This paper investigates the existence of a uniform in time L∞ bounded weak solution for the p-Laplacian Keller-Segel system with the su- percritical diffusion exponent 1&lt

A DEGENERATE p-LAPLACIAN 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 Tao Luo)

Abstract. This paper investigates the existence of a uniform in time L^∞ bounded weak solution for the p-Laplacian Keller-Segel system with the su- percritical diffusion exponent 1< p <_d+1^3d in the multi-dimensional spaceR^d under the condition that theL

d(3−p)

p norm of initial data is smaller than a universal constant. We also prove the local existence of weak solutions and a blow-up criterion for generalL¹∩L^∞initial data.

1. Introduction. In this paper, we study the following p-Laplacian Keller-Segel model ind≥3:







∂_tu=∇ ·

|∇u|^p−2∇u

− ∇ ·(u∇v), x∈R^d, t >0,

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

(1) wherep >1. 1< p <2 is called the fastp-Laplacian diffusion, whilep >2 is called the slowp-Laplacian diffusion. Especially, thep-Laplacian Keller-Segel model turns to the original model whenp= 2.

The Keller-Segel model was firstly presented in 1970 to describe the chemotaxis of cellular slime molds [13,14]. The original model was considered in 2D,







∂_tu= ∆u− ∇ ·(u∇v), x∈R², t >0,

−∆v=u, x∈R², t >0, u(x,0) =u₀(x), x∈R².

(2)

2010Mathematics Subject Classification. Primary: 35K65, 35K92, 92C17.

Key words and phrases. Chemotaxis, fast diffusion, critical space, global existence, monotone operator, non-Newtonian filtration.

The first author is supported by NSFC grant 11271154.

∗Corresponding author: Wenting Cong.

687

u(x, t) represents the cell density, and v(x, t) represents the concentration of the chemical substance which is given by the fundamental solution

v(x, t) = Φ(x)∗u(x, t), where

Φ(x) =

−_2π¹ log|x|, d= 2,

1 d(d−2)α(d)

1

|x|^d−2, d≥3,

α(d) is the volume of the d-dimensional unit ball. In this model, cells are attracted by the chemical substance and also able to emit it.

One natural extension of the original Keller-Segel model is the degenerate Keller- Segel model in the multi-dimension withm >1,







∂_tu= ∆u^m− ∇ ·(u∇v), x∈R^d, t >0,

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

(3) which has been widely studied [2,4,7,8,15,22,23,24,25]. Another natural exten- sion is the degeneratep-Laplacian Keller-Segel model in the multi-dimension since the porous medium equation and the p-Laplacian equation are all called nonlinear diffusion equations. Work in these two models has frequent overlaps both in phe- nomena to be described, results to be proved and techniques to be used. The porous medium equation and thep-Laplacian equation are different territories with some important traits in common. The evolutionp-Laplacian equation is also called the non-Newtonian filtration equation which describes the diffusion with the diffusiv- ity depending on the gradient of the unknown. The comprehensive and systematic study for these two equations can be found in V´azquez [27], DiBenedetto [10] and Wu, Zhao, Yin and Li [28].

In thep-Laplacian Keller-Segel model, the exponent pplays an important role.

Whenp=_d+1^3d , if (u, v) is a solution of (1), constructing the following mass invariant scaling foruand a corresponding scaling forv

uλ(x, t) =λu

λ^1dx, λt , vλ(x, t) =λ^1−2dv

λ^1dx, λt

, (4)

then (uλ, vλ) is also a solution for (1) and hencep= _d+1^3d is referred to the critical exponent. For the general exponentp, (uλ, vλ) satisfies the following equation

( u_t=λ(¹⁺¹_d)^p−3∇ ·

|∇u|^p−2∇u

− ∇ ·(u∇v),

−∆v=u.

(5) If 1 +_d¹

p−3<0 which is called the supercritical case, the aggregation dominates the diffusion for high density(largeλ) which leads to the finite-time blow-up, and the diffusion dominates the aggregation for low density(smallλ) which leads to the infinite-time spreading. If 1 +¹_d

p−3 > 0 which is called the subcritical case, the aggregation dominates the diffusion for low density(small λ) which prevents spreading, while the diffusion dominates the aggregation for high density(largeλ) which prevents blow-up. At the end of Section 5, we have the theorem of the existence of a global weak solution for (1) in the subcritical case.

