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

José Carrillo Antonio; Ke Lin

doi:10.1515/anona-2020-0189

Open Access Published by De Gruyter July 2, 2021

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

José Carrillo Antonio and Ke Lin

From the journal Advances in Nonlinear Analysis

https://doi.org/10.1515/anona-2020-0189

Abstract

We consider a degenerate chemotaxis model with two-species and two-stimuli in dimension d ≥ 3 and find two critical curves intersecting at one 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 on and 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.

Keywords: Degenerate parabolic system; chemotaxis; variational methods; global existence; blow up

MSC 2010: 35K65; 92C17; 35J20; 35A01; 35B44

1 Introduction

The interaction motion of two cell populations in breast cancer cell invasion models in ℝ^d (d ≥ 3) has been described by the following chemotaxis system with two chemicals and nonlinear diffusion (cf. [20,30])

ut=∇um1−∇.(u∇v)x∈Rd,t>0,−∇=w,x∈Rd,t>0,wt=∇wm2−∇.(w∇z),x∈Rd,t>0,−∇z=u,x∈Rd,t>0,ux,0=uox,wx,o=wox,x∈Rd, (1.1)

where m₁, m₂ > 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

u0∈L1(ℝd;(1+|x|2)dx)∩L∞(ℝd),∇u0m1∈L2(ℝd) and u0≥0,w0∈L1(ℝd;(1+|x|2)dx)∩L∞(ℝd),∇w0m2∈L2(ℝd) and w0≥0 (1.2)

Since the solutions to the Poisson equations can be written by the Newtonian potential such as

v(x,t)=K∗w=cd∫Rdw(y,t)|x−y|d−2dy,z(x,t)=K∗u=cd∫Rdu(y,t)|x−y|d−2dy

with K(x)=cd|x|d−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

{ut=Δum1−∇.(u∇K∗w),x∈Rd,t>0wt=Δwm2−∇.(w∇K∗u),x∈Rd,t>0,u(x,0)=u0(x),w(x,0)=w0(x),x∈Rd, (1.3)

where it follows that the solutions obey the mass conservation

M1:=∫ℝdu(x,t)dx=∫ℝdu0(x)dx and M2:=∫ℝdw(x,t)dx=∫ℝdw0(x)dx.

The associated free energy functional 𝓕 for (1.1) or (1.3) is given by

F[u(t),w(t)]=1m1−1∫ℝdum1dx+1m2−1∫ℝdwm2dx−cdH[u,w],

which is non-increasing with respect to time since for smooth cases it satisfies the following decreasing property

ddtF[u(t),w(t)]=−∫Rdu|m1m1−1∇um1−1−∇v|2dx−∫Rdw|m2m2−1∇wm2−1−∇z|2dx,

where

H[u,w]=∬ℝd×ℝdu(x,t)w(y,t)|x−y|d−2dxdy.

The chemotaxis system consisting of only one population and one chemical signal is the well-known KellerSegel model by taking into account volume filling constraints (see [9, 28, 38]) reading as

ut=Δum1−∇⋅(u∇K∗u),x∈Rd,t>0,u(x,0)=u0(x),x∈Rd, (1.4)

which has been immensely investigated over the last decades. See [3,13, 23,28, 39] for the biological motivations and a complete overview of mathematical results for related more general aggregation-diffusion models. Here the diffusion exponent m₁ is taken to be supercritical 0 < m₁ < m_c := 2 - 2/d, critical m₁ = m_c and subcritical m₁ > mc if d > 3. The critical number mc is chosen to produce a balance between diffusion and potential drift in mass invariant scaling. For the subcritical m₁ > 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 m₁ = 1, [17] for m₁ = 2d/(d + 2), [16] for 2d/(d + 2) < m₁ < m_c, and [43] for 1 < m₁ < 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 m₁ = m_c is investigated in [6,44] showing the existence of a sharp mass constant M* allowing for a dichotomy: if || u || ₁ = M₁ < M^* the solutions exist for all time, whereas if M₁ > 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 m₁ = 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 M₁ = 8π. Sufficient conditions for nonlinear diffusion m₁ > 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 exist a lot of 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 m₁ > mc 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 it is shown that such 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 m₁ = mc, the free energy functional doses not admit global minimizers except for the critical mass case M₁ = M* introduced above [10]. The minimizers were used in [6] to describe the infinite time blow-up profile. For the nonlineardiffusion in two dimension, the long time asymptotics of solutions is fully characterized in [14] based on the unique existence of radial minimizer [12]. We refer to [5] for a discussion on the existence of many stationary states for m₁ = 1 and d =2 in the critical case M₁ = 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 8n in (1.4) ([8,19]), the critical curve M₁M₂ - 4π(M₁ + M₂) = 0 for two species was discovered in [22]: solutions exist globally if M₁M₂ - 4π(M₁ + M₂) = 0 and blow up occurs if M₁M₂ - 4π(M₁ + M₂) = 0. The key tool for the proof of the global existence part is using the Moser-Trudinger inequality [42]. One can use partial results in [42] to check that mimimizers indeed exist in the case M₁M₂ - 4n(M₁ + M₂) = 0. We also mention that such nonlinear system (1.1) and the one 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 «-component multi-populations chemotaxis system, in [26, 27] the authors have made considerable progress on these aspects and obtain the global arguments in the 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) or (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 throughout the paper and assume that A = (a₁, a₂) ≤ (ȥ)B = (b₁, b₂) means that a₁ ≤ (ȥ)b₁ and b₁ ≤ (ȥ)b₂, respectively. If (u, w) is a solution of (1.3), then for any λ > 0 the following scaling

uλ(x,t)=λm2u(λm1+m2−m1m22x,λm1t),wλ(x,t)=λm1w(λm1+m2−m1m22x,λm2t)

is also a solution. Obviously, each of (u, w) preserves the L¹-norm if m := (m₁, m₂) satisfy

m1m2+2m1/d=m1+m2, (1.5)

and

m1m2+2m2/d=m1+m2, (1.6)

respectively, whereas the above scaling becomes mass invariant for both u and w if and only if m = (mc, mc). 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 m1m2+2m1/d>m1+m2 or m1m2+2m2/d>m1+m2 , weak solutions globally exist 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, the two lines will intersect at the point (m_c, m_c). Therefore, we consider the (m₁, m₂) ∈ (1, ∞)² parameter range divided by the following three critical cases (red curve in Figure 1):

LineL1:m1m2+2m1/d=m1+m2 with m1∈mc,d/2,m2∈1,mc;LineL2:m1m2+2m2/d=m1+m2 with m1∈1,mc,m2∈mc,d/2;The intersection pointI:=(mc,mc).

Fig 1

Parameter lines determining the critical regimes

Based on the above discussion, we say that m = (m₁, m₂) is subcritical if

m1m2+2m1/d>m1+m2orm1m2+2m2/d>m1+m2,

and m = (m_i, m₂) is supercritical if

m1m2+2m1/d<m1+m2 and m1m2+2m2/d<m1+m2.

Notice that this corresponds to being above (subcritical) or below (supercritical) the red curves in Figure 1. We also Define subsets of L¹(ℝ^d) as

SM1:={f≥0:f∈L1(ℝd)∩Lm1(ℝd) and ‖f‖1=M1}

and

SM2:={g≥0:g∈L1(ℝd)∩Lm2(ℝd) and ‖g‖1=M2}

Now the definition of weak solution for (1.1) or (1.3) is give as

Definition 1.1

Let m₁, m₂ > 1, d ≥ 3 and T > 0. Suppose the initial data (u₀, w₀) satisfies some classical regularities (1.2). Then (u, w) of nonnegative functions defined in ℝ^d × (0, T) is called a weak solution if

(u,w)∈C[0,T);L1ℝd∩L∞ℝd×(0,T)2,um1,wm2∈L20,T;H1ℝd2;
(u,w)satisfies∫0T∫ℝduϕ1tdxdt+∫ℝdu0(x)ϕ1(x,0)dx=∫0T∫ℝd(∇um1−u∇v)⋅∇ϕ1dxdt,∫0T∫ℝdwϕ2tdxdt+∫ℝdw0(x)ϕ2(x,0)dx=∫0T∫ℝd(∇wm2−w∇z)⋅∇ϕ2dxdt,

for any test functions ϕ1∈D(ℝd×[0,T)) and ϕ2∈D(ℝd×[0,T)) with v = K * w and z = K * u.

For a given weak solution, we also define:

Definition 1.2

Let T > 0. Then (u, w) is called a free energy solution with some regular initial data (u₀, w₀) on (0, T) if (u, w) is a weak solution and moreover satisfies (u(2m1-1)/2,w(2m2-1)/2)∈(L2(0,T;H1(ℝd)))2 and

F[u(t),w(t)]+∫0t∫ℝdu|m1m1−1∇um1−1−∇v|2dxds+∫0t∫ℝdw|m2m2−1∇wm2−1−∇z|2dxds≤F[u0,w0] (1.7)

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 L₁ and L₂ is:

Theorem 1.3

Let m₁, m₂ > 1. Suppose that the initial data (u₀, w₀) with u01=M1,w01=M2 fulfill(1.2). Then

If m is subcritical, there exists a global free energy solution.
If m is supercritical, then one can construct large initial data ensuring blow up in finite time.

On the lines L₁, L₂ and intersection point I, our second main result is as follows.

Theorem 1.4

Let m₁, m₂ > 1. Suppose that the initial data (u₀, w₀) with u01=M1,w01=M2 fulfill(1.2).

Then

If m is I, then there exists a number M_c > 0 such that if M₁M₂ < M_c², solutions globally exist and if M1M2/(M1mc+M2mc)>Mc2/d/2 , there exists a finite time blow-up solution. Moreover, non-zero global minimizers of F exist in SM1×SM2 at the crossing point M = (M_c, M_c).
If m is on L₁, there exists a number M_2c > 0 with the following properties: if M₂ < M_2c, solutions globally exist and inff∈SM1infg∈SM2F[f,g]=0 , but there exist no non-zero global minimizers of F in SM1×SM2 . In addition, blow-up solution exists if
∫Rdu0m1/m2dxm2/m1∫Rdw0dx∫Rdu0m1/m2dxm2+∫Rdw0dxm2>N0withsomeN0>0.

If m is on L₂, there exists M_1c > 0 with the similar properties for M₁ and blow-up solution exists if
∫Rdu0dx∫Rdw0m2/m1dxm1/m2∫Rdu0dxm1+∫Rdw0m2/m1dxm1>N0.
A simultaneous blow-up phenomenon exists if m is critical.

We summarize our second main result on the intersection point I, see Figure 2. The blue curve M1M2=Mc2 intersects with the green curve M1M2/(M1mc+M2mc)=Mc2/d/2 at the point J = (M_c, M_c). Theorem 1.4 implies that below the curve M1M2=Mc2 solutions globally exist and above the curve M1M2/(M1mc+M2mc)=Mc2/d/2 blow up happens.

Fig 2

Parameter lines on intersection point I.

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 L₁ and L₂ 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 a 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 nonzero minimizers if m is critical. Finally, we prove that the solutions are global 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.

