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Research article

SoftVoting6mA: An improved ensemble-based method for predicting DNA N6-methyladenine sites in cross-species genomes


  • The DNA N6-methyladenine (6mA) is an epigenetic modification, which plays a pivotal role in biological processes encompassing gene expression, DNA replication, repair, and recombination. Therefore, the precise identification of 6mA sites is fundamental for better understanding its function, but challenging. We proposed an improved ensemble-based method for predicting DNA N6-methyladenine sites in cross-species genomes called SoftVoting6mA. The SoftVoting6mA selected four (electron–ion-interaction pseudo potential, One-hot encoding, Kmer, and pseudo dinucleotide composition) codes from 15 types of encoding to represent DNA sequences by comparing their performances. Similarly, the SoftVoting6mA combined four learning algorithms using the soft voting strategy. The 5-fold cross-validation and the independent tests showed that SoftVoting6mA reached the state-of-the-art performance. To enhance accessibility, a user-friendly web server is provided at http://www.biolscience.cn/SoftVoting6mA/.

    Citation: Zhaoting Yin, Jianyi Lyu, Guiyang Zhang, Xiaohong Huang, Qinghua Ma, Jinyun Jiang. SoftVoting6mA: An improved ensemble-based method for predicting DNA N6-methyladenine sites in cross-species genomes[J]. Mathematical Biosciences and Engineering, 2024, 21(3): 3798-3815. doi: 10.3934/mbe.2024169

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  • The DNA N6-methyladenine (6mA) is an epigenetic modification, which plays a pivotal role in biological processes encompassing gene expression, DNA replication, repair, and recombination. Therefore, the precise identification of 6mA sites is fundamental for better understanding its function, but challenging. We proposed an improved ensemble-based method for predicting DNA N6-methyladenine sites in cross-species genomes called SoftVoting6mA. The SoftVoting6mA selected four (electron–ion-interaction pseudo potential, One-hot encoding, Kmer, and pseudo dinucleotide composition) codes from 15 types of encoding to represent DNA sequences by comparing their performances. Similarly, the SoftVoting6mA combined four learning algorithms using the soft voting strategy. The 5-fold cross-validation and the independent tests showed that SoftVoting6mA reached the state-of-the-art performance. To enhance accessibility, a user-friendly web server is provided at http://www.biolscience.cn/SoftVoting6mA/.



    The Forchheimer model equation describes the flow in a polar medium, which is widely used in fluid mechanics (see [1,2]). Increasing scholars have studied the spatial properties of the solution to the fluid equation defined on a semi-infinite cylinder, and a large number of results have emerged (see [3,4,5,6,7,8,9]).

    In 2002, Payne and Song [3] have studied the following Forchheimer model

    b|u|ui+(1+γT)ui=p,i+giT, in Ω×{t>0}, (1.1)
    ui,i=0, in Ω×{t>0}, (1.2)
    tT+uiT,i=ΔT, in Ω×{t>0}, (1.3)

    where i=1,2,3. ui,p,T represent the velocity, pressure, and temperature of the flow, respectively. gi is a known function. Δ is the Laplace operator, γ>0 is a constant, and b is the Forchheimer coefficient. For simplicity, we assume that

    gigi1.

    In (1.1)–(1.3), Ω is defined as

    Ω={(x1,x2,x3)|(x1,x2)D, x30},

    where D is a bounded simply-connected region on (x1,x2)-plane.

    In this paper, the comma is used to indicate partial differentiation and the usual summation convection is employed, with repeated Latin subscripts summed from 1 to 3, e.g., ui,jui,j=3i,j=1(uixj)2. We also use the summation convention summed from 1 to 2, e.g., uα,βuα,β=2α,β=1(uαxβ)2.

    The Eqs (1.1)–(1.3) also satisfy the following initial-boundary conditions

    ui(x1,x2,x3,t)=0, T(x1,x2,x3,t)=0, on D×{x3>0}×{t>0}, (1.4)
    ui(x1,x2,0,t)=fi(x1,x2,t), on D×{t>0}, (1.5)
    T(x1,x2,0,t)=H(x1,x2,t), on D×{t>0}, (1.6)
    T(x1,x2,x3,0)=0, (x1,x2,x3)Ω, (1.7)
    |u|,|T|=O(1), |u3|,|T|,|p|=o(x13), as x3. (1.8)

    where fi and H are differentiable functions.

    In this paper, we will study the structural stability of Eqs (1.1)–(1.8) on Ω by using the spatial decay results obtained in [3]. Since the concept of structural stability was proposed by Hirsch and Smale [10], the structural stability of various types of partial differential equations defined in a bounded domain has received sufficient attention(see [11,12,13,14,15,16,17,18,19]). Some perturbations are inevitable in the process of model establishment and simplification, so it is necessary to study that whether such small perturbations of the equations themselves will cause great changes in the solutions. This gives rise to the phenomenon of structural stability.

    If the bounded domain is replaced by a semi-infinite pipe, the structural stability of the partial differential equations is very interesting and has begun to attract attention. Li and Lin [20] considered the continuous dependence on the Forchheimer coefficient of Forchheimer equations in a semi-infinite pipe. Different from the studies of[11,12,13,14,15,16,17,18,19], we should consider not only the time variable but also the space variable. Therefore, the methods in the literature cannot be directly applied to the semi-infinite region. Compared with [3], we not only reconfirmed the spatial decay result of [3], but also proved the structural stability of the solution to b and γ.