In the supercritical case, there is aL^q space, whereq= ^d(3−p)_p . Theq is crucial when studying the existence and blow-up results of the p-Laplacian Keller-Segel

model and almost all the results are related to the initial data ku0(·)k_Lq(R^d). Also considering model (1), if (u, v) is a solution, then

uλ(x, t) =λu

λ^3−pp x, λt , vλ(x, t) =λ^3−6pv

λ^3−pp x, λt ,

is also a solution of (1). Furthermore, the scaling ofu(x, t) preserves theL^q norm ku_λk_Lq =kuk_Lq. For 1< p < _d+1^3d , if ku₀k_Lq(R^d)< C_d,p, whereC_d,p is a universal constant depending ondand p, then we will show that there exists a global weak solution. Since the initial condition u₀ ∈L¹₊∩L^∞(R^d), we can prove that weak solutions are bounded uniformly in time by using the bootstrap iterative method(See [3], [19]). With no restriction of the L^q norm on initial data, we prove the local existence of a weak solution. This result also provides a natural blow-up criterion for 1< p < _d+1^3d that all kuk_Lh(R^d) blow up at exactly the same time forh∈(q,+∞).

In the subcritical casep > _d+1^3d , there exists a global weak solution of (1) without any restriction of the size of initial data.

In the process of proving the existence of a global weak solution of (1), we combine the Aubin-Lions Lemma with the monotone operator theory. The theory of monotone operators was proposed by Minty [20,21]. Then the theory was used to obtain the existence results for quasi-linear elliptic and parabolic partial differential equations by Browder [5,6], Leray and Lions [17], Hartman and Stampacchia [12], DiBenedetto and Herrero [11].

The paper is organized as follows. In Section 2, we define a weak solution, intro- duce a Sobolev inequality with the best constant and some lemmas. In Section 3, we give thea prioriestimates of our weak solution. In Section 4, we prove the 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 global weak solution. Finally, in Section 6, we discuss the local existence of weak 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) in this paper.

Definition 2.1. (Weak solution) Let u0 ∈L¹₊∩L^∞(R^d) be initial data and T ∈ (0,∞). v(x, t) is given by the fundamental solution

v(x, t) = 1 d(d−2)α(d)

Z

R^d

u(y, t)

|x−y|^d−2 dy.

Then (u, v) is a weak solution to (1) ifusatisfies (i) Regularity:

u∈L^∞ 0, T;L¹₊(R^d)

∩L^p 0, T;W^1,p(R^d)

∩L²

0, T;L^d+22d (R^d) ,

∂tu∈L^p−1p

0, T;W^−2,p−1p (R^d) . (ii) ∀ψ(x, t)∈C_c^∞ [0, T)×R^d

, Z T

0

Z

R^d

u(x, t)ψt(x, t)dxdt= Z T

0

Z

R^d

|∇u(x, t)|^p−2∇u(x, t)· ∇ψ(x, t)dxdt

− 1 2dα(d)

Z T 0

Z

R^d

Z

R^d

∇ψ(x, t)− ∇ψ(y, t)

·(x−y)

|x−y|²

u(x, t)u(y, t)

|x−y|^d−2 dxdydt

− Z

R^d

u0(x)ψ(x,0)dx. (6)

The following lemma is a Sobolev inequality with the best constant which was identified by Talenti [26] and Aubin [1].

Lemma 2.2. (Sobolev inequality) Let 1 < p < d. If the function u ∈W^1,p(R^d), then

kuk_Lp∗(R^d)≤K(d, p)k∇uk_Lp(R^d), (7) wherep∗= _d−p^dp and

K(d, p) =π⁻¹²d^−1p p−1

d−p 1−_p¹"

Γ(1 +^d₂)Γ(d) Γ(^d_p)Γ(1 +d−^d_p)

#¹_d

. (8)

Next two lemmas are proposed by Bian and Liu [2].

Lemma 2.3. 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 2.4. 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 ¹_a

+

β(a−1)(t−t0) ⁻a−1¹

, for t > t0.

With the additional condition thaty(0) is bounded, we have Lemma 2.5which can be proved by contradiction arguments.

Lemma 2.5. Assume y(t) ≥0 is a C¹ function for t >0 satisfying y⁰(t) ≤γ− βy(t)^a forγ >0, β >0 anda >0. If y(0) is bounded, then

y(t)≤max y(0), γ

β ¹_a!

, t >0.