2 Approximated system

As mentioned in the introduction, we first consider an approximated system

u ϵ t x , t = Δ ( u ε + ε ) m 1 − ∇ ⋅ u ϵ ∇ v ϵ , x ∈ R d , t > 0 , v ϵ = K ∗ w ϵ , x ∈ R d , t > 0 , w ϵ t x , t = Δ ( u ε + ε ) m 2 − ∇ ⋅ w ϵ ∇ z ϵ , x ∈ R d , t > 0 , z ϵ = K ∗ u ϵ , x ∈ R d , t > 0 , u ϵ x , 0 = u 0 ϵ x ≥ 0 , w ϵ x , 0 = w 0 ϵ x ≥ 0 , x ∈ R d (2.1)

with u0ϵ and w0ϵ being the convolution of u_o and w_o with a sequence of mollifiers and ‖u0ϵ‖1=‖u‖1=M1 and ‖w0ϵ‖1=‖w‖1=M2 . Then the uniform a priori estimate for solutions to (2.1) is given if m₁ and m₂ are suitably large, thus global weak solution or even free energy solution exists by letting ∈ tends to 0.

By virtue of the local existence of strong solution for only one population chemotaxis system (see [43, Proposition 4.1]), one obtains:

Lemma 2.1

Let m₁, m₂ > 1. Then there exists Tmaxϵ∈(0,∞] , denoting the maximal existence time such that (2.1) has a unique nonnegative strong solution (uϵ,wϵ)∈Wp2,1(QT)2 with some p > 1,where QT=ℝd×(0,T) with T∈(0,Tmaxϵ] and

Wp2,1(QT):={u∈Lp(0,T;W2,p(ℝd))∩W1,p(0,T;Lp(ℝd))}.

Moreover, if Tmaxϵ<∞ then

limt→T maxϵ‖uϵ(⋅,t)‖∞+‖wϵ(⋅,t)‖∞=∞.

Now we recall the Hardy-Littlewood-Sobolev (HLS) inequality which we frequently use later (see [31] or [32, Chapter 4]).

Lemma 2.2

Let 0 < λ < d, and let the Riesz potential I_λ(h) of a function h be defined by

Iλ(h)(x)=1|x|d−λ∗h=∫ℝdh(y)|x−y|d−λdy,x∈ℝd.

Then for h∈Lκ1(ℝd) and for κ1,κ2>1 with 1κ2=1κ1−λd , then there exists a sharp constant CHLS=CHLS(d,λ,κ1)>0 such that

‖Iλ(h)‖κ2≤CHLS‖h‖κ1.

An equivalent form of the HLS inequality can be stated that if

1p+1q=1+λd,

and h1∈Lp(ℝd),h2∈Lq(ℝd) with p, q > 1, then there exists a CHLS=CHLS(d,λ,p)>0 such that

∬ R d × R d h 1 ( x ) h 2 ( y ) | x − y | d − λ d x d y ≤ C H L S ∥ h 1 ∥ p ∥ h 2 ∥ q .

Inspired by [46], the global solvability of (2.1) can be achieved based on assumptions on the boundedness for ‖uϵ‖m1 and ‖wϵ‖m2 with some large m₁ and m₂.

Lemma 2.3

Let T∈(0,Tmaxϵ] . Assume that m satisfies

m1m2+2m1m2/d> m1+m2 (2.2)

Suppose that there exists a constant C > 0 such that (u_∈, w_∈) of (2.1) with initial data (u0ϵ,w0ϵ) being the convolution of (u₀, w₀) satisfies

∥ u ϵ ( t ) ∥ m 1 ≤ C and ∥ w ϵ ( t ) ∥ m 2 ≤ C for t ∈ ( 0 , T ) . (2.3)

Then there exists a constant C=C(d,m1,m2,u0ϵ,w0ϵ)>0 such that

‖(uϵ(t),wϵ(t))‖r≤C for r∈[1,∞) and t∈(0,T) (2.4)

and

‖(vϵ(t),zϵ(t))‖r+‖(∇vϵ(t),∇zϵ(t))‖r≤C for r∈(1,∞] and t∈(0,T). (2.5)

Proof

We split the proof into three steps.

The choices of p and q. We first show that there exist p¯>1,q¯>1,r1>1 and r₂ > 1 such that for some p>p¯ p>p¯ and q>q¯ one has
{m1+1,ifm1≥d2,m2≥d2,max{m1+1,(m1−1)(m2−1)dd−2m2,m1(d−2)2m2},ifm1≥d2,m2<d2,max{m1+1,dm12+d−2m1d−2m1,m1(d−2)2m2},ifm1<d2,m2≥d2,max{m1+1,dm12+d−2m1d−2m1,(m1−1)(m2−1)dd−2m2,m1(d−2)2m2},ifm1<d2,m2<d2, (2.6)

1r1<1−d−2(q+m2−1)d, (2.7)

1r1>max1−1m2,d−2d⋅pp+m1−1, (2.8)

1r2>d−2d⋅1p+m1−1, (2.9)

1r2<min1m1,1−d−2d⋅qq+m2−1 (2.10)

and
pm1−1r11−d2+(p+m1−1)d2m1+1m2−1+1r11−d2+(q+m2−1)d2m2<2d (2.11)

as well as
1m1−1r21−d2+(p+m1−1)d2m1+qm2−1+1r21−d2+(q+m2−1)d2m2<2d. (2.12)

In order to prove this claim let us first pick r₁ > 1 and r₂ > 1 fulhlling
r1<mindd−2,m2m2−1 (2.13)

and
r2>m1, (2.14)

and let
q:=m2(p−1)m1+1. (2.15)

In (2.15), p > m₁ + 1 implies q > m₂ + 1. The assertions in (2.6)-(2.7) and (2.9) hold by choosing sufficiently large p≥p¯ with some p¯>1 and q≥q¯ with some q¯>1 .

To see the possible choice of r₁ satisfying (2.7)-(2.8), we first observe that 1−1m2≥d−2d⋅pp+m1−1 is true for any p > 1 if m2≥d2 and 1r1>1−1m2 holds by (2.13) and 1−1m2<1−d−2(q+m2−1)d for any q > 1. Thus the asserted r₁ can be actually found. When m2<d2 , one has 1r1>d−2d⋅pp+m1−1>1−1m2 The first inequality is guaranteed by (2.13) and the second is due to
d − 2 d ⋅ p p + m 1 − 1 > 1 − 1 m 2 ⟺ 1 m 2 − 2 d p > ( m 1 − 1 ) ( m 2 − 1 ) m 2 ⟺ p > ( m 1 − 1 ) ( m 2 − 1 ) d d − 2 m 2

by (2.6) if m2<d2 . Moreover, from (2.15) and (2.6), d−2d⋅pp+m1−1<1−d−2(q+m2−1)d . Therefore, one can also choose r<1 satisfying (2.7)-(2.8) in the case m2<d2 .

Similar to the choice of r₂, if m1≥d2 then it follows from (2.14) that 1r2<1m1≤1−d−2d⋅qq+m2−1 , in which (2.9)-(2.10) can be satished due to d−2d⋅1p+m1−1<1m1 . If m1<d2 (2.6)implies d−2d⋅1p+m1−1<1−d−2d⋅qq+m2−1<1m1 and the assertion is true.

Since (2.2) ensures that
m1/m2-m1< 2m1/d-1,

then
pm1−1r11−d2+(p+m1−1)d2m1+1m2−1+1r11−d2+(q+m2−1)d2m2=pm1−1r11+(p−1)d2m1+1m2−1+1r11+(q−1)d2m2=pm1−1r11+(p−1)d2m1+1m2−1+1r11+(p−1)d2m1=p+m1m2−m1p+2m1d−1⋅2d<2d,

and
1m1−1r21−d2+(p+m1−1)d2m1+qm2−1+1r21−d2+(q+m2−1)d2m2=qm2+1m1−11+(p−1)d2m1=p+m1m2−m1p+2m1d−1⋅2d<2d,

which implies (2.11)-(2.12).
Inequalities for both u and w. For p > 1 and q > 1, we test (2.1)₁ by u∈p−1 and integrate to find that
1pddt∫Rduϵpdx=−(p−1)∫Rduϵp−2∇uϵ⋅(∇(uϵ+ϵ)m1−uϵ∇vϵ)dx≤−4m1(p−1)(p+m1−1)2∫Rd|∇uϵp+m1−12|2dx−p−1p∫RduϵpΔvϵdx=−4m1(p−1)(p+m1−1)2∫Rd|∇uϵp+m1−12|2dx+p−1p∫Rduϵpwϵdx

with -∆v_∈ = w_∈, and similarly,
1qddt∫ℝdwϵqdx≤−4m2(q−1)(q+m2−1)2∫ℝd|∇wϵq+m2−12|2dx+q−1q∫ℝduϵwϵqdx

holds by multiplying (2.1)₃ by wϵq−1 and ∆z_∈ = u_∈. Then
1pddt∫Rduϵpdx+1qddt∫Rdwϵqdx+4m1(p−1)(p+m1−1)2∫Rd|∇uϵp+m1−12|2dx+4m2(q−1)(q+m2−1)2∫Rd|∇wϵq+m2−12|2dx≤p−1p∫Rduϵpwϵdx+q−1q∫Rduϵwϵqdx, (2.16)

where
∫ℝduϵpwϵdx≤∫ℝduϵpr1dx1r1∫ℝdwϵr′1dx1r′1 (2.17)

and
∫ℝduϵwϵqdx≤∫ℝduϵr2dx1r2∫ℝdwϵqr′2dx1r′2 (2.18)

by HolderÖs inequality with r1,r2>1,r′1=r1r1−1 and r′2=r2r2−1 . We begin with estimating the right sides of (2.17)-(2.18) based on the choices of p, q, r₁ and r₂ in Step 1. The assumption (2.6) ensures
pr1>m1, (2.19)

and
pr1<(p+m1−1)dd−2 (2.20)

by (2.8). Then by a variant of the Gagliardo-Nirenberg inequality (see [45, Lemma 6]),
‖φ‖k2≤C2r+m−1‖φ‖k11−σ‖∇φr+m−12‖22σr+m−1 (2.21)

with m≥1,k1∈[1,r+m−1] and 1≤k1≤k2≤(r+m−1)dd−2 with d ≥ 3, σ=r+m−121k1−1k21d−12+r+m−12k1−1 we pick r = p, m = m₁, k₁ = m₁, k₂ = pr₁ in (2.21) and use (2.19)-(2.20) to find
∫ℝduϵpr1dx1r1=‖uϵ‖pr1p≤C‖uϵ‖m1p(1−σ)‖∇uϵp+m1−12‖2p2σp+m1−1

with
σ=p+m1−121m1−1pr11d−12+p+m1−12m1∈(0,1),

where invoking (2.3) we further obtain
∫ℝduϵpr1dx1r1≤C‖∇uϵp+m1−12‖2pm1−1r11d−12+p+m1−12m1.

Likewise, (2.7)-(2.8) warrants that
m2<r′1<(q+m2−1)dd−2,

which allows one to make use of the Gagliardo-Nirenberg inequality and the upper bound for wm2 to estimate
∫ℝdwϵr′1dx1r′1=‖wϵ‖r′1≤C‖∇wϵq+m2−12‖21m2−1r′11d−12+q+m2−12m2.

