Large time behavior of solutions in super-critical cases to degenerate Keller-Segel systems

Vol. 40, N 3, 2006, pp. 597–621 www.edpsciences.org/m2an

DOI: 10.1051/m2an:2006025

LARGE TIME BEHAVIOR OF SOLUTIONS IN SUPER-CRITICAL CASES TO DEGENERATE KELLER-SEGEL SYSTEMS

Stephan Luckhaus

and Yoshie Sugiyama

Abstract. We consider the following reaction-diffusion equation:

(KS)

⎧⎪

⎨

⎪⎩

ut=∇ ·

∇u^m−u^q−1∇v

, x∈IR^N, 0< t <∞, 0 = ∆v−v+u, x∈IR^N, 0< t <∞, u(x,0) =u0(x), x∈IR^N,

whereN≥1, m >1, q≥max{m+_N²,2}.

In [Sugiyama,Nonlinear Anal. 63(2005) 1051–1062; Submitted;J. Differential Equations(in press)]

it was shown that in the case ofq≥max{m+_N²,2}, the above problem (KS) is solvable globally in time for “smallL^N(q−m)2 data”. Moreover, the decay of the solution (u, v) inL^p(IR^N) was proved. In this paper, we consider the case of “q≥max{m+_N²,2}and smallLdata” with any fixed≥^N(q−m)₂ and show that (i) there exists a time global solution (u, v) of (KS) and it decays to 0 as ttends to∞ and (ii) a solutionuof the first equation in (KS) behaves like the Barenblatt solution asymptotically asttends to∞, where the Barenblatt solution is the exact solution (with self-similarity) of the porous medium equationut= ∆u^m withm >1.

Mathematics Subject Classification. 35B40, 35K45, 35K55, 35k65.

Received: September 22, 2005. Revised: February 27, 2006.

1. Introduction

We consider the following reaction-diffusion equation:

(KS)

⎧⎪

⎨

⎪⎩

u_t=∇ ·

∇u^m−u^q−1∇v

, x∈IR^N, 0< t <∞, 0 = ∆v−v+u, x∈IR^N, 0< t <∞, u(x,0) =u₀(x), x∈IR^N,

Keywords and phrases. Degenerate parabolic system, chemotaxis, Keller-Segel model, drift term, decay property, asymptotic behavior, Fujita exponent, porous medium equation, Barenblatt solution.

1 Departement of mathematics and computer science, Universit¨at Leipzig, Leipzig, 04109, Germany. [email protected]

2Department of Mathematics and Computer Science, Tsuda College, 2-1-1, Tsuda-chou, Kodaira-shi, Tokyo, 187-8577, Japan.

[email protected]

c EDP Sciences, SMAI 2006

Article published by EDP Sciences and available at http://www.edpsciences.org/m2anor http://dx.doi.org/10.1051/m2an:2006025

whereN ≥1, m >1, q≥max{m+_N²,2}. The initial datau₀is a non-negative function inL¹∩L^∞(IR^N)×L¹∩ H¹∩W^1,∞(IR^N), u^m₀ ∈H¹(IR^N).This equation is often called the Keller-Segel model describing the motion of the chemotaxis molds. (We refer to Keller-Segel [23].)

In this paper, we are interested in the large time behavior of solutions for (KS). Concerning the large time behavior of the heat equation, the following asymptotic profile is well known:

t→∞lim t^N2u(·, t)−M G_t(·)_L∞(IR^N) = 0, (1.1) whereG_t(x) is the heat kernel andM is the initial mass.

Also for the porous medium equation:

(P) u_t(x, t) = ∆u^m(x, t)

corresponding to the initial datau₀, the asymptotic profile was obtained in the following form:

t→∞lim t^σu(·, t)−V(·, t;u0_L¹_(IR^N₎)_L^∞_(IR^N₎ = 0 with σ= N

N(m−1) + 2, (1.2) whereV(x, t;M) is the exact solution of (P) given by

V(x, t;M) := 1 t^σ

β²M^2σ(m−1)N −σ(m−1) 2mN ·|x|²

t^2σN _m−1¹

+ (1.3)

with a constantβ such that

IR^N

β²−^σ(m−1)_2mN |y|²_m−1¹

+ dy = 1.The aboveV(x, t;M) holds the self-similarity and

IR^NV(x, t;M) dx= M for all t >0. This V(x, t;M) is called the Barenblatt solution. (See Barenblatt [2].) The asymptotic profile (1.2) was firstly proved by Kamin [20], and developed by Friedman-Kamin [10], and finally established by Vazquez [34] in the above form (1.3). (We also refer to [3, 21, 35].)