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

Theorem 3.1. Letd≥3,1< p < _d+1^3d andq=^d(3−p)_p . Under the assumption that u₀∈L¹₊(R^d)andA(d, p) =C_p,d^3−p−ku₀k^3−p_Lq >0, whereC_p,d=h

qp^p K^p(d,p)(q−2+p)^p

i_3−p¹ is a universal constant, let (u,v) be a non-negative weak solution of (1). Then u∈ L^∞ R+;L^q(R^d)

, u ∈ L^q+1 R+;L^q+1(R^d)

and ∇u^q−2+pp ∈ L^p R+;L^p(R^d) . Furthermore, following a priori estimates hold true:

(i) For 1 < p < _d+1^2d , ku(·, t)k_Lq(R^d) has the finite time extinction. The extinct timeT_ext satisfies

0< Text≤T0,

whereT0 depends ond, p, A(d, p),ku0k_L1(R^d) andku0k_Lq(R^d). (ii) For p=_d+1^2d ,ku(·, t)k_Lq(R^d) decays exponentially in time

ku(·, t)k_Lq(R^d)≤ ku₀k_Lqe^−Ct,

whereC is a constant depending on d, p, A(d, p),ku₀k_L1(R^d)andku₀k_Lq(R^d). (iii) For _d+1^2d < p < _d+1^3d ,ku(·, t)k_Lq(R^d) decays in time

ku(·, t)k_Lq(R^d)≤ ku₀k_Lq

(1 +Ct)

q−1 q(^p−2+pd)

,

whereC depends ond, p, A(d, p),ku0kL¹(R^d) andku0kL^q(R^d). And for any 1≤h≤q,ku(·, t)k_Lh(R^d)decays in time

ku(·, t)k_Lh(R^d)≤ ku0k

q(h−1) h(q−1)

L^q ku0k

q−h h(q−1)

L¹

(1 +Ct)

h−1 h(^p−2+p_d)

.

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

−(q+−1)(h−q+1)(h−1) (^p−2+pd)(^h+p−3+pd) +t⁻

h−1 p−2+p

d

! ,

whereC is a constant depending onh, d, p, A(d, p)andku0k_L1, >0satisfies

(q+)p^p

K^p(d,p)(q+−2+p)^p − ku0k^3−p_Lq ≥ ^A(d,p)₂ .

Proof. Step 1. (TheL^q estimate for 1< p < _d+1^3d ). Multiplying the first equation in problem (1) byqu^q−1 and integrating it overR^d, we obtain

d

dtku(·, t)k^q_Lq(R^d)= Z

R^d

∇ ·

|∇u|^p−2∇u

qu^q−1 dx− Z

R^d

∇ ·(u∇v)qu^q−1 dx

=−q(q−1) Z

R^d

u^q−2|∇u|^p dx+ (q−1) Z

R^d

∇u^q· ∇v dx

=− q(q−1)p^p (q−2 +p)^p

∇u^q−2+pp (t)

p L^p(R^d)

+ (q−1)kuk^q+1_Lq+1(R^d). (9)

Now we estimate the second term on the right hand side. Firstly, by using the interpolation inequality, we obtain that

kuk^q+1_Lq+1(R^d)≤ kuk

d(q−2+p) pd+pq−2d

L

(q−2+p)d d−p

kuk

q(pd+pq+p−3d) pd+pq−2d

L^q

= u^q−2+pp

dp pd+pq−2d

L

dp

d−p kuk

q(pd+pq+p−3d) pd+pq−2d

L^q

=

u^q−2+pp

p L

dp

d−pkuk^3−p_Lq , (10)

where the last equality holds since _pd+pq−2d^d = 1 and q(pd+pq+p−3d)

pd+pq−2d = 3−p from q= ^d(3−p)_p . Then using the Sobolev inequality (7), (10) turns to

kuk^q+1_Lq+1(R^d)≤K^p(d, p)

∇u^q−2+pp

p

L^pkuk^3−p_Lq , (11) whereK(d, p) is given by (8). Substituting (11) into (9), we have

d

dtkuk^q_Lq+ (q−1)

qp^p

(q−2 +p)^p −K^p(d, p)kuk^3−p_Lq

∇u^q−2+pp

p

L^p≤0. (12) Since ku0(·)k_Lq(R^d) < h _qpp

K^p(d,p)(q−2+p)^p

i_3−p¹

=: Cp,d, following two estimates hold true

ku(·, t)k_Lq(R^d)<ku0(·)k_Lq(R^d)< C_p,d, (13) (q−1)K^p(d, p)

C_p,d^3−p− ku0k^3−p_Lq

Z ∞ 0

∇u^q−2+pp

p

L^p ds≤Cp,d. Combining (11) with two estimates above, we obtain

u(x, t)∈L^∞ R+;L^q(R^d)

, (14)

u(x, t)∈L^q+1 R+;L^q+1(R^d)

, (15)

∇u^q−2+pp (x, t)∈L^p R+;L^p(R^d)

. (16)

Step 2. (TheL^q decay estimate). By using the interpolation inequality and (11), we have

ku(·, t)k^q_Lq(R^d)≤ kuk

(q+1)(q−1) q

L^q+1 kuk

1 q

L¹

≤

K^p(d, p)

∇u^q−2+pp

p L^p

kuk^3−p_Lq

^q−1_q kuk

1 q

L¹, (17) i.e.

