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Research article

The existence of a compact global attractor for a class of competition model

  • Received: 10 July 2020 Accepted: 24 September 2020 Published: 09 October 2020
  • MSC : 35A01, 35B41, 35K57, 92D40

  • This paper is concerned with the existence of a compact global attractor for a class of competition model in n?dimensional (n ≥ 1) domains. Using mathematical induction and more detailed interpolation estimates, especially Gagliardo-Nirenberg inequality, we obtain the existence of a compact global attractor, which implies the uniform boundedness of the global solutions. In particular, we get that the Shigesada-Kawasaki-Teramoto competition model has a compact global attractor for n < 10. The result of the S-K-T model extends the existence results of compact global attractor in [21] from n < 8 to n < 10, and extends the uniform boundedness results of the global solutions in [17] to the non-convex domain.

    Citation: Yanxia Wu. The existence of a compact global attractor for a class of competition model[J]. AIMS Mathematics, 2021, 6(1): 210-222. doi: 10.3934/math.2021014

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  • This paper is concerned with the existence of a compact global attractor for a class of competition model in n?dimensional (n ≥ 1) domains. Using mathematical induction and more detailed interpolation estimates, especially Gagliardo-Nirenberg inequality, we obtain the existence of a compact global attractor, which implies the uniform boundedness of the global solutions. In particular, we get that the Shigesada-Kawasaki-Teramoto competition model has a compact global attractor for n < 10. The result of the S-K-T model extends the existence results of compact global attractor in [21] from n < 8 to n < 10, and extends the uniform boundedness results of the global solutions in [17] to the non-convex domain.


    In population dynamics, N. Shegesada, K. Kawasaki and E. Teromoto [15] proposed the following quasilinear competition model with cross-diffusion,

    {ut=Δ[(d1+ρ11u+ρ12v)u]+u(a1b1uc1v),xΩ,t>0,vt=Δ[(d2+ρ21u+ρ22v)v]+v(a2b2uc2v),xΩ,t>0,uν=vν=0,xΩ,t>0,u(x,0)=u0(x)0,v(x,0)=v0(x)0,xΩ, (1.1)

    where the functions u,v are the population densities of the two competing species and the initial values u0,v0 are nonnegative functions, which are not identically zero. Ω is a bounded smooth region in Rn with ν as its unit outward normal vector to Ω. The constants aj,bj,cj, dj(j=1,2) are all positive, and the constants ρij(i,j=1,2) are nonnegative, where d1 and d2 are the random diffusion rates, ρ11, ρ22 are the self-diffusion rates which represent intraspecific population pressures, and ρ12, ρ21 are the so-called cross-diffusion rates which represent the interspecific population pressures.

    When ρij=0(i,j=1,2), (1.1) is reduces to the well-known Lotka-Volterra competition-diffusion system, which has been researched intensively. When ρ12 or ρ21 is positive, (1.1) is a strongly coupled parabolic system, which has received much attention, since it occurs frequently in biological and chemical models. H. Amann considered a general class of strongly coupled parabolic systems and established the local existence (in time) and uniqueness results in a series of papers [1,2,3]. Roughly speaking, H. Amann showed that if u0,v0 in W1,p(Ω) with p>n, then (1.1) has a unique solution u,v defined in (0,t0) with t0>0 small.

    The global existence of nonnegative solutions to (1.1) is considered under some restrictive hypotheses on the smallness of cross-diffusion pressures or on the space dimension. For the case ρ12>0,ρ21>0, if ρ11=ρ22=0, J. Kim [8] proved the global existence of classical solutions by energy method when n=1 and d1=d2. Later, S. Shim [16] improved J. Kim's results and obtained the uniform boundedness of the global solutions in time by interpolated estimates. P. Deuring [6] proved the global existence of classical solutions when n1 and ρ12,ρ21 are small enough depending on the C2,α norm of initial values u0,v0. If the self-diffusion rates ρ11 and ρ22 are not zero, A. Yagi [22] proved the global existence of solutions when n=2 and 0<ρ12<8ρ11,0<ρ21<8ρ22, he also proved the same results for ρ22=ρ21=0 and ρ11>0. In addition, Y. Li and C. Zhao [13] obtained the global existence of classical solutions when n1,d1=d2 and ρ12ρ22+ρ21ρ11=2.

