Asymptotic behavior for critical Patlak–Keller–Segel model and a repulsive–attractive aggregation equation

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Ann. I. H. Poincaré – AN 31 (2014) 81–101

www.elsevier.com/locate/anihpc

Asymptotic behavior for critical Patlak–Keller–Segel model and a repulsive–attractive aggregation equation

Yao Yao

Department of Mathematics, UCLA, Los Angeles, CA, United States

Received 18 April 2012; received in revised form 28 January 2013; accepted 7 February 2013 Available online 8 February 2013

Abstract

In this paper we study the long time asymptotic behavior for a class of diffusion–aggregation equations. Most results except the ones in Section 3.3 concern radial solutions. The main tools used in the paper are maximum principle type arguments on mass concentration of solutions, as well as energy method. For the Patlak–Keller–Segel problem with critical powerm=2−2/d, we prove that all radial solutions with critical mass would converge to a family of stationary solutions, while all radial solutions with subcritical mass converge to a self-similar dissipating solution algebraically fast. For non-radial solutions, we obtain convergence towards the self-similar dissipating solution when the mass is sufficiently small. We also apply the mass comparison method to another aggregation model with repulsive–attractive interaction, and prove that radial solutions converge to the stationary solution exponentially fast.

1. Introduction

Recently there has been a growing interest in the study of nonlocal aggregation phenomena. The most widely studied models are the Patlak–Keller–Segel (PKS) models, which describe the cell movement driven by chemotaxis [22,28,25]. In this paper, we study the PKS equation in dimensiond3 with degenerate diffusion, given by

⎧⎪

⎨

⎪⎩

u_t=u^m+ ∇ ·(u∇c), x∈R^d, t0,

c=u, x∈R^d, t0,

u(x,0)=u₀(x)∈L¹₊ R^d;

1+ |x|² dx

∩L^∞(R^d), x∈R^d,

(1.1)

herem >1 models the local repulsion of cells with anti-crowding effects[30,31]. This model is a generalization of the classical parabolic–elliptic PKS model in 2D, which has been extensively studied over the years (see the review [25,7,17,10,9]).

Throughout this paper, we focus on the critical power m=md :=2−2/d, which produces an exact balance between the diffusion term and the aggregation term when one performs a mass-invariant scaling. To study the

E-mail address:yaoyao@math.ucla.edu.

http://dx.doi.org/10.1016/j.anihpc.2013.02.002

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well-posedness of (1.1), the followingfree energyfunctional (1.2) is an important quantity, where the first term is usually referred to as theentropyand the latter term is referred to as theinteraction energy:

F(u)=

R^d

1

m−1u^m+1

2u(u∗N)

dx. (1.2)

HereN(x)= −_(d₋¹_2)σ_d|x|²⁻^dis the Newtonian potential ind3, withσdthe surface area of the sphereS^d⁻¹inR^d. In[8, Definition 3.1], a notion of free energy solution is defined, which is a weak solution to (1.1) such that the free energy is non-increasing in time and satisfies some inequalities. Indeed, it is formally shown that (1.1) is the gradient flow forFwith respect to the Wasserstein metric (see for example[1]and[14]).

The key observation in[8]is the sharp Hardy–Littlewood–Sobolev inequality, which bounds the interaction term in the free energy functional by the entropy term:

R^d

u(u∗N) dx

C^∗u^2/d_L1(R^d)u^m_Lm(R^d), (1.3)

whereC^∗is a constant only depending on the dimensiond. This inequality suggests that the behavior of the solution depends on its mass. Making use of this inequality, it is proved in[8](and generalized by[4]for more general kernels) that there exists a critical massM_conly depending ond, such that all the solutions with massM < M_cexist globally in time, whereas for all supercritical massM > Mcone can construct a solution which blows up in finite time. Although the global existence/blow-up results are well understood, the asymptotic behavior of the solution has not been fully investigated: this motivates our study. In this paper we study the asymptotic behavior of solutions with massMMc, and try to answer the open questions raised in[8]and[7].

◦ Critical case:

WhenM=M_c, it is proved in[8]that the global minimizers of the free energy functionalFhave zero free energy, and are given by the one-parameter family

u_R(x)= 1 R^du₁ x

R

(1.4) subject to translations. Hereu1is the unique radial classical solution to

m

m−1u^m₁⁻¹+u1=0 inB(0,1),withu1=0 on∂B(0,1). (1.5) It was unknown that whether this family of stationary solutions attract some solutions.

In Section 3.1, we prove the convergence of radial solutions towards this family of stationary solution (see Theo- rem3.2) using a combination of mass comparison and energy method. The mass comparison property is a version of comparison principle on the mass distribution of solutions (see Proposition2.4), which has been introduced in[23].

Although it only works for radial solutions, it provides more delicate control than the energy method: it has been recently used in[3]to prove finite time blow-up of solutions with supercritical mass for a diffusion–aggregation model where Virial identity does not apply. We mention that the mass comparison property has been previously observed for the porous medium equation[32]and PKS models with linear diffusion[21,6].

