We also prove the local existence of weak entropy solutions and a blow-up criterion for general L1∩L∞initial data

PARABOLIC-PARABOLIC KELLER-SEGEL MODEL

Wenting Cong^∗

School of Mathematics Jilin University Changchun 130012, China

and

Department of Physics and Department of Mathematics Duke University

Durham, NC 27708, USA

Jian-Guo Liu

Department of Physics and Department of Mathematics Duke University

Durham, NC 27708, USA

(Communicated by Yuan Lou)

Abstract. This paper investigates the existence of a uniform in time L^∞ bounded weak entropy solution for the quasilinear parabolic-parabolic Keller- Segel model with the supercritical diffusion exponent 0< m <2−²

d in the multi-dimensional spaceR^d under the condition that the L

d(2−m)

2 norm of initial data is smaller than a universal constant. Moreover, the weak entropy solutionu(x, t) satisfies mass conservation whenm >1−²_d. We also prove the local existence of weak entropy solutions and a blow-up criterion for general L¹∩L^∞initial data.

1. Introduction. We study the following quasilinear parabolic-parabolic Keller- Segel model ind≥3:







∂tu= ∆u^m− ∇ ·(u∇v), x∈R^d, t >0,

∂tv= ∆v−v+u, x∈R^d, t >0, u(x,0) =u0(x), v(x,0) = 0, x∈R^d,

(1) where the diffusion exponentm is taken to be supercritical in this paper, i.e. 0<

m <2−²_d.

The Keller-Segel model was firstly presented in 1970 to describe the chemotaxis of cellular slime molds [11][14]. u(x, t) represents the cell density, andv(x, t) represents the concentration of the chemical substance. In this model, cells are attracted by the chemical substance and also able to emit it. Without loss of generality, we supposev(x,0) = 0 which is reasonable with the meaning that there is no chemical substance at the beginning, and then it is generated by cells.

2010Mathematics Subject Classification. Primary: 35K65, 35K59, 92C17.

Key words and phrases. Chemotaxis, fast diffusion, critical space, semi-group theory, global existence.

The first author is supported by NSFC grant 11271154.

∗Corresponding author: Wenting Cong.

307

For 1< m <2−²_d, the associate free energy of problem (1) involves a conservative variational functionuand a non-conservative variational functionv,

F u(·, t), v(·, t)

= 1

m−1 Z

R^d

u^m dx− Z

R^d

uv dx+1 2

Z

R^d

|∇v|² dx+1 2

Z

R^d

v² dx.

Model (1) can be recast into the following mixed conservative and non-conservative gradient flow

ut=∇ ·

u∇δF δu

, vt=−δF δv.

This mixed variational structure is known as the Le Chˆaterlier Principle and it formally possesses the following entropy-dissipation equality

d dtF(t) +

Z

R^d

u

∇ m

m−1u^m−1−v

2

dx+ Z

R^d

|∂tv|² dx= 0.

In the original parabolic-parabolic Keller-Segel model (m= 1, d= 2), there exists a critical mass 8πfor the initial datau₀(x). If the initial massR

R²u₀(x)dx=M <

8π, there exists a global weak non-negative solution [5].

By a natural extension to the quasilinear parabolic-parabolic Keller-Segel model, the diffusion exponent m plays an important role. 0 < m < 1 is called the fast diffusion and m >1 is called the slow diffusion to describe the limiting behaviors of the diffusivity coefficient in the diffusion term ∆u^m=∇ ·(mu^m−1∇u).

When 0 < m < 2− ²_d which is called the supercritical case, the aggregation dominates the diffusion for the high density (largeλ) which leads to the finite-time blow-up [3,4,9,18], and the diffusion dominates the aggregation for the low density (small λ) which leads to the infinite-time spreading [1,18, 20]. Whilem > 2−²_d which is called the subcritical case, the aggregation dominates the diffusion for the low density (smallλ) which prevents spreading, while the diffusion dominates the aggregation for the high density (largeλ) which prevents blow-up [12,19,20].

The model (1) has been widely studied in the slow diffusion case. Sugiyama [19,20] proved the global in time existence of weak solutions without any restriction on the size of the initial date form≥2. Then Ishida and Yokota [12] improved the global existence result fromm≥2 tom >2−²_d. For the blow-up result in the slow diffusion case, Ishida and Yokota [13] proved that every radially symmetric energy solution with large negative initial energy blows up in either finite or infinite time when 1≤m <2−²_d. However, in the fast diffusion case, i.e. 0< m <1, few work has been done for the parabolic-parabolic Keller-Segel model.