    For the case of ρ21=0, (1.1) becomes the following system

    {ut=Δ[(d1+ρ11u+ρ12v)u]+u(a1b1uc1v),xΩ,t>0,vt=Δ[(d2+ρ22v)v]+v(a2b2uc2v),xΩ,t>0,uν=vν=0,xΩ,t>0,u(x,0)=u0(x)0,v(x,0)=v0(x)0,xΩ. (1.2)

    Y. Lou, W. Ni and Y. Wu [14] established a global existence of classical solutions to (1.2) for n2 and ρ11 is merely assumed nonnegative but ρ12 and ρ22 are allowed to be positive, which is the only available result for smooth solutions with ρ11=0. When ρ11 is positive, Y. Choi, R. Lui and Y. Yamada [4,5] obtained some results on the global existence of the solutions to (1.2) with the restrictions n<6 and ρ22>0. P. Tuoc [20] showed the global existence of solutions for n<10. The global existence of solutions for arbitrary n under some restrictions on coefficients are investigated (see [7,9,11,19]). For the uniform boundedness of the global solutions, D. Le, L. Nguyen and T. Nguyen [12] using the semi-group techniques obtained the global attractor for n<6, which implies the uniform boundedness of the global solutions. Q. Xu and Y. Zhao [21] obtained the global attractor for n<8. And Y. Tao and M. Winkler [17] showed the boundedness of the solutions for n<10 when ΩRn is a bounded convex domain with smooth boundary.

    In this paper, we considered the following more general strongly coupled parabolic system

    {ut=(P(u,v)u+Q(u,v)v)+uf(u,v),xΩ,t>0,vt=(R(v)v)+vg(u,v),xΩ,t>0,uν=vν=0,xΩ,t>0,u(x,0)=u0(x)0,v(x,0)=v0(x)0,xΩ. (1.3)

    Due to the absence of the cross-diffusion term in the vequation, the diffusion matrix of (1.3) is triangular. H. Amann [3] showed that if one can obtain uL,vL, then the solution of (1.3) exists globally in time. D. Le [10] proved that if uLn,vL, then the system (1.3) has a global attractor with finite Hausdorff dimension, which attracts all the solutions of (1.3). D. Le, L. Nguyen and T. Nguyen [12] improved the results of that in [10]. In order to state their results, we first introduce the following definition.

    Definition 1.1. (see [10], Definition 2.1) Assume that there exists a solution (u,v) of system (1.3) defined on a subinterval I of R+. Let P 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

    ω(t)C0,tI.

    Furthermore, if I=(0,), then 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

    limtsupω(t)C.

    If ωP and I=(0,), one says ω is ultimately uniformly bounded.

    In [12], D. Le, L. Nguyen and T. Nguyen suppose that

    (H1) There exist a continuous function Φ and positive constant d such that the differentiable functions P,Q,R satisfying

    P(u,v)d(1+u)>0,|Q(u,v)|Φ(v)u,R(v)d>0,u,v0.

    (H2) There exists a nonnegative continuous function C(v) such that

    |f(u,v)|C(v)(1+u),g(u,v)upC(v)(1+up+1),u,v0,p>0.

    Under the above hypotheses (H1) and (H2), the authors proved the following results.

    Lemma 1.2. (see [12], Theorem 2.4) Assume (H1) and (H2) hold. Let (u,v) be a nonnegative solution to (1.3) with its maximal existence interval I. If uq,r,[t,t+1]×Ω=(t+1tu(,s)rq,Ωds)1/r (as a function in t) is in P for some q,r satisfying

    1r+n2q=1χ,q[n2(1χ),],r[11χ,]with someχ(0,1), (1.4)

    then there exists an absorbing ball where all solutions will enter eventually. Thus, if the system (1.3) is autonomous then there is a compact global attractor with finite Hausdorff dimension in B, which attracts all solutions, with

    B={(u,v)W1,p0(Ω)×W1,p0(Ω):u(x)0,v(x)0,xΩ}.

    In this paper, we impose some conditions on the functions P,Q,R,f,g in system (1.3) as follows.

