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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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(a1−b1u−c1v),x∈Ω,t>0,vt=Δ[(d2+ρ21u+ρ22v)v]+v(a2−b2u−c2v),x∈Ω,t>0,∂u∂ν=∂v∂ν=0,x∈∂Ω,t>0,u(x,0)=u0(x)⩾0,v(x,0)=v0(x)⩾0,x∈Ω, | (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 n⩾1 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 n⩾1,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(a1−b1u−c1v),x∈Ω,t>0,vt=Δ[(d2+ρ22v)v]+v(a2−b2u−c2v),x∈Ω,t>0,∂u∂ν=∂v∂ν=0,x∈∂Ω,t>0,u(x,0)=u0(x)⩾0,v(x,0)=v0(x)⩾0,x∈Ω. | (1.2) |
Y. Lou, W. Ni and Y. Wu [14] established a global existence of classical solutions to (1.2) for n⩽2 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 Ω∈Rn 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),x∈Ω,t>0,vt=∇⋅(R(v)∇v)+vg(u,v),x∈Ω,t>0,∂u∂ν=∂v∂ν=0,x∈∂Ω,t>0,u(x,0)=u0(x)⩾0,v(x,0)=v0(x)⩾0,x∈Ω. | (1.3) |
Due to the absence of the cross-diffusion term in the v−equation, the diffusion matrix of (1.3) is triangular. H. Amann [3] showed that if one can obtain u∈L∞,v∈L∞, then the solution of (1.3) exists globally in time. D. Le [10] proved that if u∈Ln,v∈L∞, 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,∀t∈I. |
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
limt↓∞supω(t)⩽C∞. |
If ω∈P 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,v⩾0. |
(H2) There exists a nonnegative continuous function C(v) such that
|f(u,v)|⩽C(v)(1+u),g(u,v)up⩽C(v)(1+up+1),∀u,v⩾0,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 ‖u‖q,r,[t,t+1]×Ω=(∫t+1t‖u(⋅,s)‖rq,Ωds)1/r (as a function in t) is in P for some q,r satisfying
1r+n2q=1−χ,q∈[n2(1−χ),∞],r∈[11−χ,∞]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,∀x∈Ω}. |
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 v⩾0, 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)=a−buα−f1(u,v),g(u,v)⩽φ(v)(1−cu)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 n⩽2, but we need the following corresponding assumptions in (B1)–(B2) for n>2,
(B1) For 0<β⩽n−26,
(i) if 0<α⩽n+4n−2β, then n2−2(1+4β)n+4β<0;
(ii) if n+4n−2β<α<β+1, then n<2(α+3β);
(iii) if α⩾β+1, then n<2(2α+2β−1).
(B2) For β>n−26, then
(i) if 0<α⩽β+1, then n2−2(1+4β)n+4β<0;
(ii) if β+1<α<n+4n−2β, then n2−2(α+3β)n+4(α−1)<0;
(iii) if α⩾n+4n−2β, then n<2(2α+2β−1).
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)ds⩽r0,∫t+τth(s)ds⩽h0,∫t+τty(s)ds⩽c0,∀t⩾t0, |
with τ,r0,h0 and c0 some positive constants. Then it holds that
y(t+τ)⩽(c0τ+h0)er0,∀t⩾t0. |
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 v−equation 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+1t∫Ωuα+1dxds∈P. | (2.3) |
Proof. Integrating the u−equation of (1.3) by parts and noting the condition of f(u,v) in (A2), we get
ddt∫Ωudx⩽a∫Ωudx−b∫Ωuα+1dx, | (2.4) |
which together with H¨older inequality ‖u‖L1(Ω)⩽‖u‖Lα+1(Ω)‖1‖Lα+1α(Ω) gives