    We also introduce the notations:

    Ωz={(x1,x2,x3)|(x1,x2)D,x3z0},
    Dz={(x1,x2,x3)|(x1,x2)D,x3=z0},

    where z is a running variable along the x3 axis.

    First, to obtain the main result, we shall make frequent use of the following three inequalities.

    Lemma 2.1.(see[21]) If ϕ is a Dirichlet integrable function on Ω and Ωϕdx=0, then there exists a Dirichlet integrable function w=(w1,w2,w3) such that

    wi,i=ϕ, in Ω, wi=0, on Ω,

    and a positive constant k1 depends only on the geometry of Ω such that

    Ωwi,jwi,jdxk1Ω(wi,i)2dx.

    Lemma 2.2.(see [3,4]) If ϕ|D=0, then

    λDϕ2dADϕ,αϕ,αdA,

    where λ is the smallest positive eigenvalue of

    Δ2ϑ+λϑ=0, in D, ϑ=0, on D.

    Here Δ2 is a two-dimensional Laplace operator.

    Now, we give a lemma which has been proved by Horgan and Wheeler [4] and has been used by Payne and Song [6].

    Lemma 2.3.(see [3,4]) If ϕ is a Dirichlet integrable function and ϕ|D=0,ϕ (as x3),

    Ωz|ϕ|4dxk2(Ωzϕ,jϕ,jdx)2,

    where k2>0.

    Lemma 2.4. If ϕC10(Ω), then

    Ωz|ϕ|6dxΛ(Ωzϕ,iϕ,idx)3,

    where [22,23] have proved that the optimal value of Λ is determined to be Λ=127(34)4.

    Using the maximum principle for the temperature T, we can have the following lemma which has been used in Song [5].

    Lemma 2.5. Assume that HL(Ω), then

    supΩ×{t>0}|T|TM,

    where TM=supΩ×{t>0}H.

    Second, we list some useful results which have been derived by Payne and Song [3].

    Payne and Song have established a function

    P(z,t)=t0Ωz(ξz)T,iT,idxdη+a1t0Ωz|u|3dxdη+a2t0Ωz(1+γT)|u|2dxdη, (2.1)

    where a1 and a2 are positive constants. From Eqs (3.27) and (3.36) of [3], we know that

    P(z,t)P(0,t)ezk3, P(0,t)k4(t), (2.2)

    where k3 is a positive constant and k4(t) is a function related to the boundary values.

    Combining Eqs (2.1) and (2.2), we have the following lemma.

    Lemma 2.6. Assume that HL(Ω) and DfdA=0, then

    a1t0Ωz|u|3dxdη+a2t0Ωz(1+γT)|u|2dxdηk4(t)ezk3.

    In order to derive the main result, we need bounds for ||u||2L2(Ω) and ||u||3L2(Ω).

    Lemma 2.7. Assume that fiH1(Ω),H,˜HL(Ω), Df3dA=0 and fα,αγf3=0 then

    bΩ|u|3dx+Ω|u|2dxk5(t),

    where k6(t) is a positive function.

    Proof. To deal with boundary terms, we set S=(S1,S2,S3), where

    Si=fieγ1x3, γ1>0. (2.3)

    Using Eq (1.1), we have

     Ω[b|u|ui+(1+γT)ui+p,igiT](uiSi)dx=0.

    Using the divergence theorem, we have

    bΩ|u|3dx+Ω(1+γT)|u|2dx=bΩ|u|uiSidx+Ω(1+γT)uiSidxΩgiTuidx+ΩgiTSidx. (2.4)

    Using the Hölder inequality and Young's inequality, we have

    bΩ|u|uiSidxb(Ω|u|3dx)23(Ω|S|3dx)1323bε1Ω|u|3dx+13bε21Ω|S|3dx, (2.5)
    Ω(1+γT)uiSidx14Ω(1+γT)|u|2dx+(1+γTM)Ω|S|2dx, (2.6)
    ΩgiTuidxTM(Ω(1+γT)|u|2dxΩgigidx)1214Ω(1+γT)|u|2dx+TMγΩgigidx, (2.7)
    ΩgiTSidxTMΩ|giSi|dx. (2.8)

    Inserting Eqs (2.5)–(2.8) into Eq (2.4) and choosing that ε1=34, we obtain

    bΩ|u|3dx+Ω(1+γT)|u|2dx23bε21Ω|S|3dx+2(1+γTM)Ω|S|2dx+2TMγΩgigidx+2TMΩ|giSi|dx. (2.9)

    After choosing

    k5(t)=23bε21Ω|S|3dx+2(1+γTM)Ω|S|2dx+2TMγΩgigidx+2TMΩ|giSi|dx, (2.10)

    we can complete the proof of Lemma 2.7.

    In this section, we derive an important lemma which leads to our main result.