Regarding to the Keller-Segel model (KS), for the semilinear case: m = 1 of parabolic-parabolic type, Nagai-Syukuinn-Umesako [26] showed the similar asymptotic profile to (1.1). (we also refer to Biler-Cannone- Guerra-Karch [5].) Their argument is based on the representation formula. On the other hand, as for our problem (KS), there is no representation formula for solutionusincem >1. In addition, differently from the porous medium equation (P), comparison principles do not hold. Therefore, we can not employ the method by Kamin [20], Friedman-Kamin [10], Nagai-Syukuinn-Umesako [26] to our problem directly.

Our aim of this paper is to prove the following (I)-(III) without “comparison principles and representation formula”:

In the case ofm >1 andq≥max{m+_N²,2};

(I) (KS) is globally solvable for the smallL data with any fixed≥ ^N(q−m)₂ ; (II) the solution (u, v) of (KS) decays to 0 inL^p(IR^N)(1< p <∞).

We also assume thatq > m+_N². Then,

(III) the solutionuto the first equation in (KS) satisfies the following asymptotic profile:

t→∞lim t^{N(m−1)+2+εN (1−1p)}u(·, t)−V(·, t;u0_L¹)_L^p_(Bt)= 0, ε >0 and 1< p <∞ (1.4) for allR >0, whereB_t=B_t(ε, R) :={x∈IR^N; |x| ≤Rt^{N(m−1)+2+ε1} }.

In what follows, we give the definition of a weak solution (u, v) for (KS).

Definition 1. Letm >1 q≥2 and letu₀∈L¹∩L^∞(IR^N) with u^m₀ ∈H¹(IR^N) andu₀≥0. A pair (u, v) of non-negative functions defined in IR^N×[0, T) is called a weak solution of (KS) on [0, T) if

(i) u∈L^∞(0, T;L¹∩L^∞(IR^N)),∇u^m∈L²(0, T;L²(IR^N));

(ii) v∈L^∞(0, T;H¹(IR^N));

(iii) (u, v) satisfies the equations in the sense of distribution,i.e.,that _∞

0

IR^N

∇u^m· ∇ϕ−u^q−1∇v· ∇ϕ−uϕ_t

dxdt =

IR^N

u₀(x)ϕ(x,0) dx,

IR^N

(∇v· ∇ψ+vψ−uψ) (t)dx = 0 for a.a. t∈(0, T) for all functionsϕ∈C₀^∞(IR^N ×[0, T)) andψ∈C₀^∞(IR^N).

In the first theorem, we show the existence and decay property of a solution (u, v) for (KS) with small initial data.

Theorem 1.1 (decay property). Let 1 ≤p <∞, N ≥1, m >1, q ≥max{m+_N²,2}, ≥ ^N(q−m)₂ (≥1).

Suppose that the initial data u₀ is non-negative everywhere. Then, there exist an absolute constantM and a positive numberε depending only onM, p, N, m, such that ifu₀∈L¹∩L(IR^N)satisfies that

u0_L1(IR^N)=M, u0_L(IR^N)≤ε, (1.5) then (KS) has a weak solution (u, v) on [0,∞) with the following decay property: there exists a constant C_p depending only onp,u0_Lp(IR^N) together withN, m, q, M,u0_L(N+2)q(IR^N) such that

u(t)_L^p_(IR^N₎+v(t)_L^p_(IR^N₎≤C_p(1 +t)^−d for all 0< t <∞, (1.6) where

d=σ

1−1 p

, σ= N

N(m−1) + 2· Remark 1.

(i) The decay rateddepends onm, N but not onq.

(ii) The above decay rate seems to be optimal. In fact, form= 1, we find thatσ= ^N₂ whose decay rated coincides with theL¹-L^p estimate for the linear heat equation.