◦ Subcritical case:

When 0< M < M_c, the free energy solution exists globally in time, as long as its initialL^m-norm is finite[8,4].

Moreover it has been proved in[8]that there exists a dissipating self-similar solution, with the same scaling as the porous medium equation. However it was unknown whether this self-similar solution would attract all solutions in the intermediate asymptotics.

In Section 3.2, we prove algebraic convergence of radial solutions towards this self-similar dissipating solution, given that the initial data is bounded and compactly supported. This is done by constructing explicit barriers in the mass comparison sense.

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For general (non-radial) solutions, the asymptotic behavior of solution to (1.1) is unknown for all mass sizes when m=2−2/d. In Section 3.3, we prove that when the mass is sufficiently small, every solution to (1.1) with compactly supported initial data converges to the self-similar dissipating solution. We mention that for the PKS equation in 2D with linear diffusion, similar results for small-mass solutions are obtained by[11], and recently the small-mass assumption is removed by[15], where they prove that all solutions with subcritical mass converge to the self-similar solution with exponential rate after proper rescaling. Both of these results are based on a spectral gap method, hence cannot be generalized to (1.1) due to the nonlinear diffusion term.

When dealing with non-radial solutions with small mass, our key result is the uniqueness of stationary solution (in rescaled variables), which is proved using a maximum principle type argument (see Theorem3.11). We point out that although it is well known that the global minimizer to the rescaled energy (3.15) is unique [8], there are few results concerning the uniqueness of stationary solutions for nonlocal PDEs, except in the following special cases: the stationary solution to a similar equation is proved to be unique by[12]in the 1D case, and stationary solution to the 2D Navier–Stokes equation (in rescaled variable) is proved to be unique by[20], where their proof is based on the fact that all stationary solutions have the same second moment, thus cannot be applied to (1.1).

In Section 4 we generalize the mass comparison methods to an aggregation equation

u_t− ∇ ·(u∇K∗u)=0, (1.6)

and prove that the radial solution converges towards the stationary solution exponentially fast for a class of kernelK.

Eq. (1.6) appears in various contexts as a mathematical model for biological aggregations[27,31]. In order to capture the biologically relevant features of solutions, it is desirable for the interaction kernelKto be repulsive for short-range interactions and attractive for long-range. Aggregation equations with such kinds of kernel are studied in[2,26,19, 18,24]. In this paper we adopt the kernelK proposed in[19], which has a repulsion component in the form of the Newtonian potentialN and an attraction component satisfying the power law:

K(x)=N(x)+1

q|x|^q, (1.7)

here whenq =0 the second term is replaced by ln|x|. We assume that the attraction part is less singular than the repulsion part at the origin, i.e.q >2−d. In addition, we assumeq2, i.e. the long-range attraction does not grow more than linearly as the distance goes to infinity. Note that in[18]they mostly focus on the caseq2.

The existence and uniqueness of weak solution is established in[18]forq >2−d. In addition, they proved that for any mass there is a unique radial stationary solution which is continuous in its support. While numerical results in[18]suggest that this stationary solution should be a global attractor, there is no rigorous proof. We prove that when 2−d < q2, the mass comparison argument can be easily generalized to (1.6), then we construct barriers in the mass comparison sense to prove that all radial solution converges to the stationary solution exponentially fast.

Outline of the paper.In Section 2 we state a mass comparison result for a general diffusion–aggregation equation.

In Section 3 we apply it to the PKS model with critical power, and obtain some asymptotic results for radial solutions with critical and subcritical mass sizes respectively. In Section 4 we demonstrate that the mass comparison can also be applied to an aggregation model with repulsive–attractive interaction, and prove the asymptotic convergence of solution towards the stationary solution.

1.1. Summary of results

By constructing explicit barriers in the mass comparison sense and using energy method, we obtain the following results for radial solutions of (1.1):

Theorem 1.1.Supposed3andm=2−2/d. Letu(x, t )be the free energy solution to(1.1)with massAand initial datau₀, whereu₀is continuous, radially symmetric and compactly supported. Then the following results hold:

• If 0< A < M_c,u(·, t )converges to the dissipating self-similar solutionu_A ast → ∞, where the Wasserstein distance betweenu(·, t )andu_Adecays algebraically fast ast→ ∞.(Corollary3.8)

• IfA=M_candu₀satisfies∇u^m₀ ∈L²(R^d)in addition to the assumptions above, thenu(·, t )→u_R₀ inL^∞(R^d) ast→ ∞for someR₀>0, whereu_R₀is a stationary solution defined in(1.4).(Theorem3.2)

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For general (possibly non-radial) initial data, we use a maximum principle type method to prove that when the mass is sufficiently small, every compactly supported stationary solution must be radially symmetric. This leads to the following asymptotic convergence result:

Theorem 1.2.Supposed3andm=2−2/d. Letu(x, t )be the free energy solution to(1.1)with mass0< A <