ddt∫Ωudx⩽a∫Ωudx−b|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 n⩾1, the solution v of (1.3) satisfies
∫t+1t∫Ω|∇v|4dxds∈P. | (2.6) |
Proof. In order to prove (2.6), we first show
∫t+1t∫Ω|∇⋅(R(v)∇v)|2dxds∈P, | (2.7) |
then we prove
∫t+1t∫Ω|R(v)∇v|4dxds∈P. | (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
12ddt∫Ωv2dx+∫ΩR(v)|∇v|2dx=∫Ωv2g(u,v)dx. |
Integrating the above equation over [t,t+1], we obtain
d∫t+1t∫Ω|∇v|2dxds⩽12‖v(t)‖2L2(Ω)+C1∫t+1t∫Ω(1−cu)α+12dxds, |
by R(v)⩾d in (A1), g(u,v)⩽φ(v)(1−cu)α+12 in (A2) and the fact (2.1). Therefore, it is known by (2.1) and (2.2) that
∫t+1t∫Ω|∇v|2dxds∈P. | (2.9) |
Next, we multiply the v−equation 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=−12ddt∫ΩR2(v)|∇v|2dx+∫ΩR(v)vtvg(u,v)dx⩽−12ddt∫ΩR2(v)|∇v|2dx+d2∫Ωv2tdx+C22∫Ω(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
ddt∫ΩR2(v)|∇v|2dx+d∫Ωv2tdx⩽C2∫Ω(1+u)α+1dx. | (2.10) |
In view of (2.1), (2.3), (2.9) and using the uniform Gronwall inequality on
ddt∫ΩR2(v)|∇v|2dx⩽C2∫Ω(1+u)α+1dx, |
we obtain
∫ΩR2(v)|∇v|2dx∈P. | (2.11) |
Moreover, integrate (2.10) over [t,t+1] to know
∫t+1t∫Ωv2t(x,s)dxds∈P. | (2.12) |
By the v−equation of (1.3) and noting (2.1), we have
∫Ω|∇⋅(R(v)∇v)|2dx=∫Ω[vt−vg(u,v)]2dx⩽2∫Ωv2tdx+C3+C4∫Ωuα+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|2ξ⋅∇vdx=−∫Ωv∇⋅(R(v)|ξ|2ξ)dx=−∫ΩvR′(v)R(v)|ξ|4dx−∫ΩvR(v)|ξ|2∇⋅ξdx−2∫ΩvR(v)ξ⋅(∇ξ⋅ξ)dx⩽−∫ΩvR(v)|ξ|2∇⋅ξdx−2∫ΩvR(v)ξ⋅(∇ξ⋅ξ)dx. |
By H¨older inequality, we can get
−∫ΩvR(v)|ξ|2∇⋅ξdx⩽‖vR(v)‖L∞(Ω)‖ξ‖2L4(Ω)‖∇⋅ξ‖L2(Ω), |
and
−2∫ΩvR(v)ξ⋅(∇ξ⋅ξ)dx⩽‖vR(v)‖L∞(Ω)‖ξ‖2L4(Ω)‖∇ξ‖L2(Ω), |
thus
‖ξ‖2L4(Ω)⩽‖vR(v)‖L∞(Ω)(‖∇⋅ξ‖L2(Ω)+2‖∇ξ‖L2(Ω)). | (2.13) |
Now, we will prove
‖∇ξ‖L2(Ω)⩽C6‖∇⋅ξ‖L2(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)T∇v+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 ‖∇2v‖L2(Ω)⩽C5‖△v‖L2(Ω), we can prove (2.14) holds.
Therefore, in virtue of (2.13) and (2.14), we obtain
‖ξ‖2L4(Ω)⩽(1+2C6)‖vR(v)‖L∞(Ω)‖∇⋅ξ‖L2(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
‖u‖Lˉp(Ω)∈P, | (2.16) |
∫t+1t∫Ωuˉp+αdxds∈P, | (2.17) |
and
∫t+1t∫Ωuˉp+β−2|∇u|2dxds∈P, | (2.18) |
for ˉp satisfying (i) ˉp=α+2β or (ii) ˉp>α+2β and (n−2)ˉp⩽3nβ.
Proof. Multiplying the first equation in (1.3) by up−1 with p>1, and integrating on Ω by parts, we have
1pddt∫Ωupdx=−∫Ω∇up−1⋅[P(u,v)∇u+Q(u,v)∇v]dx+∫Ωupf(u,v)dx=−(p−1)∫Ωup−2P(u,v)|∇u|2dx−(p−1)∫Ωup−2Q(u,v)∇u⋅∇vdx+∫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)dx⩽a∫Ωupdx−b∫Ωup+αdx, |
and
|−∫Ωup−2Q(u,v)∇u⋅∇vdx|⩽∫Ωup−2|Q(u,v)||∇u⋅∇v|dx⩽‖ϕ(v)‖∞∫Ω|up−1∇u⋅∇v|dx=‖ϕ(v)‖∞∫Ω|up+β−22∇u⋅up−β2∇v|dx⩽d2∫Ωup+β−2|∇u|2dx+C7p−1∫Ωup−β|∇v|2dx, |
by Hölder inequality.
Combining these estimates and P(u,v)⩾duβ in (A1), then
1pddt∫Ωupdx+d(p−1)2∫Ωup+β−2|∇u|2dx+b∫Ωup+αdx⩽a∫Ωupdx+C7∫Ωup−β|∇v|2dx. | (2.19) |
Case I: p⩽α+2β. In this case, we have 2(p−β)⩽p+α.