∇u^q−2+pp

p

L^p≥ kuk

q2 q−1−3+p L^q

K^p(d, p)ku0k

1 q−1

L¹

= kuk^q_Lq

¹⁺

p−2+p d q−1

K^p(d, p)ku0k

1 q−1

L¹

, (18)

sinceku(·, t)k_L1 ≤ ku0k_L1. Substituting (18) into (12) yields that d

dtku(·, t)k^q_Lq+(q−1)A(d, p) ku0k

1 q−1

L¹

kuk^q_Lq

1+^p−2+

p d

q−1 ≤0, (19)

where we denoteA(d, p) :=C_p,d^3−p− ku₀k^3−p_Lq .

Next we discuss the inequality (19) in three different situations.

(a) If 1 +^p−2+

p d

q−1 >1, i.e. _d+1^2d < p < _d+1^3d , we can prove thatku(·, t)k_Lq(R^d)decays in time

ku(·, t)k_Lq(R^d)≤ ku₀k_Lq

(1 +Ct)

q−1 q(^p−2+pd)

, (20)

whereC= ^A(d,p)(^p−2+pd)(^ku0k^q_Lq)

p−2+p q−1d

ku₀k

1 q−1 L1

. (b) If 1 +^p−2+

p d

q−1 = 1, i.e. p= _d+1^2d ,ku(·, t)k_Lq(R^d)decays exponentially in time ku(·, t)kL^q(R^d)≤ ku0kL^q(R^d)e^−Ct,

whereC= (q−1)A(d,p) qku0k^1/(q−1)_{L1 (Rd}

)

.

(c) If 0 < 1 + ^p−2+_q−1^pd < 1, i.e. 1 < p < _d+1^2d , ku(·, t)k_Lq(R^d) has the finite time extinction. The extinct time Text satisfies 0 < Text ≤T0, whereT0 =

ku0k⁻

q(^p−2+pd)

q−1

Lq(Rd) ku0k^1/(q−1)_{L1 (Rd}

)

−A(d,p)(^p−2+p_d) .

Step 3. (TheL^h decay estimate for any 1≤h≤q when _d+1^2d < p < _d+1^3d ). Using the interpolation inequality and (20), we have

ku(·, t)k_Lh(R^d)≤ ku(·, t)k

q(h−1) h(q−1)

L^q(R^d)ku(·, t)k

q−h h(q−1)

L¹(R^d)≤ ku0k

q(h−1) h(q−1)

L^q ku0k

q−h h(q−1)

L¹

(1 +Ct)

h−1 h(^p−2+pd)

. (21)

Step 4. (The hyper-contractive property for anyq < h <∞when _d+1^2d < p < _d+1^3d ).

L^restimates withr=q+forsmall enough. SinceA(d, p) =C_p,d^3−p−ku0k^3−p_Lq

whereC_p,d=h

qp^p K^p(d,p)(q−2+p)^p

i_3−p¹

, there exists >0 such that (q+)p^p

K^p(d, p)(q+−2 +p)^p − ku0k^3−p_Lq ≥ A(d, p)

2 . (22)

In the same way of obtaining (9)-(11), we obtain d

dtku(·, t)k^r_Lr(R^d)=− r(r−1)p^p (r−2 +p)^p

∇u^r−2+pp (t)

p

L^p+ (r−1)kuk^r+1_Lr+1, (23) and

kuk^r+1_Lr+1(R^d)≤ kuk

d(r−2+p)(r+1−q) rd+pd+pq−qd−2d

L

(r−2+p)d d−p

kuk

q(pd+pr+p−3d) rd+pd+pq−qd−2d

L^q

=

u^r−2+pp

dp(r+1−q) rd+pd+pq−qd−2d

L

dp

d−p kuk

q(pd+pr+p−3d) rd+pd+pq−qd−2d

L^q

= u^r−2+pp

p L

dp

d−pkuk^3−p_Lq

≤K^p(d, p)