M_c/2being sufficiently small, and the initial datau₀is continuous and compactly supported. Then we have

tlim→∞u(·, t )−uA

p=0 for all1p∞,

whereu_Ais the self-similar dissipating solution defined in(3.17).(Corollary3.13)

In Section 4 we generalize the mass comparison methods to the repulsive–attractive aggregation equation (4.1), and obtain the following asymptotic convergence result for 2−d < q2:

Theorem 1.3.Assume2−d < q2. Letube a free energy solution to(1.6)with initial datau₀and massA, where u₀∈L¹(R^d)∩L^∞(R^d)is non-negative, radially symmetric and compactly supported. In addition, we assume that u₀ is strictly positive in a neighborhood of 0. Let u_s be the unique stationary solution with mass A, as given by Proposition4.2. Then ast→ ∞,u(·, t )converges tou_s exponentially fast in Wasserstein distance.(Theorem4.5) 2. Mass comparison for radial solutions

In this section we consider the following type of diffusion–aggregation equation u_t=c₁u^m+ ∇ ·

u∇

u∗(c₂N+c₃K)+V

, (2.1)

whereN(x)= −_(d₋¹_2)σ_d|x|²⁻^dis the Newtonian potential ind3, withσdthe surface area of the sphereS^d⁻¹inR^d. We make the following assumptions on the kernelK, the potentialV and the coefficients:

(C) c₁, c₃0, andc₂∈Rcan be of any sign.

(K1) Kis radially symmetric.

(K2) K∈L¹(R^d),K0 and is radially decreasing.

(V1) V ∈C²(R^d)is radially symmetric.

For a radially symmetric functionu(x, t ), we define its mass functionM(r, t;u)by M(r, t;u):=

B(0,r)

u(x, t ) dt, (2.2)

and we may write it asM(r, t )if the dependence on the functionuis clear. The following lemma describes the PDE satisfied by the mass function.

Lemma 2.1(Evolution of mass function). Letu(x, t )be a non-negative smooth radially symmetric solution to(2.1).

LetM(r, t )=M(r, t;u)be as defined in(2.2). ThenM(r, t )satisfies

∂M

∂t =c1σ_dr^d⁻¹∂_r ∂_rM σ_dr^d⁻¹

m

+∂_rMc₂M+c₃M˜

σ_dr^d⁻¹ +∂_rM∂_rV , (2.3)

where

M(r, t˜ ;u):=

B(0,r)

u∗Kdx. (2.4)

Proof. Due to divergence theorem and radial symmetry ofu, we have

∂M

∂t =σ_dr^d⁻¹

c₁∂_ru^m+u

∂_r

u∗(c₂N+c₃K)

+∂_rV

. (2.5)

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Note that radial symmetry ofualso gives u(r)= ∂rM

σ_dr^d⁻¹. (2.6)

It remains to write∂r(u∗N)and∂r(u∗K)in terms ofM. For∂r(u∗N), divergence theorem gives

∂_r(u∗N)=

B(0,r)u∗Ndx

σ_dr^d⁻¹ =M(r, t )

σ_dr^d⁻¹. (2.7)

We can similarly obtain

∂r(u∗K)=

B(0,r)u∗Kdx

σ_dr^d⁻¹ =M(r, t )˜

σ_dr^d⁻¹, (2.8)

whereM(r, t˜ ;u):=

B(0,r)u∗Kdx. Plug (2.6), (2.7) and (2.8) into Eq. (2.5), and then we can obtain (2.3). 2 Definition 2.2.Letu₁andu₂be two non-negative radially symmetric functions inL¹(R^d). We sayu₁isless concen- trated thanu₂, oru₁≺u₂, if

B(0,r)

u₁(x) dx

B(0,r)

u₂(x) dx for allr0.

Definition 2.3.Let u₁(x, t) be a non-negative, radially symmetric function inL¹(R^d)∩L^∞(R^d), which is C¹ in its positive set. We sayu₁ is asupersolution of (2.1)in the mass comparison senseifM₁(r, t):=M(r, t;u₁)is a supersolution of (2.3), i.e.M₁(r, t)andM˜₁(r, t):= ˜M(r, t;u₁)satisfy

∂M₁

∂t c1σ_dr^d⁻¹∂_r ∂_rM₁ σ_dr^d⁻¹

m

+∂_rM1

c₂M₁+c₃M˜₁

σ_dr^d⁻¹ +∂_rM1∂_rV , (2.9)

in the positive set ofu₁, whereM(r, t˜ ;u₁)is as defined in (2.4).

Similarly we can define asubsolutionof (2.1) in the mass comparison sense.

Proposition 2.4 (Mass comparison). Suppose m >1, and c1, c2, c3, V ,K satisfy the assumptions (C), (K1), (K2), (V1). Let u1(x, t) be a supersolution and u2(x, t)be a subsolution of (2.1)in the mass comparison sense for t∈ [0, T]. Further assume that bothui are bounded, and ui’s preserve their mass over time, i.e.,

u1(·, t ) dx and

u2(·, t ) dx stay constant for all 0t T . Then their mass functions are ordered for all times: i.e., if u1(x,0)u2(x,0), then we haveu1(x, t)u2(x, t)for allt∈ [0, T].