Then
(∫Rduϵpr1dx)1r1(∫Rdwϵr1′dx)1r1′≤C‖∇uϵp+m1−12‖2pm1−1r11d−12+p+m1−12m1⋅‖∇wϵq+m2−12‖21m2−1r1′1d−12+q+m2−12m2=C‖∇uϵp+m1−12‖2pm1−1r11d−12+p+m1−12m1⋅‖∇wϵq+m2−12‖21m2−1+1r11d−12+q+m2−12m2. (2.22)

To estimate the right side of (2.18), we use (2.10) and (2.9) to obtain
m1<r2<(p+m1−1)dd−2.

Then the Gagliardo-Nirenberg inequality implies
∫ℝduϵr2dx1r2≤C‖∇uϵp+m1−12‖21m1−1r21d−12+p+m1−12m1

by (2.3). We also obtain
m2<qr′2<(q+m2−1)dd−2

by (2.10) and (2.15), and choose r=q,m=m2,k1=m2,k2=qr'2 in (2.21) to see that
(∫Rdwϵqr2′dx)1r2′=‖wϵ‖qr2′q≤C‖wϵ‖m2q(1−σ)‖∇wϵq+m2−12‖2q2σq+m2−1≤C‖∇wϵq+m2−12‖2qm2−1r2′1d−12+q+m2−12m2=C‖∇wϵq+m2−12‖2qm2−1+1r21d−12+q+m2−12m2

with
σ=q+m2−121m2−1qr′21d−12+q+m2−12m2.

Then
(∫Rduϵr2dx)1r2(∫Rdwϵqr2′dx)1r2′≤C‖∇uϵp+m1−12‖21m1−1r21d−12+p+m1−12m1‖∇wϵq+m2−12‖2qm2−1+1r21d−12+q+m2−12m2,

which combines with (2.16) and (2.22) ensures that
1pddt∫ℝduϵpdx+1qddt∫ℝdwϵqdx+4m1(p−1)(p+m1−1)2∫ℝd|∇uϵp+m1−12|2dx+4m2(q−1)(q+m2−1)2∫ℝd|∇wϵq+m2−12|2dx≤p−1p∫ℝduϵpr1dx1r1∫ℝdwϵr′1dx1r′1+q−1q∫ℝduϵr2dx1r2∫ℝdwϵqr′2dx1r′2≤C‖∇uϵp+m1−12‖2pm1−1r11d−12+p+m1−12m1⋅‖∇wϵq+m2−12‖21m2−1+1r11d−12+q+m2−12m2+C‖∇uϵp+m1−12‖21m1−1r21d−12+p+m1−12m1⋅‖∇wϵq+m2−12‖2qm2−1+1r21d−12+q+m2−12m2. (2.23)
Boundedness for u_∈ and W_∈ in L^p - and L^q - spaces. Let γ1>0,γ2>0 be such that γ1+γ2<2 . For ∈ > 0, a direct application of Young’s inequality implies that
αγ1βγ2≤ϵ(α2+β2)+C. (2.24)

(2.24)

o

From Step 1, there exist some p>p¯ and q>q¯ with some p¯>1 1 and q¯>1 1 such that
pm1−1r11d−12+p+m1−12m1+1m2−1+1r11d−12+q+m2−12m2<2

and
1m1−1r21d−12+p+m1−12m1+qm2−1+1r21d−12+q+m2−12m2<2,

where
1pddt∫ℝduϵpdx+1qddt∫ℝdwϵqdx+2m1(p−1)(p+m1−1)2∫ℝd|∇uϵp+m1−12|2dx+2m2(q−1)(q+m2−1)2∫ℝd|∇wϵq+m2−12|2dx≤C (2.25)

by (2.23)-(2.24). One may invoke the Gagliardo-Nirenberg inequality with u1=M1 and w1=M2 and Young’s inequality to obtain
1p∫ℝduϵpdx=1p‖uϵ‖pp≤C‖∇uϵp+m1−12‖2p−11d−12+p+m1−12≤2m1(p−1)(p+m1−1)2∫ℝd|∇uϵp+m1−12|2dx+C

and
1q∫ℝdwϵqdx≤2m2(q−1)(q+m2−1)2∫ℝd|∇wϵq+m2−12|2dx+C

by the fact that
p−11d−12+p+m1−12<2 and q−11d−12+q+m2−12<2.

Writing y(t)=1p∫ℝduϵpdx+1q∫ℝdwϵqdx , we obtain from (2.25) that
y′(t)+y(t)≤Cfort∈(0,T).

Then
‖uϵ(t)‖p≤C and ‖wϵ(t)‖q≤Cfort∈(0,T),

which implies that (2.4) holds for any r>max{p¯,q¯} . Together with an interpolation with L¹ any intermediate space can be obtained.
The regularities of v and z. As
vϵ=K∗wϵ=cd∫ℝdwϵ(y)|x−y|d−2dy,zϵ=cd∫ℝduϵ(y)|x−y|d−2dy,

an application of the HLS inequality ensures that for r∈(d/(d−1),∞) , we have
‖|∇vϵ|‖r≤Cd(d−2)I1(wϵ)r≤Cwϵdr/(d+r),‖|∇zϵ|‖r≤Cuϵdr/(d+r). (2.26)

Furthermore, observing that the Calderon-Zygmund inequality yields the existence of a constant C = C(r) > 0 with r e (1, ∞) such that
‖∂xi∂xjvϵ‖r≤Cwϵr,‖∂xi∂xjzϵ‖r≤Cuϵr,(1≤i,j≤d),

we combine this with (2.4), (2.26) and the Morrey’s inequality to see that
‖(vϵ(t),zϵ(t))‖r+‖(∇vϵ(t),∇zϵ(t))‖r≤C for r∈(1,∞] and t∈(0,T).

Thus we finish our proof.

Upon the boundedness arguments in Lemma 2.3, we obtain a global weak solution by letting a subsequence of ∈ approaches to 0.

Lemma 2.4

Under the same assumption in Lemma 2.3, there exists C > 0 independent of e such that strong solution (u_∈, W_∈) of (2.1) satisfies

‖(uϵ(t),wϵ(t))‖∞≤C for all t∈(0,T). (2.27)

Moreover, there exists a global weak solution (u, w) of (1.1) which also satisfies a uniform estimate.

Proof

Relying on Lemma 2.3, we apply the Moser’s iteration technique to obtain a priori estimate for the solution in L^∞. Then this local solution can be extended globally in time from the extensibility criterion in Lemma 2.1, which indeed establishes (2.27), see [45, Proposition 10]. Moreover, testing the first equation in (2.1) by ∂ t ( u ϵ + ϵ ) m 1 and integrating over ℝ^d, we make use of Young’s inequality to see that

We then test the the first equation in (2.1) by u_∈ to find that

12ddt∫ℝduϵ2dx+4m1(m1+1)2∫ℝd∇(uϵ+ϵ)m1+122dx≤C(m1)‖uϵ‖L∞(QT)2∫ℝdΔvϵ2dx. (2.29)

Here, we obtain from (2.28), (2.29) and the regularities of initial data and (u, v, w, z) obtained in Lemma 2.3 and (2.27) that

∫ℝd∇(uϵ+ϵ)m12≤C,

where C does not depend on ∈. Then ∇uϵm1∈L∞((0,T);L2(ℝd)), and ∇wϵm2∈L∞((0,T);L2(ℝd)) is ture by similar procedure. All in all, there exists (u, v, w, z) with the regularities given in Definition 1.1 such that, up to a subsequence, ϵn→0

uϵn→ustrongly inC([0,T);Llocp(ℝd))and a.e. inℝd×(0,T),∇uϵnm1⇀∇um1weakly-∗inL∞((0,T);L2(ℝd)),wϵn→wstrongly inC([0,T);Llocp(ℝd))and a.e. inℝd×(0,T),∇wϵnm2⇀∇wm2weakly-∗inL∞((0,T);L2(ℝd)),vϵn(t)→v(t)strongly inLlocr(ℝd)and a.e. in(0,T),∇vϵn(t)→∇v(t)strongly inLlocr(ℝd)and a.e. in(0,T),Δvϵn(t)⇀Δv(t)weakly inLlocr(ℝd)and a.e. in(0,T),zϵn(t)→z(t)strongly inLlocr(ℝd)and a.e. in(0,T),∇zϵn(t)→∇z(t)strongly inLlocr(ℝd)and a.e. in(0,T),Δzϵn(t)⇀Δz(t)weakly inLlocr(ℝd)and a.e. in(0,T),

where p∈(1,∞), r∈(1,∞] and T∈(0,∞) . Since the above convergence can be found in [45, Section 4],we omit the main proof here. Therefore, we have a global weak solution (u, v, w, z) over ℝ^d × (0, T) with T > 0.

The weak solution obtained in Lemma 2.4 is also a free energy solution given in Definition 1.2. The proof comes from [43].

Lemma 2.5

Consider a global weak solution in Lemma 2.4, then it is also a global free energy solution (u, w) of (1.1) given in Definition 1.2.

Proof

Define a weight function

ψ x = 1 , for 0 ≤ | x | ≤ 1 , 1 − 2 ( | x | − 1 ) 2 , for 1 < | x | ≤ 3 2 , 2 ( 2 − | x | ) 2 , for 3 2 < | x | < 2 , 0 , for 3 2 < | x | < 2 , for | x | ≥ 2 ,

and define ψ_l by ψl(x):=ψ|x|l for any x ∈ ℝ^d1 and l = 1, 2,3,... Evidently,

|∇ψl(x)|≤Cl(ψl(x))12and|Δψl(x)|≤Cl2

is valid with some C > 0. Denote

F[uϵ(t),wϵ(t)]:=1m1−1∫ℝd(uϵ+ϵ)m1ψl(x)dx+1m2−1∫ℝd(wϵ+ϵ)m2ψl(x)dx−∫ℝduϵvϵdx=1m1−1∫ℝd(uϵ+ϵ)m1ψl(x)dx−∫ℝduϵvϵdx+1m2−1∫ℝd(wϵ+ϵ)m2ψl(x)dx−∫ℝdwϵzϵdx+∫ℝd∇vϵ⋅∇zϵdx.

