1. Introduction and Statement of Main Result
Shigesada et al. [1] introduced the following competition model to describe the spatial segregation of two competing species under inter- and intraspecies population pressures:
(1)ut=Δ[(d1+α11u+α12v)u]+u(a1-b1u-c1v),hhhhhhhhhhhhhhhhhhhhhhhx∈Ω⊂ℝn, t>0,vt=Δ[(d2+α21u+α22v)v]+v(a2-b2u-c2v),hhhhhhhhhhhhhhhhhhhhhhhx∈Ω⊂ℝn, t>0,∂u∂n=∂v∂n=0, x∈∂Ω, t>0,
where Ω is a bounded smooth region in ℝn with n as its unit outward normal vector to the smooth boundary ∂Ω. u and v are the population densities of the two competing species. The constants aj, bj, cj, and dj (j=1,2) are all positive, and constants αij (i,j=1,2) are nonnegative. d1 and d2 are the random diffusion rates, α11 and α22 are the self-diffusion rates which represent intraspecific population pressures, and α12 and α21 are the so-called cross-diffusion rates which represent the interspecific population pressures.

If αij=0 (i,j=1,2), system (1) is reduced to the classical Lotka-Volterra competition model with diffusion; it has been extensively studied in the past few decades. When initial value is nonnegative and bounded, it is easy to prove that (1) has a unique uniformly bounded global solution.

For α11=0, the global existence of solutions has been widely investigated by many authors. When n=1, d1=d2, α12>0, α21>0, and α11=α22=0 hold, Kim [2] proved the global existence of classical solutions by energy method. For n≥1, α11=α22=0, Deuring [3] proved the global existence of solutions if α12 and α21 are small enough depending on the C2,α norm of initial values u0,v0. Choi et al. [4] improved Deuring's result and proved the global existence of solutions if the cross-diffusion coefficients are small depending only on the L∞ norm of initial value v0. By applying more detailed interpolated estimates, especially Gagliardo-Nirenberg inequality, Shim [5] improved Kim and Deuring’s results and established the uniform bounds of the global existence of solutions in time. For n=2, Lou et al. [6] established the unique global existence of solutions for α21=0, α12>0, α11=0, and α22≥0.

For α11>0, (1) can be written as
(2)ut=Δ[(d1+α11u+α12v)u]+u(a1-b1u-c1v),hhhhhhhhhhhhhhhhhhhhhhhx∈Ω⊂ℝn, t>0,vt=Δ[(d2+α22v)v]+v(a2-b2u-c2v),hhhhhhhhhhhhhhhhhx∈Ω⊂ℝn, t>0,∂u∂n=∂v∂n=0, x∈∂Ω, t>0.
Equation (2) has been investigated by many authors; we state the results as follows.

For n=2, either 8α11>α12>0, 8α22>α21>0 or α22=α21=0, α11>0; Yagi [7] proved the global existence of solutions. For α11>0, α22>0, and α21=0, Kuiper and Dung [8] established the uniform bounds of global solutions for any n when ∥v∥L∞(Ω) and ∥u∥Lp(Ω) (p>n) are uniformly bounded. Choi et al. [9] applied more detailed interpolated estimates and energy methods to prove the global existence of solutions for n<6, α11>0, and α22>0.

Le and his collaborators [10] have shown the existence of a global attractor for (2) in case n≤5. Le and Nguyen [11] constructed a special test function to prove the global existence of solutions for any dimension n under some certain restrictions on coefficients. Tuôc [12] improved the results of Le and Nguyen by a nontrivial application of maximum principle. Recently, Tuoc [13] has established the L4-estimate of ∇v; then by an iteration method, they show u∈Lr for any r≥1 and n<10, which implies the global existence of solutions.

In this paper, we consider the uniform bounds of the global existence of solutions in time of system (2) for α21=0, α11>0, and α22>0. In Section 2, we show some preliminary knowledge used in this paper. In Section 3, we follow the arguments of Le et al. and improve their results. We will prove the uniform bounds of the global existence of solutions in time of system (2) for n<8.

The main result in this paper is as follows.

Theorem 1.
Assume n<8 holds; for any p0>n, system (2) has a global attractor with finite Hausdorff dimension in the space 𝒳 defined by
(3)𝒳={(u,v)∈W1,p0(Ω)×W1,p0(Ω): u(x)≥0, v(x)≥0, ∀x∈ΩW1,p0}.

2. Preliminary Results
System (2) can be written in the divergence form as
(4)∂u∂t=∇[(d1+2α11u+α12v)∇u+α12u∇v]+u(a1-b1u-c1v), x∈Ω, t>0,∂v∂t=∇[(d2+2α22v)∇v]+v(a2-b2u-c2v), x∈Ω, t>0,∂u∂n=∂v∂n=0, x∈∂Ω.