Proof. LetM_i(r, t)be the mass function foru_i, wherei=1,2. We claim thatM₁(r, t)M₂(r, t)for allr0 and t∈ [0, T], which proves the proposition.

For the boundary conditions ofM_i, note that

⎧⎪

⎨

⎪⎩

M1(0, t )=M2(0, t )=0 for allt∈ [0, T],

rlim→∞

M1(r, t)−M2(r, t)

=

R^d

u1(x,0)−u2(x,0)

dx0 for allt∈ [0, T]. As for initial data, we haveM1(r,0)M2(r,0)for allr0.

For givenλ >0, we define w(r, t):=

M2(r, t)−M1(r, t) e⁻^λt,

whereλis a large constant to be determined later. Suppose the claim is false, thenwattains a positive maximum at some point(r₁, t₁)in the domain(0,∞)×(0, T]. Moreover, since the mass of bothu₁andu₂are preserved over time and thus are ordered, we know that(r₁, t₁)must lie inside the positive set for bothu₁andu₂, whereM_i’s areC_x,t^2,1.

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Sincewattains a maximum at(r₁, t₁), the following inequalities hold at(r₁, t₁):

w_t0 ⇒ ∂_t(M₂−M₁)λ(M₂−M₁), (2.10)

w_r =0 ⇒ ∂_rM₁=∂_rM₂>0, (2.11)

w_rr0 ⇒ ∂_rrM₁∂_rrM₂. (2.12)

Now we will analyze the terms on the right-hand side of (2.9) one by one. For the first term, (2.11) and (2.12) imply that

c₁∂_r ∂_rM₂ σ_dr^d⁻¹

m

−c₁∂_r ∂_rM₁ σ_dr^d⁻¹

m

0 at(r₁, t₁). (2.13)

For the term coming from Newtonian potential, we have

∂rM1

c₂(M₂−M₁)

σ_dr^d⁻¹ =c2u1(r1, t1)(M2−M1)umax|c2|(M2−M1), (2.14) whereu_max:=max{sup_Rd×[0,T]u₁,sup_Rd×[0,T]u₂}is finite by assumption onu₁andu₂.

We next claim

∂_rM₁c3(M˜2− ˜M1)(r1, t1)

σ_dr^d⁻¹ u_maxc₃K1(M₂−M₁)(r₁, t₁). (2.15)

To prove the claim, note thatM˜₂− ˜M₁can be rewritten as M˜2(r1, t1)− ˜M1(r1, t1)=

R^d

(u2−u1)∗K

χB(0,r₁)dx

=

R^d

(u2−u1)(χB(0,r₁)∗K) dx. (2.16)

Note thatK0 is radially decreasing due to assumption (K2), thusχ_B(0,r₁₎∗Kis non-negative, radially decreasing and has maximum less than or equal to K1. Therefore we can use a sum of bump function to approximate χ_B(0,r₁₎∗K, where the sum of the height is less thanK1. Hence

M˜₂(r₁, t₁)− ˜M₁(r₁, t₁)K1sup

x

(M₂−M₁)(x, t₁)= K1(M₂−M₁)(r₁, t₁), which proves the claim (2.15). Finally, for the last term in (2.9), as a result of (2.11) we have

∂_rM₂∂V −∂_rM₁∂V =0. (2.17)

Now we subtract (2.9) with the corresponding equation for the subsolution. Due to the inequalities (2.13), (2.14), (2.15) and (2.17), we obtain that

∂_t(M₂−M₁)u_max

|c₂| +c₃K1

(M₂−M₁).

Hence if we choose λ > u_max(|c₂| +c₃K1) in the beginning of the proof, the inequality above will contra- dict (2.10). 2

Remark 2.5.If bothu₁andu₂are supported in some compact setB(0, R)for 0tT, then the assumption (K2) can be replaced by (K2) as follows:

(K2) K∈L¹_loc(R^d),K0 and is radially decreasing.

The proof under condition (K2) is almost the same, except that there is some change in (2.15). Note that in this case (2.16) still holds, and we can bound the maximum ofχ_B(0,r₁₎∗Kby

B(0,R)Kdx. This yields an inequality similar to (2.15), with theK1in the right-hand side replaced by

B(0,R)Kdx.

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3. Application to critical Patlak–Keller–Segel models

3.1. Convergence towards stationary solution for critical mass

In this subsection, we prove that every radial solution with massM_cand continuous, compactly supported initial data will be eventually attracted to some stationary solution within the family (1.4).

If the initial data is bounded above and has the critical massMc, then it is proved in[8]that the free energy solution to (1.1) exists globally in time. In the next lemma we prove the solution has a global (in time)L^∞bound. In addition, if the initial data is radially symmetric and compactly supported, then the support of the solution would stay uniformly bounded in time.

