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Benchmarking alternative interpretable machine learning models for corporate probability of default

  • In this study we investigate alternative interpretable machine learning ("IML") models in the context of probability of default ("PD") modeling for the large corporate asset class. IML models have become increasingly prominent in highly regulated industries where there are concerns over the unintended consequences of deploying black box models that may be deemed conceptually unsound. In the context of banking and in wholesale portfolios, there are challenges around using models where the outcomes may not be explainable, both in terms of the business use case as well as meeting model validation standards. We compare various IML models (deep neural networks and explainable boosting machines), including standard approaches such as logistic regression, using a long and robust history of corporate borrowers. We find that there are material differences between the approaches in terms of dimensions such as model predictive performance and the importance or robustness of risk factors in driving outcomes, including conflicting conclusions depending upon the IML model and the benchmarking measure considered. These findings call into question the value of the modest pickup in performance with the IML models relative to a more traditional technique, especially if these models are to be applied in contexts that must meet supervisory and model validation standards.

    Citation: Michael Jacobs, Jr. Benchmarking alternative interpretable machine learning models for corporate probability of default[J]. Data Science in Finance and Economics, 2024, 4(1): 1-52. doi: 10.3934/DSFE.2024001

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  • In this study we investigate alternative interpretable machine learning ("IML") models in the context of probability of default ("PD") modeling for the large corporate asset class. IML models have become increasingly prominent in highly regulated industries where there are concerns over the unintended consequences of deploying black box models that may be deemed conceptually unsound. In the context of banking and in wholesale portfolios, there are challenges around using models where the outcomes may not be explainable, both in terms of the business use case as well as meeting model validation standards. We compare various IML models (deep neural networks and explainable boosting machines), including standard approaches such as logistic regression, using a long and robust history of corporate borrowers. We find that there are material differences between the approaches in terms of dimensions such as model predictive performance and the importance or robustness of risk factors in driving outcomes, including conflicting conclusions depending upon the IML model and the benchmarking measure considered. These findings call into question the value of the modest pickup in performance with the IML models relative to a more traditional technique, especially if these models are to be applied in contexts that must meet supervisory and model validation standards.



    A taxis is the movement of an organism in response to a stimulus such as chemical signal or the presence of food. Taxes can be classified based on the types of stimulus, such as chemotaxis, prey-taxis, galvanotaxis, phototaxis and so on. According to the direction of movements, the taxis is said to be attractive (resp. repulsive) if the organism moves towards (resp. away from) the stimulus. In the ecosystem, a widespread phenomenon is the prey-taxis, where predators move up the prey density gradient, which is often referred to as the direct prey-taxis. However some predators may approach the prey by tracking the chemical signals released by the prey, such as the smell of blood or specific odo, and such movement is called indirect prey-taxis (cf. [1]). Since the pioneering modeling work by Kareiva and Odell [2], prey-taxis models have been widely studied in recent years (cf. [3,4,5,6,7,8,9,10,11,12]), followed by numerous extensions, such as three-species prey-taxis models (cf. [13,14,15]) and predator-taxis models (cf. [16,17]). The indirect prey-taxis models have also been well studied (cf. [18,19,20]).

    Recently, a predator-prey model with attraction-repulsion taxis mechanisms was proposed by Bell and Haskell in [21] to describe the interaction between direct prey-taxis and indirect chemotaxis, where the direct prey-taxis describes the predator's directional movement towards the prey density gradient, while the indirect chemotaxis models a defense mechanism in which the prey repels the predator by releasing odour chemicals (like a fox breaking wind in order to escape from hunting dogs). The model reads as

    {ut=dΔu+u(a1a2ua3v),xΩ, t>0,vt=(v+χvwξvu)+ρv(1v)+ea3uv,xΩ, t>0,wt=ηΔw+ruγw,xΩ, t>0,uν=vν=wν=0,xΩ,t>0,(u,v,w)(x,0)=(u0,v0,w0)(x),xΩ, (1.1)

    where the unknown functions u(x,t), v(x,t) and w(x,t) denote the densities of the prey, predator and prey-derived chemical repellent, respectively, at position xΩ and time t>0. Here, ΩRn is a bounded domain (habitat of species) with smooth boundary Ω, and ν is the unit outer normal vector of Ω. The parameters d, η, χ, ξ, a1, a2, a3, e, ρ, r, γ are all positive, where χ>0 and ξ>0 denote the (attractive) prey-taxis and (repulsive) chemotaxis coefficients, respectively. The predator v is assumed to be a generalist, so that it has a logistic growth term ρv(1v) with intrinsic growth rate ρ>0. More modeling details with biological interpretations are referred to in [21]. We remark that the predator-prey model with attraction-repulsion taxes has some similar structures to the so-called attraction-repulsion chemotaxis model proposed originally in [22], where the species elicit both attractive and repulsive chemicals (see [23,24,25,26] and references therein for some mathematical studies).

    The initial data satisfy the following conditions:

    v0C0(¯Ω),u0,w0W1,(Ω), and u0, v0, w00 in ¯Ω. (1.2)

    In [21], the global existence of strong solutions to (1.1) was established in one dimension (n=1), and the existence of nontrivial steady state solutions alongside pattern formation was studied by the bifurcation theory. The main purpose of this paper is to study the global dynamics of (1.1) in higher dimensional spaces, which are usually more physical in the real world. Specifically, we shall show the existence of global classical solutions in all dimensions and explore the global stability of constant steady states, by which we may see how parameter values play roles in determining these dynamical properties of solutions.

    The first main result is concerned with the global existence and boundedness of solutions to (1.1). For the convenience of presentation, we let

    K1=max{a1a2,u0L(Ω)},  K2=max{a1K1+a2K21,a3K1} (1.3)

    and

    K3(z)=23z12zdz(n+2(z1)K22z+1)z+12((z1)(4z2+n)K21)z12+23zz2d1z((z1)ξ2z+1)z+1z((4z2+n)K21)1z. (1.4)

    Then, the result on the global boundedness of solutions to (1.1) is stated as follows.

    Theorem 1.1 (Global existence). Let ΩRn(n1) be a bounded domain with smooth boundary and parameters d, η, χ, ξ, a1, a2, a3, e, ρ, r, γ be positive. If

    ρ{>0,n2,2K3([n2]+1)[n2]+1,n>2,

    where K3(p) is defined in (1.4), then for any initial data (u0,v0,w0) satisfying (1.2), the system (1.1) admits a unique classicalsolution (u,v,w) satisfying

    u, v,wC0(¯Ω×[0,+))C2,1(¯Ω×(0,+)),

    and u,v,w>0 in Ω×(0,+). Moreover, there exists a constant C>0 independent of t such that

    u(,t)W1,(Ω)+v(,t)L(Ω)+w(,t)W1,(Ω)Cfor all t>0.

    Our next goal is to explore the large-time behavior of solutions to (1.1). Simple calculations show the system (1.1) has four possible homogeneous equilibria as classified below:

    {(0,0,0), (0,1,0), (a1a2,0,ra1γa2),if a1a3,(0,0,0), (0,1,0), (a1a2,0,ra1γa2),(u,v,w),if a1>a3,

    with

    u=ρ(a1a3)ρa2+ea23,v=ea1a3+ρa2ρa2+ea23,w=rρ(a1a3)γ(ρa2+ea23) (1.5)

    where the trivial equilibrium (0,0,0) is called the extinction steady state, (0,1,0) is the predator-only steady state, and (u,v,w) is the coexistence steady state. We shall show that if a1>a3, then the coexistence steady state is globally asymptotically stable with exponential convergence rate, provided that ξ and χ are suitably small, while if a1a3, the predator-only steady state is globally asymptotically stable with exponential or algebraic convergence rate when ξ and χ are suitably small. To state our results, we denote

    Γ=4dρ(a1a3)K21(ea1a3+ρa2),Φ=2a2ρa23+e,Ψ=γηa23K21(ρa2+ea23)dρ2r2(a1a3) (1.6)

    and

    A=ξ24d,B=ea2a1,D=16ηγa1r2, (1.7)

    where K1 is defined in (1.3). Then, the global stability result is stated in the following theorem.