(iii) Concerning the following Cauchy problem (PS)

u_t= ∆u^m+u^q x∈IR^N, t >0, u(x,0) =u₀(x), x∈IR^N

in the case of “m ≥1 and q > m+_N²”, Kawanago [22] obtained the decay estimate under smallness assumption for u0

L^N(q−m)2 . In Remark 4.1 in [22], he mentioned that p₀ := ^N(q−m)₂ is the special exponent to obtain the decay property for (PS). Regarding to (KS), we show that ifu_0L(IR^N₎<<1 for any fixed number≥^N(q−m)₂ (≥1), then the decay property is obtained,i.e.,that the exponentp₀ is not special for (KS).

For any positive numbersε, R, we defineB_tby

B_t:={x∈IR^N; |x|< Rt^{N(m−1)+2+ε1} }. (1.7)

We introduce the self-similar solutionV(x, t;M) of Barenblatt [2]:

V(x, t;M) := 1 t^σ

β²M^2σ(m−1)N −σ(m−1) 2mN ·|x|²

t^2σN _m−1¹

+

with a constantβ such that

IR^N

β²−σ(m−1)

2mN |y|²_m−1¹

+ dy = 1.

It is easily verified that

IR^N

V(x, t;M) dx=M.

We now give the asymptotic profile in the following theorem.

Theorem 1.2 (asymptotic profile). Let the same assumption as that in Theorem 1.1 hold. In addition, let q > m+_N². Then, the weak solutionuobtained in Theorem 1.1 satisfies that

t→∞lim t^{N(m−1)+2+εN (1−1p)}u(·, t)−V(·, t;u_0L¹_(IR^N₎)

L^p(Bt) = 0 with ε >0, 1< p <∞, (1.8) for allR >0, where B_t=B_t(ε, R) is the ball defined in(1.7).

Remark 2.

(i) It seems to be difficult to takeε= 0.

(ii) Theorem 1.2 implies that ∆u^mis dominant to∇(u^q−1∇v) in the case of “q > m+_N² and small initial data”.

(iii) The proof of Theorem 1.2 is based on the estimate (1.6) in Theorem 1.1.

To show the asymptotic profile for (KS) withm >1, we consider the following sequence of functions:

w_k(x, t) =k^Nu(kx, k^N(m−1)+2t) and z_k(x, t) =k^Nv(kx, k^N(m−1)+2t) fork≥1. (1.9) Then, (KS) can be rewritten as follows:

w(KS)

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

w_kt(x, t) =∇ ·

∇(wk)^m−k^−N(q−m)·(w_k)^q−1· ∇zk

, x∈IR^N, 0< t <∞, · · ·(1)_w, 0 = ∆z_k−k²z_k+k²w_k, x∈IR^N, 0< t <∞, · · ·(2)_w,

w_k(x,0) = k^Nu₀(kx), x∈IR^N.

The above system^w(KS) does not have any invariance under change of scaling. In addition, the second equation includes the scaling parameter k. However, in the case of q ≥m+_N², we obtain the L^∞(IR^N)-bound forw_k independently of k. Next, we prove that (w_k)^m is bounded inH¹(δ, T;L²(IR^N))∩L^∞(δ, T;H¹(IR^N)) for all 0 < δ < T <∞. In this point, we have the difficulty such asw_k(0)_L^p_(IR^N₎= (k^N(1−1p)u_0L^p_(IR^N₎) depends onkfor allp∈(1,∞]. Under this difficulty, to obtain the boundness inH¹(δ, T;L²(IR^N))∩L^∞(δ, T;H¹(IR^N)) independent of k, we use the cut-off function which attains 0 att= 0 and hasC^∞-regularity. As a result, we obtain the desired bounds independent ofk(see Sect. 5.1 in this paper) and show thatw_k converges a function U. Simultaneously, we find thatU satisfies (P) in a distribution sense since the power ofkin the coefficient of the perturbation term is negative. (See Sect. 5.2)

Furthermore, we verify the following key fact:

(H) U(·, t)∈L¹(IR^N) and U(·, t)_L1(IR^N)=u0_L1(IR^N).