    Assume that (ui,T,p) is a solution of Eqs (1.1)–(1.8) when b=b. If we let

    Di=uiui, Σ=TT, π=pp, ˜b=bb,

    then (Di,Σ,π) satisfies

    [b1|u|uib2|u|ui]+(1+γT)Di+γΣui=π,i+giΣ, in Ω×{t>0}, (3.1)
    Di,i=0, in Ω×{t>0}, (3.2)
    tΣ+uiΣ,i+DiT,i=ΔΣ, in Ω×{t>0}, (3.3)
    Di=0,Σ=0, on D×{x3>0}×{t>0}, (3.4)
    Di=0,Σ=0, on D×{t>0}, (3.5)
    Σ(x1,x2,x3,0)=0, in Ω (3.6)
    |u|,|Σ|=O(1),|D3|,|Σ|,|π|=o(x13), as x3. (3.7)

    We can have the following lemma.

    Lemma 3.1. Assume that (Di,Σ,π) is a solution to Eqs (3.1)–(3.6) with Df3dA=0,HL(Ω) and the boundary data (e.g., H) satisfies Eq (3.21), then

    Φ(z,t)n6[zΦ(z,t)]+n7(t)˜b2ezk3,

    where n6 is the maximum of n6(t) and n6(t),n7(t) will be defined in Eq (3.39).

    Proof. We define an auxiliary function

    Φ1(z,t)=t0ΩzeωηπD3dxdη, (3.8)

    where ω>0.

    Using the divergence theorem and Eq (3.1), we have

    Φ1(z,t)=t0Ωzeωη(ξz)π,iDidxdη=t0Ωzeωη(ξz)Di[b1|u|uib2|u|ui]dxdη+t0Ωzeωη(ξz)(1+γT)DiDidxdη+γt0Ωzeωη(ξz)DiΣuidxdηt0Ωzeωη(ξz)DigiΣdxdη. (3.9)

    Since

    Di[b1|u|uib2|u|ui]=˜b2Di[|u|ui+|u|ui]+b1+b22Di[|u|ui|u|ui]=˜b2[|u|ui+|u|ui]Di+b1+b24[|u|+|u|]DiDi+b1+b24[|u||u|]2[|u|+|u|],

    from Eq (3.9) we have

    Φ1(z,t)=b1+b24t0Ωzeωη(ξz)[|u|+|u|]DiDidxdη+t0Ωzeωη(ξz)(1+γT)DiDidxdη+˜b2t0Ωzeωη(ξz)[|u|ui+|u|ui]Didxdη+b1+b24t0Ωzeωη(ξz)[|u||u|]2[|u|+|u|]dxdη+γt0Ωzeωη(ξz)DiΣuidxdηt0Ωzeωη(ξz)DigiΣdxdη. (3.10)

    Using the Hölder inequality, Young's inequality and Lemma 2.6, we obtain

    ˜b2t0Ωzeωη[|u|ui+|u|ui]Didxdη˜b2(t0Ωzeωη|u|DiDidxdη)12(t0Ωzeωη|u|3dxdη)12+˜b2(t0Ωzeωη|u|DiDidxdη)12(t0Ωzeωη|u|3dxdη)12b1+b216t0Ωzeωη[|u|+|u|]DiDidxdη4˜b2b1+b2t0Ωzeωη[|u|3+|u|3]dxdηb1+b216t0Ωzeωη[|u|+|u|]DiDidxdη8˜b2a1(b1+b2)k4(t)ezk3. (3.11)

    Using the Hölder inequality, Young's inequality and Lemmas 2.3 and 2.7, we obtain

    γt0ΩzeωηDiΣuidxdηγt0eωη(Ωz|u|DiDidx)12(Ωz|u|2dx)14(ΩzΣ4dx)14dηγ4k5(t)k2t0eωη(Ωz|u|DiDidx)12(ΩzΣ,iΣ,idx)12dηb1+b216t0Ωzeωη|u|DiDidxdη4γ2k5(t)k2b1+b2t0ΩzeωηΣ,iΣ,idxdη, (3.12)
    t0ΩzeωηDigiΣdxdη12t0Ωzeωη(1+γT)DiDidxdη12γt0ΩzeωηΣ2dxdη. (3.13)

    Calculating the differential of Eq (3.10) and then inserting Eqs (3.11)–(3.13) into Eq (3.10), we have

    zΦ1(z,t)b1+b28t0Ωzeωη[|u|+|u|]DiDidxdη+12t0Ωzeωη(1+γT)DiDidxdη4γ2k5(t)k2b1+b2t0ΩzeωηΣ,iΣ,idxdη12γt0ΩzeωηΣ2dxdη8˜b2a1(b1+b2)k4(t)ezk3, (3.14)

    where we have dropped the fourth term of Eq (3.10).