Lemma 3.1. Suppose d 3 and m = 2 − 2/d. Consider the problem (1.1) with initial data u₀ ∈ L¹₊(R^d;(1+|x|²) dx)∩L^∞(R^d), whereu₀is continuous and has critical massM_c. Then theL^∞-norm of the weak so- lutionu(x, t )is globally bounded, i.e. there existsK₁>0depending onu₀_∞andd, such thatu(·, t )L^∞(R^d)K₁ for allt0.

Ifu₀is radially symmetric and compactly supported in addition to the assumptions above, then there exists some R₂>0, such that{u(·, t ) >0} ⊆B(0, R2)for allt0, whereR₂depend ond andu₀.

Proof. In order to bound theL^∞-norm ofu(·, t ), we first consider Eq. (1.1) with symmetrized initial data, which is described below. Letu(¯ ·, t )be the solution to (1.1) with initial datau^∗₀, whereu^∗₀(·)is the radial decreasing rearrange- ment ofu0. Here theradial decreasing rearrangementof a non-negative functionf is defined as

f^∗(x):=

∞ 0

χ_{_{f >t}_}∗(x) dt. (3.1)

Sinceu¯has a radially symmetric initial data and has massM_c, due to[8], we readily obtain thatu¯ exists globally in time, andu¯is radially symmetric for allt0. We first prove that there is a globalL^∞bound foru.¯

Since ¯u(·,0)∞= u₀∞<∞, we can chooseR₁>0 depending onu₀∞, whereR₁is sufficiently small such thatu^∗₀≺u_R₁, whereu_R₁ is as defined in (1.4). Then the mass comparison result in Proposition2.4yields that

¯

u(·, t )≺u_R₁ for allt0. (3.2)

Now we go back to the original solutionu, and compareuwithu. It is proved in Theorem 6.3 of¯ [23]that u^∗(·, t )≺ ¯u(·, t ) for allt0.

Combining the above two inequalities together, we readily obtain that u^∗(·, t )≺u_R₁ for allt0,

which implies thatu^∗(0, t )uR₁(0)fort0. Note thatu^∗(·, t )is radially decreasing for allt0 by definition, and uR₁ is radially decreasing due to[8]. Hence the above inequality implies that

u(·, t )

∞=u^∗(·, t )

∞uR₁∞=R⁻₁^du1(0) fort0, (3.3)

thusuhas a globalL^∞boundR⁻₁^du1(0), whereu1is as defined in (1.4).

Next we hope to show that ifu0is radially symmetric and compactly supported in addition to the conditions above, the support ofu(·, t )will stay in some compact set for all time. We first prove it for the case whereu0(0) >0. Due to the continuity ofu0, we haveu0is uniformly positive in a neighborhood of 0. This enables us to chooseR2>0 sufficiently large such thatu0u_R₂, whereu_R₂ is as defined in (1.4). Proposition2.4again gives usu(·, t )u_R₂ for allt0, which implies that

suppu(·, t )⊆suppu_R₂=B(0, R2) for allt0.

Ifu₀(0)=0, we claim that after some finite timet₁,u(0, t1)becomes positive, andu(·, t₁)has a compact support, wheret₁depends ond andu₀. Then we can taket₁as the starting time and argue as in the caseu₀(0) >0.

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Now we will prove the claim. This is done by performing the mass comparison betweenuandw, wherewis the solution to the porous medium equation

wt=w^m inR^d× [0,∞), (3.4)

with initial dataw(·,0)=u₀. It can be readily checked thatuis a supersolution of (3.4) in the mass comparison sense, hence Proposition2.4implies thatu(·, t )w(·, t )for allt0, which yields

u(0, t )w(0, t ) and suppu(·, t )⊆suppw(·, t ) for allt0. (3.5) For the porous medium equation (3.4), it is well known that the solution will eventually converge to the self-similar Barenblatt profile (see[32]for example), which has a positive density at 0 for allt0. Hence there exists somet₁0 such thatw(0, t1) >0. In addition,w(·, t₁)has a compact support, due to the finite speed of propagation property of porous medium equation[32]. Therefore (3.5) yields thatu(0, t1)w(0, t1) >0 andu(·, t₁)is compactly supported, which prove the claim. 2

Next we prove that under the conditions in Lemma3.1, every radial solution converges to some stationary solution in the family (1.4) ast→ ∞. To do this we investigate the free energy functional (1.2), and make use of the following result proved in [8]and[4]: Letube a free energy solution (for definition, see[8, Definition 3.1]) to (1.1), then it satisfies the following energy dissipation inequality for almost everyt during its existence:

F u(t )

+ t 0

R^d

u m

m−1∇u^m⁻¹+ ∇N∗u

²dx dtF(u₀). (3.6)

Theorem 3.2.Supposed3andm=2−2/d. Letu(x, t )be the free energy solution to(1.1)with critical massM_c and nonnegative initial datau₀, whereu₀is continuous, radially symmetric and compactly supported, and satisfies

∇u^m₀ ∈L²(R^d). Then there existsR₀>0 depending onu₀ andd, such thatu(·, t )→u_R₀ inL^∞(R^d)ast→ ∞, whereu_R₀is as defined in(1.4).