To this end, we invent the crucial lemma (Lem. 2.4) in Section 2:

“w_k →U strongly inL¹_loc(IR^N) (1.10)

together with some additional assumption on w_k, we have in fact, that

IR^N

(w_k−U) dx→0 ask→ ∞.” (1.11) We ensure the sufficient conditions in the above lemma (Lem. 2.4). (See Sect. 5.3.) As a result, by virtue of (1.11) and “the mass conservation law and theL¹-scaling invariance for the initial data ofw_k”,i.e.,that

IR^N

w_k(x, t) dx =

IR^N

w_k(x,0) dx =

IR^N

u₀ dx for allt≥0, (1.12)

the above (H) is verified. Once (H) is verified, we can apply Theorem 1.1 in Vazquez [34] and obtain that U(·, t)−V(·, t;u0_L¹)_Lp(IR^N)→ 0 ast→ ∞. (1.13) For any positive numbersε, R, we defineB_tby

B_t:={x∈IR^N; |x|< Rt^{N(m−1)+2+ε1} }. (1.14)

Then, taking the time variable byk^ε, and combining (1.13) with the convergence ofw_k to U, we observe that wk(·, k^ε)−V(·, k^ε;u0_L¹)_L^p_(BR)

≤ wk(·, k^ε)−U(·, k^ε)_L^p_(BR) + U(·, k^ε)−V(·, k^ε;u0_L¹)_Lp(IR^N)

→0 ask→ ∞ (1.15)

for anyp∈(1,∞) and for allR >0, whereB_R:={x∈IR^N; |x|< R}. Moreover, takingkbyk=t^{N(m−1)+2+ε1} in (1.15) and using the self-similarity of the Barenblatt solution, we conclude that

t^{N(m−1)+2+εN (1−1p)}u(·, t)−V(·, t;u0_L¹)_L^p_(Bt) → 0 ast→ ∞ (1.16) for any ε >0 and p∈ (1,∞) and for allR >0, where B_t =B_t(ε, R) is the ball defined in (1.14). Thus, we prove Theorem 1.2. (see Section 5.4.)

In the following section, we shall prepare several lemmas which will be used in the sequent sections. In Section 3, we introduce the results obtained in [30–32] concerning the existence of a time global strong solution of the approximated problem of (KS). In Section 4, we organize the proof of the decay of a solution (u, v). In Section 5, in the case of “m >1, q > m+_N²”, we prove that the solutionuof (KS) behaves like the Barenblatt solution asymptotically as t → ∞ which is the exact solution of porous medium equation: u_t = ∆u^m with m >1.

Remark 3.

(i) In our argument, any type of comparison principles is not used.

(ii) When we substitute the second equation: ∆v=v−uinto the first equation in (KS), it holds that (E) u_t = ∆u^m− ∇u^q−1· ∇v−u^q−1∆v = ∆u^m+u^q− ∇u· ∇v−u^q−1v.

The above equation (E) includes the termsu_t,∆u^mandu^q. Therefore, we observe that (PS) in Remark 1 is analogous to (E).

For (PS) with N ≥1, m, q >1, it is well known that the critical exponentq=m+_N² divides the situation into the global existence and the finite time blow-up of a solution. Indeed,

(1) whenq > m+_N², the problem (PS) is globally solvable for small initial data and evolves in a finite time blow-up for large initial data and

(2) whenq < m+_N² andq=m+_N², it is proved that (all) non-negative solutions of (PS) blow up in a finite time without any restriction on the size of the initial data. (See for example Galaktionov-Kurdyumov- Mikhailov-SamarskiiN [13], Galaktionov [12], Kawanago [22] and Mochizuki-Suzuki[24].) This exponent q=m+_N² is called the Fujita exponent [11].

For (KS) withN ≥1, m >1, q ≥2, in [30–32] the Fujita’s exponent was found. Specifically, in [32]

it was shown that

(i) when q < m+_N², the problem (KS) is globally solvable without any restriction on the size of the initial data; and

(ii) when m > 1 and q ≥ max{m+ _N²,2}, the problem (KS) is globally solvable for small L^N(q−m)2 initial data. Furthermore, the decay of solution (u, v) inL^p(IR^N)(1< p <∞) was proved.

In addition, in the case ofq= 2 with 2> m+_N²;

(iii) we [33] constructed such an initial function that a solution (u, v) blows up in a finite time.

In this paper, the case of (ii) above is considered.

We will use the simplified notations:

(1) ∂_t= _∂t^∂, ∂_i =_∂x^∂

i, ∂_ij² =∂_i∂_j,∇u=

∂₁, ∂₂,· · ·

,∇²u=

∂²₁₁, ∂₁₂² ,· · ·

; (2) · L^r = · _Lr(IR^N),(1≤r≤ ∞),

·dx:=

IR^N·dx.

