We also obtain by (2.10) and (2.15), and choose r = q, m = m 2 , k 1 = m 2 , k 2 = qr ′ 2 in (2.21) to see that (2.23) Step 3. Boundedness for u ǫ and w ǫ in L p -and L q -spaces. Let γ 1 > 0, γ 2 > 0 be such that γ 1 + γ 2 < 2. For ǫ > 0, a direct application of Young's inequality implies that Step 1, there exist some p >p and q >q with somep > 1 andq > 1 such that by (2.23)-(2.24). One may invoke the Gagliardo-Nirenberg inequality with u 1 = M 1 and w 1 = M 2 and Young's inequality to obtain Writing Then which implies that (2.4) holds.
Step 4. Improve the regularities of v and z. As an application of the HLS inequality ensures that Furthermore, observing that the Calderon-Zygmund inequality yields the existence of a constant C = C(r) > 0 such that we combine (2.4), (2.26) with the Morrey's inequality to see that Thus we finish our proof.
Upon the boundedness arguments in Lemma 2.3, we obtain a global weak solution by letting a subsequence of ǫ approaches to 0.

THE FREE ENERGY FUNCTIONAL
Now we concentrate on a deeper analysis of the energy functional F given by The estimate for H can be given as follows. then for any f ∈ L m (R d ) and Proof. Fixing m ∈ (1, d/2), using Hölder's inequality with 1 m + m−1 m = 1 and the HLS inequality with λ = 2 in Lemma 2.2, we find that Since the assumption m + m 2 ≤ mm 2 + 2mm 2 /d ensures that with m 2 > 1, the following interpolation inequality holds: We establish several variants to the HLS inequality on the lines L 1 , L 2 and the intersection point I.

Lemma 3.2.
Let m be on L 1 , and let f ∈ L m 1 (R d ) and g ∈ L 1 (R d ) ∩ L m 2 (R d ). Then If m is on L 2 , and f ∈ L 1 (R d ) ∩ L m 1 (R d ) and g ∈ L m 2 (R d ), then

In addtion, assume that m is I and
Proof. If m is on L 1 , then m 1 ∈ (m c , d/2) and using (3.5) with m = m 1 we have Therefore, C * is finite and bounded above by C HLS . It is also easy to see that C ⋆ is controlled by C HLS if m is on L 2 . Finally, with the help of the HLS inequality and Hölder's inequality, we find that The lower and upper bounds for F in the sets S M 1 × S M 2 below is given next.
If m is I, then Furthermore, Then F can be estimated as In the case M 2 ≤ M 2c , since F ≥ 0, then the infimum is nonnegative. Taking If m is on L 2 , we have (3.8) one more by the HLS inequality and Hölder's inequality, and take h i above to see (3.9).
If m is I, since One finally obtains from In addition, there exists a minimizer ( f , g) ∈ S M 1 × S M 2 of F if M 1 = M 2 = M c , and the minimizer satisfies with some R 0 > 0 and x 0 ∈ R d , where ζ is the unique positive radial classical solution to the Lane-Emden equation Proof. We claim that if C c in (3.6) is obtained by some non-zero f and g, then g = c 0 f with some c 0 . This is easily verified by the positive definite of |x − y| −(d−2) , see [32,Theorem 9.8]. In fact, suppose that there exist a pair of maximizing non- such that ≤C c f The existence of a maximizing nonnegative, radially symmetric and non-increasing f * with f * 1 = f * m c = 1 for (3.12) has been given in [6,Proposition 3.3]. So choosing g * = c 0 f * , then H[ f * , g * ] = c 0 C c and the first conclusion has been proved with c 0 = 1.
To derive minimizers for F in the situation After a careful computation we infers that The precisely description of the set of minimizers of F was derived in [6, Proposition 3.5], we omit it here and have proved the second conclusion.
On L 1 , we assert that there is no non-zero minimizer of F in S M 1 × S M 2 if M 2 = M 2c . The proof includes two steps: the first one is to derive the nonexistence of non-trivial classical solution to a Lane-Emden system (see Lemma 3.5), and the second is to make a contradiction by the achievement of Euler-Largrange equalities which consist of the Lane-Emden system on the assumption that minimizers of its free energy exist (see Theorem 3.6 ). (3.13)