∇u^r−2+pp

p L^p

kuk^3−p_Lq , (24) where the third equality holds since rd+pd+pq−qd−2d^d(r+1−q) = 1 and q(pd+pr+p−3d)

rd+pd+pq−qd−2d = 3−p, and the last inequality holds from the Sobolev inequality. Then combining

(22), (23) and (24) together, we have d

dtku(·, t)k^r_Lr+(r−1)K^p(d, p)A(d, p) 2

∇u^r−2+pp

p

L^p≤0. (25) By using the interpolation inequality and (24), we have

ku(·, t)k^r_Lr(R^d)≤ kuk

(r+1)(r−1) r

L^r+1 kuk_L^1r1

≤

K^p(d, p)

∇u^r−2+pp

p L^p

kuk^3−p_Lq

^r−1_r kuk_L^1r1

≤

K^p(d, p)

∇u^r−2+pp

p L^pkuk

r(3−p)(q−1) q(r−1)

L^r ku0k

(3−p)(r−q) q(r−1)

L¹

^r−1_r

ku0k_L^1r1, i.e.

∇u^r−2+pp

p

L^p≥ (kuk^r_Lr)¹⁺

p−2+p r−1d

K^p(d, p)ku0k

1

r−1 1+^p(r−q)_d

L¹

, (26)

sincekuk_L1(R^d)≤ ku0k_L1(R^d). Substituting (26) into (25) yields that d

dtku(·, t)k^r_Lr+β₁ kuk^r_Lr

¹⁺

p−2+p d

r−1 ≤0, β₁:= (r−1)A(d, p) 2ku0k

1

r−1 1+^p(r−q)_d

L¹

. (27)

Solving this inequality by using Lemma2.3, we have ku(·, t)k^r_Lr ≤C(r)t⁻

r−1 p−2+p

d. (28)

Hyper-contractive estimates of L^h norm for h≥r. For h≥r > q, using the interpolation inequality, Sobolev inequality and Young’s inequality together, we obtain

kuk^h+1_Lh+1(R^d)≤ kuk

d(h−2+p)(h+1−r) hd+pd+pr−rd−2d

L

(h−2+p)d d−p

kuk

r(pd+ph+p−3d) hd+pd+pr−rd−2d

L^r

=

u^h−2+pp

dp(h+1−r) hd+pd+pr−rd−2d

L

dp

d−p kuk

r(pd+ph+p−3d) hd+pd+pr−rd−2d

L^r

≤Khd+pd+pr−rd−2d^dp(h+1−r) (d, p)

∇u^h−2+pp

dp(h+1−r) hd+pd+pr−rd−2d

L^p

kuk

r(pd+ph+p−3d) hd+pd+pr−rd−2d

L^r

≤ hp^p 2(h−2 +p)^p

∇u^h−2+pp

p L^p

+C(h, r) kuk^r_Lr

^1+h−r+1r−q , (29) where

dp(h+ 1−r)

hd+pd+pr−rd−2d = pd(h+ 1−r)

d(h+ 1−r) +p(r−q)< p.

Considering (9) withh=q, we have d

dtku(·, t)k^h_Lh(R^d)=− h(h−1)p^p (h−2 +p)^p

∇u^h−2+pp (t)

p L^p(R^d)

+ (h−1)kuk^h+1_Lh+1(R^d)

≤ − h(h−1)p^p 2(h−2 +p)^p

∇u^h−2+pp (t)

p L^p(R^d)

+C(h, r) kuk^r_Lr

^1+h−r+1_r−q

. (30)

Substituting (28) into (30) yields that d

dtku(·, t)k^h_Lh(R^d)≤ − h(h−1)p^p 2(h−2 +p)^p

∇u^h−2+pp

p L^p(R^d)

+C(h, r)t

− (r−1)(h−q+1)

(^p−2+pd)^(r−q). (31)

By the same way of obtaining (26), we obtain

∇u^h−2+pp

p L^p≥

kuk^h_Lh

1+^p−2+

p h−1d

K^p(d, p)ku0k

1

h−1(^1+p(h−q)_d )

L¹

. (32)

Then (31) turns to d

dtku(·, t)k^h_Lh(R^d)≤ −β2

kuk^h_Lh

¹⁺

p−2+p h−1d

+C(h, r)t

− (r−1)(h−q+1)

(^p−2+pd)^(r−q), (33) whereβ2= ^h(h−1)pp

2(h−2+p)^pK^p(d,p)ku₀k

h−11 (^1+p(h−q)d )

L1

.