In the supercritical case 0< m <2−²_d, there is anL^p space, wherep= ^d(2−m)₂ . Thepis crucial when studying the existence and blow-up results of (1) and almost all the results are related toku0k_Lp(R^d). In fact, this criticalL^pspace is widely used in studying the parabolic-elliptic Keller-Segel models [1,2,20], especiallyp=^d₂ for the original parabolic-parabolic Keller-Segel model (m= 1) inR^d [7].

For 0 < m < 2− ²_d, if ku0k_Lp(R^d) < Cd,m, whereCd,m is a universal constant depending on d and m, then we prove that there exists a global weak solution (u, v) with the properties that u(x, t) preserves mass when 1− ²_d < m < 2− ²_d, and extincts at a finite time when 0< m < 1−²_d. Furthermore, for m >1, this weak solution is also a weak entropy solution satistying energy inequality if the initial second moment is bounded and u0 ∈ L^m(R^d). With the initial condition u0∈ L¹₊∩L^∞(R^d), we can prove that the weak solution is bounded uniformly in time by using bootstrap iterative method(See [2], [16]). With no restriction of the

L^p norm on initial data, we prove the local existence of a weak entropy solution for 1 < m <2− ²_d. This result also provides a natural blow-up criterion that all kuk_Lq(R^d) blow up at exactly the same time forq∈(p,+∞).

The results concerning the finite-time blow-up for the solutions of the Keller-Segel model in multi-dimension have only been proved for its parabolic-elliptic type until Winkler made a breakthrough in [21] to introduce a new method in fully parabolic problem whenm= 1. There is few paper containing the finite time blow-up result for the solutions whenm6= 1. This is still an open problem.

The paper is organized as follows. In Section 2, we define a weak solution and introduce some crucial inequalities about semigroup theory and some lemmas. In Section 3, we proposea prioriestimates of a weak solution. In Section 4, we prove our main theorem about uniformly in time L^∞ bound of weak solutions using a bootstrap iterative method. In Section 5, we construct a regularized problem to prove the existence of a weak solution. Finally, in Section 6, we prove the local existence of weak entropy solutions and a blow-up criterion.

2. Preliminaries. The generic constant will be denoted byC, even if it is different from line to line. At the beginning, we define a weak solution of (1).

Definition 2.1. (Weak solution) Let u₀ ∈ L¹₊(R^d) be the initial data and T ∈ (0,∞). Then (u, v) is a weak solution to (1) if it satisfies

(i) Regularity:

u∈L^∞ 0, T;L¹(R^d)

∩L² 0, T;L²(R^d)

, u^m∈L¹ 0, T;L¹(R^d) ,

∂_tu∈L^p¯

0, T;W^−2,

2(p+1) p+3

loc (R^d)

, p¯= min p+ 1

m , p+ 1

>1, v∈L^∞ 0, T;H¹(R^d)

, ∂_tv∈L²

0, T;W_loc^−2,2(R^d) . (ii) ∀ψ(x)∈C_c^∞(R^d) and any 0< t <∞,

Z

R^d

u(x, t)ψ(x)dx−

Z

R^d

u₀(x)ψ(x)dx= Z t

0

Z

R^d

u^m(x, s)∆ψ(x)dxds +

Z t 0

Z

R^d

u(x, s)∇v(x, s)· ∇ψ(x)dxds, Z

R^d

v(x, t)ψ(x)dx=− Z t

0

Z

R^d

∇v(x, s)· ∇ψ(x)dxds− Z t

0

Z

R^d

v(x, s)ψ(x)dxds +

Z t 0

Z

R^d

u(x, s)ψ(x)dxds.

We use semigroup theory in this paper. The following definition and estimates are standard(See [12,17]). Consider the following Cauchy problem:

∂_th= ∆h−h+f, x∈R^d, t >0,

h(x,0) =h₀(x), x∈R^d. (2)

Definition 2.2. LetT > 0,p≥1, h0 ∈L^p(R^d) andf ∈L² 0, T;L²(R^d) . The functionh(x, t)∈C [0, T];L²(R^d)

given by h(x, t) =e^−te^t∆h0(x) +

Z t 0

e^−(t−s)e^(t−s)∆f(x, s)ds, 0≤t≤T, (3)

is the uniquemild solution of problem (2) on [0, T]. The heat semigroup operator e^t∆ is defined by

e^t∆f

(x, t) :=G(x, t)∗f(x, t), whereG(x, t) is the heat kernel byG(x, t) = ¹

(4πt)^d2

e^−|x|

2 4t .