Proof. Due to Lemma3.1, we obtain the existence of a free energy solution globally in time, which has a globalL^∞ bound. In addition, by treatingu∗N as ana prioripotential in (1.1) and applying the continuity result in[16], we obtain thatu(x, t )is uniformly continuous in space and time in[τ,∞)for allτ >0.

Our preliminary goal is to find a time sequence{tn}which increases to infinity, such thatu(tn)uniformly converges to some stationary solution asn→ ∞. Note thatF(u(·, t ))is non-increasing for almost everyt due to (3.6), and is bounded below ast→ ∞. This enables us to find a time sequence{t_n}which increases to infinity, such that

nlim→∞

R^d

u(t_n) m

m−1∇u(t_n)^m⁻¹+ ∇N ∗u(t_n)

²dx=0. (3.7)

We will slightly abuse the notation and denoteu(tn)byun. Note that{un}is uniformly bounded and equicontinuous, hence Arzelà–Ascoli theorem enables us to find a subsequence of{un}, such that

u_n→u_∞ uniformly inn, (3.8)

whereu_∞is some radially symmetric and continuous function. Moreover, Lemma3.1implies that the support of{u_n} all stays in some fixed compact set, hence we haveu_∞is compactly supported as well, and it has massM_c. We will prove thatu_∞is indeed a stationary solution later.

We first claim that{∇u^m_n}are uniformly bounded inL²(R^d). To prove the claim, note that

R^d

∇u^m_n +un∇N∗un²dxun∞

R^d

un

m

m−1∇u^m_n⁻¹+ ∇N ∗un

²dx,

where the right-hand side is uniformly bounded for alln. In addition, since{u_n}are uniformly bounded and are all supported in someB(0, R), we know

R^du_n|∇N ∗u_n|²dx is also uniformly bounded for alln. Therefore triangle inequality yields the uniform boundedness of

R^d|∇u^m_n|²dx, which proves the claim.

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The uniform boundedness of{∇u^m_n}inL²(R^d)implies that{∇u^m_∞}is inL²(R^d)as well. Moreover, we can find a subsequence of{u_n}(which we again denote by{u_n}for the simplicity of notation), such that

∇u^m_n ∇u^m_∞ asn→ ∞weakly inL¹

R^d_L:R^d

. (3.9)

Using (3.7) and the two convergence properties (3.8) and (3.9), we can proceed in the same way as Lemma 10 in[13]

and prove thatu_∞satisfies

R^d

u_∞ m

m−1∇u^m_∞⁻¹+ ∇N ∗u_∞

²dx=0. (3.10)

Next we claim thatu_∞is a radial stationary solution to (1.1). First note thatu_∞is continuous, compactly supported (since the support ofu(·, t )is uniformly bounded int) and radially symmetric. Moreover, due to (3.10),_m^m₋₁∂ru^m_∞⁻¹=

−∂_r(N ∗u_∞)= −_σ^M(r)

dr^d−¹ 0 holds almost everywhere in the support ofu_∞, whereM(r):=

B(0,r)u_∞(x) dxis the mass function foru_∞. This implies thatu_∞(x)must be radially decreasing, hence it only has one positive component. In other words, the support ofu_∞must coincide withB(0, R0)for someR0. (3.10) then implies _m^m₋₁u^m_∞⁻¹+ N∗u_∞=Cholds inB(0, R0)for some constantC. Due to the fact thatN∗u_∞is once more differentiable thanu_∞, one can iteratively show thatu_∞is indeedC^∞inB(0, R0), hence it is a classical solution to the following equation:

m

m−1u^m_∞⁻¹+u_∞=0 inB(0, R0),

u_∞=0 on∂B(0, R0),

henceu_∞is indeed in the family (1.4).

Next we will prove thatu(·, t)→u_∞uniformly inL^∞(R^d)ast→ ∞. In order to prove this, we make use of the monotonicity of the second moment ofu(·, t )in time. By combining the following Virial identity

d dt

R^d

|x|²u(x, t ) dx=2(d−2)F u(t )

for allt (3.11)

with the fact that the minimizer ofFhas free energy 0, it is shown in[8]that M₂

u(·, t ) :=

R^d

|x|²u(x, t ) dtis non-decreasing int. (3.12)

This implies that any subsequence ofu(·, t )can converge to only one limit: if not, then we can find another sequence {t_n}increasing to infinity, such thatu(t_n)converges to another stationary solutionu_∞uniformly asn→ ∞, where u_∞ is also in the family (1.4). Since u(t_n)are uniformly bounded and uniformly compactly supported, we have M₂[u(t_n)] →M₂[u_∞]. On the other hand for the time sequence{t_n}we have M₂[u(t_n)] →M₂[u_∞], hence (3.12) implies thatu_∞andu_∞must have the same second moment. Since bothu_∞andu_∞are within the family (1.4), they can have the same second moment only if they are the same stationary solution. 2

Remark 3.3.Since the proof is done by extracting a subsequence of time, we are unable to obtain the rate of the convergence. We also point out that the above proof is for radial solution only; for general initial data the difficulty lies in the fact that we are unable to bound the solution in some compact set uniform in time.