Using Lemma2.4 with y(t) =ku(·, t)k^h_Lh(R^d),a= 1 + ^p−2+

p d

h−1 >1, β =β2 >0 andf(t) =C(h, r)t

− (r−1)(h−q+1)

(^p−2+pd)^(r−q), for anyt > t0>0, we have ku(·, t)k^h_Lh(R^d)≤C(h, r)t

−(q+−1)(h−q+1)(h−1) (^p−2+pd)(^h+p−3+pd)

0 +C(h)(t−t₀)⁻

h−1 p−2+p

d. (34)

By choosingt0= ₂^t, we obtain that for anyt >0 ku(·, t)k^h_Lh(R^d)≤C t

−(q+−1)(h−q+1)(h−1) (^p−2+p_d)(^h+p−3+p_d) +t⁻

h−1 p−2+p

d

!

, (35)

whereC is a constant depending onh, d, p, A(d, p) andku0k_L1,satisfies (22).

4. The uniformly in time L^∞ estimate of weak solutions. In this section, we prove our theorem about uniformly in timeL^∞boundness of weak solutions by using a bootstrap iterative method. At the beginning of this section, we prove the following proposition concerningL^h norm estimates of weak solutions for 1< h <

∞.

Proposition 1. Let d ≥ 3, 1 < p < _d+1^3d and q = ^d(3−p)_p . If u0 ∈ L¹₊(R^d)∩ L^h(R^d) for 1 < h < ∞ and A(d, p) = C_p,d^3−p − ku0k^3−p_Lq > 0, where Cp,d = h qp^p

K^p(d,p)(q−2+p)^p

i_3−p¹

is a universal constant, let(u, v)be a non-negative weak solu- tion of (1). Thenu(x, t)satisfies for any t >0

ku(·, t)k^h_Lh(R^d)≤Cku₀k

q(h−1) q−1

L^q(R^d), 1< h≤q, (36) whereC depends onh, q,andku₀k_L1, and

ku(·, t)k^h_Lh(R^d)≤C_u^h, q < h <∞, (37) where C_u^h is a constant depending on d, p, h,ku0k_L1 and ku0k_Lh, > 0 satisfies

(q+)p^p

K^p(d,p)(q+−2+p)^p− ku0k^3−p_Lq ≥ ^A(d,p)₂ .

Actually, the proof of Proposition1 is almost the same as the proof of Theorem 3.1, except for the different initial conditionu0∈L¹₊(R^d)∩L^h(R^d) for 1< h <∞.

Proof. Using the same method in Step 1 of Theorem3.1, we have for allt >0 ku(·, t)k_Lq(R^d)<ku₀(·)k_Lq(R^d)< C_p,d.

Then we discuss in two different situations with respect toh.

For 1< h≤q, using the interpolation inequality, we have ku(·, t)k^h_Lh(R^d)≤ ku0(·)k

q−h q−1

L¹(R^d)ku0(·)k

q(h−1) q−1

L^q(R^d). (38) Forq < h <∞, lettingr:=q+≤h <∞, there exists >0 small enough such that

(q+)p^p

K^p(d, p)(q+−2 +p)^p − ku0k^3−p_Lq ≥ A(d, p) 2 . Then (25) also holds true, i.e.

d

dtku(·, t)k^r_Lr+(r−1)K^p(d, p)A(d, p) 2

∇u^r−2+pp

p L^p

≤0.

Sinceq < r≤h, we haveu0∈L^r(R^d) and

ku(·, t)k_Lr(R^d)≤ ku0(·)k_Lr(R^d), (39) for allt >0. Combining (30), (32) and (39) together, we obtain

d

dtku(·, t)k^h_Lh(R^d)≤ −β₃ kuk^h_Lh

1+^p−2+

p d

h−1 +C(h, r) ku₀k^r_Lr

^1+h−r+1_r−q

, (40) where β3 := ^h(h−1)pp

2(h−2+p)^pK^p(d,p)ku0k

1

h−1(^1+p(h−q)d )

L1

>0. Using Lemma 2.5 withy(t) = ku(·, t)k^h_Lh, a= 1 + ^p−2+

p d

h−1 >0,β =β3>0 andγ =C(h, r) ku0k^r_Lr

^1+h−r+1r−q >0, for anyt >0, we have

ku(·, t)k^h_Lh ≤max

ku0k^h_Lh, C(h, r) ku0k^r_Lr

(h−q+1)(h−1) (h+p−3+p

d)

≤max (

ku0k^h_Lh, C(h) ku0k^h_Lh

(h−q+1)(q+−1) (h+p−3+p

d)

)

=:C_u^h, (41) wheresatisfies _Kp(d,p)(q+−2+p)^{(q+)pp p} − ku₀k^3−p_Lq ≥ ^A(d,p)₂ .