Definition 2 (see [<xref ref-type="bibr" rid="B4">10</xref>, Definition 2.1]).
Assume that there exists a solution (u,v) of system (4) defined on a subinterval I of ℝ+. Let 𝒪 be the set of function ω on I such that there exists a positive constant C0, which may generally depend on the parameters of the system and the W1,p0 norm of the initial value (u0,v0), such that
(5)ω(t)≤C0, ∀t∈I.
Furthermore, if I=(0,∞), one says that ω is in 𝒫 if ω∈𝒪 and there exists a positive constant C∞ that depends only on the parameters of the system but does not depend on the initial value of (u0,v0) such that
(6)limt↓∞ supω(t)≤C∞.
If ω∈𝒫 and I=(0,∞), one says ω is ultimately uniformly bounded.

Lemma 3 (the uniform Gronwall inequality).
Assume that u(t)≥0, a(t)≥0, and b(t)≥0 hold and that they are integrable in [t0,+∞] satisfying
(7)∫tt+ra(s)ds≤a, ∫tt+rb(s)ds≤b,∫tt+ru(s)ds≤C,
where a, b, and C are positive constants. If u′(t)≤a(t)u(t)+b(t), then one has
(8)u(t+r)≤(Cr+b)ea, ∀t≥t0.

Lemma 4 (see [<xref ref-type="bibr" rid="B4">10</xref>, Lemmas 3.2-3.3]).
For any dimension n, one has the following estimates for the solutions of system (4):
(9)∥v∥L∞(Ω)∈𝒫,(10)∥∇v(·,t)∥L2(Ω)∈𝒫,(11)∥u(·,t)∥L1(Ω)∈𝒫,(12)∫tt+1∫Ωu2(x,s)dx ds∈𝒫,(13)∫tt+1∫Ωvt2(x,s)dx ds∈𝒫.

Lemma 5 (see [<xref ref-type="bibr" rid="B4">10</xref>, Theorem 2.4]).
For the system (4), if
(14)∥u∥q,r,[t,t+1]×Ω=(∫tt+1∥u(·,s)∥q,Ωrds)1/r∈𝒫
holds, with q, r satisfying
(15)1r+n2q=1-χ,q∈[n2(1-χ),∞], r∈[11-χ,∞],
where χ∈(0,1), then there exists γ>1 such that
(16)∥v(·,t)∥Cγ(Ω)∈𝒫, ∥u(·,t)∥Cγ(Ω)∈𝒫.

3. Proof of Theorem <xref ref-type="statement" rid="thm1.1">1</xref>
Lemma 6.
For any dimension n, any solution u of (4) has the following estimate:
(17)∫tt+1∫Ω|∇v|4dx ds∈𝒫.

Proof.
Define
(18)w=(d2+α22v)v;
then w satisfies the following equation:
(19)wt=(d2+2α22v)Δw+(d2+2α22v)v(a2-b2u-c2v).
Multiplying (19) by Δw and integrating with respect to x over Ω, we have
(20)-12ddt∫Ω|∇w|2dx =∫Ω(d2+2α22v)|Δw|2dx +∫Ω(d2+2α22v)v(a2-b2u-c2v)Δw dx.
Integrating (20) over [t,t+1], we obtain
(21)12∥∇w(t)∥L2(Ω)2-12∥∇w(t+1)∥L2(Ω)2 =∫tt+1∫Ω(d2+2α22v)|Δw|2dx ds +∫tt+1∫Ω(d2+2α22v)v(a2-b2u-c2v)Δw dx ds.
In virtue of (9), there exist positive constants C1, C2, and C3 such that
(22)C1∫tt+1∫Ω|Δw|2dx ds ≤12∥∇w(t)∥L2(Ω)2-12∥∇w(t+1)∥L2(Ω)2 +∫tt+1∫Ω(C2+C3u)|Δw|dx ds.
Here (18) implies
(23)∇w=d2∇v+2α22v∇v.
By (9)-(10) and (23), we have
(24)∥∇w∥L2(Ω)∈𝒫.
Hence (22) and Hölder’s inequality imply
(25)C1∫tt+1∫Ω|Δw|2dx ds≤C4+C12∫tt+1∫Ω|Δw|2dx ds+C5∫tt+1∫Ω(C2+C3u)2dx ds.
By (12) and (25), we get
(26)∫tt+1∫Ω|Δw|2dx ds∈𝒫.