3.2. Convergence towards self-similar solution for subcritical mass, radial case

In this subsection, we prove that every radial solution with subcritical mass and compactly supported initial data would converge to some self-similar solution which is dissipating with the same scaling as the solution of the porous medium equation.

Letube the free energy solution to (1.1), with massA∈(0, Mc). Following[32]and[8], we rescaleuaccording to the scaling from porous medium equation:

μ(λ, τ )=(t+1)u(x, t ); λ=x(t+1)⁻^1/d; τ=ln(t+1). (3.13)

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Thenμ(λ,0)=u(x,0), andμ(λ, τ )solves the following rescaled equation μτ=μ^m+ ∇ · μ∇|λ|²

2d

+ ∇ ·

μ∇(μ∗N)

. (3.14)

It is pointed out in Theorem 5.2 of[8]that the free energy associated to the rescaled problem (3.14) is G

μ(·, t ) :=

R^d

m

m−1μ^m+1

2μ(N∗μ)+|λ|²μ 2d

dλ, (3.15)

and for any mass A∈(0, Mc), there is a unique minimizer μ_A of G inZA subject to translation, where ZA:=

{h∈L¹(R^d)∩L^m(R^d): h1=Mand

R^d|x|²h(x) dx∞}. In addition,μ_A is continuous, radially decreasing and has a compact support, andμ_Asatisfies

m m−1

∂

∂rμ^m_A⁻¹+ r

d +M(r;μ_A)

σ_dr^d⁻¹ =0 (3.16)

in its positive set, where the mass functionMis as defined in (2.2).

SinceμA is a stationary solution of (3.14), if we go back to the original scaling,μA gives a self-similar solution of (1.1):

u_A(x, t)=(t+1)⁻¹μ_A x (t+1)^1/d

. (3.17)

It is then asked in[8]and[7]that whether this self-similar solution attracts all global solutions.

We will first prove that all radial solutions to the rescaled equation (3.14) converge toμ_A. The following lemma constructs a family of explicit solutions to (3.14), which all converge toμ_Aexponentially fast asτ → ∞.

Lemma 3.4(A family of explicit solutions). Supposed3andm=2−2/d. For0< A < M_c, we denote byμ_Athe stationary solution of (3.14). Letμ¯ be defined as

¯

μ(λ, τ ):= 1

R^d(τ )μ_A λ R(τ )

, (3.18)

whereR(τ )solves the ODE

⎧⎨

⎩

R(τ )˙ =1 d

1 R^d −1

R, R(0)=R₀,

(3.19) whereR0>0is a constant. Then for anyR0>0,μ(λ, τ )¯ is a weak solution to(3.14).

Proof. Sinceμ¯ is a self-similar function, it can be easily verified thatμ¯ solves the following transport equation

¯

μτ+ ∇ · μ¯R(τ )˙ R(τ )λ

=0.

On the other hand, note that (3.14) can also be written as a transport equation μ_τ= ∇ ·(μv),

where

v= m

m−1∇μ^m⁻¹+λ

d +M(|λ|, τ;μ) σ_d|λ|^d⁻¹

λ

|λ|.

Therefore, to prove thatμ¯ solves (3.14), it suffices to verify that

−R(τ )˙

R(τ )r= m m−1

∂

∂rμ¯^m⁻¹

T₁

+r

d + M(r, τ; ¯μ) σ_dr^d⁻¹

T2

for 0rR(τ ). (3.20)

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Sinceμ¯ is a rescaling ofμ_A, T₁= 1

R^(m⁻^1)d⁺¹

∂

∂rμ^m_A⁻¹ r R(τ )

= 1 R^d⁻¹

∂

∂rμ^m_A⁻¹ r R(τ )

,

where in the last inequality we used the fact thatmis the critical power, i.e. m=2−2/d. For T2 in (3.20), the definition ofμ¯ gives

T₂= 1 R(τ )^d⁻¹

M(_{R(τ )}^r ;μ_A)

σ_d(_{R(τ )}^r )^d⁻¹ for 0rR(τ ).

Now recall thatμ_Asatisfies (3.16) in its positive set, which implies RHS of (3.20)= 1

R(τ )^d⁻¹ m m−1

∂

∂rμ^m_A⁻¹ r R(τ )

+ r

dR(τ )+M(_{R(τ )}^r ;μ_A) σ_d(_{R(τ )}^r )^d⁻¹

+1

d 1− 1 R^d(τ )

r

=1

d 1− 1 R^d(τ )

r

= −R(τ )˙ R(τ )r,

where the last equality comes from the definition ofR in (3.19). This verifies that (3.20) is indeed true, which completes the proof. 2

Next we use the family of explicit solution constructed above as barriers, and perform mass comparison between the real solution and the barriers.