    (A1) The functions P,Q,R are differentiable in there variables, and there exist constants β>0,b>0 and continuous function ϕ(v)0 for v0, such that

    P(u,v)duβ,|Q(u,v)|ϕ(v)u,R(v)d,R(v)0. (1.5)

    (A2) For the reaction terms (f,g), we assume that there exist positive constants a,b,c,α and nonnegative continuous functions f1(u,v),φ(v), such that

    f(u,v)=abuαf1(u,v),g(u,v)φ(v)(1cu)α+12. (1.6)

    Remark 1. Our assumptions (A1)(A2) on (P,Q,R,f,g) in this paper satisfy (H1)(H2) in [12].

    Now, we state our main results.

    Theorem 1.3. Suppose (A1)(A2) hold and (u0,v0)B with some p0>n. Then (1.3) has a compact global attractor with finite Hausdorff dimension in the space B, which attracts all the solutions, for any given α>0,β>0 and n2, but we need the following corresponding assumptions in (B1)(B2) for n>2,

    (B1) For 0<βn26,

    (i) if 0<αn+4n2β, then n22(1+4β)n+4β<0;

    (ii) if n+4n2β<α<β+1, then n<2(α+3β);

    (iii) if αβ+1, then n<2(2α+2β1).

    (B2) For β>n26, then

    (i) if 0<αβ+1, then n22(1+4β)n+4β<0;

    (ii) if β+1<α<n+4n2β, then n22(α+3β)n+4(α1)<0;

    (iii) if αn+4n2β, then n<2(2α+2β1).

    Theorem 1.4. Assume n<10 and (u0,v0)B with some p0>n, then (1.2) has a compact global attractor with finite Hausdorff dimension in the space B, which attracts all the solutions.

    Remark 2. Theorem 1.3 and Theorem 1.4 imply the uniform boundedness of the global solutions to the systems (1.3) and (1.2), respectively.

    This paper is organized as follows. In section 2, we shall prove the existence of a compact global attractor with finite Hausdorff dimension to system (1.3). As an application, we consider the Shegesada-Kawasaki-Teromoto competition model (1.2), and get the existence of a global attractor for n<10 in section 3.

    We shall first give the uniform Gronwall inequality, which will be frequently used in our proof.

    Lemma 2.1. (the uniform Gronwall inequality) (see[18] Chapt. 3, Lemma 1.1). Suppose positive Lipschitz functions y(t),r(t),h(t) defined on [t0,+] satisfy

    y(t)r(t)y(t)+h(t),

    and

    t+τtr(s)dsr0,t+τth(s)dsh0,t+τty(s)dsc0,tt0,

    with τ,r0,h0 and c0 some positive constants. Then it holds that

    y(t+τ)(c0τ+h0)er0,tt0.

    For given initial data u0(x),v0(x)B, it is standard to show that the solutions of (1.3) are still nonnegative. Then using comparison principle for parabolic equation on the vequation of (1.3), it is easy to see

    v(,t)L(Ω)P. (2.1)

    For the solution u, it is easy to get the following properties.

    Lemma 2.2. The solution u of (1.3) satisfies

    u(,t)L1(Ω)P, (2.2)

    and

    t+1tΩuα+1dxdsP. (2.3)

    Proof. Integrating the uequation of (1.3) by parts and noting the condition of f(u,v) in (A2), we get

    ddtΩudxaΩudxbΩuα+1dx, (2.4)

    which together with H¨older inequality uL1(Ω)uLα+1(Ω)1Lα+1α(Ω) gives

    ddtΩudxaΩudxb|Ω|α(Ωudx)α+1. (2.5)

    Then the comparison principle of ordinary differential equation implies (2.2) holds. Integrate (2.4) from t to t+1 and use (2.2) to yield (2.3).

    For the solution v, we will prove the following result, which plays an important role in the following estimates of u in Theorem 2.4. In the rest of our paper, Ci(i=1,2,) are some positive constants, and we will not point out them one by one.