Using Young’s inequality of the convolution and property of Gamma function, we immediately obtain that

e^t∆f _Lp(

R^d)≤Ct^−d2(¹_q⁻¹_p)kfk_Lq(R^d), ∇e^t∆f

_Lp(

R^d)≤Ct^−12−d2(¹q−¹_p)kfk_Lq(R^d),

where C is a positive constant depending onp, q and d, for any 1≤q≤p≤+∞, f ∈L^q(R^d) and allt >0.

Let 1 ≤ q ≤ p≤ ∞, ¹_q −_p¹ < ¹_d. Assuming f ∈ L^∞ 0,∞;L^q(R^d)

and h0 ∈ L^p(R^d), using two inequalities above and Bochner Theorem in [8, pp.650], we have fort∈[0,∞)

kh(·, t)k_Lp≤ kh0(·)k_Lp+C·Γ 1− 1

q−1 p

d 2

!

kfk_L∞(0,∞;L^q), (4)

k∇h(·, t)k_Lp ≤Ct^−12−d2(¹q−¹_p)kh₀(·)k_Lq+C·Γ 1 2−

1 q−1

p d

2

!

kfk_L∞(0,∞;L^q), (5) whereC is a positive constant depending onp,qandd.

Remark 1. It is well known that the mild solution defined above is also a weak solution. In fact, for any test function φ∈C_c^∞ [0, T)×R^d

, multiplyφ_t to both sides of (3) and integrate over [0, T)×R^d to obtain

Z T 0

Z

R^d

h(x, t)φ_t(x, t)dxdt=− Z

R^d

h₀(x)φ(x,0)dx

− Z T

0

Z

R^d

e^−te^t∆h0(x)

t

φ(x, t)dxdt− Z T

0

Z

R^d

f(x, t)φ(x, t)dxdt

− Z T

0

Z

R^d

Z t 0

e^−(t−s)e^(t−s)∆f(x, s)

t

dsφ(x, t)dxdt

=− Z

R^d

h₀(x)φ(x,0)dx− Z T

0

Z

R^d

f(x, t)φ(x, t)dxdt

− Z T

0

Z

R^d

(∆−Id)h(x, t)φ(x, t)dxdt

=− Z

R^d

h0(x)φ(x,0)dx− Z T

0

Z

R^d

f(x, t)φ(x, t)dxdt +

Z T 0

Z

R^d

∇h(x, t)· ∇φ(x, t)dxdt+ Z T

0

Z

R^d

h(x, t)φ(x, t)dxdt, (6) where in the last equality, we use the regularity in (5).

Then recall the following well-known maximal L^p-regularity result for the heat kernel:

Lemma 2.3. Let 1< p <+∞ andT >0. Then for each f ∈L^p 0, T;L^p(R^d) , problem (2) has a unique solution h(x, t) with h₀(x) = 0 in the L^p 0, T;L^p(R^d) sense. Moreover, there exists a positive constant C_p such that

k∆h(x, t)k

L^p 0,T;L^p(R^d)≤C_pkfk

L^p 0,T;L^p(R^d), (7) for allf ∈L^p 0, T;L^p(R^d)

.

The lemma above is a special case of the famous maximalL^p-regularity Theorem which was proved by Hieber and Pr¨uss in [10]. We can use the maximalL^p result in our paper since the space R^d and elliptical operator ∆ satisfy the conditions of the Theorem 3.1 in [10], and we considerv0(x) = 0. We also refer the readers to a thorough review on maximalL^p-regularity for parabolic equation [15].

The following four lemmas which are proved in [1] are useful for later estimations.

Lemma 2.4. Let 0< m≤2−²_d,p=^d(2−m)₂ . Then for q≥p kuk^q+1_Lq+1(

R^d)≤S_d⁻¹

∇u^m+q−12

2 L²(R^d)

kuk^2−m_Lp(

R^d), (8)

whereS_d is the sharp constant in Sobolev inequality ford≥3.