    Similarly, we have

    Φ2(z,t)=t0ΩzeωηΣΣ,3dxdη+12t0Ωzeωηu3Σ2dxdη+t0ΩzeωηD3TΣdxdηΦ21(z,t)+Φ22(z,t)+Φ23(z,t). (3.15)

    Using the divergence theorem and Eq (3.2), we have

    Φ2(z,t)=12eωtΩz(ξz)Σ2dx+t0Ωzeωη(ξz)[12ωΣ2+Σ,iΣ,i]dxdηt0Ωzeωη(ξz)DiΣ,iTdxdη. (3.16)

    Using the Hölder inequality and Lemma 2.5, we have

    t0ΩzeωηDiΣ,iTdxdη12t0ΩzeωηΣ,iΣ,idxdη12T2Mt0ΩzeωηDiDidxdη. (3.17)

    Calculating the differential of Eq (3.16) and then inserting Eq (3.17) into Eq (3.16), we have

    zΦ2(z,t)=12eωtΩzΣ2dx+t0Ωzeωη[12ωΣ2+12Σ,iΣ,i]dxdη12γT2Mt0Ωzeωη(1+γT)DiDidxdη. (3.18)

    Now, we define

    zΦ(z,t)=2γT2M[zΦ1(z,t)]+[zΦ2(z,t)]. (3.19)

    Combining Eqs (3.14) and (3.18), we have

    zΦ(z,t)b1+b24γT2Mt0Ωzeωη[|u|+|u|]DiDidxdη+12γT2Mt0Ωzeωη(1+γT)DiDidxdη+12eωtΩzΣ2dx+t0Ωzeωη[12ωΣ2+12Σ,iΣ,i]dxdη8γk5(t)k2b1+b2t0ΩzeωηΣ,iΣ,idxdη1γ2T2Mt0ΩzeωηΣ2dxdη16˜b2a1γ(b1+b2)T2Mk4(t)ezk3. (3.20)

    Choosing ω>4γ2T2M and the boundary data (e.g., H) satisfies

    8γk5(t)k2b1+b2<14, (3.21)

    from Eq (3.20) we obtain

    zΦ(z,t)b1+b24γT2Mt0Ωzeωη[|u|+|u|]DiDidxdη+12γT2Mt0Ωzeωη(1+γT)DiDidxdη+12eωtΩzΣ2dx+t0Ωzeωη[14ωΣ2+14Σ,iΣ,i]dxdη16˜b2a1γ(b1+b2)T2Mk4(t)ezk3. (3.22)

    Integrating Eq (3.22) from z to , we obtain

    Φ(z,t)b1+b24γT2Mt0Ωzeωη(ξz)[|u|+|u|]DiDidxdη+12γT2Mt0Ωzeωη(ξz)(1+γT)DiDidxdη+12eωtΩz(ξz)Σ2dx+t0Ωzeωη(ξz)[14ωΣ2+14Σ,iΣ,i]dxdη16˜b2a1γ(b1+b2)k3T2Mk4(t)ezk3. (3.23)

    We note that

    DzD3dA=DD3dA+z0DξD3x3dAdξ=z0DξDα,αdAdξ=0.

    According to Lemma 2.1, there exists a vector function w=(w1,w2,w3) such that

    wi,i=D3, in Ω;wi=0, on Ω.

    Therefore, using Eq (3.1) we obtain

    2γT2MΦ1(z,t)=2γT2Mt0Ωzeωηπwi,idxdη=2γT2Mt0Ωzeωηπ,iwidxdη=2γT2Mt0Ωzeωη{[b1|u|uib2|u|ui]+(1+γT)Di+γΣuigiΣ}widxdη. (3.24)

    Since

    [b1|u|uib2|u|ui]wi=˜b2[|u|ui+|u|ui]wi+b1+b22[|u|+|u|]Diwi+b1+b22[|u||u|](ui+ui)wi=˜b2[|u|ui+|u|ui]wi+b1+b22[|u|+|u|]Diwi+b1+b22(ujuj)(uj+uj)|u|+|u|(ui+ui)wi˜b2[|u|ui+|u|ui]wi+b1+b22[|u|+|u|]Diwi+b1+b22[|u|+|u|]|D||w|,

    we have

    t0Ωzeωη[b1|u|uib2|u|ui]widxdη˜b2t0Ωzeωη[|u|ui+|u|ui]widxdη+b1+b22t0Ωzeωη[|u|+|u|]Diwidxdη+b1+b22t0Ωzeωη[|u|+|u|]|D||w|dxdη. (3.25)

    Using the Hölder inequality, Lemmas 2.2, 2.3, 2.1, 2.7 and 2.6, and Young's inequality, we obtain

    ˜b2t0Ωzeωη[|u|ui+|u|ui]widxdη˜b2t0eωη[Ωz[|u|3+|u|3]dx]23[Ωz(wiwi)32dx]13dη˜b2t0eωη[Ωz[|u|3+|u|3]dx]23[Ωzwiwidx]16[Ωz(wiwi)2dx]16dη˜b6k226λt0eωη[Ωz[|u|3+|u|3]dx]23[Ωzwi,αwi,αdx]16[Ωzwi,jwi,jdx]13dη˜b6k226λt0eωη[Ωz[|u|3+|u|3]dx]23[Ωzwi,jwi,jdx]12dη˜b6k2k126λt0eωη[Ω[|u|3+|u|3]dx]16[Ωz[|u|3+|u|3]dx]12[ΩzD23dx]12dη˜b232k5(t)k2k143bλt0Ωzeωη[|u|3+|u|3]dxdη+12t0ΩzeωηD23dxdη˜b232k5(t)k2k1k4(t)2a3bλezk3+12t0Ωzeωη(1+γT)D23dxdη. (3.26)