    Lemma 2.3. For n1, the solution v of (1.3) satisfies

    t+1tΩ|v|4dxdsP. (2.6)

    Proof. In order to prove (2.6), we first show

    t+1tΩ|(R(v)v)|2dxdsP, (2.7)

    then we prove

    t+1tΩ|R(v)v|4dxdsP. (2.8)

    Recalling the condition of R(v)d in (A1), (2.8) ensures (2.6) holds.

    Now, we first deal with the proof of (2.7). For this purpose, multiplying the second equation of (1.3) by v and integrating by parts, we have

    12ddtΩv2dx+ΩR(v)|v|2dx=Ωv2g(u,v)dx.

    Integrating the above equation over [t,t+1], we obtain

    dt+1tΩ|v|2dxds12v(t)2L2(Ω)+C1t+1tΩ(1cu)α+12dxds,

    by R(v)d in (A1), g(u,v)φ(v)(1cu)α+12 in (A2) and the fact (2.1). Therefore, it is known by (2.1) and (2.2) that

    t+1tΩ|v|2dxdsP. (2.9)

    Next, we multiply the vequation of (1.3) by R(v)vt and integrate by parts to get

    ΩR(v)v2tdx=ΩR(v)vt(R(v)v)dx+ΩR(v)vtvg(u,v)dx=Ω(R(v)vt)(R(v)v)dx+ΩR(v)vtvg(u,v)dx=12ddtΩR2(v)|v|2dx+ΩR(v)vtvg(u,v)dx12ddtΩR2(v)|v|2dx+d2Ωv2tdx+C22Ω(1+u)α+1dx,

    here, we use H¨older inequality, the condition of g(u,v) in (A2) and (2.1).

    Due to R(v)d, thus

    ddtΩR2(v)|v|2dx+dΩv2tdxC2Ω(1+u)α+1dx. (2.10)

    In view of (2.1), (2.3), (2.9) and using the uniform Gronwall inequality on

    ddtΩR2(v)|v|2dxC2Ω(1+u)α+1dx,

    we obtain

    ΩR2(v)|v|2dxP. (2.11)

    Moreover, integrate (2.10) over [t,t+1] to know

    t+1tΩv2t(x,s)dxdsP. (2.12)

    By the vequation of (1.3) and noting (2.1), we have

    Ω|(R(v)v)|2dx=Ω[vtvg(u,v)]2dx2Ωv2tdx+C3+C4Ωuα+1dx,

    this together with (2.12) and (2.3) gives (2.7).

    Next, we will prove (2.8). Denote ξ=R(v)v and note R(v)0 in (A1) to get

    Ω|ξ|4dx=ΩR(v)|ξ|2ξvdx=Ωv(R(v)|ξ|2ξ)dx=ΩvR(v)R(v)|ξ|4dxΩvR(v)|ξ|2ξdx2ΩvR(v)ξ(ξξ)dxΩvR(v)|ξ|2ξdx2ΩvR(v)ξ(ξξ)dx.

    By H¨older inequality, we can get

    ΩvR(v)|ξ|2ξdxvR(v)L(Ω)ξ2L4(Ω)ξL2(Ω),

    and

    2ΩvR(v)ξ(ξξ)dxvR(v)L(Ω)ξ2L4(Ω)ξL2(Ω),

    thus

    ξ2L4(Ω)vR(v)L(Ω)(ξL2(Ω)+2ξL2(Ω)). (2.13)

    Now, we will prove

    ξL2(Ω)C6ξL2(Ω). (2.14)

    Noting ξ=R(v)v, then we have

    ξ=(R(v)v)=R(v)|v|2+R(v)v,ξ=(R(v)v)=R(v)(v)Tv+R(v)2v, (2.15)

    where we see v as a row vector, (v)T is the transpose of v, and 2v is a matrix.

    By (2.15) and the standard elliptic regularity 2vL2(Ω)C5vL2(Ω), we can prove (2.14) holds.

    Therefore, in virtue of (2.13) and (2.14), we obtain

    ξ2L4(Ω)(1+2C6)vR(v)L(Ω)ξL2(Ω).

    This together with (2.7) and (2.1) indicates that (2.8) holds. This completes the proof of Lemma 2.3.

    Next, we shall give the critical estimates in our paper.