Moreover, forq≥r > p, we have kuk^q+1_Lq+1≤ 2mq

Cq(m+q−1)²

∇u^q+m−12

2 L²

+C(q, r, d) (kuk^r_Lr)^δ, (9) whereδ= 1 +^1+q−r_r−p >1,

C(q, r, d) =

"

2mq q−r+ 1 + 2(r−p)/d S_d⁻¹Cq(q+m−1)²(q−r+ 1)

#−^d(q−r+1)_2(r−p)

2(r−p)

d(q−r+ 1) + 2(r−p). Lemma 2.5. Let 0< m <2−²_d, p= ^d(2−m)₂ . Then for q≥p andu∈L¹₊(R^d), we have

kuk^q_Lq(R^d)

¹⁺

m−1+ 2 q−1d

≤S_d⁻¹

∇u^q+m−12

2

L²(R^d)kuk

1

q−1(^1+2(q−p)_d )

L¹(R^d) . (10) Lemma 2.6. Assume y(t) ≥0 is a C¹ function for t >0 satisfying y⁰(t) ≤γ− βy(t)^a forγ≥0, β >0 anda >0. Then

(i) fora >1,y(t)has the following hyper-contractive property:

y(t)≤ γ

β _a¹

+ 1

β(a−1)t _a−1¹

, t >0, (ii) for a= 1,y(t)decays as

y(t)≤ γ

β +y(0)e^−βt,

(iii) fora <1, γ= 0, y(t) has the finite time extinction, which means that there exists aText satisfying0< Text≤ ^y_β(1−a)^1−a(0) such that y(t) = 0for all t > Text. Lemma 2.7. Assume f(t)≥0 is a non-increasing function for t >0,y(t)≥0 is aC¹ function fort >0 and satisfiesy⁰(t)≤f(t)−βy(t)^a for some constantsa >1 andβ >0, then for anyt0>0 one has

y(t)≤ f(t₀)

β _a¹

+

β(a−1)(t−t₀) −_a−1¹

, fort > t₀.

With the additional condition thaty(0) is bounded, we have Lemma 2.8which can be proved by contradiction arguments.

Lemma 2.8. Assume y(t) ≥0 is a C¹ function for t >0 satisfying y⁰(t) ≤γ− βy(t)^a forγ >0 andβ >0. Ify(0)is bounded, then

y(t)≤max y(0), γ

β ¹_a!

, t >0, for alla >0.

3. A priori estimates of weak solutions. In this section, we prove Theorem3.1 which is concerninga prioriestimates of weak solutions for (1).

Theorem 3.1. (A priori estimates) Letd≥3,0< m <2−²_d andp= ^d(2−m)₂ . Cp

is the positive constant in (7). Under the assumption that u₀ ∈L¹₊∩L^p(R^d)and η=C_d,m^2−m− ku₀k^2−m_Lp(

R^d)>0, where C_d,m^2−m= ^4mp

S_d⁻¹(m+p−1)²C_p is a universal constant, let (u,v) be a non-negative weak solution of (1). Then u∈L^∞ R+;L^p(R^d)

, u∈ L^p+1 R+;L^p+1(R^d)

and∇u^m+p−12 ∈L² R+;L²(R^d)

. Furthermore, the following a priori estimates hold true:

(i) For0< m <1−²_d,ku(·, t)k_Lp(R^d)has finite time extinction. The extinct time Text satisfies

0< Text≤T0,

whereT0 depends ond, m, η,ku0k_L1(R^d)andku0k_Lp(R^d). (ii) For m= 1−²_d,ku(·, t)k_Lp(R^d)decays exponentially in time

ku(·, t)k_Lp(R^d)≤ ku0k_Lp(R^d)e

− ^Cp(p−1)η

pku0k1/(p−1) L1 (Rd)

t

.

(iii) For1−²_d < m <2−²_d, the solution u(x, t)satisfies mass conservation and ku(·, t)k_Lp(R^d) decays in time

ku(·, t)k_Lp(R^d)≤ ku0k_Lp(R^d)

1 +C(d, m, η,ku₀k_L1,ku₀k_Lp)t

_p(m−1+2/d)^p−1 .

And for any 1≤q≤p,ku(·, t)k_Lq(R^d)decays in time

ku(·, t)k_Lq(R^d)≤ ku₀(·)k

p(q−1) q(p−1)

L^p(R^d)ku₀(·)k

p−q q(p−1)

L¹(R^d)

1 +C(d, m, η,ku₀k_L1,ku₀k_Lp)t

_q(m−1+2/d)^q−1 .

For any p < q <∞,u(x, t)has hyper-contractive property ku(·, t)k_Lq(R^d)≤C

t⁻

(p+−1)(q−p+1) (m−1+2/d)

q−1

q+m−2+2/d+t^{−m−1+2/dq−1 1}q

, where satisfies ^4m(p+)

S_d⁻¹(m+p+−1)²Cp+ − ku0k^2−m_Lp(R^d) ≥ ^η₂, and C is a constant depending onm, d, q, η andku0k_L1(R^d).