Since

1m1−1ddt(uϵ+ϵ)m1ψl−ddt(uϵvϵ)+uϵvϵt=∇⋅(∇(uϵ+ϵ)m1−uϵ∇vϵ)⋅(m1(uϵ+ϵ)m1−1m1−1ψl−vϵ),1m2−1ddt(wϵ+ϵ)m2ψl−ddt(wϵzϵ)+wϵzϵt=∇⋅(∇(wϵ+ϵ)m2−wϵ∇zϵ)⋅(m2(wϵ+ϵ)m2−1m2−1ψl−zϵ)

by testing (2.1)₁ by m1(uϵ+ϵ)m1−1m1−1ψl−vϵ and (2.1)₃ by m2(wϵ+ϵ)m2−1m2−1ψl−zϵ , then the derivative of F[uϵ(t),wϵ(t)] with respect to time is

ddtF[uϵ(t),wϵ(t)]=1m1−1ddt∫Rd(uϵ+ϵ)m1ψl(x)dx−ddt∫Rduϵvϵdx+∫Rduϵvϵtdx+1m2−1ddt∫Rd(wϵ+ϵ)m2ψl(x)dx−ddt∫Rdwϵzϵdx+∫Rdwϵzϵtdx=−∫Rd(∇(uϵ+ϵ)m1−uϵ∇vϵ)⋅∇(m1(uϵ+ϵ)m1−1m1−1ψl−vϵ)dx−∫Rd(∇(wϵ+ϵ)m2−wϵ∇zϵ)⋅∇(m2(wϵ+ϵ)m2−1m2−1ψl−zϵ)dx,

which can be written as

ddtF[uϵ(t),wϵ(t)]=−∫Rd[(uϵ+ϵ)∇(m1m1−1(uϵ+ϵ)m1−1−vϵ)+ϵ∇vϵ]⋅[∇(m1m1−1(uϵ+ϵ)m1−1−vϵ)ψl+(m1m1−1(uϵ+ϵ)m1−1−vϵ)∇ψl+∇vϵ(ψl−1)+vϵ∇ψl]dx−∫Rd[(wϵ+ϵ)∇(m2m2−1(wϵ+ϵ)m2−1−zϵ)+ϵ∇zϵ]⋅[∇(m2m2−1(wϵ+ϵ)m2−1−zϵ)ψl+(m2m2−1(wϵ+ϵ)m2−1−zϵ)∇ψl+∇zϵ(ψl−1)+zϵ∇ψl]dx=−∫RdI1×J1dx−∫RdI2×J2dx. (2.30)

With Uϵ:=m1m1−1(uϵ+ϵ)m1−1−vϵ , we expand the term −∫ℝdI1×J1dx to find that

−∫ℝdI1×J1dx=−∫ℝd(uϵ+ϵ)ψl|∇Uϵ|2dx−∫ℝd(uϵ+ϵ)(Uϵ+vϵ)∇Uϵ⋅∇ψldx−∫ℝd(uϵ+ϵ)(ψl−1)∇Uϵ⋅∇vϵdx−ϵ∫ℝd∇(ψlUϵ)⋅∇vϵdx−ϵ∫ℝd∇(vϵ(ψl−1))⋅∇vϵdx,

where by defining Ωl={x∈ℝd:l<|x|<2l} , upon using Young’s inequality, HÖlder’s inequality and (a+b)m≤2m(am+bm) with a, b > 0 and m > 1, with any η ∈ (0,1) we deduce from |∇ψl|≤Clψl12 and |∇ψl|=Ω¯l that

−∫Rd(uϵ+ϵ)(Uϵ+vϵ)∇Uϵ⋅∇ψldx≤η∫Rd(uϵ+ϵ)ψl|∇Uϵ|2dx+Cηl2(‖uϵ‖2m1−12m1−1+ϵ2m1−1|Ωl|),−∫Rd(uϵ+ϵ)(ψl−1)∇Uϵ⋅∇vϵdx=∫Rd(1−ψl)∇(uϵ+ϵ)m1⋅∇vϵdx+∫Rd(uϵ+ϵ)(ψl−1)|∇vϵ|2dx≤∫Rd(uϵ+ϵ)m1∇ψl⋅∇vϵdx−∫Rd(1−ψl)(uϵ+ϵ)m1Δvϵdx≤∫Rd(uϵ+ϵ)m1wϵ(1−ψl)dx+Cl∫Rd(um1+ϵm1)|∇vϵ|dx,−ϵ∫Rd∇(ψlUϵ)⋅∇vϵdx=−ϵ∫RdψlUϵwϵdx≤ϵ‖wϵ‖L1‖Uϵ‖L∞,−ϵ∫Rd∇(vϵ(ψl−1))⋅∇vϵdx≤ϵ∫Rdwϵvϵ(1−ψl)dx.

The regularities of (u_∈, v_∈, W_∈) from Lemmas 2.3-2.4 assert that

−∫RdI1×J1dx≤−(1−η)∫Rd(uϵ+ϵ)ψl|∇Uϵ|2dx+Cηl2(‖uϵ(t)‖2m1−12m1−1+ϵ2m1−1|Ωl|)+∫Rd(uϵ+ϵ)m1wϵ(1−ψl)dx+Cl∫Rd(uϵm1+ϵm1)|∇vϵ|dx+ϵ‖wϵ‖L1‖Uϵ‖L∞+ϵ∫Rdwϵvϵ(1−ψl)dx≤−(1−η)∫Rd(uϵ+ϵ)ψl|∇Uϵ|2dx+Cηl2(1+ϵ2m1−1|Ωl|)+C∫Rdwϵ(1−ψl)dx+Cl+ϵC.

Doing a similar argument for − ∫ R d I 2 × J 2 d x and integrating (2.30) over time shows that

where as ∈ tends to 0,

F[u(t),w(t)]≤F[u0,w0]−(1−η)∫0t∫Rduψl|m1m1−1∇um1−1−∇v|2−(1−η)∫0t∫Rdwψl|m2m2−1∇wm2−1−∇z|2+C∫0t∫Rd(u0+w0)(1−ψl)+CTηl2+CTlfort∈(0,T)

by the claimed convergence in Lemma 2.4 and a lower semi-continuity of the free energy dissipation. Finally as l→+∞ and η→0

F [ u ( t ) , w ( t ) ] ≤ F [ u 0 , w 0 ] − ∫ 0 t ∫ R d u m 1 m 1 − 1 ∇ u m 1 − 1 − ∇ v 2 − ∫ 0 t ∫ R d w m 2 m 2 − 1 ∇ w m 2 − 1 − ∇ z 2 for t ∈ ( 0 , T ) .

Therefore, (u, w) is a free energy solution by the definition.

3 The free energy functional

Now we concentrate on a deeper analysis of the energy functional F given by

F [ u ( t ) , w ( t ) ] = 1 m 1 − 1 ∫ R d u m 1 d x + 1 m 2 − 1 ∫ R d w m 2 d x − c d H [ u , w ]

with decay property F[u(t),w(t)]≤F[u0,w0] for t > 0, where

H [ u , w ] = ∬ R d × R d u ( x ) w ( y ) | x − y | d − 2 d x d y = ∫ R d u ( x ) I 2 ( w ) ( x ) d x = ∫ R d w ( y ) I 2 ( u ) ( y ) d y .

The estimate for H can be given as follows.

Lemma 3.1

Let η > 0, and let m₁, m₂, m > 1.If

m<d/2 and mm2+2mm2/d≥m+m2, (3.1)

then for any f ∈ L m ( R d ) a n d g ∈ L 1 ( R d ) ∩ L m 2 ( R d ) there holds

|H[f,g]|≤η‖f‖mm+Cη−1m−1‖g‖1mm2+2mm2/d−m−m2(m−1)(m2−1)‖g‖m2m2−2mm2/d(m−1)(m2−1). (3.2)

Moreover, if

m < d / 2 a n d m m 1 + 2 m m 1 / d ≥ m + m 1 , (3.3)

then for any f ∈ L 1 ( R d ) ∩ L m 1 ( R d ) a n d g ∈ L m ( R d ) there holds

|H[f,g]|≤Cη−1m−1‖f‖1m1m+2m1m/d−m1−m(m1−1)(m−1)‖f‖m1m1−2m1m/d(m1−1)(m−1)+η‖g‖mm. (3.4)

Proof

Fixing m ∈ (1, d/2), using H×lder’s inequality with 1m+m−1m=1 and the HLS inequality with λ=2 Lemma 2.2, we find that

H [ f , g ] = ∫ R d f ( x ) I 2 ( g ) ( x ) d x ≤ ∥ f ∥ m ∥ I 2 ( g ) ∥ m m − 1 ≤ C HLS ∥ f ∥ m ∥ g ∥ m d ( d + 2 ) m − d . (3.5)

Since the assumption m+m2≤mm2+2mm2/d ensures that

1<md(d+2)m−d≤m2,

then if g∈L1(ℝd)∩Lm2(ℝd) with m₂ > 1, the following interpolation inequality holds:

‖g‖md(d+2)m−d≤‖g‖1θ1‖g‖m21−θ1

with (d+2)m−dmd=θ1+1−θ1m2,θ1∈(0,1) Hence

|H[f,g]|≤CHLS‖f‖m‖g‖1θ1‖g‖m21−θ1≤η‖f‖mm+Cη−1m−1‖g‖1mm2+2mm2/d−m−m2(m−1)(m2−1)‖g‖m2m2−2mm2/d(m−1)(m2−1),

by Young’s inequality, which implies (3.2). (3.4) can be also proved if (3.3) holds.

We establish several variants to the HLS inequality on the lines L_i, L₂ and the intersection point I.

Lemma3.2

Let m be on L₁, and let f ∈ L m 1 ( R d ) a n d g ∈ L 1 ( R d ) ∩ L m 2 ( R d ) . Then

C*:=supf≠0,g≠0|H[f,g]|‖f‖m1‖g‖12/d‖g‖m21−2/d<∞.

If m is on L₂, and f ∈ L m 1 ( R d ) a n d g ∈ L 1 ( R d ) ∩ L m 2 ( R d ) , then

C⋆:=supf≠0,g≠0|H[f,g]|‖f‖12/d‖f‖m11−2/d‖g‖m2<∞.

In addtion, assume that m is I and f,g)∈L1(ℝd)∩Lmc(ℝd)2 . Then

Cc:=supf≠0,g≠0H[f,g]‖f‖11/d‖f‖mcmc/2‖g‖11/d‖g‖mcmc/2<∞. (3.6)

Proof

If m is on L¹, then m1∈(mc,d/2) and using (3.5) with m = m_i we have

|H[f,g]|≤CHLS‖f‖m1‖g‖m1d(d+2)m1−d≤CHLS‖f‖m1‖g‖12/d‖g‖m21−2/d.

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 L₂. Finally, with the help of the HLS inequality and H×lder’s inequality, we find that

|H[f,g]|≤CHLS‖f‖2dd+2‖g‖2dd+2≤CHLS‖f‖11/d‖f‖mcmc/2‖g‖11/d‖g‖mcmc/2

if m is I. Then the definition of Cc is valid.

Define

M1c=(cdC⋆)−d/2m2/(m2−1)d/2(m1−1)−d(m2−1)/(2m2),

M2c=(cdC*)−d/2m1/(m1−1)d/2(m2−1)−d(m1−1)/(2m1),

and

Mc=2/[cdCc(mc−1)]d/2.

The lower and upper bounds for F in the sets SM1×SM2 are given next.

Lemma 3.3

Let (f, g) satisfy f ∈ S M 1 a n d g ∈ S M 2 . If m is on l₁, then

cdC*m1m1−1m1−1m1m1−1m1−m1m1−1M2c2m1dm1−1−M22m1dm1−1‖g‖m2m2≤Ff,g≤2m1−1‖f‖m1m1+cdC*m1m1−1m1−1m1m1−1m1−m1m1−1M2c2m1dm1−1+M22m1dm1−1‖g‖m2m2 (3.7)

and

inff∈SM1infg∈SM2F[f,g]=0,ifM2∈(0,M2c].

If m is on L₂, then

F[f,g]≥cdC⋆m2m2−1(m2−1)m2m2−1m2−m2m2−1M1c2m2d(m2−1)−M12m2d(m2−1)‖f‖m1m1 (3.8)

and

inff∈SM1infg∈SM2F[f,g]=0,ifM1∈(0,M1c]. (3.9)

If m is I, then

F[f,g]≥(cdCc)2(mc−1)4Mc4d−M12dM22d‖g‖mcmc

F[f,g]≥(cdCc)2(mc−1)4Mc4d−M12dM22d‖f‖mcmc.