Multiplying (19) by w|∇w|2 and integrating with respect to x over Ω, we have
(27)∫Ωwtw|∇w|2dx =∫ΩΔw(d2+2α22v)w|∇w|2dx+∫Ωfw|∇w|2dx =-∫Ω∇w∇[(d2+2α22v)w|∇w|2]dx+∫Ωfw|∇w|2dx ≤-∫Ω(d2+2α22v)|∇w|4dx +∫Ω2(d2+2α22v)w|∇w|2|∇2w|dx -∫Ωw|∇w|22α22∇v∇w dx+∫Ωfw|∇w|2dx,
with f=(d2+2α22v)v(a2-b2u-c2v).

By (27), we get
(28)∫Ω(d2+2α22v)|∇w|4dx ≤-∫Ωwtw|∇w|2dx+∫Ω2(d2+2α22v)w|∇w|2|∇2w|dx -∫Ωw|∇w|22α22(∇wd2+2α22v)∇w dx +∫Ωfw|∇w|2dx.
Recall that (9) and (18) yield
(29)∥w∥L∞(Ω)∈𝒫.
It follows from (28) and (29) that
(30)∫Ω(d2+4α22v)|∇w|4dx ≤C(∫Ω|wt||∇w|2dx+∫Ω|∇w|2|∇2w|dx) +∫Ω|f||∇w|2dx.
By Young’s inequality and (30)
(31)d2∫Ω|∇w|4dx ≤C(d23C∫Ω|∇w|4dx+C6∫Ω|wt|2dx+d24C∫Ω|∇w|4dx + C7∫Ω|∇2w|2dx+d23C∫Ω|∇w|4dx + C8∫Ω|f|2dx).
Since
(32)∫Ω|∇2w|2dx≤C0∫Ω|Δw|2dx+C0∫Ωu2dx,
together with (31), we see from (31) that
(33)∫tt+1∫Ω|∇w|4dx ds ≤C(∫tt+1∫Ω|wt|2dx ds+∫tt+1∫Ω|Δw|2dx ds +∫tt+1∫Ω|u|2dx ds)≤C~,
where C~ is independent of t.

Since
(34)w=(d2+α22v)v, wt=d2vt+2α22vvt,
together with (9) and (13), we have ∫tt+1∫Ωwt2(x,s)dx ds∈𝒫. This fact, together with (12) and (26), implies ∫tt+1∫Ω|∇w(x,s)|4dx ds∈𝒫. Hence, in view of ∇v=∇w/(d2+2α22v) and (9), we get the desired result ∫tt+1∫Ω|∇v|4dx ds∈𝒫.

Lemma 7.
For any dimension n, any solution u of (4) satisfies the following estimates:
(35)∥u∥L2(Ω)∈𝒫, ∥u∥L3(Ω)∈𝒫.

Proof.
Multiplying the first equation of (4) by u and integrating, we get
(36)12ddt∫Ωu2dx =-∫Ω[(d1+2α11u+α12v)|∇u|2-α12u∇u·∇v]dx +∫Ωu2(a1-b1u-c1v)dx.
Young’s inequality and (36) imply
(37)12ddt∫Ωu2dx+∫Ωd1|∇u|2dx+2α11∫Ωu|∇u|2dx +α12∫Ωv|∇u|2dx =-α12∫Ωu∇u·∇v dx+∫Ωu2(a1-b1u-c1v)dx ≤ɛ∫Ωu|∇u|2dx+C(ɛ)∫Ωu|∇v|2dx+∫Ωa1u2dx.
Taking ɛ=α11 in (37), we have
(38)12ddt∫Ωu2dx+∫Ωd1|∇u|2dx+α11∫Ωu|∇u|2dx +α12∫Ωv|∇u|2dx ≤C(α11,α12)∫Ωu|∇v|2dx+∫Ωa1u2dx ≤C9∫Ωu2dx+C10∫Ω|∇v|4dx+∫Ωa1u2dx.
By the uniform Gronwall inequality, together with (12), (17), and (38), we obtain
(39)∥u∥L2(Ω)∈𝒫.
In virtue of (36), we have
(40)12ddt∫Ωu2dx+∫Ωd1|∇u|2dx+2α11∫Ωu|∇u|2dx +α12∫Ωv|∇u|2dx+b1∫Ωu3dx =-α12∫Ωu∇u·∇v dx+∫Ωu2(a1-c1v)dx.
Integrating (40) over [t,t+1], we get
(41)12∥u(t+1)∥L2(Ω)2-12∥u(t)∥L2(Ω)2+∫tt+1∫Ωd1|∇u|2dx ds +2α11∫tt+1∫Ωu|∇u|2dx ds+α12∫tt+1∫Ωv|∇u|2dx ds +b1∫tt+1∫Ωu3dx ds ≤ɛα12∫tt+1∫Ωu|∇u|2dx ds+C∫tt+1∫Ωu|∇v|2dx ds +∫tt+1∫Ωa1u2dx ds.
By Young’s inequality, we have
(42)∫tt+1∫Ωu|∇v|2dx ds ≤12∫tt+1∫Ωu2dx ds+12∫tt+1∫Ω|∇v|4dx ds.
Taking ɛ=α11/α12 in (41) and applying Hölder’s inequality, we see from (42) that
(43)b1∫tt+1∫Ωu3dx ds ≤12∥u(t)∥L2(Ω)2-12∥u(t+1)∥L2(Ω)2 +C11∫tt+1∫Ωu2dx ds+C12∫tt+1∫Ω|∇v|4dx ds.
By (12), (17), and (39), we get
(44)∫tt+1∫Ωu3dx ds∈𝒫.