    Using the Hölder inequality, Lemmas 2.3, 2.1 and 2.7, and Young's inequality, we obtain

    b1+b22t0Ωzeωη[|u|+|u|]Diwidxdηb1+b22t0eωη[Ωz|u|DiDidx]12[Ωz|u|2dx]14[Ωz(wiwi)2dx]14dη+b1+b22t0eωη[Ωz|u|DiDidx]12[Ωz|u|2dx]14[Ωz(wiwi)2dx]14dηb1+b224k2k5(t)t0eωη[Ωz|u|DiDidx]12[Ωzwi,jwi,jdx]12dη+b1+b224k2k5(t)t0eωη[Ωz|u|DiDidx]12[Ωzwi,jwi,jdx]12dηb1+b244k1k2k5(t)t0Ωzeωη[|u|+|u|]DiDidxdη+b1+b224k1k2k5(t)t0Ωzeωη(1+γT)D23dxdη. (3.27)

    Using the Hölder inequality, Lemmas 2.4, 2.1 and 2.7, and Young's inequality, we obtain

    b1+b22t0Ωzeωη[|u|+|u|]|D||w|dxdηb1+b22t0eωη[Ωz|DiDidx]12[Ωz|u|3dx]13[Ωz(wiwi)3dx]16dη+b1+b22t0eωη[Ωz|DiDidx]12[Ωz|u|3dx]13[Ωz(wiwi)3dx]16dη(b1+b2)3k5(t)b6Λt0eωη[ΩzDiDidx]12[Ωzwi,jwi,jdx]12dηb1+b223k1k5(t)b6Λt0Ωzeωη(1+γT)DiDidxdη. (3.28)

    Inserting Eqs (3.26)–(3.28) into Eq (3.25), we have

    t0Ωzeωη[b1|u|uib2|u|ui]widxdη˜b232k5(t)k2k1k4(t)2a3bλezk3+b1+b244k2k5(t)ε3t0Ωzeωη[|u|+|u|]DiDidxdη+[b1+b223k1k5(t)b6Λ+b1+b224k2k5(t)+12]t0Ωzeωη(1+γT)DiDidxdη. (3.29)

    Using the Hölder inequality, Young's inequality and Lemmas 2.5, 2.2, 2.1, 2.7 and 2.3, we have

    t0Ωzeωη(1+γT)Diwidxdη(1+γTM)[t0Ωzeωη(1+γT)DiDidxdηt0Ωzeωηwiwidxdη]12(1+γTM)λ[t0Ωzeωη(1+γT)DiDidxdηt0Ωzeωηwi,αwi,αdxdη]12
    \begin{align} &\leq\frac{(1+\gamma T_M)\sqrt{k_1}}{\sqrt{\lambda}}\Big[\int_0^t\int_{\Omega_z}e^{-\omega\eta}(1+\gamma T)\mathcal{D}_i\mathcal{D}_idx d\eta \int_0^t\int_{\Omega_z}e^{-\omega\eta}\mathcal{D}_{3}^2dx d\eta\Big]^\frac{1}{2} \\ &\leq\frac{(1+\gamma T_M)\sqrt{k_1}}{\sqrt{\lambda}}\int_0^t\int_{\Omega_z}e^{-\omega\eta}(1+\gamma T)\mathcal{D}_i\mathcal{D}_idx d\eta, \end{align} (3.30)
    \begin{align} & \gamma\int_0^t\int_{\Omega_z}e^{-\omega\eta}w_i\Sigma u_i^*dx d\eta \\&\leq\gamma\int_0^t e^{-\omega\eta} \Big(\int_{\Omega_z}(u_3^*)^2dx\Big)^\frac{1}{2}\Big(\int_{\Omega_z}\Sigma^4dx\Big)^\frac{1}{4}\Big(\int_{\Omega_z}(w_iw_i)^2dx\Big)^\frac{1}{4} d\eta \\ &\leq\gamma\sqrt{k_5(t)k_2}\int_0^te^{-\omega\eta} \Big(\int_{\Omega_z}\Sigma_{,i}\Sigma_{,i}dx\Big)^\frac{1}{2}\Big(\int_{\Omega_z}w_{i,j}w_{i,j}dx\Big)^\frac{1}{2} d\eta \\ &\leq\gamma\sqrt{k_5(t)k_2k_1}\int_0^te^{-\omega\eta} \Big(\int_{\Omega_z}\Sigma_{,i}\Sigma_{,i}dx\Big)^\frac{1}{2}\Big(\int_{\Omega_z}\mathcal{D}_{3}^2dx\Big)^\frac{1}{2} d\eta \\ &\leq \frac{\sqrt{\gamma k_5(t)k_2k_1}}{T_M}\Big[\frac{1}{4}\int_0^t\int_{\Omega_z}e^{-\omega\eta}e^{-\omega\eta}\Sigma_{,i}\Sigma_{,i}dxd\eta \\ &+\frac{1}{\gamma}T_M^2 \int_0^t\int_{\Omega_z}e^{-\omega\eta}(1+\gamma T)\mathcal{D}_{3}^2dxd\eta\Big], \end{align} (3.31)
    \begin{align} \int_0^t\int_{\Omega_z}e^{-\omega\eta}w_i\Sigma g_idx d\eta &\leq \sqrt{\frac{k_1\gamma}{T_M\omega}}\Big[\frac{1}{\gamma}T_M^2 \int_0^t\int_{\Omega_z}e^{-\omega\eta}(1+\gamma T)\mathcal{D}_{3}^2dxd\eta \\ &+ \frac{1}{4}\omega\int_0^t\int_{\Omega_z}e^{-\omega\eta}\Sigma^2dx d\eta\Big] . \end{align} (3.32)