    Theorem 1.2 (Global stability). Let the assumptions in Theorem 1.1 hold. Then, the following results hold.

    (1) Let a1>a3. If ξ and χ satisfy

    ξ2<Γ(Φ+Φ2e2) and χ2<Ψmaxy[a,b](Γyξ2)(y2+2Φye2)y,

    where a=max{ξ2Γ,ΦΦ2e2},b=Φ+Φ2e2, then there exist some constants T, C, α>0 such that the solution (u,v,w) obtained in Theorem 1.1 satisfies for all tT

    u(,t)uL(Ω)+v(,t)vL(Ω)+w(,t)wL(Ω)Ceαt.

    (2) Let a1a3, If ξ and χ satisfy

    ξ2<4dea2a1andχ2<D(A+B2AB),

    then there exist some constants T, C, β>0 such that the solution (u,v,w) obtained in Theorem 1.1 satisfies, for all tT,

    u(,t)L(Ω)+v(,t)1L(Ω)+w(,t)L(Ω){Ceβt if a1<a3,C(t+1)1 if a1=a3.

    Remark 1.1. In the biological view, the relative sizes of a1 and a2 determine the coexistence of the system. The results indicated that a large a1a2 facilitates the coexistence of the species.

    The rest of this paper is organized as follows. In Section 2, we state the local existence of solutions to (1.1) with extensibility conditions. Then, we deduce some a priori estimates and prove Theorem 1.1 in Section 3. Finally, we show the global convergence to the constant steady states and prove Theorem 1.2 in Section 4.

    For convenience, in what follows we shall use Ci(i=1,2,) to denote a generic positive constant which may vary from line to line. For simplicity, we abbreviate t0Ωf(,s)dxds and Ωf(,t)dx as t0Ωf and Ωf, respectively. The local existence and extensibility result of problem (1.1) can be directly established by the well-known Amman's theory for triangular parabolic systems (cf. [27,28]). Below, we shall present the local existence theorem without proof for brevity, and we refer to [21] for the proof in one dimension as a reference.

    Lemma 2.1 (Local existence and extensibility). Let ΩRn be a bounded domain with smooth boundary. The parameters d, η, χ, ξ, a1, a2, a3, e, ρ, r, γ are positive. Then, for the initial data (u0,v0,w0) satisfying (1.2), there exists Tmax(0,] such that the system (1.1) admits a unique classicalsolution (u,v,w) satisfying

    u, v, wC0(¯Ω×[0,Tmax))C2,1(¯Ω×(0,Tmax)),

    and u,v,w>0 in Ω×(0,Tmax). Moreover, we have

    either Tmax=+ or lim suptTmax(u(,t)W1,(Ω)+v(,t)L(Ω)+w(,t)W1,(Ω))=+.

    We recall some well-known results which will be used later frequently. The first one is an uniform Grönwall inequality [29].

    Lemma 2.2. Let Tmax>0, τ(0,Tmax). Suppose that c1, c2, y are three positive locally integrable functions on (0,Tmax) such that y is locally integrable on (0,Tmax) and satisfies

    y(t)c1(t)y(t)+c2(t)for all t(0,Tmax).

    If

    t+τtc1C1,t+τtc2C2,  t+τtyC3for all t[0,Tmaxτ),

    where Ci(i=1,2,3) are positive constants, then

    y(t)(C3τ+C2)eC1for all t[τ,Tmax).

    Next, we recall a basic inequality [30].

    Lemma 2.3. Let p[1,). Then, the following inequality holds:

    Ω|u|2(p+1)2(4p2+n)u2L(Ω)Ω|u|2(p1)|D2u|2

    for any uC2(ˉΩ) satisfying uν=0 on Ω, where D2u denotes the Hessian of u.

    The last one is a Gagliardo-Nirenberg type inequality shown in [31,Lemma 2.5].

    Lemma 2.4. Let Ω be a bounded domain in R2 with smooth boundary. Then, for any φW2,2(Ω) satisfying φν|Ω=0, there exists a positive constant C depending only on Ω such that

    φL4(Ω)C(Δφ12L2(Ω)φ12L2(Ω)+φL2(Ω)). (2.1)

    In this section, we establish the global boundedness of solutions to the system (1.1). To this end, we will proceed with several steps to derive a priori estimates for the solution of the system (1.1). The first one is the uniform-in-time L(Ω) boundedness of u.

    Lemma 3.1. Let (u,v,w) be the solution of (1.1) and K1 be as defined in (1.3). Then, we have

    uL(Ω)K1for all t(0,Tmax).

    Furthermore, there is a constant C>0 such that for any 0<τ<min{Tmax,1}, it follows that

    t+τt|u|2C  for all  t(0,Tmaxτ).

    Proof. The result is a direct consequence of the maximum principle applied to the first equation in (1.1). Indeed, if we let ˉu=max{a1a2,u0L(Ω)}, then ˉu satisfies

    {ˉutdΔˉu+ˉu(a1a2ˉua3v),xΩ,t>0,ˉuν=0,xΩ,t>0,ˉu(x,0)u0(x),xΩ.

    Apparently, the comparison principle of parabolic equations gives uˉu on Ω×(0,Tmax).

    Next, we multiply the first equation of (1.1) by u and integrate the result to get

    ddtΩu2+dΩ|u|2=a1Ωu2Ωu(a2u+a3v)a1K21|Ω|.

    Then, the integration of the above inequality with respect to t over (t,t+τ) completes the proof by noting that Ωu20 is bounded.

    Having at hand the uniform-in-time L(Ω) boundedness of u, the a priori estimate of w follows immediately.

    Lemma 3.2. Let (u,v,w) be the solution of (1.1). We can find a constant C>0 satisfying

    wW1,(Ω)Cfor all t(0,Tmax).

    Proof. Noting the boundedness of uL(Ω) from Lemma 3.1, we get the desired result from the third equation of (1.1) and the regularity theorem [32,Lemma 1].

    Now, the a priori estimate of v can be obtained as below.

    Lemma 3.3. Let (u,v,w) be the solution of (1.1). Then, there exists a constant C>0 such that

    ΩvCfor all t(0,Tmax), (3.1)

    and

    t+τtΩv2Cfor all t(0,Tmaxτ), (3.2)

    where τ is a constant such that 0<τ<min{Tmax,1}.

    Proof. Integrating the second equation of (1.1) over Ω by parts, using Young's inequality and Lemma 3.1, we find some constant C1>0 such that

    ddtΩv=ρΩvρΩv2+ea3Ωuv(ρ+ea3supt(0,Tmax)uL(Ω))ΩvρΩv2Ωvρ2Ωv2+C1for all t(0,Tmax). (3.3)

    Hence, (3.1) is obtained by the Grönwall inequality. Integrating (3.3) over (t,t+τ), we get (3.2) immediately.

    Due to the estimates of u and v obtained in Lemmas 3.1 and 3.3 respectively, we have the following improved uniform-in-time L2(Ω) boundedness of u and the space-time L2 boundedness of Δu when n=2.

    Lemma 3.4. Let (u,v,w) be the solution of (1.1). If n=2, then we can find a constant C>0 such that

    Ω|u|2Cfor all t(0,Tmax) (3.4)

    and

    t+τtΩ|Δu|2Cfor all t(0,Tmaxτ), (3.5)

    where τ is defined in Lemma 3.3.