Proof. Step 1. (L^pestimate for 0< m <2−²_d). Multiplying the first equation in model (1) bypu^p−1 and integrating it overR^d, we obtain

d

dtku(·, t)k^p_Lp(

R^d)=− 4mp(p−1) (m+p−1)²

∇u^m+p−12 (t)

2 L²(R^d)

−(p−1) Z

R^d

u^p∆v dx. (11) Now we estimate the second term on the right hand side. Using H¨older’s inequality, we have

−(p−1) Z

R^d

u^p∆v dx≤(p−1) Z

R^d

u^p|∆v|dx

≤(p−1)ku(t)k^p_Lp+1(

R^d)k∆v(t)k_Lp+1(R^d). (12) Define

I(t) := (p−1)ku(t)k^p_Lp+1(R^d)k∆v(t)k_Lp+1(R^d). Then (11) turns to

d

dtku(·, t)k^p_Lp(R^d)≤ − 4mp(p−1) (m+p−1)²

∇u^m+p−12 (t)

2

L²(R^d)+I(t). (13) Integrating (13) from 0 tot, it follows that

ku(·, t)k^p_Lp(R^d)≤ku0k^p_Lp(R^d)− 4mp(p−1) (m+p−1)²

Z t 0

∇u^m+p−12 (s)

2

L²(R^d)ds +

Z t 0

I(s)ds. (14)

Next, using H¨older’s inequality and Lemma2.3, we obtain Z t

0

I(s)ds≤(p−1) Z t

0

ku(s)k^p+1_Lp+1(

R^d) ds

p p+1Z t

0

k∆v(s)k^p+1_Lp+1(Rd) ds p+1¹

≤C_p(p−1) Z t

0

ku(s)k^p+1_Lp+1(

R^d)ds, (15)

whereCp is the constant in Lemma2.3. Substituting (15) into (14), we see that ku(·, t)k^p_Lp(R^d)≤ ku0k^p_Lp(R^d)− 4mp(p−1)

(m+p−1)² Z t

0

∇u^m+p−12 (s)

2 L²(R^d)

ds +Cp(p−1)

Z t 0

ku(s)k^p+1_Lp+1(R^d) ds. (16) From Lemma2.4withq=p, then (16) turns to

ku(·, t)k^p_Lp(R^d)≤ ku0k^p_Lp(R^d)

−S⁻¹_d (p−1)Cp

Z t 0

C_d,m^2−m− ku(s)k^2−m_Lp

∇u^m+p−12

2

L² ds, (17) where

C_d,m^2−m= 4mp S_d⁻¹(m+p−1)²Cp

. (18)

By contradiction arguments, we can prove that for allt >0,

ku(·, t)k_Lp(R^d)<ku0k_Lp(R^d)< C_d,m. (19)

Therefore, combining (17) and (19), we obtain ku(·, t)k^p_Lp(

R^d)+η(p−1)Cp

S_d

Z t 0

∇u^m+p−12 (s)

2 L²(R^d)

ds≤ ku0k^p_Lp(

R^d)< Cd,m, i.e.

u∈L^∞ R+;L^p(R^d)

, ∇u^m+p−12 ∈L² R+;L²(R^d) . In the same time, from Lemma2.4, we have

u∈L^p+1 R+;L^p+1(R^d) .

Step 2. (L^p decay estimates). From the fact ku(·, t)k_L1(R^d) ≤ ku0k_L1(R^d) and Lemma2.5withq=p, we have

∇u^p+m−12

2 L²(R^d)≥

kuk^p_Lp(R^d)

¹⁺

m−1+ 2d p−1

S_d⁻¹ku0k

1 p−1

L¹(R^d)

. (20)

Substituting (20) into (17), we see that ku(·, t)k^p_Lp(

R^d)≤ ku0k^p_Lp−Cp(p−1)η ku0k

1 p−1

L¹(R^d)

Z t 0

ku(s)k^p_Lp(

R^d)

1+^{m−1+ 2}_p−1^d

ds. (21) Define

y(t) =ku(·, t)k^p_Lp(

R^d)− ku₀k^p_Lp(

R^d)+C_p(p−1)η ku0k

1 p−1

L¹(R^d)

Z t 0

ku(s)k^p_Lp(

R^d)

1+^{m−1+ 2}_p−1^d

ds.

For any small0>0, we have

y(t+₀) =ku(·, t+₀)k^p_Lp(R^d)− ku0k^p_Lp(R^d)

+Cp(p−1)η ku0k

1 p−1

L¹(R^d)

Z t+₀ 0

ku(s)k^p_Lp(

R^d)

1+^{m−1+ 2}_p−1^d

ds.