Next, we prove the uniformly in timeL^∞ boundness ofu(x, t) by using a boot- strap iterative technique [3,19] with Proposition1 and an additional initial condi- tionu₀∈L^∞(R^d).

Theorem 4.1. Let d≥3,1< p < _d+1^3d andq= ^d(3−p)_p . Ifu0∈L¹₊(R^d)∩L^∞(R^d) andA(d, p) =C_p,d^3−p−ku0k^3−p_Lq >0, whereCp,d=h _qpp

K^p(d,p)(q−2+p)^p

i3−p¹

is a universal constant, let(u, v)be a non-negative weak solution of (1). Then for anyt >0,

ku(·, t)k_L∞(R^d)≤C(d, p, K0), whereK0= max

1, ku0k_L1(R^d), ku0k_L∞(R^d)

. Proof. We denote

h_k = 3^k+d(3−p)

p + 1, fork≥1.

Multiplying the first equation in (1) byhku^hk−1and integrating, we have d

dtku(·, t)k^h_L^k_hk(

R^d)=−h_k(h_k−1)p^p (hk−2 +p)^p

∇u^hk−2+pp (t)

p L^p

+ (h_k−1)kuk^h_L^k_hk⁺¹+1. (42)

Step 1. (TheL^hk estimate for 1< p≤2) Taking 0< C1 ≤ _2(h^hk(hk−1)pp

k−2+p)^p is a fixed constant, then (42) turns to

d

dtku(·, t)k^h_L^k_hk(

R^d)≤ −2C1

∇u^hk−2+pp (t)

p

L^p+hkkuk^h_L^k_hk⁺¹+1. (43) Using the interpolation inequality and Sobolev inequality together, we obtain

kuk^h_L^k_hk⁺¹+1(R^d)≤ kuk^(hk+1)θ

L

(hk−2+p)d d−p

kuk^{(hk+1)(1−θ)}

L^hk−1

= u^hk

−2+p p

p(hk+1)θ hk−2+p

L

dp

d−p kuk^{(hk+1)(1−θ)}

L^hk−1

≤K

p(hk+1)θ hk−2+p(d, p)

∇u^hk−2+pp

p(hk+1)θ hk−2+p

L^p kuk^{(hk+1)(1−θ)}

L^hk−1 , (44) where

θ= d(hk−2 +p)(hk−h_k−1+ 1) (hk+ 1) (hk−2 +p)d−h_k−1(d−p), 1−θ= h_k−1(hkp+pd−3d+p)

(hk+ 1) (hk−2 +p)d−h_k−1(d−p). Sinceh_k−1= 3^k−1+^d(3−p)_p + 1>^d(3−p)_p , it is easy to see that ^p(h_h ^k+1)θ

k−2+p < p. Then using Young’s inequality and (44), we have

hkkuk^h_L^k_hk⁺¹+1(R^d)≤1 aδ^a₁

∇u^hk−2+pp

p L^p+1

bδ^−b₁ K

p(hk+1)θb

hk−2+p (d, p)h^b_kkuk^{(hk+1)(1−θ)b}

L^hk−1

≤C1

∇u^hk−2+pp

p

L^p+C2(hk)h^b_kkuk^{(hk+1)(1−θ)b}

L^hk−1 , (45)

where

a=hk−2 +p

(h_k+ 1)θ =d(hk−h_k−1+ 1) +h_k−1p+pd−3d d(h_k−h_k−1+ 1) >1, b= hk−2 +p

hk−2 +p−(hk+ 1)θ = d(hk−hk−1+ 1) +hk−1p+pd−3d h_k−1p+pd−3d >1, δ1= (C1a)^1a, C2(hk) = 1

b(C1a)^−baK

p(hk+1)θb hk−2+p (d, p).

We can see that C2(hk) is uniformly bounded since a → ^2d+p_2d and b → ^2d+p_p as k→ ∞. Substituting (45) into (43) yields to

d

dtku(·, t)k^hk

L^hk(R^d)≤ −C₁

∇u^hk−2+pp (t)

p L^p

+C₂(h_k)h^b_k kuk^hk−1

L^hk−1

γ1

, (46) where

γ₁=(h_k+ 1)(1−θ)b hk−1

=h_kp+pd−3d+p hk−1p+pd−3d <3.