    Proof. Integrating the first equation of (1.1) by parts and using Lemma 3.1, we find a constant C1>0 such that

    ddtΩ|u|2=2Ωuut=2ΩutΔu=2ΩΔu(dΔu+a1ua2u2a3uv)2dΩ|Δu|2+C1Ω(v+1)|Δu|for all t(0,Tmax). (3.6)

    The Gagliardo-Nirenberg inequality in Lemma 2.4, Young's inequality and Lemma 3.1 yield some constants C2,C3>0 satisfying

    Ω|u|2=u2L2(Ω)C2(ΔuL2(Ω)uL2(Ω)+u2L(Ω))d2Ω|Δu|2+C3

    and

    C1Ω(v+1)|Δu|d2Ω|Δu|2+C3Ωv2+C3for all t(0,Tmax),

    which along with (3.6) imply

    ddtΩ|u|2+Ω|u|2+dΩ|Δu|2C3Ωv2+2C3for all t(0,Tmax). (3.7)

    Then, applications of Lemma 2.2, 3.1 and 3.3 give (3.4). Finally, (3.5) can be obtained by integrating (3.7) over (t,t+τ).

    Now, the uniform-in-time boundedness of v in L2(Ω) can be established when n=2.

    Lemma 3.5. Let (u,v,w) be the solution of (1.1). If n=2, then there exists a constant C>0 such that

    Ωv2Cfor all t(0,Tmax).

    Proof. Multiplying the second equation of (1.1) by v, integrating the result by parts and using Young's inequality, we have

    ddtΩv2+2Ω|v|2=2χΩvvw+2ξΩvuv+2ρΩv22ρΩv3+2ea3Ωuv2Ω|v|2+2χ2w2L(Ω)Ωv2+2ξ2Ωv2|u|2+2ρΩv22ρΩv3+2ea3uL(Ω)Ωv2,

    which along with Lemma 3.1 and Lemma 3.2 gives some constant C1>0 such that

    ddtΩv2+Ω|v|22ξ2Ωv2|u|2+C1Ωv22ρΩv3for all t(0,Tmax). (3.8)

    Using Lemmas 3.1 and 3.3, Hölder's inequality, Lemma 2.4 and Young's inequality, we find some constants C2,C3,C4>0 such that

    2ξΩv2|u|22ξv2L4(Ω)u2L4(Ω)C2(v12L2(Ω)v12L2(Ω)+vL2(Ω))2(Δu12L2(Ω)u12L(Ω)+uL(Ω))2C3(vL2(Ω)vL2(Ω)ΔuL2(Ω)+vL2(Ω)vL2(Ω)+ΔuL2(Ω)v2L2(Ω)+v2L2(Ω))v2L2(Ω)+C4(1+Δu2L2(Ω))v2L2(Ω)for all t(0,Tmax). (3.9)

    Furthermore, Young's inequality yields some constant C5>0 such that

    C1Ωv22ρΩv3C5for all t(0,Tmax). (3.10)

    Substituting (3.9) and (3.10) into (3.8), we get

    ddtΩv2C4(1+Δu2L2(Ω))v2L2(Ω)+C5for all t(0,Tmax),

    which alongside Lemma 2.2, Lemma 3.3 and Lemma 3.4 completes the proof.

    To get the global existence of solutions in any dimensions, we derive the following functional inequality which gives an a priori estimate on u.

    Lemma 3.6. Let (u,v,w) be the solution of (1.1) and q2. If n1, then there exists a constant C>0 such that

    ddtΩ|u|2q+dqΩ|u|2(q1)|D2u|2q(n+2(q1))K22dΩ(v2+1)|u|2(q1)+Cfor all t(0,Tmax),

    where K2 is defined in (1.3).

    Proof. From the first equation of (1.1) and the fact 2uΔu=Δ|u|22|D2u|2, it follows that

    ddtΩ|u|2q=2qΩ|u|2(q1)uut=2qΩ|u|2(q1)u(dΔu+a1ua2u2a3uv)=dqΩ|u|2(q1)Δ|u|22dqΩ|u|2(q1)|D2u|2+2qΩ|u|2(q1)u(a1ua2u2a3uv)

    which implies

    ddtΩ|u|2q+2dqΩ|u|2(q1)|D2u|2=dqΩ|u|2(q1)Δ|u|2+2qΩ|u|2(q1)u(a1ua2u2a3uv)=:I1+I2for all t(0,Tmax). (3.11)

    Now, we estimate the right hand side of (3.11). Choosing s(0,12) and

    θ=12s+12nq121nq(0,1),

    we get

    12s+12n=θ(121n)+(1θ)q,

    which, along with the Gagliardo-Nirenberg inequality, Young's inequality and the embedding

    Ws+12,2(Ω)Ws,2(Ω)L2(Ω),

    gives some constants C1, C2, C3, C4>0 such that

    Ω|u|2(q1)|u|2νC1Ω|u|2q=C1|u|q2L2(Ω)C2|u|q2Ws+12,2(Ω)C3|u|q2θL2(Ω)|u|q2(1θ)L1q(Ω)+C3|u|q2L1q(Ω)2(q1)q2|u|q2L2(Ω)+C4for all t(0,Tmax).

    Therefore, it holds that

    I1=dqΩ|u|2(q1)|u|2νdqΩ|u|2(q1)|u|22d(q1)qΩ||u|q|2+C4dq4d(q1)qΩ||u|q|22d(q1)qΩ||u|q|2+C4dqfor all t(0,Tmax).

    Owning to the fact |Δu|n|D2u|, Young's inequality and Lemma 3.1, we have

    I2=2q(q1)Ω(a1ua2u2a3uv)|u|2(q2)|u|2u2qΩ(a1ua2u2a3uv)|u|2(q1)Δu2q(q1)K2Ω(v+1)|u|2(q2)||u|2||u|+2qnK2Ω(v+1)|u|2(q1)|D2u|qd(q1)2Ω|u|2(q2)||u|2|2+2q(q1)K22dΩ(v2+1)|u|2(q1)+dqΩ|u|2(q1)|D2u|2+qnK22dΩ(v2+1)|u|2(q1)=2d(q1)qΩ||u|q|2+dqΩ|u|2(q1)|D2u|2+q(n+2(q1))K22dΩ(v2+1)|u|2(q1)for all t(0,Tmax),

    where K2 is defined in (1.3). Hence, substituting the estimates I1 and I2 into (3.11), we finish the proof of the lemma.

    Now, we show the following functional inequality to derive the a priori estimate on v in any dimensions.

    Lemma 3.7. Let (u,v,w) be the solution of (1.1) and q2. If n1, we can find a constant C>0 such that

    ddtΩvq+2(q1)qΩ|vq2|2+ρqΩvq+1q(q1)ξ2Ωvq|u|2+CΩvq

    for all t(0,Tmax).

    Proof. Utilizing the second equation of (1.1) and integration by parts, we get

    ddtΩvq=qΩvq1vt=qΩvq1((v+χvwξvu)+v(ρ(1v)+ea3u))=q(q1)Ωvq2|v|2χq(q1)Ωvq1wv+ξq(q1)Ωvq1uv+ρqΩvqρqΩvq+1+ea3qΩuvq. (3.12)

    Now, we estimate the right hand side of (3.12). An application of Young's inequality and Lemma 3.2 yields some constant C1>0 such that

    χq(q1)Ωvq1wvχq(q1)supt(0,Tmax)wL(Ω)Ωvq1|v|q(q1)4Ωvq2|v|2+C1Ωvq

    and

    ξq(q1)Ωvq1uvq(q1)4Ωvq2|v|2+q(q1)ξ2Ωvq|u|2,

    which along with (3.12), Lemma 3.1 and the fact

    vq2|v|2=4q2|vq2|2

    gives a constant C2>0 such that

    ddtΩvq+2(q1)qΩ|vq2|2q(q1)ξ2Ωvq|u|2+(ρq+C1)ΩvqρqΩvq+1+ea3qΩuvqq(q1)ξ2Ωvq|u|2ρqΩvq+1+C2Ωvqfor all t(0,Tmax).

    Hence, we finish the proof of the lemma.

    Combining Lemmas 3.6 and 3.7, we have the following inequality which can help us to achieve the global existence of solutions in any dimensions.

    Lemma 3.8. Let (u,v,w) be the solution of (1.1) and p2. If n1, we can find a constant C>0 such that

    ddt(Ω|u|2p+Ωvp)+2(p1)pΩ|vp2|2+Ω|u|2p+Ωvp(K3(p)ρp2)Ωvp+1+Cfor all t(0,Tmax),

    where K3(p) is defined in (1.4).