Then from two equations above, we obtain that y(t+0)−y(t) =ku(·, t+0)k^p_Lp(

R^d)− ku(·, t)k^p_Lp(

R^d)

+C_p(p−1)η ku0k

1 p−1

L¹(R^d)

Z t+₀ t

ku(s)k^p_Lp(R^d)

¹⁺

m−1+ 2d

p−1 ds. (22)

In the similar way of obtaining (21), integrating fromttot+0instead of integrating from 0 tot, we see that

ku(·, t+0)k^p_Lp− ku(·, t)k^p_Lp+Cp(p−1)η ku0k

1 p−1

L¹(R^d)

Z t+₀ t

ku(s)k^p_Lp

¹⁺

m−1+ 2

p−1d ds≤0.

It means thaty(t) is a non-increasing function in time, i.e.

d

dtku(·, t)k^p_Lp(

R^d)≤ −Cp(p−1)η ku0k

1 p−1

L¹(R^d)

ku(·, t)k^p_Lp

¹⁺

m−1+ 2d

p−1 . (23)

Then we have the conclusion that

(a) for 1−_d² < m <2−²_d,ku(·, t)k_Lp(R^d) decays in time ku(·, t)k_Lp(R^d)≤ ku0k_Lp(R^d)

1 +C(d, m, η,ku0kL¹,ku0kL^p)t

_p(m−1+2/d)^p−1

, (24)

whereC(d, m, η,ku0kL¹,ku0kL^p) =^{Cpη(m−1+2/d)}(ku0k^p_Lp)^{m−1+2/dp−1}

ku0k

p−11 L1

, (b) form= 1−_d², ku(·, t)k_Lp(R^d)decays exponentially in time

ku(·, t)k_Lp(R^d)≤ ku0k_Lp(R^d)e

− ^Cp(p−1)η

pku0k1/(p−1) L1 (Rd)

t

,

(c) for 0< m <1−²_d,ku(t)k_Lp(R^d) has finite time extinction. The extinct time Text satisfies 0< Text≤T0, where T0= ^ku0k

−p(m−1+2/d) p−1

Lp(Rd) ku0k^1/(p−1)_{L1 (Rd}

)

−Cpη(m−1+2/d) .

Step 3. (Hyper-contractive estimate for anyp < q <∞with 1−²_d < m <2−²_d).

L^r estimate with r:=p+ for small enough.

SinceC_d,m^2−m− ku0k^2−m_Lp(

R^d)=η, there exists >0 such that 4m(p+)

S_d⁻¹(m+p+−1)²Cp+

− ku0k^2−m_Lp(R^d)≥ η

2. (25)

In the similar way of obtaining (23), we obtain d

dtku(·, t)k^r_Lr(R^d)≤ −β

ku(t)k^r_Lr(R^d)

1+^m−1+2/d_r−1

, β= η(r−1)Cr

2ku₀k

1

r−1(1+2/d) L¹(R^d)

.

Since 1−²_d < m <2−²_d, from Lemma2.6, we have

ku(·, t)k^r_Lr(R^d)≤C(d, m, η, r,ku0k_L1)t^{−m−1+2/dr−1} . (26) Hyper-contractive estimates ofL^q norm for q≥r.

Combining (9) and (16) withq=p, we have ku(·, t)k^q_Lq(R^d)≤ ku0k^q_Lq(R^d)− 2mq(q−1)

(m+q−1)² Z t

0

∇u^m+q−12 (s)

2 L²(R^d)

ds +C(q, r, d)

Z t 0

ku(·, t)k^r_Lr(R^d)

^δ

ds, (27)

whereδ= 1 +^1+q−r_r−p . Substituting (26) into (27), we obtain ku(·, t)k^q_Lq(

R^d)≤ku0k^q_Lq(

R^d)− 2mq(q−1) (m+q−1)²

Z t 0

∇u^m+q−12 (s)

2 L²(R^d)

ds +C(d, m, η, q,ku0k_L1)

Z t 0

s⁻

(r−1)δ

m−1+2/d ds. (28)

Then in the similar way of obtaining (23), (28) turns to d

dtku(·, t)k^q_Lq ≤ −βˆ

kuk^q_Lq(R^d)

¹⁺

m−1+ 2 q−1d

+C(d, m, η, q,ku0kL¹)t⁻

(r−1)(q−p+1) (m−1+2/d)(r−p),

(29)

where ˆβ = ^2mq(q−1)

(m+q−1)²S_d⁻¹ku0k

1

q−1(^1+2(q−p)d )

L1 (Rd)

. Using Lemma 2.7 and choosing t0 = ₂^t, we obtain that for anyt >0

ku(·, t)k^q_Lq(

R^d)≤C t⁻

(p+−1)(q−p+1)(q−1)

(m−1+2/d)(q+m−2+2/d) +t^{−m−1+2/dq−1}

, (30)

whereC is a constant depending onm, d, q, η andku₀k_L1(R^d), satisfies (25).