(3) Q_T := IR^N×(0, T), B_R:={x∈IR^N;|x|< R}.

(4) When the weak derivatives∇u,∇²uand∂_tuare inL^p(Q_T) for somep≥1, we say thatu∈W_p^2,1(Q_T), i.e.,

W_p^2,1(Q_T) :=

u∈L^p(0, T;W^2,p(IR^N))∩W^1,p(0, T;L^p(IR^N));

u_Wp^2,1_(QT):=u_L^p_(QT)+∇u_L^p_(QT)+∇²u_L^p_(QT)+∂tu_L^p_(QT)<∞ .

2. Preliminary lemmas

The following lemma gives us a version of Gagliardo-Nirenberg inequality. (See [33], Lem. 2.4. and Nakao [27]) Lemma 2.1. Let N ≥1,m ≥1, a > 2, u∈L^q1(IR^N) with q₁ ≥1 andu^r+m−12 ∈H¹(IR^N) with r >0. Let q₁∈[1, r+m−1],q₂∈[^r+m−1₂ ,^a(r+m−1)₂ ]and

⎧⎨

⎩

1≤q₁≤q₂≤ ∞ when N = 1, 1≤q₁≤q₂<∞ when N = 2, 1≤q₁≤q₂≤^(r+m−1)N_N−2 when N ≥3.

(2.1)

Then, it holds that

u_L^q2(IR^N)≤C^r+m−12 u^1−Θ_Lq1(IR^N)· ∇u^r+m−12 _L^r+m−12^2Θ(IR^N) (2.2) with

Θ = r+m−1

2 ·1

q₁ − 1 q₂

·1 N −1

2+r+m−1 2q₁

₋₁

, (2.3)

where

C depends only on Nanda when q₁≥^r+m−1₂ ,

C=c₀^1β with c₀ depending only on Nanda when 1≤q₁< ^r+m−1₂ , (2.4) and

β := q₂−^r+m−1₂ q₂−q₁

2q₁ r+m−1 +

1− 2q₁ r+m−1

2N N+ 2

· (2.5)

The following inequalities are well known. (For instance, see Duoandikoetxea [9], p. 110 and Brezis [6], IX.12.) Lemma 2.2. Let w∈W^2,r(IR^N). Then, the following inequalities hold:

∇²w_Lr(IR^N) ≤C r²

r−1 ₂

∆w_Lr(IR^N) for 1< r <∞, (2.6) w_L∞(IR^N) ≤ 2r

r−Nw_W1,r(IR^N) for r > N, (2.7)

whereC is a positive constant depending only on N.

We prepare a technical lemma which is used often when establishing the uniform bound of a solution w_k for (1)_w in^w(KS).

Lemma 2.3. Let η=η(r)be as

η(r) :=

⎧⎨

⎩

1 for r≥1,

exp(1−¹_r) for 0≤r≤1,

0 for r≤0.

We define a sequence η_δ(t)of cut-off functions byη_δ(t) :=η(_δ^t). Then, it holds that

0<t<∞sup t^−pη_δ(t) ≤ p^p

δ^pe^p−1 and sup

0<t<∞t^−p∂_tη_δ(t) ≤ (p+ 2)^p+2

(δe)^p+1 for any p≥1. (2.8) Proof of Lemma 2.3. For anyp≥1, we have

0<t<∞sup t^−pη_δ(t) = sup

0<t<∞

e

t^pe^δt = e δ^p · sup

0<y<∞

y^p e^y ≤ e

δ^p ·p^p

e^p = p^p

δ^pe^p−1· (2.9) By the definition ofη andη_δ, we see that

η(r) =

0 for r≥1,

1

r²·exp(1−¹_r) for 0≤r≤1, (2.10)

and

∂_tη_δ(t) =

0 for t≥δ,

δ

t² ·exp(1−^δ_t) for 0≤t≤δ. (2.11) From (2.11), we observe that

0<t<∞sup t^−p∂_tη_δ(t) = sup

0<t≤δ

δe

t^p+2e^δt = δe

δ^p+2 · sup

1≤y<∞

y^p+2

e^y ≤ δe

δ^p+2 ·(p+ 2)^p+2

e^p+2 = (p+ 2)^p+2 (δe)^p+1 ·

Thus, we complete the proof of Lemma 2.3.