    Proof. Combining Lemmas 3.6 and 3.7, we see for any p=q2 there exists a constant C1>0 such that for all t(0,Tmax)

    ddt(Ω|u|2p+Ωvp)+2(p1)pΩ|vp2|2+dpΩ|u|2(p1)|D2u|2+ρpΩvp+1p(n+2(p1))K22dΩv2|u|2(p1)+p(p1)ξ2Ωvp|u|2+C1Ω|u|2(p1)+C1Ωvp+C1. (3.13)

    Now, we estimate the right hand side of the above inequality. Indeed, owing to Lemma 2.3 and Young's inequality, for all t(0,Tmax), we have

    p(n+2(p1))K22dΩv2|u|2(p1)dp8(4p2+n)u2L(Ω)Ω|u|2(p+1)+2p+1(dp(p+1)8(p1)(4p2+n)u2L(Ω))p12(p(n+2(p1))K22d)p+12Ωvp+1dp4Ω|u|2(p1)|D2u|2+23p12pdp(n+2(p1)K22p+1)p+12((p1)(4p2+n)K21)p12Ωvp+1

    and

    p(p1)ξ2Ωvp|u|2dp8(4p2+n)u2L(Ω)Ω|u|2(p+1)+pp+1(dp(p+1)8(4p2+n)u2L(Ω))1p(p(p1)ξ2)p+1pΩvp+1dp4Ω|u|2(p1)|D2u|2+23pp2d1p((p1)ξ2p+1)p+1p((4p2+n)K21)1pΩvp+1,

    where K1 and K2 are defined in (1.3). Similarly, we can find a constant C2>0 such that

    C1Ω|u|2(p1)dp8(4p2+n)u2L(Ω)Ω|u|2(p+1)+C2dp4Ω|u|2(p1)|D2u|2+C2for all t(0,Tmax).

    Substituting the above estimates into (3.13), we get

    ddt(Ω|u|2p+Ωvp)+2(p1)pΩ|vp2|2+dp4Ω|u|2(p1)|D2u|2+ρpΩvp+1K3(p)Ωvp+1+C1Ωvp+C1+C2for all t(0,Tmax), (3.14)

    where K3(p) is given in (1.4). Furthermore, we can use Young's inequality and Lemma 2.3 to get a constant C3>0 such that

    (C1+1)Ωvpρp2Ωvp+1+C3,

    and

    Ω|u|2pdp8(4p2+n)u2L(Ω)Ω|u|2(p+1)+C3dp4Ω|u|2(p1)|D2u|2+C3for all t(0,Tmax),

    which together with (3.14) finishes the proof.

    Next, we shall deduce a criterion of global boundedness of solutions for the system (1.1) inspired by an idea of [33].

    Lemma 3.9. Let n1. If there exist M>0 and p0>n2 such that

    Ωvp0Mfor all t(0,Tmax), (3.15)

    then Tmax=+. Moreover, there exists C>0 such that

    u(,t)W1,(Ω)+v(,t)L(Ω)+w(,t)W1,(Ω)Cfor all t>0.

    Proof. We divide the proof into two steps.

    Step 1: We claim that there exists a constant C1>0 such that

    Ωv2p0C1for all t(0,Tmax).

    Indeed, due to Lemma 3.8, for any p=2p0, there exists a constant C2>0 such that

    ddt(Ω|u|4p0+Ωv2p0)+2p01p0Ω|vp0|2+Ω|u|4p0+Ωv2p0(K3(2p0)ρp0)Ωv2p0+1+C2for all t(0,Tmax). (3.16)

    Let

    θ=nn+22p0+22p0+1(0,1).

    Then, 2p0+12p0θ<1 due to p0>n2. By the Gagliardo-Nirenberg inequality, Young's inequality and (3.15), we can find some constants C3,C4>0 such that

    (K3(2p0)ρp0)Ωv2p0+1=(K3(2p0)ρp0)vp02p0+1p0L2p0+1p0(Ω)C3(vp02p0+1p0(1θ)L1(Ω)vp02p0+1p0θL2(Ω)+vp02p0+1p0L1(Ω))C3(M2p0+1p0(1θ)vp02p0+1p0θL2(Ω)+M2p0+1p0)2p01p0Ω|vp0|2+C4for all t(0,Tmax),

    which along with (3.16) implies

    ddt(Ω|u|4p0+Ωv2p0)+Ω|u|4p0+Ωv2p0C2+C4for all t(0,Tmax).

    Therefore, the claim follows from the Grönwall inequality applied to the above inequality.

    Step 2: Thanks to the regularity theorem [32,Lemma 1], we can find a constant C5>0 such that uL(Ω)C5 due to 2p0>n. With (3.12) and Lemmas 3.1 and 3.2, we get a constant C6>0 such that for any p2

    ddtΩvp+p(p1)Ωvp2|v|2p(p1)(C6χ+C5ξ)Ωvp1|v|+p(ρ+ea3K1)Ωvp. (3.17)

    Thanks to Young's inequality, we find a constant C_7 > 0 such that

    \begin{equation*} p(p-1)(C_6\chi+C_5\xi)\int_\Omega v^{p-1}|\nabla v|\leqslant\frac{p(p-1)}{2}\int_\Omega v^{p-2}|\nabla v|^2+C_7p(p-1)\int_\Omega v^p, \end{equation*}

    which together with (3.17) implies

    \begin{equation} \frac{d}{dt}\int_\Omega v^p+p(p-1)\int_\Omega v^p+\frac{2(p-1)}{p}\int_\Omega|\nabla v^{\frac{p}{2}}|^2\leqslant p(p-1)C_8\int_\Omega v^p, \end{equation} (3.18)

    with C_8 = C_7+\rho+ea_3K_1+1 . Applying 1+p^n\leqslant(1+p)^n and the following inequality [34]

    \|f\|_{L^2}^2\leqslant\varepsilon\|\nabla f\|_{L^2}^2+C_9(1+\varepsilon^{-\frac{n}{2}})\|f\|_{L^1}^2,

    with f = v^{\frac{p}{2}} and \varepsilon = \frac{2}{p^2C_8} , we find a constant C_{10} > 0 such that

    \begin{equation} p(p-1)C_8\int_\Omega u^p\leqslant\frac{2(p-1)}{p}\int_\Omega \left|\nabla u^{\frac{p}{2}}\right|^2+C_{10}p(p-1)(1+p^n)\left(\int_\Omega u^{\frac{p}{2}}\right)^2. \end{equation} (3.19)

    Substituting (3.19) into (3.18), we have

    \begin{equation*} \frac{d}{dt}\int_\Omega u^p+p(p-1)\int_\Omega u^p\leqslant C_{10}p(p-1)(1+p)^n\left(\int_\Omega u^{\frac{p}{2}}\right)^2. \end{equation*}

    Then, employing the standard Moser iteration in [35] or a similar argument as in [34], we can prove that there exists a constant C_{11} > 0 such that

    \begin{equation*} \|v\|_{L^\infty(\Omega)}\leqslant C_{11}\quad\text{for all}\ t\in(0, T_{max}). \end{equation*}

    Thus, with the help of Lemma 3.2, we finish the proof.

    Now, utilizing the criterion in Lemma 3.9, we prove the global existence and boundedness of solutions for the system (1.1).