Step 4. (L^q decay estimate for any 1 ≤q ≤ pwith 1−_d² < m < 2−_d² ). For 1−²_d < m <2−²_d, by using interpolation inequality and (24), we obtain that for anyt >0,

ku(·, t)k_Lq(R^d)≤ ku0(·)k

p(q−1) q(p−1)

L^p(R^d)ku0(·)k

p−q q(p−1)

L¹(R^d)

1 +C(d, m, η,ku0k_L1,ku0kL^p)t

_q(m−1+2/d)^q−1

. (31)

Step 5. (Mass conservation foru(x, t) when 1−²_d < m <2−²_d).

We take a cut-off function 0≤ψ1(x)≤1, satisfying ψ1(x) =

1, if|x| ≤1, 0, if|x| ≥2, whereψ1(x)∈C_c^∞(R^d).

Define ψ_R(x) := ψ_{1 R}^x

, then we know that lim

R→∞ψ_R(x) = 1, |∇ψ_R(x)| ≤ ^C_R¹ and|∆ψ_R(x)| ≤ ^C_R²₂ forx∈R^d, where C₁ andC₂are positive constants.

From the definition of weak solution foruand taking ψR(x)∈ C_c^∞(R^d) as test function, we have

Z

R^d

u(x, t)ψR(x)dx− Z

R^d

u0(x)ψR(x)dx= Z t

0

Z

R^d

u^m(x, s)∆ψR(x)dxds +

Z t 0

Z

R^d

u(x, s)∇v(x, s)· ∇ψR(x)dxds. (32) For 1− ²_d < m < 1, we can estimate the first term on RHS by using H¨older’s inequality

Z t 0

Z

R^d

u^m(x, s)∆ψR(x)dxds≤ C2

R² Z t

0

Z

B2R

u^m(x, s)dxds

≤ C

R^2−d(1−m) Z t

0

ku(·, s)k^m_L1(B_2R) ds

≤C ku0k_L1(R^d)

R^2−d(1−m) t. (33)

Using young’s inequality, the second term on RHS of (32) goes to Z t

0

Z

R^d

u(x, s)∇v(x, s)· ∇ψ_R(x)dxds≤ C1

R Z t

0

Z

B2R

u(x, s)|∇v(x, s)|dxds

≤ C R

Z t 0

Z

B_2R

u²(x, s)dxds+C R

Z t 0

Z

B_2R

|∇v(x, s)|² dxds. (34)

Recalling the second equation of (1) vt= ∆v−v+u, multiplying it by −∆v and integrating from 0 totand over R^d, we have

1 2 Z

R^d

|∇v(x, t)|² dx+ Z t

0

Z

R^d

|∆v(x, s)|² dxds+ Z t

0

Z

R^d

|∇v(x, s)|²dxds

≤ Z t

0

Z

R^d

|∆v(x, s)|u(x, s)dxds

≤ Z t

0

Z

R^d

|∆v(x, s)|²dxds+ Z t

0

Z

R^d

u²(x, s)dxds

≤C Z t

0

Z

R^d

u²(x, s)dxds, (35)

where the last inequality can be obtained from (7).

From (34) and (35), by using interpolation inequality, H¨older’s inequality and u∈L^p+1 R+;L^p+1(R^d)

, we have Z t

0

Z

R^d

u(x, s)∇v(x, s)· ∇ψR(x)dxds≤ C R

Z t 0

Z

B_2R

u²(x, s)dxds

≤ C R

Z t 0

ku(·, s)k

p−1 p

L¹(B_2R)ku(·, s)k

p+1 p

L^p+1(B_2R)ds

≤ C p,ku₀k_L1(R^d)

R

Z t 0

ku(·, s)k^p+1_Lp+1(B_2R) ds p¹

t^p−1p

≤ C p,ku0k_L1(R^d)

R t^p−1p . (36)

Therefore, collecting (32), (33) and (36) together, it shows that

Z

R^d

u(x, t)ψ_R(x)dx− Z

R^d

u₀(x)ψ_R(x)dx

≤ C(ku0k_L1)

R^2−d(1−m)t+C(p,ku0k_L1) R t^p−1p . Since 2−d(1−m)>0 from 1−²_d < m <1, we have

Z

R^d

u(x, t)dx= Z

R^d

u0(x)dx, as R→ ∞, by the dominated convergence theorem.