We present the crucial lemma which will play an important role when showing the asymptotic profile.

Lemma 2.4. Let N ≥ 1 and g belong to L¹(IR^N) and assume that {w_k} is a sequence of non-negative L¹-functions in IR^N satisfying

w_k→g strongly in L¹_loc(IR^N). (2.12)

We also assume that for any fixed numberδ >0, there exist f_δ ∈L¹(IR^N)andk₀∈INsuch that

IR^N

[w_k−f_δ]⁺ dx < δ

6 for all k > k₀, (2.13)

where[s]⁺= max(s,0). Then, we have convergence:

IR^N

(w_k−g) dx→0 as k→ ∞. (2.14)

Proof of Lemma 2.4. Letδ be any fixed positive number and Ω be a domain in IR^N. Then, Ω can be written as the union

Ω = Ω₁∪Ω₂, where

Ω₁ :={Ω⊂IR^N;w_k(x)−f_δ(x)≥0} and Ω₂ := {Ω⊂IR^N;w_k(x)−f_δ(x)<0}.

Therefore, it holds that

Ω

w_k−f_δ dx ≤

Ω1

|wk−f_δ|dx +

Ω2

|wk−f_δ| dx

=

Ω1

(w_k−f_δ) dx +

Ω2

(f_δ−w_k) dx

=

IR^N

[w_k−f_δ]⁺ dx +

Ω

|f_δ|dx. (2.15)

On the other hand, sinceg, f_δ∈L¹(IR^N), there exists a domainK_δ ⊂IR^N depending onδsuch that

IR^N\Kδ

|g|dx≤ δ

6 and

IR^N\Kδ

|fδ|dx ≤ δ

6· (2.16)

Taking Ω by Ω = IR^N\K_δ in (2.15), from (2.13) and (2.16), we observe that for any fixed numberδ >0, there existsK_δ ⊂IR^N andk₀∈IN such that

IR^N\Kδ

(w_k−f_δ) dx≤

IR^N

[w_k−f_δ]⁺ dx +

IR^N\Kδ

|fδ|dx

≤ δ

3 for allk > k₀. (2.17)

Consequently, by (2.12), (2.16) and (2.17), we see that for any fixed numberδ >0, there exists ˜k₀(≥k₀)∈IN such that

IR^N

(w_k(x)−g(x)) dx≤

Kδ

(w_k(x)−g(x)) dx +

IR^N\Kδ

(w_k(x)−g(x)) dx

≤ w_k−gL¹(Kδ) +

IR^N\Kδ

(w_k(x)−f_δ(x)) dx +

IR^N\Kδ

|f_δ(x)|+|g(x)|dx

≤ δ 3 +δ

3 +δ 6+δ

6 =δ

for anyk >k˜₀. We thus conclude (2.14) and complete the proof of Lemma 2.4.

3. Approximated problem

The first equation of (KS) is a quasi-linear parabolic equation of degenerate type. Therefore, we can not expect the problem (KS) to have a classical solution at the point where the first solutionuvanishes. In order to justify all the formal arguments, we need to introduce the following approximated equation of (KS):

(KS)_ε

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

u_εt(x, t) =∇ ·

∇(u_ε+ε)^m−(u_ε+ε)^q−2u_ε∇v_ε

, (x, t)∈IR^N ×(0, T), · · ·(1), 0(x, t) = ∆v_ε−v_ε+u_ε, (x, t)∈IR^N ×(0, T), · · ·(2),

u_ε(x,0) =u_0ε(x), x∈IR^N,

whereq >1 andεis a positive parameter andu_0εis an approximation for the initial datau₀ such that (A.1): 0≤u_0ε∈L¹∩W^2,p(IR^N) for all

p∈[_N^N₋₁, N+ 3],

p∈[2,3], for allε∈(0,1], (A.2): u0εL^p≤ u0L^p for allp∈[1,∞], for allε∈(0,1], (A.3): ∇u_0εL²≤ ∇u₀L² for allε∈(0,1], (A.4): u_0ε→u₀ for somep∈[1,∞), asε→0.

We call (u_ε, v_ε) a strong solution of (KS)_ε if it belongs to W_p^2,1×W_p^2,1(Q_T) for some p ≥ 1 and the equa- tions (1), (2) in (KS)_εare satisfied almost everywhere.