Furthermore,

inff∈SM1infg∈SM2F[f,g]=0,ifM1M2∈(0,Mc2]. (3.10)

Proof

By Lemma 3.2, H satisfies

|H[f,g]|≤C∗‖f‖m1‖g‖12d‖g‖m21−2d≤1cd(m1−1)‖f‖m1m1+C∗(cdC∗)1m1−1(m1−1m1)m1m1−1‖g‖12m1d(m1−1)‖g‖m2(1−2d)m1m1−1=1cd(m1−1)‖f‖m1m1+C∗(cdC∗)1m1−1(m1−1m1)m1m1−1‖g‖12m1d(m1−1)‖g‖m2m2.

Then F can be estimated by

F[f,g]=1m1−1‖f‖m1m1+1m2−1‖g‖m2m2−cdH[f,g]≥1m2−1‖g‖m2m2−(cdC∗)m1m1−1(m1−1m1)m1m1−1‖g‖12m1d(m1−1)‖g‖m2m2=(cdC∗)m1m1−1(m1−1m1)m1m1−1(M2c2m1d(m1−1)−M22m1d(m1−1))‖g‖m2m2

and

F[f,g]≤2m1−1‖f‖m1m1+(cdC∗)m1m1−1(m1−1m1)m1m1−1(M2c2m1d(m1−1)+M22m1d(m1−1))‖g‖m2m2.

In the case M₂ < M_2C, since F > 0, then the infimum is nonnegative. Taking

h1(x,t)=M1(4πt)d2e−|x|24t and h2(x,t)=M2(4πt)d2e−|x|24t,

it is obvious that h i ∈ L 1 ( R d ) with ‖ h i ‖ 1 = M i , i = 1 , 2 satisfy

‖hi‖mimi=O(t−d(mi−1)2),

which implies that hi∈SMi and that F[h₁, h₂] tends to 0 as t→∞ . Therefore,

inff∈SM1infg∈SM2F[f,g]=0.

If m is on L₂, we have (3.8) by the HLS inequality and H×lder’s inequality, and take h_i above to see (3.9). If m is I, since

|H[f,g]|≤CcM11dM21d‖f‖mcmc2‖g‖mcmc2≤1cd(mc−1)‖f‖mcmc+M12dM22dcd(mc−1)Mc4d‖g‖mcmc

|H[f,g]|≤M12dM22dcd(mc−1)Mc4d‖f‖mcmc+1cd(mc−1)‖g‖mcmc

by Young’s inequality, then F satisfies

F[f,g]≥1mc−1‖f‖mcmc+1mc−1‖g‖mcmc−cdCcM11dM21d‖f‖mcmc2‖g‖mcmc2≥(cdCc)2(mc−1)4(4(cdCc(mc−1))2−M12dM22d)‖g‖mcmc

F[f,g]≥(cdCc)2(mc−1)4(4(cdCc(mc−1))2−M12dM22d)‖f‖mcmc.

One Finally obtains from

F[f,g]≤2mc−1‖f‖m1m1+(cdCc)2(mc−1)4(4(cdCc(mc−1))2+M12dM22d)‖g‖mcmc

that (3.10) is true by taking f = h₁ and g = h₂.

The characterization of non-zero minimizers of F in SM1×SM2 on critical lines and point is the goal in this subsection. If m is I, the existence of global minimizers is guaranteed in the follow. The proof is inspired by [6, Proposition 3.5].

Theorem 3.4

Let m be I. Then there exist a pair of nonnegative, radially symmetric and non-increasing functions ( (f*,g*)∈L1(ℝd)∩Lmc(ℝd)2 such that

H[f*,g*]=Cc.

In addition, there exists a minimizer ( (f,g)∈SM1×SM2 of F if M₁ = M₂ = M_c, which satisfies

1 R 0 d ζ x − x 0 R 0 d / ( d − 2 ) , i f x ∈ B x 0 , R 0 , 0 , i f x ∈ R d ∖ B x 0 , R 0

with some R_o > 0 and x_o ∈ ℝ^d, where ζ is the unique positive radial classical solution to the Lane-Emden equation

− Δ ζ = m c − 1 m c ζ 1 / ( m c − 1 ) , x ∈ B ( 0 , 1 ) , ζ = 0 , x ∈ ∂ B ( 0 , 1 ) .

Proof

We claim that if C_c in (3.6) is obtained by some non-zero f and g, then g = c₀f with some c₀. This is easily verified by the positive definiteness of |x−y|−(d−2) , see [32, Theorem 9.8]. In fact, suppose that there exist a pair of maximizing nonnegative functions (f,g)∈L1(ℝd)∩Lmc(ℝd)2 such that

H[f,g]=Cc‖f‖11d‖f‖mcmc2‖g‖11d‖g‖mcmc2.

Then by [32, Theorem 9.8] and the HLS inequality,

H[f,g]≤H[f,f]⋅H[g,g]≤Cc‖f‖11d‖f‖mcmc2‖g‖11d‖g‖mcmc2. (3.11)

However, (3.11) is an equality if and only if g = c₀f with some constant c₀. Note that

Cc=supf≠0H[f,f]‖f‖12d‖f‖mcmc,f∈L1(ℝd)∩Lmc(ℝd). (3.12)

The existence of a maximizing nonnegative, radially symmetric and non-increasing f^* with ‖f*‖1=‖f*‖mc=1 for (3.12) has been given in [6, Proposition 3.3]. So choosing g^* = f^*, then H[f*,g*]=Cc and the first conclusion has been proved.

To derive minimizers for F in the situation M₁ = M₂ = M_c, with f := M_cf^* and g := M_cf^* we have (f,g)∈SM1×SM2 with ‖f‖1=‖f‖mc=Mc,‖g‖1=‖g‖mc=Mc . After a careful computation we infers that by the definition of M_c and thus (f, g) is a non-zero global minimizer of F in SM1×SM2 . The precise 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 L₁, we assert that there is no non-zero minimizer of F in SM1×SM2 if M₂ = 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-Lagrange equalities which consist of the Lane-Emden system on the assumption that minimizers of its free energy exist (see Theorem 3.6).

Lemma 3.5

Let M₁, M₂, ρ > 0,and let m_l > l and m₂ > l. Consider α Lane-Emden system

− Δ ϑ ( x ) = m 1 − 1 m 1 ς 1 m 2 − 1 ( x ) , x ∈ Ω 1 = R d , − Δ ς ( x ) = m 2 − 1 m 2 ϑ 1 m 1 − 1 ( x ) , x ∈ Ω 2 = B ( 0 , ρ ) , ς ( x ) = 0 , x ∈ R d ∖ Ω 2 . (3.13)

Then (3.13) does not admit any nonnegative and non-trivial classical solution ( ϑ , ς ) ∈ L 1 / ( m 1 − 1 ) ( R d ) ∩ L m 1 / ( m 1 − 1 ) ( R d ) × L 1 / ( m 2 − 1 ) ( R d ) ∩ L m 2 / ( m 2 − 1 ) ( R d ) with ‖ ϑ 1 / ( m 1 − 1 ) ‖ 1 = M 1 a n d ‖ ς 1 / ( m 2 − 1 ) ‖ 1 = M 2 provided that m is on L₁.

Proof

Let

q:=1m1−1∈2d−2,dd−2.

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 is not yet known as far as we know. Here, we assert that there exists no non-trivial classical solution for (3.13) if m is on L₁.

Consider the following properties: Suppose that ω∈C2(ℝd) is non-trivial and satisfies Δw≤0,x∈ℝd . Then

ω(x)≥C|x|2−d,|x|≥1 (3.14)

by the strong maximum principle (see [40, Proposition 3.4]). Relying on the finiteness of ‖ϑ‖q , we have the following contradiction: For R > l,

M1≥∫B(0,R)ϑq=cd∫0R∫Sd−1ϑq(r,θ)rd−1dS(θ)dr,

where one combines with the fact that Δϑ≤0 for x ∈ Ω₁ = ℝ^d and (3.14) to see that

M 1 ≥ C ∫ 1 R r d − 1 + q ( 2 − d ) d r = C ∫ 1 R r d m 1 + 2 − 2 d m 1 − 1 − 1 d r = C ( m 1 − 1 ) d m 1 + 2 − 2 d R d m 1 + 2 − 2 d m 1 − 1 − 1 → ∞ as R → ∞

due to m₁ > m_c = 2 - 2/d. So (3.13) has no non-trivial and nonnegative classical solution.

Theorem 3.6

Let m be on L₁. For all M₂ < M_2c, then F does not admit any non-zero minimizer in SM1×SM2 .

Proof

The left inequality in (3.7) in Lemma 3.3 makes sure that there exists no minimizer if M₂ < M_2c. Thus we only consider M₂ = M_2c and prove it by contradiction.

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*)∈SM1×SM2 is a minimizer of F in the sense that F[f*,g*]=0 . Then
1m1−1‖f*‖m1m1+1m2−1‖g*‖m2m2=cdH[f*,g*]≤cdC*‖f*‖m1‖g*‖12/d‖g*‖m21−2/d≤1m1−1‖f*‖m1m1+cdC*m1m1−1m1−1m1m1m1−1‖g*‖12m1d(m1−1)‖g*‖m21−2dm1m1−1=1m1−1‖f*‖m1m1+1m2−1M2c−2m1d(m1−1)‖g*‖12m1d(m1−1)‖g*‖m2m2=1m1−1‖f*‖m1m1+1m2−1M2c−2m1d(m1−1)M22m1d(m1−1)‖g*‖m2m2=1m1−1‖f*‖m1m1+1m2−1‖g*‖m2m2 (3.15)

by the HLS inequality, Young’s inequality, the definition of M_2c and M₂ = M_2c. As a consequence of (3.15), we obtain that
‖f*‖m1m1=1m2−1M2c−2m1d(m1−1)‖g*‖12m1d(m1−1)‖g*‖m2m2=1m2−1M2c−2m1d(m1−1)M22m1d(m1−1)‖g*‖m2m2=1m2−1‖g*‖m2m2 (3.16)

and
H[f*,g*]=C*‖f*‖m1‖g*‖12/d‖g*‖m21−2/d=m1cd(m1−1)(m2−1)‖g*‖m2m2.
The Euler-Lagrange equalities. Let f and g be symmetric rearrangement of f^* and g^*. Then (f, g) ∈ SM1×SM2 satisfies
‖f‖m1m1=‖f*‖m1m1=1m2−1‖g*‖m2m2=1m2−1‖g‖m2m2 (3.17)

and
H[f,g]≥H[f*,g*]

by (3.16) and the Riesz rearrangement properties [31, Lemma 2.1]. Obviously, F[f, g] = 0 and (f, g) is also a minimizer of F. Note that
cdH[f,g]=m1m1−1‖f‖m1m1=m1(m1−1)(m2−1)‖g‖m2m2. (3.18)

Given Ω 10 = { x ∈ R d : f ( x ) = 0 } and Ω 1 + = { x ∈ R d : f ( x ) > 0 } and introduce ϕ1∈C0∞(ℝd) with with ϕ1(x)=ϕ1(−x) and
ψ1(x)=f(x)M1ϕ1(x)−1M1∫ℝdf(x)ϕ1(x)dx.