    Theorem 2.4. The solution u of (1.3) satisfies

    uLˉp(Ω)P, (2.16)
    t+1tΩuˉp+αdxdsP, (2.17)

    and

    t+1tΩuˉp+β2|u|2dxdsP, (2.18)

    for ˉp satisfying (i) ˉp=α+2β or (ii) ˉp>α+2β and (n2)ˉp3nβ.

    Proof. Multiplying the first equation in (1.3) by up1 with p>1, and integrating on Ω by parts, we have

    1pddtΩupdx=Ωup1[P(u,v)u+Q(u,v)v]dx+Ωupf(u,v)dx=(p1)Ωup2P(u,v)|u|2dx(p1)Ωup2Q(u,v)uvdx+Ωupf(u,v)dx.

    Recalling the condition of f(u,v) in (A2) and |Q(u,v)|ϕ(v)u in (A1), we have

    Ωupf(u,v)dxaΩupdxbΩup+αdx,

    and

    |Ωup2Q(u,v)uvdx|Ωup2|Q(u,v)||uv|dxϕ(v)Ω|up1uv|dx=ϕ(v)Ω|up+β22uupβ2v|dxd2Ωup+β2|u|2dx+C7p1Ωupβ|v|2dx,

    by Hölder inequality.

    Combining these estimates and P(u,v)duβ in (A1), then

    1pddtΩupdx+d(p1)2Ωup+β2|u|2dx+bΩup+αdxaΩupdx+C7Ωupβ|v|2dx. (2.19)

    Case I: pα+2β. In this case, we have 2(pβ)p+α.

    Applying Hölder inequality and Young's inequality to the last term of (2.19), we have

    C7Ωupβ|v|2dxb2Ωu2p2βdx+C8Ω|v|4dxb2Ωup+αdx+C8Ω|v|4dx+C9.

    Consequently, (2.19) becomes

    1pddtΩupdx+d(p1)2Ωup+β2|u|2dx+b2Ωup+αdxaΩupdx+C8Ω|v|4dx+C9. (2.20)

    Obviously, (2.20) entails

    1pddtΩupdxaΩupdx+C8Ω|v|4dx+C9. (2.21)

    For the above inequality (2.21), if we can show

    t+1tΩupdxdsP, (2.22)

    then (2.6) and the uniform Gronwall inequality yield

    uLp(Ω)P. (2.23)

    Furthermore, integrating (2.20) from t to t+1, we can obtain

    t+1tΩup+αdxdsP, (2.24)

    and

    t+1tΩup+β2|u|2dxdsP. (2.25)

    Now, we will use mathematical induction to prove that (2.22) holds for p=α+2β. There exists some kN+ such that 1α+2βkαα+1. Denote p0=α+2βkα,pm=pm1+α(m=1,2,,k). On one hand, using (2.3) and Hölder inequality, it is easy to see that (2.22) holds for p=p0. On the other hand, we suppose (2.22) holds for p=pm1, then (2.24) means that (2.22) holds for p=pm1+α=pm. Hence the mathematical induction ensures that (2.22) holds for p=pk=α+2β.

    Therefore, (2.23)–(2.25) hold for p=α+2β, which implies (2.16)–(2.18) hold for ˉp=α+2β.

    Case II: p>α+2β. In this case, we assume (n4)p(3n4)β.

    Let wp=up+β2 and denote wp as w sometimes for simplicity, then (2.19) can be written as

    1pddtΩw2pp+βdx+2d(p1)(p+β)2Ω|w|2dx+bΩw2(p+α)p+βdxaΩw2pp+βdx+C7Ωw2(pβ)p+β|v|2dxaΩw2pp+βdx+C7w2(pβ)p+βL2(Ω)v2L4(Ω)=aw2pp+βL2pp+β(Ω)+C7w2(pβ)p+βL4(pβ)p+β(Ω)v2L4(Ω), (2.26)

    by the Hölder inequality.

    The conditions p>α+2β and (n4)p(3n4)β implies

    2(p+α)p+β<4(pβ)p+β2nn2,

    here 2nn2 can be replaced by + for n=2.

    It is known by Gagliardo-Nirenberg inequality that

    wL4(pβ)p+β(Ω)C10w1θL2pp+β(Ω)wθL2(Ω)+C10wL1(Ω), (2.27)

    with

    θ=n(p+β)(p2β)2(pβ)(2p+nβ).