For the strong solution, we consider the spaceW(Q_T) defined by W(Q_T) :=W₁(Q_T)×W₂(Q_T)

:=

⎧⎨

⎩

W^2,1_N

N−1

W_N^2,1₊₃(Q_T)

×W_N+2^2,1 (Q_T) forN ≥2,

W₃^2,1(Q_T)×W₃^2,1(Q_T) forN = 1.

In [30–32], the following proposition concerning the existence of the strong solution was proved:

Proposition 3.1 (time local existence,[30–32]). Let N ≥ 1, m >1. Suppose that (A.1) is satisfied. Then, there exists a number T₁ = T₁(ε,u0ε_W2,N+2(IR^N), m, N) > 0 such that (KS)_ε has the unique non-negative strong solution (u_ε, v_ε)belonging toW(Q_T1).

Proposition 3.2 (extension criterion,[30–32]). Let the same assumption as that in Proposition 3.1 hold and letT >0. Suppose that(u_ε, v_ε)is a strong solution of (KS)_ε in the classW(Q_T). If it holds that

sup

0<t<Tu_ε(t)_L^∞_(IR^N₎<∞,

then there is T> T such that (u_ε, v_ε)can be a strong solution of (KS)_ε inW(Q_T).

4. Proof of Theorem 1.1

For the rigorous proof, we multiply (1) in (KS)_ε by (u_ε+ε)^r−1. For the sake of simplicity, we multiply u_εt=∇ ·

∇(uε+ε)^m− ∇u^q−1_ε ∇vε

byu^r−1_ε ,wherer >1, and integrate it over IR^N. Then, we have d

dtuε^r_Lr ≤ −mr(r−1)

u^r+m−3_ε |∇uε|² dx+r(r−1)

u^q−1_ε · ∇vε·u^r−2_ε ∇uε dx

=− 4mr(r−1)

(r+m−1)²∇uε^{r+m−12 2}_L2 + r(r−1) r+q−2

∇u^r+q−2_ε · ∇vε dx

=− 4mr(r−1)

(r+m−1)²∇uε^{r+m−12 2}_L2 − r(r−1) r+q−2

u^r+q−2_ε ·∆v_ε dx

≤ − 4mr(r−1)

(r+m−1)²∇uε^{r+m−12 2}_L2 + r(r−1)

r+q−2uε^r+q−1_Lr+q−1 for allr∈(1,∞). (4.1) By takingq₁= ^N(q−m)₂ ,q₂=r+q−1,a= 2 + _N² in Lemma 2.1, we have

uε^r+q−1_Lr+q−1 ≤c

2(N+2) N ·_r+m−1^r+q−1

0 uε^q−m

L^N(q−m)2 · ∇uε^{r+m−12 2}_L2 forr≥ N(q−m)

2 (4.2)

for some absolute constantc₀, where we used 1

β = N+ 2

N · r+m−1

r−m+ 2q−1 ≤ N+ 2 N bym≤q−_N² < q.

Combining (4.1) with (4.2), we obtain d

dtuε^r_Lr ≤r(r−1) r+q−2c

2(N+2) N ·_r+m−1^r+q−1

0 uε^q−m

L^N(q−m)2 − 4mr(r−1) (r+m−1)²

∇uε^{r+m−12 2}_L2. (4.3)

By the H¨older inequality, u0

L^N(q−m)2 ≤ u0^γ_L1u0^1−γ_L for someγ =γ(m, q, N, ). Therefore, when we take u₀L is sufficiently small for any fixed ≥ ^N(q−m)₂ , it holds thatu₀

L^N(q−m)2 is small.

On the other hand, uε(t)

L^N(q−m)2 ∈ C([0, T]). Therefore, using this continuity and (4.3) with r =

N(q−m)

2 ,we find that there exists a short interval [0,t₁] such that _dt^du_ε(t)L^r ≤ 0 for t ∈ [0, t₁], and

u_ε(t)

L^N(q−m)2 ≤ u₀

L^N(q−m)2 for t ∈ [0, t₁]. Since this implies that u_ε(t₁)

L^N(q−m)2 ≤ u₀

L^N(q−m)2 , we can repeat this procedure. In consequence, we obtain

uε(t)

L^N(q−m)2 ≤ u0

L^N(q−m)2 for t∈[0, T]. (4.4)