Applying Hölder inequality and Young's inequality to the last term of (2.19), we have
C7∫Ωup−β|∇v|2dx⩽b2∫Ωu2p−2βdx+C8∫Ω|∇v|4dx⩽b2∫Ωup+αdx+C8∫Ω|∇v|4dx+C9. |
Consequently, (2.19) becomes
1pddt∫Ωupdx+d(p−1)2∫Ωup+β−2|∇u|2dx+b2∫Ωup+αdx⩽a∫Ωupdx+C8∫Ω|∇v|4dx+C9. | (2.20) |
Obviously, (2.20) entails
1pddt∫Ωupdx⩽a∫Ωupdx+C8∫Ω|∇v|4dx+C9. | (2.21) |
For the above inequality (2.21), if we can show
∫t+1t∫Ωupdxds∈P, | (2.22) |
then (2.6) and the uniform Gronwall inequality yield
‖u‖Lp(Ω)∈P. | (2.23) |
Furthermore, integrating (2.20) from t to t+1, we can obtain
∫t+1t∫Ωup+αdxds∈P, | (2.24) |
and
∫t+1t∫Ωup+β−2|∇u|2dxds∈P. | (2.25) |
Now, we will use mathematical induction to prove that (2.22) holds for p=α+2β. There exists some k∈N+ such that 1⩽α+2β−kα⩽α+1. Denote p0=α+2β−kα,pm=pm−1+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=pm−1, then (2.24) means that (2.22) holds for p=pm−1+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 (n−4)p⩽(3n−4)β.
Let wp=up+β2 and denote wp as w sometimes for simplicity, then (2.19) can be written as
1pddt∫Ωw2pp+βdx+2d(p−1)(p+β)2∫Ω|∇w|2dx+b∫Ωw2(p+α)p+βdx⩽a∫Ωw2pp+βdx+C7∫Ωw2(p−β)p+β|∇v|2dx⩽a∫Ωw2pp+βdx+C7‖w2(p−β)p+β‖L2(Ω)‖∇v‖2L4(Ω)=a‖w‖2pp+βL2pp+β(Ω)+C7‖w‖2(p−β)p+βL4(p−β)p+β(Ω)‖∇v‖2L4(Ω), | (2.26) |
by the Hölder inequality.
The conditions p>α+2β and (n−4)p⩽(3n−4)β implies
2(p+α)p+β<4(p−β)p+β⩽2nn−2, |
here 2nn−2 can be replaced by +∞ for n=2.
It is known by Gagliardo-Nirenberg inequality that
‖w‖L4(p−β)p+β(Ω)⩽C10‖w‖1−θL2pp+β(Ω)‖∇w‖θL2(Ω)+C10‖w‖L1(Ω), | (2.27) |
with
θ=n(p+β)(p−2β)2(p−β)(2p+nβ). |
Using (2.27) and Young's inequality, we have
C7‖w‖2(p−β)p+βL4(p−β)p+β(Ω)‖∇v‖2L4(Ω)⩽C11‖w‖2(p−β)(1−θ)(p+β)L2pp+β(Ω)‖∇w‖2θ(p−β)p+βL2(Ω)‖∇v‖2L4(Ω)+C11‖w‖2(p−β)p+βL1(Ω)‖∇v‖2L4(Ω)⩽ε‖∇w‖2L2(Ω)+Cε‖w‖m1L2pp+β(Ω)‖∇v‖m2L4(Ω)+C11‖w‖4(p−β)p+βL1(Ω)+C11‖∇v‖4L4(Ω), |
with
m1=2(p−β)(1−θ)p+β−θ(p−β),m2=2(p+β)p+β−θ(p−β). |
Let ε=d(p−1)(p+β)2, then (2.26) becomes
1pddt∫Ωw2pp+βdx+d(p−1)(p+β)2∫Ω|∇w|2dx+b∫Ωw2(p+α)p+βdx⩽a‖w‖2pp+βL2pp+β(Ω)+Cε‖w‖m1L2pp+β(Ω)‖∇v‖m2L4(Ω)+C11‖w‖4(p−β)p+βL1(Ω)+C11‖∇v‖4L4(Ω). | (2.28) |
Let y(t)=‖w‖2pp+βL2pp+β(Ω),h(t)=C11‖w‖4(p−β)p+βL1(Ω)+C11‖∇v‖4L4(Ω), then we have
1pddty(t)⩽ay(t)+Cεy(t)(p+β)m12p‖∇v‖m2L4(Ω)+h(t). | (2.29) |
For the case of y(t)⩽1, obviously we have
y(t)=∫Ωw2pp+βdx∈P. | (2.30) |
Since p>α+2β, (2.30) implies ‖w‖L1(Ω)∈P by Hölder inequality. Let
(n−2)p⩽3nβ, |
then a direct calculation shows that m2⩽4. Consequently, (2.6) and Hölder inequality give
‖∇v‖m2L4(Ω)∈P. | (2.31) |
Furthermore, integrating (2.28) from t to t+1 yields
∫t+1t∫Ωw2(p+α)p+βdxds∈P, | (2.32) |
and
∫t+1t∫Ω|∇w|2dxds∈P. | (2.33) |
For the case of y(t)>1, denote r(t)=a+Cε‖∇v‖m2L4(Ω), then
1pddty(t)⩽r(t)y(t)+h(t), | (2.34) |
here, we used the fact
(p+β)m12p=(3n−4)β−(n−4)p(4p−np+4nβ)<1. |
It is easy to see that (2.31) implies ∫t+1tr(s)ds∈P for (n−2)p⩽3nβ, thus if we can show