    Using (2.27) and Young's inequality, we have

    C7w2(pβ)p+βL4(pβ)p+β(Ω)v2L4(Ω)C11w2(pβ)(1θ)(p+β)L2pp+β(Ω)w2θ(pβ)p+βL2(Ω)v2L4(Ω)+C11w2(pβ)p+βL1(Ω)v2L4(Ω)εw2L2(Ω)+Cεwm1L2pp+β(Ω)vm2L4(Ω)+C11w4(pβ)p+βL1(Ω)+C11v4L4(Ω),

    with

    m1=2(pβ)(1θ)p+βθ(pβ),m2=2(p+β)p+βθ(pβ).

    Let ε=d(p1)(p+β)2, then (2.26) becomes

    1pddtΩw2pp+βdx+d(p1)(p+β)2Ω|w|2dx+bΩw2(p+α)p+βdxaw2pp+βL2pp+β(Ω)+Cεwm1L2pp+β(Ω)vm2L4(Ω)+C11w4(pβ)p+βL1(Ω)+C11v4L4(Ω). (2.28)

    Let y(t)=w2pp+βL2pp+β(Ω),h(t)=C11w4(pβ)p+βL1(Ω)+C11v4L4(Ω), then we have

    1pddty(t)ay(t)+Cεy(t)(p+β)m12pvm2L4(Ω)+h(t). (2.29)

    For the case of y(t)1, obviously we have

    y(t)=Ωw2pp+βdxP. (2.30)

    Since p>α+2β, (2.30) implies wL1(Ω)P by Hölder inequality. Let

    (n2)p3nβ,

    then a direct calculation shows that m24. Consequently, (2.6) and Hölder inequality give

    vm2L4(Ω)P. (2.31)

    Furthermore, integrating (2.28) from t to t+1 yields

    t+1tΩw2(p+α)p+βdxdsP, (2.32)

    and

    t+1tΩ|w|2dxdsP. (2.33)

    For the case of y(t)>1, denote r(t)=a+Cεvm2L4(Ω), then

    1pddty(t)r(t)y(t)+h(t), (2.34)

    here, we used the fact

    (p+β)m12p=(3n4)β(n4)p(4pnp+4nβ)<1.

    It is easy to see that (2.31) implies t+1tr(s)dsP for (n2)p3nβ, thus if we can show

    t+1ty(s)dsP,t+1th(s)dsP, (2.35)

    then using the uniform Gronwall inequality to the inequality (2.34), we obtain (2.30). Similarly, integrate (2.28) over [t,t+1] to obtain (2.32) and (2.33).

    Now, we prove (2.35) for (n2)p=3nβ. In order to get t+1th(s)dsP, recalling (2.6), the key step is to deal with

    t+1tw4(pβ)p+βL1(Ω)dsP, (2.36)

    since the Minkowski's inequality ensures

    t+1th(s)dsC11t+1tw4(pβ)p+βL1(Ω)ds+C11t+1tv4L4(Ω)ds.

    Now, we will prove t+1ty(s)dsP and (2.36) by mathematical induction, simultaneously. For the case of n>2, there exists some ˉkN+ such that α+2β<3nβn2ˉkα2α+2β. Denote q0=3nβn2ˉkα,qm=qm1+α(m=1,2,,ˉk). Since we have proved (2.17) for ˉp=α+2β, it is easy to get t+1ty(s)dsP for p=q0 by Hölder inequality. In addition, noting that q0+β22α+3β2<α+2β and (2.16) holds for ˉp=α+2β, using Hölder inequality we obtain wq0L1(Ω)P, which indicates that the result (2.36) is true for p=q0. On the other hand, assume t+1ty(s)dsP and (2.36) hold for p=qm1, then (2.32) holds for p=qm1. According to the definition wp=up+β2, it is easy to see

    (wqm1)2(qm1+α)qm1+β=(wqm)2qmqm+β,

    thus t+1ty(s)dsP for p=qm. Moreover, the assumption implies (2.30) holds for p=qm1 and hence

    wqmL1(Ω)Cwqm12qm1qm1+βL2qm1qm1+β(Ω),

    by qm1>α+2β and Hölder inequality with some C>0. And thus (2.36) holds for p=qm.