    Inserting Eqs (3.29)–(3.32) into Eq (3.24), we obtain

    \begin{align} \frac{2}{\gamma}T_M^2\Phi_{1}(z, t)&\leq n_1(t)\widetilde{b}^2e^{-\frac{z}{k_3}} +n_2(t)\cdot\frac{(b_1+b_2)T_M^2}{4\gamma}\int_{0}^{t}\int_{\Omega_z}e^{-\omega\eta}\Big[|\mathit{\boldsymbol{u}}| +|\mathit{\boldsymbol{u}}^*|\Big]\mathcal{D}_i\mathcal{D}_i dx d\eta \\ &+n_3(t)\cdot\frac{T_M^2}{2\gamma}\int_{0}^{t}\int_{\Omega_z}e^{-\omega\eta} (1+\gamma T)\mathcal{D}_i\mathcal{D}_i dxd\eta \\ &+n_4(t)\cdot\frac{1}{4}\int_0^t\int_{\Omega_z}e^{-\omega\eta}e^{-\omega\eta}\Sigma_{,i}\Sigma_{,i}dxd\eta +n_5(t)\cdot \frac{1}{4}\omega\int_0^t\int_{\Omega_z}e^{-\omega\eta}\Sigma^2dx d\eta. \end{align} (3.33)

    where

    \begin{align} n_1(t)& = \frac{2}{\gamma}T_M^2\frac{\sqrt[3]{2k_5(t)k_2}k_1k_4(t)}{2a\sqrt[3]{b\lambda}}, n_2(t) = 2\sqrt[4]{k_2k_5(t)}, \\ n_3(t)& = 2\Big[\frac{b_1+b_2}{2}\sqrt[3]{\frac{k_1k_5(t)}{b}}\sqrt[6]{\Lambda}+\frac{b_1+b_2}{2}\sqrt[4]{k_2k_5(t)}+\frac{1}{2}\Big] \\ &+2\frac{(1+\gamma T_M)\sqrt{k_1}}{\sqrt{\lambda}}+\frac{2\sqrt{\gamma k_5(t)k_2k_1}T_M}{\gamma}+\frac{1}{\gamma}T_M^2\sqrt{\frac{k_1\gamma}{T_M\omega}}, \\ n_4(t)& = \frac{2\sqrt{\gamma k_5(t)k_2k_1}T_M}{\gamma}, n_5(t) = \frac{1}{\gamma}T_M^2\sqrt{\frac{k_1\gamma}{T_M\omega}}. \end{align}

    Now, we begin to derive a bound of \Phi_2(z, t) which has been defined in Eq (3.15). Using the Hölder inequality, Young's inequality, Lemmas 2.7 and 2.3, we have

    \begin{align} \Phi_{21}(z, t)&\leq\frac{1}{\sqrt{\omega}}\Big[\frac{1}{4}\int_0^t\int_{\Omega_z}e^{-\omega\eta}\Sigma_{,3}^2dx d\eta +\frac{1}{4}\omega\int_0^t\int_{\Omega_z}e^{-\omega\eta}\Sigma^2dx d\eta\Big], \end{align} (3.34)
    \begin{align} \Phi_{22}(z, t)&\leq\frac{1}{2}\int_0^t e^{-\omega\eta} \Big(\int_{\Omega_z}u_3^2dx\Big)^\frac{1}{2}\Big(\int_{\Omega_z}\Sigma^4dx\Big)^\frac{1}{2} d\eta \\ &\leq\sqrt{2k_5(t)k_2}\cdot\frac{1}{4}\int_0^t \int_{\Omega_z}e^{-\omega\eta}\Sigma_{,i}\Sigma_{,i}dx d\eta, \end{align} (3.35)
    \begin{align} \Phi_{23}(z, t)&\leq\sqrt{\frac{2\gamma}{\omega}}\Big[\frac{1}{2\gamma}T_M^2\int_0^t\int_{\Omega_z} e^{-\omega\eta}(1+\gamma T)\mathcal{D}_3^2dxd\eta+ \frac{1}{4}\omega\int_0^t\int_{\Omega_z}e^{-\omega\eta}\Sigma^2dx d\eta\Big]. \end{align} (3.36)

    Inserting Eqs (3.34)–(3.36) into Eq (3.15), we obtain

    \begin{align} \Phi_2(z, t)&\leq\Big[\frac{1}{\sqrt{\omega}}+\sqrt{2k_5(t)k_2} \Big]\cdot\frac{1}{4}\int_0^t \int_{\Omega_z}e^{-\omega\eta}\Sigma_{,i}\Sigma_{,i}dx d\eta \\ &+\Big[\frac{1}{\sqrt{\omega}}+\sqrt{\frac{2\gamma}{\omega}}\Big]\cdot\frac{1}{4}\omega\int_0^t\int_{\Omega_z}e^{-\omega\eta}\Sigma^2dx d\eta \\ &+\sqrt{\frac{2\gamma}{\omega}}\cdot\frac{1}{2\gamma}T_M^2\int_0^t\int_{\Omega_z} e^{-\omega\eta}(1+\gamma T)\mathcal{D}_3^2dxd\eta. \end{align} (3.37)