Proposition 3.5. Supposed 3 and m=2−2/d. Let μ(λ, τ ) be a radially symmetric weak solution to (3.14) with mass0< A < M_c, where the initial dataμ(·,0)is nonnegative, continuous and compactly supported. Then as τ→ ∞, the mass function ofμconverges to the mass function ofμ_Aexponentially, i.e.

sup

r

M(r, τ;μ)−M(r, τ;μ_A)Ce⁻^τ,

whereμAis as defined in(3.16), andCdepends ond, Aandμ(·,0).

Proof. Without loss of generality we assume that μ(0,0) >0. (When μ(0,0)=0, from the same discussion in Lemma3.1,μ(0, τ )will become positive after some finite time.) Then we can findR₀₁ sufficiently small and R₀₂sufficiently large, such that

1

R₀₂^d μ_A · R02

≺μ(·,0)≺ 1

R₀₁^d μ_A · R01

,

where in the first inequality we used thatμ(0,0) >0, and in the second inequality we usedμ(·,0)_∞<∞.

Letμ₁(λ, τ )andμ₂(λ, τ )be defined as in (3.18), withR(0)equal toR₀₁andR₀₂respectively. Then Lemma3.4 says that bothμ₁andμ₂are solutions to (3.14). Note that (3.14) is a special case of (2.1), hence the mass comparison result in Proposition2.4holds here as well, which gives

μ2(·, τ )≺μ(·, τ )≺μ1(·, τ ) for allτ 0, or in other words,

M(·, τ;μ₂)M(·, τ;μ)M(·, τ;μ₁) for allτ 0.

It remains to show that sup

r

M(r, τ;μ_i)−M(r;μ_A)Ce⁻^τ fori=1,2.

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Recall that bothμ_i’s are scalings ofμ_Awith scaling coefficientR_i(τ ), hence M(r, τ;μi)−M(r;μA)=

M r R_i(τ );μA

−M(r;μA)

. (3.21)

Sinceμ_Ais bounded and compactly supported, it suffices to show thatR_i(τ )→1 exponentially asr→ ∞. Recall thatR˙_i=¹_d( ¹

R_i^d −1)Riwith initial dataR0ifori=1,2, a simple calculation reveals that|R_i(τ )−1|C_ie⁻^τ, where C_i depends onR_0i. This implies that the right-hand side of (3.21) decays likee⁻^τ, which completes the proof. 2

Making use of the explicit barriers μ₁ andμ₂ constructed in the proof of Proposition3.5, we get exponential convergence ofμ/Atowards theμ_A/Ain thep-Wasserstein metric, which is defined below. Note that the Wasserstein metric is natural for this problem, since as pointed out in[1]and[14], Eq. (1.1) is formally a gradient flow of the free energy (1.2) with respect to the 2-Wasserstein metric.

Definition 3.6. Let μ₁ and μ₂ be two (Borel) probability measures on R^d with finite p-th moment. Then the p-Wasserstein distancebetweenμ₁andμ₂is defined as

Wp(μ1, μ2):= inf

π∈P(μ1,μ2) R^d×R^d

|x−y|^pπ(dx dy) _p¹

,

whereP(μ1, μ2)is the set of all probability measures onR^d×R^dwith first marginalμ1and second marginalμ2. Corollary 3.7.Letd3, andm=2−_d². Letμ(λ, τ )andμ_Abe as given in Proposition3.5. Then for allp >1, we have

W_p μ(·, τ ) A ,μA

A

Ce⁻^τ, whereCdepends ondandμ(·,0).

Proof. The proof is very similar to the proof of Corollary 5.8 in[23], and we will briefly sketch it below for the sake of self-consistency.

First we state a useful property of Wasserstein distance [33]: for two probability densities f₀, f₁ on R^d, the p-Wasserstein distance between them coincides with the solution of Monge’s optimal mass transportation problem.

Namely,

W_p(f₁, f₀)= inf

T#f0=f1

R^d

f₀(x)x−T (x)^pdx ¹

p

, (3.22)

whereT is a “push-forward map” fromR^d toR^d, andT #f₀=f₁stands for “the mapT transportsf₀ontof₁”, in the sense that for all bounded continuous functionhonR^d,

R^d

h(x)f₁(x) dx=

R^d

h T (x)

f₀(x) dx.

Without loss of generality we assume the massA=1. In order to prove thatWp(μ(·, τ ), μA)is small, it suffices to show that there is a push-forward mapT (·, τ )satisfyingT (·, τ )#μ(·, τ )=μA, such that|T (x, τ )−x|Ce⁻^τ holds for allxandτ >0 for someC.

The exact formula forT (x, τ )is a little bit complicated, so we will first look for a push-forward mapT˜that pushes the “subsolution”μ₂to the “supersolution”μ₁, whereμ₂ andμ₁are defined in the proof of Proposition3.5. Note that for eachτ,μ₂andμ₁are exact scalings of each other:

μ1(·, τ )=R₂(τ )^d

R₁(τ )^d μ1 ·R₂(τ ) R₁(τ )

,