    Proof of Theorem 1.1. If n\leqslant2 , then the conclusion of the theorem can be obtained by Lemmas 3.3, 3.5 and 3.9. If n\geqslant3 and

    \rho\geqslant \frac{2K_3\left(\left[\frac{n}{2}\right]+1\right)}{\left[\frac{n}{2}\right]+1},

    then according to Lemma 3.8, by fixing p = [\frac{n}{2}]+1 we can find a constant C_1 > 0 such that

    \frac{d}{dt}\left(\int_{\Omega}|\nabla u|^{2\left[\frac{n}{2}\right]+2}+\int_{\Omega}v^{\left[\frac{n}{2}\right]+1}\right)+\int_{\Omega}|\nabla u|^{2\left[\frac{n}{2}\right]+2}+\int_{\Omega} v^{\left[\frac{n}{2}\right]+1} \leqslant C_1\quad\text{for all}\ t\in(0, T_{max}),

    which along with the Grönwall inequality gives a constant C_2 > 0 ,

    \int_{\Omega} v^{\left[\frac{n}{2}\right]+1} \leqslant C_2\quad\text{for all}\ t\in(0, T_{max}).

    Together with Lemma 3.9, we finish the proof by Lemma 2.1.

    In this section, we will employ suitable Lyapunov functionals to study the large-time behavior of u , v and w . We first improve the regularity of the solution.

    Lemma 4.1. There exist constants \theta_1, \theta_2, \theta_3\in(0, 1) and C > 0 such that

    \|u\|_{C^{2+\theta_1, 1+\frac{\theta_1}{2}}(\overline{\Omega}\times[t, t+1])}+\|v\|_{C^{2+\theta_2, 1+\frac{\theta_2}{2}}(\overline{\Omega}\times[t, t+1])}+\|w\|_{C^{2+\theta_3, 1+\frac{\theta_3}{2}}(\overline{\Omega}\times[t, t+1])}\leqslant C\quad\mathit{\text{for all}}\ t > 1.

    In particular, one can find C > 0 such that

    \|\nabla u\|_{L^\infty(\Omega)}+\|\nabla v\|_{L^\infty(\Omega)}+\|\nabla w\|_{L^\infty(\Omega)}\leqslant C\quad\mathit{\text{for all}}\ t > 1.

    Proof. The conclusion is a consequence of the regularity of parabolic equations in [36].

    We split our analysis into two cases: a_1 > a_3 and a_1\leqslant a_3 .

    We know that there are four homogeneous equilibria (0, 0, 0) , (0, 1, 0) , \left(\frac{a_1}{a_2}, 0, \frac{ra_1}{\gamma a_2}\right) and (u_*, v_*, w_*) when a_1 > a_3 , where u_*, v_* and w_* are defined in (1.5). In this case, we shall prove the coexistence steady state (u_*, v_*, w_*) is globally exponentially stable under certain conditions. Define an energy functional for (1.1) as follows:

    \mathcal{F}(t) = \varepsilon_1\int_\Omega\left(u-u_*-u_*\ln\frac{u}{u_*}\right)+\int_\Omega\left(v-v_*-v_*\ln\frac{v}{v_*}\right)+\frac{\varepsilon_2}{2}\int_\Omega\left(w-w_*\right)^2,

    where \varepsilon_1 and \varepsilon_2 are to be determined below.

    Proof of Theorem 1.2–(1). We complete the proof in four steps.

    Step 1: The parameters \varepsilon_1 and \varepsilon_2 can be chosen in the following way. First, we recall from (1.5) and (1.6) that

    \begin{equation} \Gamma = \frac{4du_*}{K_1^2v_*}, \Phi = \frac{2 a_2 \rho}{a_3^2}+e, \Psi = \frac{\gamma\eta a_3^2K_1^2}{d\rho^2 r^2u_*}. \end{equation} (4.1)

    Let

    f(y) = \frac{\Psi(\Gamma y-\xi^2)(-y^2+2\Phi y-e^2)}{y}, \quad y > 0.

    It is clear that f\in C^0((0, +\infty)) . Then, if

    \frac{\xi^2}{\Gamma} < \Phi+\sqrt{\Phi^2-e^2},

    the following holds:

    \begin{equation} \frac{\xi^2K_1^2 v_*}{4du_*} < \frac{2a_2\rho}{a_3^2}+e+\frac{2}{a_3}\sqrt{a_2\rho\left(\frac{a_2\rho}{a_3^2}+e\right)}. \end{equation} (4.2)

    Under (4.2), we let a = \max\left\{\frac{\xi^2}{\Gamma}, \Phi-\sqrt{\Phi^2-e^2}\right\} and b = \Phi+\sqrt{\Phi^2-e^2} with a < b . Then, f(y) is continuous on [a, b] with f(a) = f(b) = 0 , and consequently f(y) must attain the maximum at some point, say \varepsilon_1 , in (a, b) , namely f(\varepsilon_1) = \max \limits_{y\in [a, b]} f(y) . Then, a < \varepsilon_1 < b , or equivalently (see (4.1))

    \begin{equation} \max\left\{\frac{\xi^2u^2 v_*}{4du_*}, \frac{2a_2\rho}{a_3^2}+e-\frac{2}{a_3}\sqrt{a_2\rho\left(\frac{a_2\rho}{a_3^2}+e\right)}\right\} < \varepsilon_1 < \frac{2a_2\rho}{a_3^2}+e+\frac{2}{a_3}\sqrt{a_2\rho\left(\frac{a_2\rho}{a_3^2}+e\right)}. \end{equation} (4.3)

    Next, we assume \chi > 0 is suitably small such that

    \begin{align*} \chi^2 < f(\varepsilon_1) = &\frac{\gamma\eta a_3^2K_1^2}{d\rho r^2u_*\varepsilon_1}\left(\frac{4du_*\varepsilon_1}{v_*K_1^2}-\xi^2\right)\left(-\varepsilon_1^2+2\left(\frac{2a_2\rho}{a_3^2}+e\right)\varepsilon_1-e^2\right)\\ = &\frac{4\gamma\eta}{d\rho r^2u_*v_*\varepsilon_1}\left(4du_*\varepsilon_1-\xi^2v_*K_1^2\right)\left(a_2\rho\varepsilon_1-\frac{a_3^2(\varepsilon_1-e)^2}{4}\right), \end{align*}

    which implies

    \frac{d\chi^2u_*v_*^2\varepsilon_1}{\eta\left(4du_*v_*\varepsilon_1-\xi^2v_*^2K_1^2\right)} < \frac{4\gamma}{\rho r^2}\left(a_2\rho\varepsilon_1-\frac{a_3^2(\varepsilon_1-e)^2}{4}\right).

    Hence, there exists a constant \varepsilon_2 > 0 such that

    \begin{equation*} \label{24} \frac{d\chi^2u_*v_*^2\varepsilon_1}{\eta\left(4du_*v_*\varepsilon_1-\xi^2v_*^2K_1^2\right)} < \varepsilon_2 < \frac{4\gamma}{\rho r^2}\left(a_2\rho\varepsilon_1-\frac{a_3^2(\varepsilon_1-e)^2}{4}\right) \end{equation*}

    which along with Lemma 3.1 yields

    \begin{equation} \frac{d\chi^2u_*v_*^2\varepsilon_1}{\eta\left(4du_*v_*\varepsilon_1-\xi^2v_*^2u^2\right)} < \varepsilon_2 < \frac{4\gamma}{\rho r^2}\left(a_2\rho\varepsilon_1-\frac{a_3^2(\varepsilon_1-e)^2}{4}\right). \end{equation} (4.4)

    Step 2: We claim

    \|u-u_*\|_{L^{\infty}(\Omega)}+\|v-v_*\|_{L^{\infty}(\Omega)}+\|w-w_*\|_{L^{\infty}(\Omega)} \rightarrow0\quad\text{as}\ t\rightarrow +\infty.