R^d

Then for f∈SM1 and fix ϵ∈(0,ϵ0:=M1(2‖ϕ1‖∞)−1) , there holds
‖f+ϵψ1‖1=M1

and
f + ϵ ψ 1 = f 1 + ϵ M 1 ϕ 1 ( x ) − 1 M 1 ∫ R d f ( x ) ϕ 1 ( x ) d x ≥ f 1 − 2 ‖ ϕ 1 ‖ ∞ ϵ M 1 ≥ 0 ,

which implies that f+ϵψ1∈SM1 . Moreover, supp ( (ψ1)⊂Ω¯1+ Then
F[f+ϵψ1,g]−F[f,g]ϵ=1m1−1∫Ω1+(f+ϵψ1)m1−fm1ϵ−∫ℝdK∗g(x)ψ1(x)dx.

fli₊ R^d

According to F [ f + ϵ ψ 1 , g ] ≥ F [ f , g ] , as ϵ → 0 , Lebesgue’s dominated convergence theorem shows that
∫ℝdm1m1−1fm1−1(x)−K∗g(x)ψ1(x)dx≥0.

By replacing -ψ₁ by ψ₁, one also obtains from above to see that
∫ℝdm1m1−1fm1−1(x)−K∗g(x)ψ1(x)dx=0,

where
0=1M1∫ℝdm1m1−1fm1−1(x)−K∗g(x)f(x)ϕ1(x)dx−1M12∫ℝdf(x)ϕ1(x)dx⋅∫ℝdm1m1−1fm1(x)−K∗f(x)g(x)dx=1M1∫ℝdm1m1−1fm1−1(x)−K∗g(x)f(x)ϕ1(x)dx

by (3.18). For any choice of symmetric test function ϕ1∈C0∞(ℝd) , we also obtain
m1m1−1fm1−1(x)−K∗g(x)=0a.e.inΩ¯1+. (3.19)

Now we intend to extent above equality to the whole space. Denote ϕ1∈C0∞(ℝd) with ϕ 1 ( x ) = ϕ 1 ( − x ) and ϕ₁ ≥ 0. Define
ψ1(x)=ϕ1−f(x)M1∫ℝdϕ1(x)dx. (3.20)

Then for f∈SM1 and fix ϵ∈0,M1‖ϕ1‖∞|supp(ϕ1)|−1 , we have f+ϵψ1∈SM1 due to ‖f+ϵψ1‖1=M1 and
f+ϵψ1≥f1−ϵM1∫ℝdϕ1(x)dx≥0inΩ¯1+

and outside Ω¯1+ since ϕ₁ > 0. Following above similar arguments, one has
∫ℝdm1m1−1fm1−1(x)−K∗g(x)ψ1(x)dx≥0,

in which we make use of the definition of ψ₁ in (3.20) to see that
∫ℝdm1m1−1fm1−1(x)−K∗g(x)ϕ1(x)dx≥0.

Then
m1m1−1fm1−1(y)−K∗g(x)≥0a.e. inℝd.

_m^m-₁ f^m1—1(y) — K * g(x) > 0 a.e. in R^d.

Hence for almost every x ∈ Ω₁₀,
m1m1−1fm1−1(y)=0=K∗g(x),

which together with (3.19) implies that
m1m1−1fm1−1=K∗g(x)a.e.inℝd. (3.21)

For g, arguing similarly as above and we Define Ω 20 = { x ∈ R d : g ( x ) = 0 } and Ω 2 + = { x ∈ R d : g ( x ) > 0 } and introduce φ2∈C0∞(ℝd) with ϕ2(x)=ϕ2(−x) and
ψ2(x)=g(x)M2ϕ2(x)−1M2∫ℝdg(x)ϕ2(x)dx.

Then for g∈SM2 and fix ϵ∈(0,M2(2‖ϕ2‖∞)−1) , there holds g+ϵψ2∈SM2 .Then
F[f,g+ϵψ2]−F[f,g]ϵ=1m2−1∫Ω2+(g+ϵψ2)m2−gm2ϵdy−∫ℝdK∗f(y)ψ2(y)dy,

where by Lebesgue’s dominated convergence theorem again,
∫ℝdm2m2−1gm2−1(y)−K∗f(y)ψ2(y)dy≥0,

and replacing -ψ₂ by ψ₂, it follows that
∫ℝdm2m2−1gm2−1(y)−K∗f(y)ψ2(y)dy=0.

Then (3.17) and (3.18) imply that
0=1M2∫ℝdm2m2−1gm2−1(y)−K∗f(y)g(y)ϕ2(y)dy−1M22∫ℝdg(y)ϕ2(y)dy⋅∫ℝdm2m2−1gm2(y)−K∗f(y)g(y)dy=1M2∫ℝdm2m2−1gm2−1(y)−K∗f(y)g(y)ϕ2(y)dy+2m1M22(d−2m1)‖g‖m2m2∫ℝdg(y)ϕ2(y)dy=1M2∫ℝdm2m2−1gm2−1(y)−K∗f(y)+2m1‖g‖m2m2M2(d−2m1)g(y)ϕ2(y)dy

on L₁. Therefore,
m2m2−1gm2−1−K∗f+2m1M2(d−2m1)‖g‖m2m2=0a.e.inΩ¯2+. (3.22)

To extend the whole space, we repeat the previous argument for f. Denote ϕ2∈C0∞(ℝd) with ϕ2(x)=ϕ2(−x) and ϕ2≥0 . Define
ψ2(x)=ϕ2−g(x)M2∫ℝdϕ2(x)dx.

Then for g∈SM2 and fix ϵ∈0,M2‖ϕ2‖∞|supp(ϕ2)|−1 we have g+ϵψ2∈SM2 due to g + ϵ ψ 2 1 = M 2 and g + ϵ ψ 2 ≥ 0 , as well as
∫ℝdm2m2−1gm2−1(y)−K∗f(y)ψ2(y)dy≥0.

Then taking account of the definition of ψ₂, we see that
m2m2−1gm2−1(y)−K∗f(y)+2m1M2(d−2m1)‖g‖m2m2≥0a.e. inℝd,

which together with (3.22) implies that
m2m2−1gm2−1=K∗f−2m1M2(d−2m1)‖g‖m2m2+a.e.inℝd. (3.23)

Since g is radially symmetric and non-increasing, there exists ρ ∈ (0, ∞] such that
Ω2+⊂B(0,ρ) and Ω20⊂ℝd\B(0,ρ),

and from (3.23) we obtain
m2m2−1gm2−1=K∗f−2m1M2(d−2m1)‖g‖m2m2a.e.inB(0,ρ).

Hence such symmetric non-increasing minimizer (f,g)∈SM1×SM2 of F satisfies the following Euler-Lagrange equalities
m1m1−1fm1−1(x)=K∗g(x)a.e.inℝd,m2m2−1gm2−1(x)=K∗f(x)−2m1M2(d−2m1)‖g‖m2m2a.e.inB(0,ρ). (3.24)
The regularities of minimizer. From (3.24)₁, one invokes the HLS inequality in Lemma 2.2 to see for g∈L1(ℝd)∩Lm2(ℝd) that
f∈Lp(ℝd)withp∈d(m1−1)d−2,d(m1−1)m2d−2m2,

where once more using the HLS inequality again, one concludes that
K∗f∈Lq(ℝd)withq∈d(m1−1)d−2m1d(m1−1)m2d−2m1m2,ifd>2m1m2,d(m1−1)d−2m1,∞,ifd≤2m1m2.

In particular, K∗f∈Lm2m2−1(ℝd) since m1+m2=2m1/d+m1m2≤2m1m2/d+m1m2 and
m2m2−1∈d(m1−1)d−2m1,d(m1−1)m2d−2m1m2+.

Consequently, gm2−1∈Lm2m2−1(ℝd) , which excludes ρ = ∞ in (3.24)2. Henceρ < ∞ and
m2m2−1gm2−1(x)=K∗f(x)−2m1M2(d−2m1)‖g‖m2m2,if|x|<ρ,0,if|x|>ρ

by the monotonicity of g. Moreover, a bootstrap argument ensures that
(f,g)∈(L∞(ℝd))2.

Letting ϑ := f m 1 − 1 a n d ς := g m 2 − 1 , we readily infer from (3.24)₁ that
ϑ(x)=m1−1m1K∗ς1m2−1(x)a.e.inℝd,

and invoke [21, Theorem 9.9] to have ϑ∈W2,r(B(0,ρ)) with r ∈ (m₁, ∞) and -Δϑ=m1-1m1ς1m2-1 a.e. x∈ℝd . Furthermore, from the expression for ς such as
m2−1m2K∗ϑ1m1−1(x)−2m1(m2−1)m2M2(d−2m1)‖ς‖m2/(m2−1)m2/(m2−1),x∈B(0,ρ),

by means of the regularity of ∂ and [21, Lemma 4.2], we obtain ς∈C2(B(0,ρ)) with −Δς=m2−1m2ϑ1m1−1 in B(0, ρ) and [21, Lemma 4.1] ensures that ς∈C1(ℝd) . Then ς(x)=0 if |x|=ρ and ζ is a classical solution to
−Δς(x)=m2−1m2ϑ1m1−1(x),x∈B(0,ρ),ς(x)=0,x∈∂B(0,ρ). (3.25)

With the smoothness of 9, [21, Lemma 4.2] applies so as to assert that ϑ∈C2(ℝd) and
−Δϑ(x)=m1−1m1ς1m2−1(x),x∈ℝd. (3.26)
Contradiction. (3.25)-(3.26) 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 L₁, which makes a contradiction.

Remark 3.7

Let m be on L₂, there exists no non-zero minimizer for F in SM1×SM2 with M₁ ≤ M_1c

4 The global existence

This section deals with the global solvability of (1.1) in the subcritical case. We first present a local existence and extensibility criterion of free energy solutions to (1.1). Note that this theorem also provides the simultaneous blow-up argument in Section 5.

Theorem 4.1

Let m₁, m₂ > 1. Under assumption (1.2) on the initial data (u₀, w₀) with u 0 1 = M 1 , w 0 1 = M₂, then there exists Tmax∈(0,∞] and a free energy solution (u, w) over ℝ^d × (0, T_max) of (1.1) such that either Tmax=∞ or Tmax<∞ and

limt→Tmax‖u(⋅,t)‖∞+‖w(⋅,t)‖∞=∞. (4.1)

Moreover, let m be subcritical or critical. Then if Tmax<∞

limt→Tmax‖u(⋅,t)‖m1=limt→Tmax‖w(⋅,t)‖m2=∞. (4.2)

Proof

For (u₀, w₀) 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 uniformly bound in the 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 one other with some constants.

Since

1m1−1∫ℝdum1+1m2−1∫ℝdwm2≤cdH[u,w]+F[u0,w0], (4.3)

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

|H[f,g]|≤η‖f‖mm+Cη−1m−1‖g‖1mm2+2mm2/d−m−m2(m−1)(m2−1)‖g‖m2m2−2mm2/d(m−1)(m2−1) (4.4)

for some f∈Lm(ℝd) and g∈L1(ℝd)∩Lm2(ℝd) with η > 0. If m₁ < d/2, choosing m = m₁ in (4.4), then

1m1−1∫ℝdum1+1m2−1∫ℝdwm2≤cdη‖u‖m1m1+cdCη−1m1−1M2m1m2+2m1m2/d−m1−m2(m1−1)(m2−1)‖w‖m2m2−2m1m2/d(m1−1)(m2−1)+F[u0,w0]≤cdη‖u‖m1m1+cdCη−1m1−1‖w‖m2m2+C

by Young’s inequality, since

m2−2m1m2/d(m1−1)(m2−1)≤m2

if m1m2+2m1/d≥m1+m2 holds. Taking η small enough, we have

∥ u ( t ) ∥ m 1 m 1 ≤ C ∥ w ( t ) ∥ m 2 m 2 + C for t ∈ ( 0 , T max ) (4.5)

and if η is sufficiently large, we see that

∥ w ( t ) ∥ m 2 m 2 ≤ C ′ ∥ u ( t ) ∥ m 1 m 1 + C ′ for t ∈ ( 0 , T max ) . (4.6)

Therefore, (4.2) holds by (4.1), (4.5)-(4.6). In fact, suppose that T_max < ∞ and (4.2) does not hold. Then the finiteness of ‖u(t)‖m1 or ‖w(t)‖m2 ensures both norms are finite by means of (4.5)-(4.6), which actually implies that T_max = ∞ due to Lemma 2.4. This is a contradiction.