Next, we estimate

∇u^hk−2+pp (t)

p

L^p. By using the interpolation inequality and Sobolev inequality, we have

kuk^h_L^k_hk(

R^d)≤ kuk^hkβ

L

(hk−2+p)d d−p

kuk^hk(1−β)

L^hk−1

= u^hk

−2+p p

phkβ hk−2+p

L

dp

d−p kuk^hk(1−β)

L^hk−1

≤K^{hkphkβ−2+p}(d, p)

∇u^hk−2+pp

phkβ hk−2+p

L^p

kuk^hk(1−β)

L^hk−1 , (47)

where

β= d(h_k−2 +p)(h_k−h_k−1) h_k (h_k−2 +p)d−h_k−1(d−p), 1−β= h_k−1(hkp+pd−2d)

hk (hk−2 +p)d−h_k−1(d−p). Since it is easy to see that _h^phkβ

k−2+p < p, then using Young’s inequality, we have kuk^h_L^k_hk(

R^d)≤ 1 a⁰δ₂^a0

∇u^hk−2+pp

p L^p

+ 1

b⁰δ₂^−b0K ^phkβb

0

hk−2+p(d, p)kuk^hk(1−β)b0

L^hk−1

≤C1

∇u^hk−2+pp

p L^p

+C3(hk) kuk^hk−1

L^hk−1

γ2

, (48)

where

a⁰ =hk−2 +p

hkβ =d(hk−hk−1) +hk−1p+pd−2d d(hk−h_k−1) >1, b⁰= hk−2 +p

h_k−2 +p−h_kβ =d(hk−h_k−1) +h_k−1p+pd−2d h_k−1p+pd−2d >1, δ2= (C1a⁰)^a10, C3(hk) = 1

b⁰(C1a⁰)^−b

0

a0K^phkβb

0 hk−2+p(d, p), γ₂=h_k(1−β)b⁰

hk−1

= h_kp+pd−2d hk−1p+pd−2d <3.

We can also check that C3(hk) is uniformly bounded as k → ∞. Combining (46) and (48) together, we have

d

dtkuk^h_L^k_hk ≤ −kuk^h_L^k_hk+C₂(h_k)h^b_k kuk^hk−1

L^hk−1

γ1

+C₃(h_k) kuk^hk−1

L^hk−1

γ2

. (49) SinceC2(hk) and C3(hk) are both uniformly bounded ask→ ∞, we can choose a constant C4 >1 which is an upper bound ofC2(hk) andC3(hk). Then byhk >1 andb >1, we have for any t >0,

d

dtkuk^h_L^k_hk ≤ −kuk^h_L^k_hk+C4h^b_k

kuk^hk−1

L^hk−1

γ1

+ kuk^hk−1

L^hk−1

γ2

. (50) Step 2. (TheL^hk estimate for 2< p < _d+1^3d ) By changing form of (42), we have

d dt

h

(hk−2 +p)^p−2kuk^h_L^k_hki

=−hk(hk−1)p^p (hk−2 +p)²

∇u^hk−2+pp (t)

p L^p

+ (hk−1)(hk−2 +p)^p−2kuk^h_L^k_hk⁺¹+1

≤ −2C₁⁰ ∇u^hk

−2+p p (t)

p L^p

+C₅h²_kkuk^h_L^k_hk⁺¹+1, (51) where 0 < C₁⁰ ≤ ^h_2(h^k(hk−1)pp

k−2+p)² is a fixed constant and C₅ is also a fixed constant satisfying (hk−1)(hk−2 +p)^p−2≤C5h²_k sincehk >1 andp <3. Using Young’s inequality and (44), we have

C5h²_kkuk^h_L^k_hk⁺¹+1 ≤1 aδ^a₃

∇u^hk−2+pp

p L^p+1

bδ^−b₃ (C5)^bK

p(hk+1)θb

hk−2+p h^2b_k kuk^{(hk+1)(1−θ)b}

L^hk−1

≤C₁⁰

∇u^hk−2+pp (t)

p

L^p+C₂⁰(hk)h^2b_k kuk^{(hk+1)(1−θ)b}

L^hk−1 , (52)

where

a=h_k−2 +p

(hk+ 1)θ =d(h_k−h_k−1+ 1) +h_k−1p+pd−3d d(hk−hk−1+ 1) >1,