    Indeed, using the equations in system (1.1) along with integration by parts, we have

    \begin{aligned} &\frac{d}{dt}\int_\Omega\left(u-u_*-u_*\ln\frac{u}{u_*}\right) = \int_\Omega \frac{u-u_*}{u}u_t\\ = &-du_*\int_\Omega\frac{|\nabla u|^2}{u^2}+\int_\Omega(u-u_*)(a_1-a_2u-a_3v)\\ = &-du_*\int_\Omega\frac{|\nabla u|^2}{u^2}-a_2\int_\Omega(u-u_*)^2-a_3\int_\Omega(u-u_*)(v-v_*). \end{aligned}

    Similarly, we obtain

    \begin{aligned} &\frac{d}{dt}\int_\Omega\left(v-v_*-v_*\ln\frac{v}{v_*}\right) = \int_\Omega \frac{v-v_*}{v}v_t\\ = &-v_*\int_\Omega\frac{|\nabla v|^2}{v^2}-\chi v_*\int_\Omega\frac{\nabla v\cdot\nabla w}{v}+\xi v_*\int_\Omega\frac{\nabla u\cdot\nabla v}{v}+\int_\Omega(v-v_*)(\rho-\rho v+ea_3u)\\ = &-v_*\int_\Omega\frac{|\nabla v|^2}{v^2}-\chi v_*\int_\Omega\frac{\nabla v\cdot\nabla w}{v}+\xi v_*\int_\Omega\frac{\nabla u\cdot\nabla v}{v}\\ &\quad-\rho\int_\Omega(v-v_*)^2+ea_3\int_\Omega(u-u_*)(v-v_*) \end{aligned}

    and

    \begin{align*} \frac{d}{dt}\int_\Omega\left(w-w_*\right)^2 = &2\int_\Omega\left(w-w_*\right)w_t = 2\int_\Omega\left(w-w_*\right)\left(\eta \Delta w+ru-\gamma w\right)\\ = &-2\eta\int_\Omega|\nabla w|^2+2r\int_\Omega(u-u_*)(w-w_*)-2\gamma\int_\Omega(w-w_*)^2\quad\text{for all}\ t > 0. \end{align*}

    Then, it follows that

    \begin{align*} \frac{d}{dt}\mathcal{F}(t) = &-du_*\varepsilon_1\int_\Omega\frac{|\nabla u|^2}{u^2}-v_*\int_\Omega\frac{|\nabla v|^2}{v^2}-\eta \varepsilon_2\int_\Omega|\nabla w|^2\\ &\quad-\chi v_*\int_\Omega\frac{\nabla v\cdot\nabla w}{v}+\xi v_*\int_\Omega\frac{\nabla u\cdot\nabla v}{v}\\ &\quad-a_2\varepsilon_1\int_\Omega(u-u_*)^2-\rho\int_\Omega(v-v_*)^2-\gamma\varepsilon_2\int_\Omega(w-w_*)^2\\ &\quad-a_3(\varepsilon_1-e)\int_\Omega(u-u_*)(v-v_*)+r\varepsilon_2\int_\Omega(u-u_*)(w-w_*)\\ = &:-X^TSX-Y^TTY, \end{align*}

    where X = (\nabla u, \nabla v, \nabla w) , Y = (u-u_*, v-v_*, w-w_*) , and

    S = \left[\begin{array}{ccc} \frac{du_*\varepsilon_1}{u^2} & -\frac{\xi v_*}{2v} & 0\\ -\frac{\xi v_*}{2v} & \frac{v_*}{v^2} & \frac{\chi v_*}{2v}\\ 0 & \frac{\chi v_*}{2v} & \eta\varepsilon_2 \end{array}\right], \quad T = \left[\begin{array}{ccc} a_2\varepsilon_1 & \frac{a_3(\varepsilon_1-e)}{2} & -\frac{r\varepsilon_2}{2}\\ \frac{a_3(\varepsilon_1-e)}{2} & \rho & 0\\ -\frac{r\varepsilon_2}{2} & 0 & \gamma\varepsilon_2 \end{array}\right].

    Note that (4.3) yields

    \frac{du_*v_*\varepsilon_1}{u^2v^2}-\frac{\xi^2v_*^2}{4v^2} > \frac{v_*^2}{4v^2}\bigg(\frac{4d u_*\varepsilon}{K_1^2}-\xi^2\bigg) > 0,

    and (4.4) gives

    \frac{\eta du_*v_*\varepsilon_1\varepsilon_2}{u^2v^2}-\frac{d\chi^2u_*v_*^2\varepsilon_1}{4u^2v^2}-\frac{\eta \xi^2v_*^2\varepsilon_2}{4v^2} > 0.

    The above results indicate that matrix S is positive definite. Using (4.3) and (4.4) again, we observe that

    a_2\rho\varepsilon_1-\frac{a_3^2(\varepsilon_1-e)^2}{4} > 0,

    and

    a_2\rho\gamma\varepsilon_1\varepsilon_2-\frac{\rho r^2\varepsilon_2^2}{4}-\frac{a_3^2\gamma(\varepsilon_1-e)^2\varepsilon_2}{4} > 0,

    which imply that matrix T is positive definite. Therefore, one can choose a constant C_1 > 0 such that

    \begin{equation} \frac{d}{dt}\mathcal{F}(t)\leqslant-C_1\left(\int_\Omega(u-u_*)^2+\int_\Omega(v-v_*)^2+\int_\Omega(w-w_*)^2\right)\quad\text{for all}\ t > 0. \end{equation} (4.5)

    Integrating the above inequality with respect to time, we get a constant C_2 > 0 satisfying

    \int_{1}^{+\infty}\int_\Omega(u-u_*)^2+\int_{1}^{+\infty}\int_\Omega(v-v_*)^2+\int_{1}^{+\infty}\int_\Omega(w-w_*)^2\leqslant C_2,

    which together with the uniform continuity of u, v and w due to Lemma 4.1 yields

    \begin{equation} \int_\Omega(u-u_*)^2+\int_\Omega(v-v_*)^2+\int_\Omega(w-w_*)^2\rightarrow0, \quad\text{as}\ t\rightarrow +\infty. \end{equation} (4.6)

    By the Gagliardo-Nirenberg inequality, we can find a constant C_3 > 0 such that

    \begin{equation} \|u-u_*\|_{L^{\infty}(\Omega)} \leqslant C_3\|u-u_*\|_{W^{1, \infty}(\Omega)}^{\frac{n}{n+2}}\|u-u_*\|_{L^{2}(\Omega)}^{\frac{2}{n+2}}, \end{equation} (4.7)
    \begin{equation} \|v-v_*\|_{L^{\infty}(\Omega)} \leqslant C_3\|v-v_*\|_{W^{1, \infty}(\Omega)}^{\frac{n}{n+2}}\|v-v_*\|_{L^{2}(\Omega)}^{\frac{2}{n+2}} \end{equation} (4.8)

    and

    \begin{equation} \|w-w_*\|_{L^{\infty}(\Omega)} \leqslant C_3\|w-w_*\|_{W^{1, \infty}(\Omega)}^{\frac{n}{n+2}}\|w-w_*\|_{L^{2}(\Omega)}^{\frac{2}{n+2}}\quad\text{for all}\ t > 1, \end{equation} (4.9)

    which along with (4.6) and Lemma 4.1 prove the claim.

    Step 3: From the L'Hôpital rule, it holds that for any s_0 > 0

    \lim\limits_{s\rightarrow s_0}\frac{s-s_0-s_0\ln\frac{s}{s_0}}{(s-s_0)^2} = \lim\limits_{s\rightarrow s_0}\frac{1-\frac{s_0}{s}}{2(s-s_0)} = \lim\limits_{s\rightarrow s_0}\frac{1}{2s} = \frac{1}{2s_0},

    which gives a constant \eta > 0 such that for all |s-s_0|\leqslant\eta

    \begin{equation} \frac{1}{4s_0}(s-s_0)^2\leqslant s-s_0-s_0\ln\frac{s}{s_0}\leqslant\frac{1}{s_0}(s-s_0)^2. \end{equation} (4.10)

    By (4.6), there exists T_1 > 1 such that

    \begin{equation*} \|u-u_*\|_{L^\infty(\Omega)}+\|v-v_*\|_{L^\infty(\Omega)}+\|w-w_*\|_{L^\infty(\Omega)}\leqslant\eta\quad\text{for all}\ t\geqslant T_1. \end{equation*}