Next we prove ∥u∥L3(Ω)∈𝒫. Multiplying (4) by u2 and integrating with respect to x over Ω, we get
(45)13ddt∫Ωu3dx+2∫Ωd1u|∇u|2dx+4α11∫Ωu2|∇u|2dx +2α12∫Ωuv|∇u|2dx =-2∫Ωα12u2∇v·∇u dx+∫Ωu3(a1-b1u-c1v)dx.
Apply the following inequalities:
(46)∫Ωv2dx≤ɛ(∫Ω|∇v|2dx+∥v∥L1(Ω)2)+Cɛ-n/2∥v∥L1(Ω)2,∫Ωu2∇u·∇v dx≤ε1∫Ωu2|∇u|2dx+C(ε1)∫Ωu2|∇v|2dx,∫Ωu2|∇v|2dx≤12∫Ωu4dx+12∫Ω|∇v|4dx.
Use (46) with v=u2 to get
(47)∫Ωu4dx≤ɛ{∫Ω|∇(u2)|2dx+∥u2∥L1(Ω)2}+Cɛ-n/2∥u2∥L1(Ω)2=ɛ{4∫Ωu2|∇u|2dx+∥u∥L2(Ω)4}+Cɛ-n/2∥u∥L2(Ω)4.
Choosing small positive numbers ε and ε1 in the above inequalities, we get
(48)13ddt∫Ωu3dx+2∫Ωd1u|∇u|2dx+α11∫Ωu2|∇u|2dx +2α12∫Ωuv|∇u|2dx ≤C(∫Ωu2dx)2+C∫Ω|∇v|4dx+∫Ωa1u3dx.
By (17), (39), (44), (48), and uniform Gronwall’s inequality, we get the desired result
(49)∥u∥L3(Ω)∈𝒫.

Proof of Theorem <xref ref-type="statement" rid="thm1.1">1</xref>.
It follows from (48) that
(50)α11∫tt+1∫Ωu2|∇u|2dx ds ≤13∥u(t)∥L3(Ω)3-13∥u(t+1)∥L3(Ω)3 +C∫tt+1(∫Ωu2dx)2ds+C∫tt+1∫Ω|∇v|4dx ds +∫tt+1∫Ωa1u3dx ds.
In virtue of (17), (35), (44), and (50), we obtain
(51)∫tt+1∫Ωu2|∇u|2dx ds∈𝒫.
For l=2, v=ul, we see ∫tt+1∫Ω|∇v|2dx ds=4∫tt+1∫Ωu2|∇u|2dx ds∈𝒫.

Let w=v-∫Ωv dx; then Gagliardo-Nirenberg inequality gives
(52)∥w∥L2*(Ω)≤C∥∇w∥L2(Ω),
which implies
(53)∥v∥L2*(Ω)≤C(∥∇v∥L2(Ω)+∥v∥L1(Ω)),
with 2*=2n/(n-2).

For r=2l, q=2*l, in virtue of (53), we have
(54)∫tt+1∥u∥Lq(Ω)rds=∫tt+1∥v∥L2*(Ω)2ds≤C(∫tt+1∥∇v∥L2(Ω)2ds+sup[t,t+1]∥v∥L1(Ω)2).
Note
(55)∥v∥L1(Ω)2=∥u(·,t)∥Ll(Ω)l=∥u(·,t)∥L2(Ω)2∈𝒫;
thus ∫tt+1∥u∥Lq(Ω)γds∈𝒫, with q, r satisfying
(56)1-χ∶=1r+n2q=1l(12+n2·2*)=n4l.
Let
(57)A=q-n2(1-χ)=q-2l, B=r-11-χ=2l-4ln,
when l=2 holds; in order to satisfy (15) in Lemma 5, we need to check χ∈(0,1), A≥0, and B≥0. By (56), we have the following results:
(58)n=3 χ=58 A=8 B=43,(59)n=4 χ=12 A=4 B=2,(60)n=5 χ=38 A=83 B=125,(61)n=6 χ=14 A=2 B=83,(62)n=7 χ=18 A=85 B=207.
Since Cυ×Cυ (υ>1) is compact in 𝒳, by the attractor theory in [14], we complete the proof of Theorem 1.