∫t+1ty(s)ds∈P,∫t+1th(s)ds∈P, | (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 (n−2)p=3nβ. In order to get ∫t+1th(s)ds∈P, recalling (2.6), the key step is to deal with
∫t+1t‖w‖4(p−β)p+βL1(Ω)ds∈P, | (2.36) |
since the Minkowski's inequality ensures
∫t+1th(s)ds⩽C11∫t+1t‖w‖4(p−β)p+βL1(Ω)ds+C11∫t+1t‖∇v‖4L4(Ω)ds. |
Now, we will prove ∫t+1ty(s)ds∈P and (2.36) by mathematical induction, simultaneously. For the case of n>2, there exists some ˉk∈N+ such that α+2β<3nβn−2−ˉkα⩽2α+2β. Denote q0=3nβn−2−ˉkα,qm=qm−1+α(m=1,2,⋅⋅⋅,ˉk). Since we have proved (2.17) for ˉp=α+2β, it is easy to get ∫t+1ty(s)ds∈P for p=q0 by Hölder inequality. In addition, noting that q0+β2⩽2α+3β2<α+2β and (2.16) holds for ˉp=α+2β, using Hölder inequality we obtain ‖wq0‖L1(Ω)∈P, which indicates that the result (2.36) is true for p=q0. On the other hand, assume ∫t+1ty(s)ds∈P and (2.36) hold for p=qm−1, then (2.32) holds for p=qm−1. According to the definition wp=up+β2, it is easy to see
(wqm−1)2(qm−1+α)qm−1+β=(wqm)2qmqm+β, |
thus ∫t+1ty(s)ds∈P for p=qm. Moreover, the assumption implies (2.30) holds for p=qm−1 and hence
‖wqm‖L1(Ω)⩽C‖wqm−1‖2qm−1qm−1+βL2qm−1qm−1+β(Ω), |
by qm−1>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=3nβn−2. Therefore, (2.30), (2.32) and (2.33) hold for p=3nβn−2, which implies (2.16)–(2.18) hold for ˉp⩽3nβn−2. Similarly, we can prove (2.16)–(2.18) hold for any positive constant ˉp>α+2β if n⩽2.
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+1t‖u‖sLs(Ω)ds=∫t+1t∫Ωuˉp+αdxds∈P. |
Define
1−χ=1s+n2s=n+22s=n+22(ˉp+α), |
A=s−n2(1−χ)=2sn+2>0,B=s−11−χ=nsn+2>0. |
By Lemma 1.2, we also need χ∈(0,1), which is equivalent to
n<2(ˉp+α−1). | (2.37) |
It is known by Gagliardo-Nirenberg inequality that
‖w‖L2∗(Ω)⩽C12(‖∇w‖L2(Ω)+‖w‖L1(Ω)), |
with 2∗=2n/(n−2).
Let l=ˉp+β2 and r=2l,q=2∗l, then w=ul and
∫t+1t‖u‖rLq(Ω)ds=∫t+1t‖w‖2L2∗(Ω)ds⩽2C12[∫t+1t‖∇w‖2L2(Ω)ds+sup[t,t+1]‖w‖2L1(Ω)]. |
The estimate ‖w‖L1(Ω)∈P comes from (2.16) by Hölder inequality, which together with (2.18) indicates
∫t+1t‖u‖rLq(Ω)ds∈P. |
Let
1−χ=1r+n2q=1l(12+n2⋅2∗)=n4l=n2(ˉp+β), |
A=q−n2(1−χ)=2(ˉp+β)n−2>0,B=r−11−χ=n−2n(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+α−1) 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 (n−2)ˉp⩽3nβ in Theorem 2.4, hence we can assign any positive number to ˉp for n⩽2, but we choose ˉp=3nβn−2 if 0<α⩽n+4n−2β, 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(a1−b1u−c1v),x∈Ω,t>0,vt=Δ[(d2+ρ22v)v]+v(a2−b2u−c2v),x∈Ω,t>0,∂u∂ν=∂v∂ν=0,x∈∂Ω,t>0,u(x,0)=u0(x)⩾0,v(x,0)=v0(x)⩾0,x∈Ω. | (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)=a1−b1u−c1v,g(u,v)=a2−b2u−c2v. |
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 n⩽2 directly. Moreover, a simple computation shows (B2) (i) in Theorem 1.3 holds for 2<n<8 and (B1) (i) holds for 8⩽n<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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