    Therefore, by (4.10), we get

    \begin{equation} \frac{1}{4u_*}\int_\Omega(u-u_*)^2\leqslant\int_\Omega \left(u-u_*-u_*\ln\frac{u}{u_*}\right)\leqslant\frac{1}{u_*}\int_\Omega(u-u_*)^2\quad\text{for all}\ t\geqslant T_1 \end{equation} (4.11)

    and

    \begin{equation} \frac{1}{4v_*}\int_\Omega(v-v_*)^2\leqslant\int_\Omega \left(v-v_*-v_*\ln\frac{v}{v_*}\right)\leqslant\frac{1}{v_*}\int_\Omega(v-v_*)^2\quad\text{for all}\ t\geqslant T_1. \end{equation} (4.12)

    Step 4: From (4.11) and (4.12), it follows that

    \mathcal{F}(t)\leqslant\max\left\{\frac{\varepsilon_1}{u_*}, \frac{1}{v_*}, \frac{\varepsilon_2}{2}\right\}\left(\int_\Omega(u-u_*)^2+\int_\Omega(v-v_*)^2+\int_\Omega(w-w_*)^2\right),

    which alongside (4.5) yields a constant C_4 > 0 such that

    \frac d{dt}\mathcal{F}(t)\leqslant-C_4\mathcal{F}(t)\quad\text{for all}\ t\geqslant T_1.

    This immediately gives a constant C_5 > 0 such that

    \begin{equation*} \mathcal{F}(t)\leqslant C_5e^{-C_4 t}\quad\text{for all}\ t\geqslant T_1. \end{equation*}

    Hence, utilizing (4.11) and (4.12) again, one obtains a constant C_6 > 0 such that

    \begin{equation*} \int_\Omega(u-u_*)^2+\int_\Omega(v-v_*)^2+\int_\Omega(w-w_*)^2\leqslant C_6e^{-C_4 t}\quad\text{for all}\ t\geqslant T_1. \end{equation*}

    Finally, by (4.7)–(4.9) and Lemma 4.1, we get the decay rates of \|u-u_*\|_{L^\infty(\Omega)} , \|v-v_*\|_{L^\infty(\Omega)} and \|w-w_*\|_{L^\infty(\Omega)} , as claimed in Theorem 1.2–(1).

    In this case, there are three homogeneous equilibria (0, 0, 0) , (0, 1, 0) and \left(\frac{a_1}{a_2}, 0, \frac{ra_1}{\gamma a_2}\right) , and we shall show that the steady state (0, 1, 0) is global asymptotically stable, where the convergence rate is exponential if a_1 < a_3 and algebraic if a_1 = a_3 . Define an energy functional for (1.1) as follows:

    G(t) = e\int_\Omega u+\frac{\zeta_1}{2}\int_\Omega u^2+\int_\Omega\left(v-1-\ln v\right)+\frac{\zeta_2}{2}\int_\Omega w^2,

    where \zeta_1 and \zeta_2 will be determined below.

    Proof of Theorem 1.2–(2). We divide the proof into five steps.

    Step 1: We shall choose the appropriate parameters \zeta_1 and \zeta_2 . By the definitions of A and B in (1.7), since A < B , we have

    \begin{equation} \left(\frac{\xi^2}{4d}\right)^2 < \frac{\xi^2ea_2}{4da_1} < \left(\frac{ea_2}{a_1}\right)^2. \end{equation} (4.13)

    Let

    g(y) = \frac{16\eta\gamma}{dr^2}\frac{(dy-\frac{\xi^2}{4})(ea_2-a_1y)}{y}, \quad \frac{\xi^2}{4d} < y < \frac{ea_2}{a_1}.

    Then, g\in C^1\left(\left(\frac{\xi^2}{4d}, \frac{ea_2}{a_1}\right)\right) , and g(y) > 0 in \left(\frac{\xi^2}{4d}, \frac{ea_2}{a_1}\right) . We further observe that

    g\left(\frac{\xi}{2}\sqrt{\frac{ea_2}{da_1}}\right) = D\left(A+B-2\sqrt{AB}\right)

    which along with \chi^2 < D\left(A+B-2\sqrt{AB}\right) implies

    \chi^2 < g\left(\frac{\xi}{2}\sqrt{\frac{ea_2}{da_1}}\right).

    By the definition of g , one has

    g'(y_0) = \frac{16\eta\gamma}{dr^2}\left(-da_1+\frac{\xi^2ea_2}{4y_0^2}\right) = 0,

    which alongside (4.13) gives y_0 = \frac{\xi}{2}\sqrt{\frac{ea_2}{da_1}}\in\left(\frac{\xi^2}{4d}, \frac{ea_2}{a_1}\right) . Thus, g(y) is increasing in \left(\frac{\xi^2}{4d}, \frac{\xi}{2}\sqrt{\frac{ea_2}{da_1}}\right) and decreasing in \left(\frac{\xi}{2}\sqrt{\frac{ea_2}{da_1}}, \frac{ea_2}{a_1}\right) . We can find a constant \zeta_1 > 0 such that

    \begin{equation} \frac{\xi}{2}\sqrt{\frac{ea_2}{da_1}} < \zeta_1 < \frac{ea_2}{a_1} \end{equation} (4.14)

    and

    0 = g\left(\frac{ea_2}{a_1}\right) < \chi^2 < g(\zeta_1) < g\left(\frac{\xi}{2}\sqrt{\frac{ea_2}{da_1}}\right).

    With the definition of g , we get

    \frac{d\chi^2\zeta_1}{4\eta(d\zeta_1-\frac{\xi^2}{4})} < \frac{4\gamma}{r^2}(ea_2-a_1\zeta_1),

    which implies that there exists \zeta_2 > 0 such that

    \begin{equation} \frac{d\chi^2\zeta_1}{4\eta(d\zeta_1-\frac{\xi^2}{4})} < \zeta_2 < \frac{4\gamma}{r^2}(ea_2-a_1\zeta_1). \end{equation} (4.15)

    One can verify that

    \begin{equation} \eta d\zeta_1\zeta_2-\frac{d\chi^2}{4}\zeta_1-\frac{\eta \xi^2}{4}\zeta_2 > 0, \end{equation} (4.16)

    and

    \begin{equation} (ea_2-a_1\zeta_1)\rho\gamma\zeta_2-\frac{\rho r^2}{4}\zeta_2^2 > 0. \end{equation} (4.17)

    Thanks to (4.13) and (4.14), one obtains

    \begin{equation} \frac{\xi^2}{4d} < \zeta_1 < \frac{ea_2}{a_1}. \end{equation} (4.18)

    Step 2: We claim

    \begin{equation} \|u\|_{L^{\infty}(\Omega)}+\|v-1\|_{L^{\infty}(\Omega)}+\|w\|_{L^{\infty}(\Omega)} \rightarrow0\quad\text{as}\ t\rightarrow +\infty. \end{equation} (4.19)

    Indeed, if (u, v, w) is the solution of system (1.1), then we get

    \begin{gather} \frac{d}{dt}\int_\Omega u = a_1\int_\Omega u-a_2\int_\Omega u^2-a_3\int_\Omega uv, \end{gather} (4.20)
    \begin{gather} \frac{d}{dt}\int_\Omega u^2 = 2\int_\Omega uu_t = -2d\int_\Omega|\nabla u|^2+2a_1\int_\Omega u^2-2a_2\int_\Omega u^3-2a_3\int_\Omega u^2v, \end{gather} (4.21)
    \begin{gather} \begin{aligned} &\frac{d}{dt}\int_\Omega\left(v-1-\ln v\right) = \int_\Omega \frac{v-1}{v}v_t\\ = &-\int_\Omega\frac{|\nabla v|^2}{v^2}-\chi \int_\Omega\frac{\nabla v\cdot\nabla w}{v}+\xi \int_\Omega\frac{\nabla u\cdot\nabla v}{v}+\int_\Omega(v-1)(\rho-\rho v+ea_3u)\\ = &-\int_\Omega\frac{|\nabla v|^2}{v^2}-\chi \int_\Omega\frac{\nabla v\cdot\nabla w}{v}+\xi\int_\Omega\frac{\nabla u\cdot\nabla v}{v}-\rho\int_\Omega(v-1)^2+ea_3\int_\Omega uv-ea_3\int_\Omega u \end{aligned} \end{gather} (4.22)

    and

    \begin{equation} \frac{d}{dt}\int_\Omega w^2 = 2\int_\Omega ww_t = -2\eta\int_\Omega|\nabla w|^2+2r\int_\Omega uw-2\gamma\int_\Omega w^2\quad\text{for all}\ t > 0. \end{equation} (4.23)