For 1≤m <2−²_d, also using interpolation inequality and H¨older’s inequality, we have the following estimate

Z t 0

Z

R^d

u^m(x, s)∆ψR(x)dxds≤ C2

R² Z t

0

Z

B2R

u^m(x, s)dxds

≤ C2

R² Z t

0

ku(·, s)k

p−m+1 p

L¹(B_2R)ku(·, s)k

(m−1)(p+1) p

L^p+1(B_2R) ds

≤C ku0k_L1(R^d)

R²

Z t 0

ku(·, s)k^p+1_Lp+1(B_2R)ds

m−1

p Z t

0

1ds

p−m+1 p

≤C m, p,ku0k_L1(R^d)

R² t^p−m+1p . (37)

Then from (36) and (37), we have

Z

R^d

u(x, t)ψR(x)dx− Z

R^d

u0(x)ψR(x)dx

≤ C p,ku₀k_L1(R^d)

R t^p−1p +C m, p,ku0k_L1(R^d)

R² t^p−m+1p , (38) i.e. R

R^du(x, t) dx=R

R^du0(x)dx, as R→ ∞. Therefore, for 1−_d² < m < 2−²_d, we have mass conservation for u.

4. The uniformly in timeL^∞ estimate of weak solutions. In this section, we prove our main theorem about uniformly in time L^∞ boundness of weak solution by using a bootstrap iterative method.

At the beginning of this section, we prove the following proposition concerning L^q norm estimates of the weak solution for 1< q <∞.

Proposition 1. Letd≥3,0< m <2−²_d andp=^d(2−m)₂ . Ifu0∈L¹₊(R^d)∩L^q(R^d) for1< q <∞ andη =C_d,m^2−m− ku0k^2−m_Lp(R^d)>0, whereC_d,m^2−m = ^4mp

S_d⁻¹(m+p−1)²Cp is a universal constant, let(u, v)be a non-negative weak solution of (1). Thenu(x, t) satisfies for any t >0

ku(·, t)k^q_Lq(

R^d)≤C(p, q,ku0k_L1)ku0k

p(q−1) p−1

L^p(R^d), 1< q≤p, (39) whereC depends onp, q and|u0k_L1(R^d),

ku(·, t)k^q_Lq(

R^d)≤C_u^q, p < q <∞, (40) where C_u^q is a constant depending on d, m, q,ku0k_L1(R^d) andku0k_Lq(R^d), satisfies

4m(p+)

S_d⁻¹(m+p+−1)²C_p+− ku₀k^2−m_Lp(

R^d)≥ ^η₂. Furthermore, for anyt >0

kv(·, t)k_W1,∞(R^d)≤C_v^∞, (41) whereC_v^∞ is a positive constant depending onC_u^d+1.

Proof. Actually, the proof of Proposition 1 is almost the same as the proof of Theorem 3.1, except for the different initial condition u0 ∈ L¹₊(R^d)∩L^q(R^d) for 1< q <∞. Step 1 isL^q estimate foru(x, t) and Step 2 is the uniform estimate for v(x, t). We omit some details which are similar to the proof of Proposition 1 in [2].

Step 1. (L^q estimate for u(x, t))We have obtained the uniform L^p estimate for 0< m <2−²_d in (19)

ku(·, t)k_Lp(R^d)<ku0k_Lp(R^d)< Cd,m, for allt >0.

Then for 1< q≤p, using interpolation inequality, we have ku(·, t)k^q_Lq(

R^d)≤ ku0k

p−q p−1

L¹(R^d)ku0k

p(q−1) p−1

L^p(R^d), (42) which is (39) by taking C(p, q,ku0k_L1) = ku0k

p−q p−1

L¹(R^d). For p < r ≤ q, it is not hard to see thatku(·, t)k_Lr(R^d) ≤ ku0k_Lr(R^d) for any t >0. By the similar way of obtaining (29), we have

d

dtku(·, t)k^q_Lq(

R^d)≤ −β˜ kuk^q_Lq(

R^d)

1+^{m−1+ 2}_q−1^d

+C(q, r, d)

ku₀k^r_Lr(R^d)

δ

, (43)