Substituting (4.4) into (4.3), we have d

dtu_ε^r_L^r ≤r(r−1) r+q−2c

2(N+2) N ·_r+m−1^r+q−1

0 u₀^q−m

L^N(q−m)2 − 4mr(r−1) (r+m−1)²

∇u_ε^{r+m−12 2}_L2

≤ − 2mr(r−1)

(r+m−1)²∇uε^{r+m−12 2}_L2 (4.5)

≤0 forr∈N(q−m)

2 ,∞

. By applying Moser’s iteration technique, we obtain

sup

0<t<Tu_ε(t)_L^∞_(IR^N₎ < ∞. (4.6) (see [32], Lem. 15 or [33], Sect. 5). Combining (4.6) with Propositions 3.1 and 3.2, we prove the following lemma:

Lemma 4.1. Let N ≥1, m >1, q ≥m+^N₂, ≥ ^N(q−m)₂ (≥1), T > 0 and suppose that (A.1) is satisfied.

Then, there exist an absolute constant M and a positive number εˆdepending only on M, N, m, such that if u₀∈L¹∩L(IR^N)satisfies that

u0_L1(IR^N)=M, u0_L(IR^N)≤ε,ˆ (4.7) then(KS)_ε has the strong solution(u_ε, v_ε)in the class obtained inW(Q_T) with the following property:

d

dtuε^r_Lr+ 2mr(r−1)

(r+m−1)²∇uε^{r+m−12 2}_L2 ≤ 0 for all t∈(0, T)and r∈N(q−m)

2 ,∞

.

From Lemma 4.1, we are going to show (1.6) in Theorem 1.1.

Lemma 2.1 with a= 3 and (A.2) gives that

uεL^r ≤c^{β11·r+m−12} u0^1−θ_L1 ¹· ∇uε^r+m−12 _L^r+m−12^2θ1 for any r∈[2,∞), (4.8) wherec depends only onN, and

β₁ := N N+ 2

(r+m−2 + _N²)(r−m+ 1) (r−1)(r+m−1) , θ₁ := r+m−1

2 ·

1−1 r

· 1 1

N −1

2 +r+m−1 2

·

Here and in what follows,c denotes a general constant (not necessarily the same at different occurrences) but which depends only onN.

Noting that 1

θ₁ ≤ 2, 1−θ₁

θ₁ (r+m−1) ≤ N+ 2

N for r∈[q,∞),

1

β₁ ≤ 2(N+ 2)

N for r∈[3(m−1),∞),

we obtain

uε_L^r+m−1r^θ1 ≤cu0_L^N+2N1 · ∇uε^{r+m−12 2}_L2 for anyr∈

max{q,3(m−1)},∞

. (4.9)

By (4.9), we easily see that

C_m,r· uε^r·λ_Lr ≤ 2mr(r−1)

(r+m−1)²∇uε^{r+m−12 2}_L2 form >1− 2

N, (4.10)

where

λ:= r+m−1

θ₁·r = 1 + m−1 +_N² r−1 > 1, C_m,r := 2mr(r−1)

(r+m−1)² ·(cu_0L¹)^−N+2N . By combining (4.10) with Lemma 4.1,

d

dtuε(t)^r_Lr+C_m,ruε(t)^r·λ_Lr ≤0 forr∈

(N+ 2)q,∞

. (4.11)

Let us denoteuε(t)^r_Lr byX(t). Then, (4.11) gives X(t)

X(t)^λ +C_m,r = 1 1−λ·

X(t)^−λ+1

+C_m,r ≤0. (4.12)

From (4.12), we obtain

X(t)≤ 1

(λ−1)C_m,r·t+X(0)^−λ+1 _λ−1¹

≤ 1

min

(λ−1)C_m,r, u0ε^r(−λ+1)_Lr

_λ−1¹ · 1 (1 +t)^λ−11

= max

(λ−1)C_{m,r −λ−1}¹

, u0ε^r_Lr

·(1 +t)^−λ−11 . (4.13)

This means that

u_ε(t)L^r ≤max

(λ−1)C_m,r

_−λ−1¹ _·¹_r

, u_0εL^r

·(1 +t)^{−λ−11 ·1r}

≤C˜_0,r(1 +t)^{−(m−1)N+2N ·(1−1r)}, (4.14)