    Above all, for the case of n>2, we have proved (2.35) with p=3nβn2. Therefore, (2.30), (2.32) and (2.33) hold for p=3nβn2, which implies (2.16)–(2.18) hold for ˉp3nβn2. Similarly, we can prove (2.16)–(2.18) hold for any positive constant ˉp>α+2β if n2.

    This complete the proof of Theorem 2.4.

    Next, we will use Lemma 1.2 and Theorem 2.4 to give the proof of Theorem 1.3.

    proof of Theorem 1.3. Let s=ˉp+α, then by (2.17),

    t+1tusLs(Ω)ds=t+1tΩuˉp+αdxdsP.

    Define

    1χ=1s+n2s=n+22s=n+22(ˉp+α),
    A=sn2(1χ)=2sn+2>0,B=s11χ=nsn+2>0.

    By Lemma 1.2, we also need χ(0,1), which is equivalent to

    n<2(ˉp+α1). (2.37)

    It is known by Gagliardo-Nirenberg inequality that

    wL2(Ω)C12(wL2(Ω)+wL1(Ω)),

    with 2=2n/(n2).

    Let l=ˉp+β2 and r=2l,q=2l, then w=ul and

    t+1turLq(Ω)ds=t+1tw2L2(Ω)ds2C12[t+1tw2L2(Ω)ds+sup[t,t+1]w2L1(Ω)].

    The estimate wL1(Ω)P comes from (2.16) by Hölder inequality, which together with (2.18) indicates

    t+1turLq(Ω)dsP.

    Let

    1χ=1r+n2q=1l(12+n22)=n4l=n2(ˉp+β),
    A=qn2(1χ)=2(ˉp+β)n2>0,B=r11χ=n2n(ˉp+β)>0.

    By Lemma 1.2, we also need χ(0,1), which means

    n<2(ˉp+β). (2.38)

    Comparing (2.37) and (2.38), we choose n<2(ˉp+α1) if α>β+1, otherwise, we choose n<2(ˉp+β). In addition, recall ˉp satisfies (i) ˉp=α+2β or (ii) ˉp>α+2β and (n2)ˉp3nβ in Theorem 2.4, hence we can assign any positive number to ˉp for n2, but we choose ˉp=3nβn2 if 0<αn+4n2β, otherwise, we choose ˉp=α+2β for n>2. Consequently, combining these analysis we can obtain Theorem 1.3.

    In this part, we will consider the boundedness of the global solutions to the following S-K-T model

    {ut=Δ[(d1+ρ11u+ρ12v)u]+u(a1b1uc1v),xΩ,t>0,vt=Δ[(d2+ρ22v)v]+v(a2b2uc2v),xΩ,t>0,uν=vν=0,xΩ,t>0,u(x,0)=u0(x)0,v(x,0)=v0(x)0,xΩ. (3.1)

    Proof of Theorem 1.4. Comparing with the divergence form of system (1.3), we have

    P(u,v)=d1+2ρ11u+ρ12v,Q(u,v)=ρ12u,R(v)=d2+2ρ22v,
    f(u,v)=a1b1uc1v,g(u,v)=a2b2uc2v.

    It is easy to see that P,Q,R and f,g satisfy the conditions in (A1) and (A2), respectively, with α=β=1.

    Theorem 1.3 gives Theorem 1.4 for n2 directly. Moreover, a simple computation shows (B2) (i) in Theorem 1.3 holds for 2<n<8 and (B1) (i) holds for 8n<10. This completes the proof of Theorem 1.4.

    Remark 3. Our result implies the uniform boundedness of the global solutions to the system (3.1). This result extends the existence results of global attractor in [21] from n<8 to n<10, and extends the uniform boundedness results of the global solutions in [17] to the non-convex domain.

    The author is greatly indebted to Professor Yaping Wu for her valuable suggestions and helpful discussions. And the author is very grateful for the anonymous referees for their valuable comments and many useful suggestions which helped to improve the exposition of the current paper. The work is supported by NSFC (No. 11801314).

    The author declares no conflicts of interest in this paper.



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