    Then, combining (4.20), (4.21), (4.22) and (4.23), we have from the definition of G(t) that

    \begin{equation} \begin{aligned} \frac{d}{dt}G(t)\leqslant&-d\zeta_1\int_\Omega|\nabla u|^2-\int_\Omega\frac{|\nabla v|^2}{v^2}-\eta\zeta_2\int_\Omega|\nabla w|^2\\ &\quad-\chi \int_\Omega\frac{\nabla v\cdot\nabla w}{v}+\xi\int_\Omega\frac{\nabla u\cdot\nabla v}{v}+e(a_1-a_3)\int_\Omega u\\ &\quad-(ea_2-a_1\zeta_1)\int_\Omega u^2-\rho\int_\Omega(v-1)^2-\gamma\zeta_2\int_\Omega w^2+r\zeta_2\int_\Omega uw\\ = &:-X^TPX-Y^TQY+e(a_1-a_3)\int_\Omega u, \end{aligned} \end{equation} (4.24)

    where X = (\nabla u, \nabla v, \nabla w) , Y = (u, v-1, w) ,

    P = \left[\begin{array}{ccc} d\zeta_1 & -\frac{\xi}{2v} & 0\\ -\frac{\xi}{2v} & \frac{1}{v^2} & \frac{\chi}{2v}\\ 0 & \frac{\chi}{2v} & \eta\zeta_2 \end{array}\right]\quad\text{and}\quad Q = \left[\begin{array}{ccc} ea_2-a_1\zeta_1 & 0 & -\frac{r\zeta_2}{2}\\ 0 & \rho & 0\\ -\frac{r\zeta_2}{2} & 0 & \gamma\zeta_2 \end{array}\right].

    It can be checked that (4.16) and (4.18) ensure that the matrix P is positive definite while (4.17) and (4.18) guarantee that the matrix Q is positive definite. Thus, there is a constant C_1 > 0 such that if a_1 < a_3 , then

    \begin{equation} \frac{d}{dt}G(t)\leqslant-C_1\left(\int_\Omega u+\int_\Omega u^2+\int_\Omega(v-1)^2+\int_\Omega w^2\right)\quad\text{for all}\ t > 0, \end{equation} (4.25)

    and if a_1 = a_3 , then

    \begin{equation} \frac{d}{dt}G(t)\leqslant-C_1\left(\int_\Omega u^2+\int_\Omega(v-1)^2+\int_\Omega w^2\right)\quad\text{for all}\ t > 0. \end{equation} (4.26)

    Integrating the above inequalities with respect to time, we find a constant C_2 > 0 satisfying

    \int_1^{+\infty}\int_\Omega u^2+\int_1^{+\infty}\int_\Omega(v-1)^2+\int_1^{+\infty}\int_\Omega w^2\leqslant C_2,

    which together with the uniform continuity of u, v and w due to Lemma 4.1 yields

    \begin{equation} \int_\Omega u^2+\int_\Omega(v-1)^2+\int_\Omega w^2\rightarrow0, \quad\text{as}\ t\rightarrow +\infty. \end{equation} (4.27)

    Thus, (4.19) is obtained by the Gagliardo-Nirenberg inequality and Lemma 4.1.

    Step 3: By the L'Hôpital rule, we get

    \lim\limits_{s\rightarrow 1}\frac{s-1-\ln s}{(s-1)^2} = \lim\limits_{s\rightarrow 1}\frac{1-\frac{1}{s}}{2(s-1)} = \lim\limits_{s\rightarrow 1}\frac{1}{2s} = \frac{1}{2},

    which gives a constant \varepsilon > 0 such that

    \begin{equation} \frac{1}{4}(s-1)^2\leqslant s-1-\ln s\leqslant(s-1)^2 \ \ \text{for all}\ \ |s-1|\leqslant\varepsilon. \end{equation} (4.28)

    By (4.19), there exists T_1 > 0 such that

    \begin{equation} \|u\|_{L^\infty(\Omega)}+\|v-1\|_{L^\infty(\Omega)}+\|w\|_{L^\infty(\Omega)}\leqslant\varepsilon\quad\text{for all}\ t\geqslant T_1. \end{equation} (4.29)

    Therefore, it follows from (4.28) that

    \begin{equation} \frac{1}{4}\int_\Omega(v-1)^2\leqslant\int_\Omega (v-1-\ln v)\leqslant \int_\Omega(v-1)^2\quad\text{for all}\ t\geqslant T_1. \end{equation} (4.30)

    Step 4: If a_1 < a_3 , from the definition of G(t) and (4.30), one has

    G(t)\leqslant\max\left\{e, \frac{\zeta_1}{2}, \frac{\zeta_2}{2}, 1\right\}\left(\int_\Omega u+\int_\Omega u^2+\int_\Omega(v-1)^2+\int_\Omega w^2\right),

    which along with (4.25) yields a constant C_3 > 0 such that

    \frac d{dt}G(t)\leqslant-C_3G(t)\quad\text{for all}\ t\geqslant T_1.

    This gives a constant C_4 > 0 such that

    \begin{equation*} G(t)\leqslant C_4e^{-C_3 t}\quad\text{for all}\ t\geqslant T_1. \end{equation*}

    Hence, utilizing (4.30) again, we find a constant C_5 > 0 such that

    \begin{equation*} \int_\Omega u^2+\int_\Omega(v-1)^2+\int_\Omega w^2\leqslant C_5e^{-C_3 t}\quad\text{for all}\ t\geqslant T_1. \end{equation*}

    Then, by the Gagliardo-Nirenberg inequality and Lemma 4.1, we get the exponential convergence for \|u\|_{L^{\infty}(\Omega)}+\|v-1\|_{L^{\infty}(\Omega)}+\|w\|_{L^{\infty}(\Omega)} .

    Step 5: If a_1 = a_3 , we use (4.29), (4.30) and Young's inequality to find a constant C_6 > 0 :

    \begin{align*} G^2(t)\leqslant& C_6\left(\int_\Omega u+\int_\Omega u^2+\int_\Omega(v-1)^2+\int_\Omega w^2\right)^2\\ \leqslant&C_6(\varepsilon+1)^2\left(\int_\Omega u+\int_\Omega(v-1)+\int_\Omega w \right)^2\\ \leqslant&3C_6(\varepsilon+1)^2|\Omega|\left(\int_\Omega u^2+\int_\Omega(v-1)^2+\int_\Omega w^2 \right)\quad\text{for all}\ t\geqslant T_1, \end{align*}

    which alongside (4.26) implies some constant C_7 > 0

    \frac{d}{dt}G(t)\leqslant-C_7G^2(t)\quad\text{for all}\ t\geqslant T_1.

    Solving the above inequality directly yields a constant C_8 > 0 such that

    G(t)\leqslant C_8(t+1)^{-1}\quad\text{for all}\ t\geqslant T_1.

    Similar to the case a_1 < a_3 , we can use (4.30), the Gagliardo-Nirenberg inequality and Lemma 4.1 to get the convergence rate of \|u\|_{L^{\infty}(\Omega)}+\|v-1\|_{L^{\infty}(\Omega)}+\|w\|_{L^{\infty}(\Omega)} .

    The author warmly thanks the reviewers for several inspiring comments and helpful suggestions. The research of the author was supported by the National Nature Science Foundation of China (Grant No. 12101377) and the Nature Science Foundation of Shanxi Province (Grant No. 20210302124080).

    The author declares there is no conflict of interest.



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