However, if m₁ ≥ d/2, we pick m ∈ (1, d/2) such that

m2m2+2/d−1<m<d/2,

and next take interpolation inequality to find that

‖u‖mm≤‖u‖1m1−mm1−1‖u‖m1m1(m−1)m1−1.

Upon

m2−2mm2/d(m−1)(m2−1)<m2,

llull^m <11 ull ^m1-1 Hull^m1-1 l\ llm \| ^y1 \| \m1

then (4.4) implies that

| H [ u , w ] | ≤ η ‖ u ‖ 1 m 1 − m m 1 − 1 ‖ u ‖ m 1 m 1 ( m − 1 ) m 1 − 1 + C η − 1 m − 1 ‖ w ‖ 1 m m 2 + 2 m m 2 / d − m − m 2 ( m − 1 ) ( m 2 − 1 ) ‖ w ‖ m 2 m 2 − 2 m m 2 / d ( m − 1 ) ( m 2 − 1 ) = η M 1 m 1 − m m 1 − 1 ‖ u ‖ m 1 m 1 ( m − 1 ) m 1 − 1 + C η − 1 m − 1 M 2 m m 2 + 2 m m 2 / d − m − m 2 ( m − 1 ) ( m 2 − 1 ) ‖ w ‖ m 2 m 2 − 2 m m 2 / d ( m − 1 ) ( m 2 − 1 ) ≤ η ‖ u ‖ m 1 m 1 + η − 1 m − 1 ‖ w ‖ m 2 m 2 + C (4.7)

with ‖u‖1=M1 and ‖w‖1=M2 . Hence (4.5)-(4.6) are valid by picking suitable η > 0. By the same token, the case m1m2+2m2/d≥m1+m2 is also true for both m₂ < d/2 and m₂ > d/2. The proof is finished.

The global existence result in the subcritical case is the subject of our next theorem.

Theorem 4.2

Let m₁, m₂ > 1. Suppose that the initial data (u₀, w₀) with ‖u0‖1=M1,‖w0‖1=M2 fulfills (1.2). Then if m is subcritical, (1.1) has a global free energy solution given in Definition 1.2.

Remark 4.3

If m₁ > d/2 or m₂ > d/2, the conclusion in Theorem 4.2 holds for all m₂ > 1 or m₁ > 1.

Proof

In the case m 1 m 2 + 2 m 1 / d > m 1 + m 2 and m₁ < d/2, since m2−2m1m2/d(m1−1)(m2−1)<m2 , then Lemma 3.1 warrants that

|H[u,w]|≤12cd(m1−1)‖u‖m1m1+C‖w‖1m1m2+2m1m2/d−m1−m2(m1−1)(m2−1)‖w‖m2m2−2m1m2/d(m1−1)(m2−1)≤12cd(m1−1)‖u‖m1m1+12cd(m2−1)‖w‖m2m2+C

by Young’s inequality. Then substituting (4.3) into above, we have

1m1−1∫ℝdum1dx+1m2−1∫ℝdwm2dx≤12(m1−1)∫ℝdum1dx+12(m2−1)∫ℝdwm2dx+C.

As a corollary,

‖u‖m1≤C and ‖w‖m2≤C. (4.8)

If mi m1≥d2 we recalculate (4.7) carefully and also have (4.8), in which the global existence of free energy solution is immediate from Theorem 4.1. The other case m 1 m 2 + 2 m 2 / d > m 1 + m 2 is similar.

Also on the critical lines, we obtain global existence results reading as

Theorem 4.4

Let m be on L₁, and let (u, w) be a free energy solution of (1.1) with (u₀, w₀) satisfying (1.2) on [0, T_max) with T_max given in Theorem 4.1. If

M2<M2c, (4.9)

then T_max = ∞. The subcritical condition (4.9) will be replaced by M₁ < M_1c on L₂. Moreover, if m is I, one has T_max =∞ if M1M2<Mc2

Proof

We just infer from (1.7) and Lemma 3.3 that

cdC*m1m1−1m1−1m1m1−1m1−m1m1−1M2c2m1dm1−1−M22m1dm1−1‖w‖m2m2≤Fu,w≤Fu0,w0.

Due to (4.9), there exists C > 0 such that for all t ∈ [0, T_max) we have ‖w‖m2≤C . Then the extensibility criterion in Theorem 4.1 makes sure that T_max = ∞. The other cases can be similarly obtained.

5 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 estimate of solutions can be achieved in a straightforward computation.

Lemma 5.1

Let (u₀, w₀) satisfy (1.2), and let (u, w) be a free energy solution of (1.1) on [0, T_max) with T_max (0, ∞]. Then

d d t I ( t ) = G ( t ) f o r a l l t ∈ ( 0 , T max ) ,

where

I(t):=∫ℝd|x|2u(x,t)+w(x,t)dx

and

G ( t ) := 2 d ∫ R d u m 1 ( x , t ) d x + 2 d ∫ R d w m 2 ( x , t ) d x − 2 c d ( d − 2 ) ∬ R d × R d u ( x , t ) w ( y , t ) | x − y | d − 2 d x d y .

Proof

We present a formal computation for the proof of this lemma. Otherwise, one can easily invoke some localisation arguments in [8, Lemma 2.1] or [43, Lemma 6.2] to give a complete rigorous proof. We differentiate the second moment to see that

d d t ∫ R d | x | 2 ( u ( x , t ) + w ( x , t ) ) d x = ∫ R d | x | 2 ( Δ u m 1 − ∇ ⋅ ( u ∇ v ) ) d x + ∫ R d | x | 2 ( Δ w m 2 − ∇ ⋅ ( w ∇ z ) ) d x = 2 d ∫ R d u m 1 ( x , t ) d x + 2 d ∫ R d w m 2 ( x , t ) d x + 2 ∬ R d × R d [ x ⋅ ∇ K ( x − y ) ] u ( x , t ) w ( y , t ) d x d y + 2 ∬ R d × R d [ x ⋅ ∇ K ( x − y ) ] u ( y , t ) w ( x , t ) d x d y .

With K(x)=cd1|x|d−2 we have

2 ∬ R d × R d [ x ⋅ ∇ K ( x − y ) ] u ( x , t ) w ( y , t ) d x d y = − 2 c d ( d − 2 ) ∬ R d × R d ( x − y ) ⋅ x | x − y | d u ( x , t ) w ( y , t ) d x d y = − 2 c d ( d − 2 ) ∬ R d × R d | x | 2 | x − y | d u ( x , t ) w ( y , t ) d x d y + 2 c d ( d − 2 ) ∬ R d × R d x ⋅ y | x − y | d u ( x , t ) w ( y , t ) d x d y = − c d ( d − 2 ) ∬ R d × R d | x | 2 | x − y | d u ( x , t ) w ( y , t ) d x d y − c d ( d − 2 ) ∬ R d × R d | y | 2 | x − y | d u ( y , t ) w ( x , t ) d x d y + 2 c d ( d − 2 ) ∬ R d × R d x ⋅ y | x − y | d u ( x , t ) w ( y , t ) d x d y

and

2 ∬ R d × R d [ x ⋅ ∇ K ( x − y ) ] u ( y , t ) w ( x , t ) d x d y = − c d ( d − 2 ) ∬ R d × R d | x | 2 | x − y | d u ( y , t ) w ( x , t ) d x d y − c d ( d − 2 ) ∬ R d × R d | y | 2 | x − y | d u ( x , t ) w ( y , t ) d x d y + 2 c d ( d − 2 ) ∬ R d × R d x ⋅ y | x − y | d u ( x , t ) w ( y , t ) d x d y .

Combining above equations, it follows that

d d t ∫ R d | x | 2 ( u ( x , t ) + w ( x , t ) ) d x = 2 d ∫ R d u m 1 ( x , t ) d x + 2 d ∫ R d w m 2 ( x , t ) d x − c d ( d − 2 ) ∬ R d × R d | x | 2 + | y | 2 | x − y | d u ( x , t ) w ( y , t ) d x d y − c d ( d − 2 ) ∬ R d × R d | x | 2 + | y | 2 | x − y | d u ( y , t ) w ( x , t ) d x d y + 4 c d ( d − 2 ) ∬ R d × R d x ⋅ y | x − y | d u ( x , t ) w ( y , t ) d x d y = 2 d ∫ R d u m 1 ( x , t ) d x + 2 d ∫ R d w m 2 ( x , t ) d x − 2 c d ( d − 2 ) ∬ R d × R d u ( x , t ) w ( y , t ) | x − y | d − 2 d x d y ,

which readily implies the lemma.

We construct initial data which ensures the non-positivity of G(0).

Lemma 5.2

Let m be critical or super-critical. There exists initial data (u₀, w₀) satisfying (1.2), and fulfilling

∫ R d u 0 1 / ι 1 d x ι 1 ∫ R d w 0 1 / ι 2 d x ι 2 ∫ R d u 0 1 / ι 1 d x ι 1 m 1 + ∫ R d w 0 1 / ι 2 d x ι 2 m 2 > N 0 , i f m 1 m 2 + 2 max { m 1 , m 2 } / d ≤ m 1 + m 2 < m 1 m 2 + 2 / d m 1 m 2 , 2 N 0 , i f m 1 + m 2 ≥ m 1 m 2 + 2 / d m 1 m 2 , (5.1)

and

G(0)<0, (5.2)

where

ι 1 := 2 m 2 ( m 1 + m 2 − m 1 m 2 ) d a n d ι 2 := 2 m 1 ( m 1 + m 2 − m 1 m 2 ) d , (5.3)

N0=d/cd2−2/d21+2/d(d−2)1+ι11+ι2

and G is given in Lemma 5.1.

Proof

Consider the following functions having the same compact support as initial data of form

u0(x)=A1−|x|dad+ι1,x∈ℝd,w0(x)=B1−|x|dad+ι2,x∈ℝd, (5.4)

where A, B > 0 denote the maximum of initial data and a > 0 denotes the size of the corresponding supports. Such constructions in (5.4) are inspired by [44, Section 6] which deals with one population Keller-Segel system.

In the Case 1: m1m2+2max{m1,m2}/d≤m1+m2<m1m2+2m1m2/d , one has

∫ℝdu0m1dx=Am1∫ℝd1−|x|dad+2m1m2(m1+m2−m1m2)ddx=Am1∫ℝd1−|x|dad+2m1m2(m1+m2−m1m2)d−11−|x|dad+dx≤Am1∫ℝd1−|x|dad+dx=cdadAm1/(2d) (5.5)

and

∫ℝdw0m2dx≤cdadBm2/(2d).