    Combining Eqs (3.19), (3.22), (3.33) and (3.37), we obtain

    \begin{align} \Phi(z, t)&\leq n_6(t)\Big[-\frac{\partial}{\partial z}\Phi(z, t)\Big]+n_7(t)\widetilde{b}^2e^{-\frac{z}{k_3}}, \end{align} (3.38)

    where

    \begin{align} n_6(t)& = \max\Big\{n_2(t), n_3(t)+\sqrt{\frac{2\gamma}{\omega}}, n_5(t)+\frac{1}{\sqrt{\omega}}+\sqrt{\frac{2\gamma}{\omega}}, n_4(t)+\frac{1}{\sqrt{\omega}}+\sqrt{2k_5(t)k_2}\Big\}, \\ n_7(t)& = \frac{16}{a_1\gamma(b_1+b_2)}T_M^2k_4(t)n_6(t)+n_1(t). \end{align} (3.39)

    In this section, we will analysis Lemma 3.1 to derive the following theorem.

    Theorem 4.1. Let (u_i, T, p) and (u_i^*, T^*, p^*) be solutions of the Eqs (1.1)–(1.8) in \Omega , corresponding to b_1 and b_2 , respectively. If \int_Df_3dA = 0 , Equation (3.21) holds and f_{\alpha, \alpha}-\gamma_1f_3 = 0, H\in L^\infty(\Omega\times\{t > 0\}) , then

    (u_i, T)\rightarrow (u_i^*, T^*),\ as\ b_1\rightarrow b_2.

    Specifically, either the inequality

    \begin{align} \frac{b_1+b_2}{4\gamma}T_M^2&\int_{0}^{t}\int_{\Omega_z}e^{-\omega\eta}(\xi-z)\Big[|\mathit{\boldsymbol{u}}|+|\mathit{\boldsymbol{u}}^*|\Big]\mathcal{D}_i\mathcal{D}_idx d\eta \\ &+ \frac{1}{2\gamma}T_M^2\int_{0}^{t}\int_{\Omega_z}e^{-\omega\eta}(\xi-z)(1+\gamma T)\mathcal{D}_i\mathcal{D}_idx d\eta \\ &+\frac{1}{2}e^{-\omega t}\int_{\Omega_z}(\xi-z)\Sigma^2dx+\int_{0}^{t}\int_{\Omega_z}e^{-\omega\eta}(\xi-z)\Big[ \frac{1}{4}\omega\Sigma^2+\frac{1}{4}\Sigma_{,i}\Sigma_{,i}\Big]dxd\eta \\ &\leq \frac{16\widetilde{b}^2}{a_1\gamma(b_1+b_2)}k_3T_M^2k_4(t)e^{-\frac{z}{k_3}}+ \widetilde{b}^2n_7(t)e^{-\frac{1}{n_6^*}z} +\widetilde{b}^2\frac{n_7(t)}{n_6^*}ze^{-\frac{1}{n_6^*}z} \end{align}

    holds, or the inequality

    \begin{align} \frac{b_1+b_2}{4\gamma}T_M^2&\int_{0}^{t}\int_{\Omega_z}e^{-\omega\eta}(\xi-z)\Big[|\mathit{\boldsymbol{u}}|+|\mathit{\boldsymbol{u}}^*|\Big]\mathcal{D}_i\mathcal{D}_idx d\eta \\ &+ \frac{1}{2\gamma}T_M^2\int_{0}^{t}\int_{\Omega_z}e^{-\omega\eta}(\xi-z)(1+\gamma T)\mathcal{D}_i\mathcal{D}_idx d\eta \\ &+\frac{1}{2}e^{-\omega t}\int_{\Omega_z}(\xi-z)\Sigma^2dx+\int_{0}^{t}\int_{\Omega_z}e^{-\omega\eta}(\xi-z)\Big[ \frac{1}{4}\omega\Sigma^2+\frac{1}{4}\Sigma_{,i}\Sigma_{,i}\Big]dxd\eta \\ &\leq \frac{16\widetilde{b}^2}{a_1\gamma(b_1+b_2)}k_3T_M^2k_4(t)e^{-\frac{z}{k_3}}+ \\ &\widetilde{b}^2n_7(t)e^{-\frac{1}{n_6^*}z} +\widetilde{b}^2\frac{n_7(t)}{n_6^*(\frac{1}{n_6^*}-\frac{1}{k_3})}b_3(t)[e^{-\frac{1}{k_3}z}-e^{-\frac{1}{n_6^*}z}] \end{align}

    holds.

    Proof. Using Lemma 3.1, we have

    \begin{align} \frac{\partial }{\partial z}\Big\{\Phi(z, t)e^{\frac{1}{n_6^*}z}\Big\}\leq \widetilde{b}^2\frac{n_7(t)}{n_6^*}e^{(\frac{1}{n_6^*}-\frac{1}{k_3})z},\ z\geq 0. \end{align} (4.1)

    Now, we consider (4.1) for two cases.