Then (3.13) does not admit any nonnegative and non-trivial classical solution
The existence/nonexistence of solutions to the general form of Lane-Emden system has been investigated in [37,40,41], for example. However, the solvability of (3.13) involving both whole space and bounded domains has not yet known as far as we know. We assert that there exists no non-trivial classical solution for (3.13) if m is on L 1 . Consider the following properties: Suppose that ω ∈ C 2 (R d ) is non-trivial and satisfies ∆w ≤ 0, x ∈ R d . Then by the strong maximum principle (see [40,Proposition 3.4]). Relying on the finite of ϑ q , we have the following contradiction: For R > 1, where one combines with the fact that ∆ϑ ≤ 0 for x ∈ Ω 1 = R d and (3.14) to see that (3.13) has no non-trivial and nonnegative classical solution. Step 1. Necessary conditions for global minimizers of F . We assume that minimizers exist and try to present some basic properties of them. Suppose that ( f * , g * ) ∈ S M 1 × S M 2 is a minimizer of F in the sense that F [ f * , g * ] = 0. Then by the HLS inequality, Young's inequality, the definition of M 2c and M 2 = M 2c . As a consequence of (3.15), we obtain that and Step 2. The Euler-Lagrange equalities. Let f and g be symmetric rearrangement of f * and g * .
Then such symmetric non-increasing minimizer ( f , g) ∈ S M 1 × S M 2 of F satisfies the following Euler-Lagrange equalities (3.20) Step 3.
where once more using the HLS inequality again, one concludes that In particular, K * f ∈ L m 2 by the monotonicity of g. Moreover, a bootstrap argument ensures that Letting ϑ := f m 1 −1 and ς := g m 2 −1 , we readily infer from (3.20) 1 that and invoke [21,Theorem 9.9] to have ϑ ∈ W 2,r (B(0, ρ)) with r ∈ (m 1 , ∞) and x ∈ R d . Furthermore, from the expression for ς such as by means of the regularity of ϑ and [21,Lemma 4.2], we obtain ς ∈ C 2 (B(0, ρ)) With the smoothness of ς, [21,Lemma 4.2] applies so as to assert that ϑ ∈ C 2 (R d ) and Step 4. Contradiction. (3.21)-(3.22) consist of the Lane-Emden system (3.13). However, it has been proved that there exists no non-trivial classical solution of (3.13) if m is on L 1 , which makes a contradiction.

THE GLOBAL EXISTENCE
This section deals with the global solvability of (1.1) in subcritical case. We first present a local existence and extensibility criterion of free energy solution to (1.1). Note that this theorem also provides simultaneous blow-up argument in Section 5.  Proof. For (u 0 , w 0 ) satisfying (1.2), local existence and (4.1) can be proved by approximation arguments (similar to those in the proof of Theorem 1.1 in [43] for instance). To see (4.2), since the solution is globally solved if both u m 1 and w m 2 are uniform bound in subcritical or critical case due to Lemmas 2.3-2.5, then it is sufficient to show that the two terms u m 1 and w m 2 are governed by each other with some constants. Since then it needs to control the term H at the right side of (4.3). For m ∈ (1, d/2) satisfying (3.1), Lemma 3.1 yields that Taking η small enough, we have and if η is sufficiently large, we see that Therefore, (4.2) holds by (4.1), (4.5)-(4.6).
and next take interpolation inequality to find that Upon with u 1 = M 1 and w 1 = M 2 . Hence (4.5)-(4.6) are valid by picking suitable η > 0. By the same token, the case m 1 m 2 + 2m 2 /d ≥ m 1 + m 2 is also true for both m 2 < d/2 and m 2 ≥ d/2. The proof is finished.
The global existence result in subcritical case is the subject of our next theorem.
by Young's inequality. Then substituting (4.3) into above, we have As a corollary, Due to (4.9), there exists C > 0 such that for all t ∈ [0, T max ) we have w m 2 ≤ C . Then the extensibility criterion in Theorem 4.1 makes sure that T max = ∞. The other cases can be similarly obtained.

BLOW UP
Our last section concerns finite-time blow-up phenomenon when m is critical or super-critical. These results actually show that lines L i , i = 1, 2 are optimal in view of the global existence for sub-critical case. The following second moment of solutions can be achieved in a straightforward computation. where Proof. We differentiate the second moment to see that Combining above equations, it follows that which readily implies the lemma.
We construct initial data which ensures the nonnegativity of G(0).
Proof. Consider the following functions having the same compact support as initial data of form with ι 1 := 2m 2 (m 1 + m 2 − m 1 m 2 )d and ι 2 := 2m 1 (m 1 + m 2 − m 1 m 2 )d , (5.4) where A, B > 0 denote the maximum of the supports and a > 0 denotes the size of the supports of initial data. Such constructions in ( AB.

(5.6)
Since by (5.5)-(5.6), to show (5.2), it only needs to show the right side of (5.7) is negative such that AB A m 1 + B m 2 a 2 > N 1 (5.8) with in the Case 1, whereas the right side will be replaced by 2N 1 in the Case 2. Since then (5.8) can be rewritten as which yields G(0) < 0 with N 0 = N 2 .