For the Case 2: m1+m2>m1m2+2m1m2/d,

∫ℝdu0m1dx≤Am1∫|x|<a1dx=cdadAm1/d,∫ℝdw0m2dx≤cdadBm2/d.

The coupled term can be estimated as

∬ R d × R d u 0 ( x ) w 0 ( y ) | x − y | d − 2 d x d y ≥ min | x | , | y | ≤ a | x − y | − ( d − 2 ) ∫ R d u 0 ( x ) d x ⋅ ∫ R d w 0 ( x ) d x ≥ a − ( d − 2 ) ∫ R d A 1 − | x | d a d + ι 1 d x ⋅ ∫ R d B 1 − | x | d a d + ι 2 d x = c d 2 a d + 2 d 2 ( 1 + ι 1 ) ( 1 + ι 2 ) A B . (5.6)

Since

G(0)≤cdadAm1+cdadBm2−2cd3ad+2(d−2)d2(1+ι1)(1+ι2)AB (5.7)

by (5.5)-(5.6), to show (5.2), it only needs to show the right side of (5.7) is negative in the sense

ABAm1+Bm2a2>N1 (5.8)

with

N1=d22cd2(d−2)1+ι11+ι2

in the Case 1, whereas the right side will be replaced by 2N₁ in the Case 2.

Since

∫ℝdu01/ι1dx=A1/ι1∫ℝd1−|x|dad+dx=cdadA1/ι1/(2d),∫ℝdw01/ι2dx=B1/ι2∫ℝd1−|x|dad+dx=cdadB1/ι2/(2d)

imply that

A=2d∫ℝdu01/ι1dx/cdι1a−ι1d,B=2d∫ℝdw01/ι2dx/cdι2a−ι2d,

then (5.8) can be rewritten as

A B A m 1 + B m 2 a 2 = 2 d ∫ R d u 0 1 / ι 1 d x / c d ι 1 2 d ∫ R d w 0 1 / ι 2 d x / c d ι 2 2 d ∫ R d u 0 1 / ι 1 d x / c d ι 1 m 1 + 2 d ∫ R d w 0 1 / ι 2 d x / c d ι 2 m 2 = 2 d / c d 2 / d ∫ R d u 0 1 / ι 1 d x ι 1 ∫ R d w 0 1 / ι 2 d x ι 2 ∫ R d u 0 1 / ι 1 d x ι 1 m 1 + ∫ R d w 0 1 / ι 2 d x ι 2 m 2 > N 1 or 2 N 1 for the C a s e 2 .

Therefore, we have

∫ R d u 0 1 / ι 1 d x ι 1 ∫ R d w 0 1 / ι 2 d x ι 2 ∫ R d u 0 1 / ι 1 d x ι 1 m 1 + ∫ R d w 0 1 / ι 2 d x ι 2 m 2 > c d / 2 d 2 / d N 1 , if m 1 m 2 + 2 max { m 1 , m 2 } / d ≤ m 1 + m 2 < m 1 m 2 + 2 m 1 m 2 / d , 2 c d / 2 d 2 / d N 1 , if m 1 + m 2 ≥ m 1 m 2 + 2 m 1 m 2 / d ,

which yields G(0) < 0

The blow-up results state that

Theorem 5.3

Let m be critical or super-critical. Then one can find some initial data (u₀, w₀) satisfying (1.2) such that free energy solution (u, w) of (1.1) with (u,w)∣t=0=(u0,w0) blows up in finite time.

Proof

For a given initial data(u₀, w₀) in (5.4) satisfying (5.1), then G(0) < 0 from Lemma 5.2. By the continuity argument, there exists T_* > 0 such that

G ( t ) < G ( 0 ) / 2 for all t ∈ [ 0 , T ∗ ] ,

where from Lemma 5.1, one obtains ddtI(t)<G(0)/2 for all t∈[0,T*] . Integrating by parts, it follows that

I(T*)<I(0)+G(0)T*/2. (5.9)

I(0)=∫ℝd|x|2A1−|x|dad+ι1+B1−|x|dad+ι2dx=A∫|x|≤a|x|21−|x|dadι1dx+B∫|x|≤a|x|21−|x|dadι2dx=cdA∫0a1−rdadι1rd+1dr+cdB∫0a1−rdadι2rd+1dr=(cdad+2AN2)/d+(cdad+2BN3)/d (5.10)

with i_l, i₂ given in (5.3) and

N 2 := ∫ 0 1 1 − r ι 1 r 2 / d d r < ∞ and N 3 := ∫ 0 1 1 − r ι 2 r 2 / d d r < ∞ ,

then inserting (5.7) and (5.10) into (5.9), the right side of (5.9) should be negative if we may fix small a > 0 such that

T*2⋅2cd3ad+2(d−2)d2(1+ι1)(1+ι2)AB−cdadAm1−cdadBm2≥(cdad+2AN2)/d+(cdad+2BN3)/d.

More precisely, if

dT*2⋅[21+2/d(d−2)(1+ι1)(1+ι2)cdd2−2/d⋅∫ℝdu01/ι1dxι1∫ℝdw01/ι2dxι2−∫ℝdu01/ι1dxm1ι1−∫ℝdw01/ι2dxm2ι2]≥2d/cd(1−m1)ι1∫ℝdu01/ι1dxι1adι2N2+2d/cd(1−m2)ι2∫ℝdw01/ι2dxι2adι1N3,

this leads to a contradiction after time T^* since I(t) is always nonnegative for all t > 0. Hence the solutions blow up in finite time.

If m is I, Theorem 5.3 shows that the blow up condition (5.1) can be written as

M1M2M1mc+M2mc>12(d−2)⋅2dcdmc, (5.11)

since

ddtI(t)=G(t)=2(d−2)F[u(t),w(t)]≤2(d−2)F[u0,w0]=G(0)<0

if (5.11) holds, then the second moment will be negative after some time and it contradicts the non-negativity of u and w.

We improve blow-up arguments if m is I by using a different method and summarize the blows up results on the lines L₁, L₂ and intersection point I as

Theorem 5.4

Let m be critical. Suppose that (u, w) is a free energy solution of (1.1) with u01=M1,w01= M₂ fulfilling (1.2).

If m is on L₁, for sufficiently small size of the supports of (u₀, w₀) one asserts that blow up happens if

∫ R d u 0 m 1 / m 2 d x m 2 / m 1 ∫ R d w 0 d x ∫ R d u 0 m 1 / m 2 d x m 2 + ∫ R d w 0 d x m 2 > N 0

with N₀ given in Lemma 5.2.

If m is on L₂, for sufficiently small size of the supports of (u₀, w₀) blow-up solution can be constructed if

∫ R d u 0 d x ∫ R d w 0 m 2 / m 1 d x m 1 / m 2 ∫ R d u 0 d x m 1 + ∫ R d w 0 m 2 / m 1 d x m 1 > N 0 .

If m is I, blow up occurs if

M1M2/(M1mc+M2mc)>Mc2/d/2.

Finally, let (u, w) blow up in finite time T_max. Then T_max < ∞ implies that

limt→Tmax‖u‖m1=limt→Tmax‖w‖m2=∞.

Proof

The asserted blow-up conditions on the lines L_i and L₂ just follow from Lemma 5.2 and Theorem 5.3. If m is I, note that for any M 1 ∗ and M 2 ∗ > 0 such that

M1*M2*/(M1*mc+M2*mc)=Mc2/d/2, (5.12)

there exists nonnegative function (u^*, w^*) with ∥ u ∗ ∥ 1 = M 1 ∗ , ∥ w ∗ ∥ 1 = M 2 ∗ fulfilling F[u^*, w^*] = 0.

This can be seen by the fact that C_c in (3.6) is

Cc=supf≠0H[f,f]‖f‖12/d‖f‖mcmc,f∈L1(ℝd)∩Lmc(ℝd)

from Theorem 3.4. From [6, Proposition 3.3], for any M1*>0 there exists nonnegative, radially symmetric and non-increasing function u*∈L1(ℝd)∩Lmc(ℝd) with u*1=M1* such that

‖u*‖mcmc=Cc−1‖u*‖1−2/dH[u*,u*]. (5.13)

Define w*=M2*/M1*u* . Then w*∈L1(ℝd)∩Lmc(ℝd) with ‖w*‖1=M2* and

F[u*,w*]=0

by (5.12) and the definition of M_c. Then

cdH[u*,w*]=cdM2*/M1*H[u*,u*]=1mc−11+M2*M1*mc‖u*‖mcmc.

Given u0=M1/M1*u* and w0=M2/M2*w* with ‖u0‖1=M1 and ‖w0‖1=M2 , then

F[u0,w0]=1mc−1‖u0‖mcmc+1mc−1‖w0‖mcmc−cdH[u0,w0]=1mc−1M1M1*mc+M2M1*mc−M1M2M1*M2*1+M2*M1*mc‖u*‖mcmc<0,

since

M1M2/(M1mc+M2mc)>M1*M2*/(M1*mc+M2*mc)=Mc2/d/2.

If (u, w) is the corresponding free energy solution with initial data (u_o, w_o), then

F [ u ( t ) , w ( t ) ] ≤ F [ u 0 , w 0 ] < 0 , t > 0

by the decreasing property of F. From Lemma 5.1, it follows that blow up occurs.

To see the simultaneous blow-up phenomenon, from extensibility criterion in Theorem 4.1 we have

C ∥ w ( t ) ∥ m 2 m 2 + C ≤ ∥ u ( t ) ∥ m 1 m 1 ≤ C ′ ∥ w ( t ) ∥ m 2 m 2 + C ′ for t ∈ ( 0 , T max )

with some C > 0 and C'>0 if m is critical. Then all assertions have been proved.

Acknowledgements

Acknowledgment. JAC was supported by the Advanced Grant Nonlocal-CPD (Nonlocal PDEs for Complex Particle Dynamics: Phase Transitions, Patterns and Synchronization) of the European Research Council Executive Agency (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 883363). JAC was also partially supported by the EPSRC grant number EP/P031587/1. JAC acknowledges support through the Changjiang Visiting Professorship Scheme of the Chinese Ministry of Education. KL is partially supported by NSFC (Grant No. 11601516) and by Sichuan Science and Technology Program (Grant No. 2020YJ0060).

Professor José A. Carrillo was a member of the Editorial Advisory Board of ANONA although had no effect on the final decision for the article.

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Received: 2020-12-19

Accepted: 2021-04-26

Published Online: 2021-07-02

This work is licensed under the Creative Commons Attribution 4.0 International License.

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

Abstract

1 Introduction

Definition 1.1

Definition 1.2

Theorem 1.3

Theorem 1.4

2 Approximated system

Lemma 2.1

Lemma 2.2

Lemma 2.3

Proof

Lemma 2.4

Proof

Lemma 2.5

Proof

3 The free energy functional

Lemma 3.1

Proof

Lemma3.2

Proof

Lemma 3.3

Proof

Theorem 3.4

Proof

Lemma 3.5

Proof

Theorem 3.6

Proof

Remark 3.7

4 The global existence

Theorem 4.1

Proof

Theorem 4.2

Remark 4.3

Proof

Theorem 4.4

Proof

5 Blow up

Lemma 5.1

Proof

Lemma 5.2

Proof

Theorem 5.3

Proof

Theorem 5.4

Proof

Acknowledgements

References

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