    Ⅰ. If n_6^* = k_3 , we integrate Eq (4.1) from 0 to z to obtain

    \begin{align} \Phi(z, t)&\leq \Phi(0, t)e^{-\frac{1}{n_6^*}z} +\widetilde{b}^2\frac{n_7(t)}{n_6^*}ze^{-\frac{1}{n_6^*}z}. \end{align} (4.2)

    Ⅱ. If n_6^*\neq k_3 , we integrate Eq (4.1) from 0 to z to obtain

    \begin{align} \Phi(z, t)&\leq \Phi(0, t)e^{-\frac{1}{n_6^*}z} +\widetilde{b}^2\frac{n_7(t)}{n_6^*(\frac{1}{n_6^*}-\frac{1}{k_3})}b_3(t)[e^{-\frac{1}{k_3}z}-e^{-\frac{1}{n_6^*}z}]. \end{align} (4.3)

    From Eqs (4.2) and (4.3), to obtain the main result, we can conclude that we have to derive a bound for \Phi(0, t) . We choose z = 0 in Lemma 3.1 to obtain

    \begin{align} \Phi(0, t)\leq n_6^*\Big[-\frac{\partial \Phi}{\partial z}(0, t)\Big]+\widetilde{b}^2n_7(t). \end{align} (4.4)

    Clearly, if we want to derive a bound for \Phi(0, t) , we only need derive a bound for -\frac{\partial \Phi}{\partial z}(0, t) . To do this, choosing z = 0 in Eq (3.19) and combining Eqs (3.8) and (3.15), we have

    \begin{align} -\frac{\partial \Phi}{\partial z}(0, t)& = \frac{2}{\gamma}T_M^2\int_{0}^{t}\int_{D}e^{-\omega\eta}\pi\mathcal{D}_3dAd\eta -\int_{0}^{t}\int_{D}e^{-\omega\eta}\Sigma\Sigma_{,3}dAd\eta \\ &+\frac{1}{2}\int_{0}^{t}\int_{D}e^{-\omega\eta}u_3\Sigma^2dAd\eta +\int_0^t\int_{D}e^{-\omega\eta}\mathcal{D}_{3}T^*\Sigma dAd\eta. \end{align} (4.5)

    In light of the boundary conditions (3.4)–(3.6), from Eq (4.5) we can know that

    \begin{align} -\frac{\partial \Phi}{\partial z}(0, t) = 0. \end{align} (4.6)

    Inserting Eq (4.6) into Eq (4.4), we obtain

    \begin{align} \Phi(0, t)\leq \widetilde{b}^2n_7(t). \end{align} (4.7)

    Therefore, from Eqs (4.2), (4.3) and (4.7) we have

    \begin{align} \Phi(z, t)&\leq \widetilde{b}^2n_7(t)e^{-\frac{1}{n_6^*}z} +\widetilde{b}^2\frac{n_7(t)}{n_6^*}ze^{-\frac{1}{n_6^*}z},\ \text{if}\ n_6^* = k_3, \end{align} (4.8)
    \begin{align} \Phi(z, t)&\leq \widetilde{b}^2n_7(t)e^{-\frac{1}{n_6^*}z} +\widetilde{b}^2\frac{n_7(t)}{n_6^*(\frac{1}{n_6^*}-\frac{1}{k_3})}b_3(t)[e^{-\frac{1}{k_3}z}-e^{-\frac{1}{n_6^*}z}],\ \text{if }\ n_6^*\neq k_3. \end{align} (4.9)

    Combining Eqs (3.24), (4.8) and (4.9) we can complete the proof of Theorem 4.1.

    Remark 4.1 Theorem 1 shows that the small perturbation of Forchheimer coefficient will not cause great changes to the solution of Eqs (1.1)–(1.8). Meanwhile, Theorem 1 also shows that the solutions of Eqs (2.12)–(2.21) decay exponentially as the space variable z\rightarrow \infty .

    This section shows how to use the prior estimates in Section 2 and the method in Section 3 to derive the continuous dependence of the solution on \gamma . Assume that (u_i^*, T^*, p^*) is a solution of Eqs (1.1)–(1.8) with \gamma = \gamma^* .

    If we also let

    \mathcal{D}_i = u_i-u_i^*,\ \Sigma = T-T^*,\ \pi = p-p^*,\ \widetilde{\gamma} = \gamma-\gamma^*,

    then (\mathcal{D}_i, \Sigma, \pi) satisfies

    \begin{align} b[|\mathit{\boldsymbol{u}}|u_i-|\mathit{\boldsymbol{u}}^*|u_i^*]+\widetilde{\gamma}Tu_i+\gamma_2\Sigma u_i+(1+\gamma T^*)\mathcal{D}_i+\gamma\Sigma u_i^* = -\pi_{,i}+g_i\Sigma,\ &in\ \Omega\times\{t > 0\}, \end{align} (5.1)
    \begin{align} \mathcal{D}_{i,i} = 0,\ &in\ \Omega\times\{t > 0\}, \end{align} (5.2)
    \begin{align} \partial_{t}\Sigma+u_i\Sigma_{,i}+\mathcal{D}_i T^*_{,i} = \Delta\Sigma, \ &in\ \Omega\times\{t > 0\}, \end{align} (5.3)
    \begin{align} \mathcal{D}_i = 0, \Sigma = 0,\ on\ \partial D\times\{x_3 > 0\}&\times\{t > 0\}, \end{align} (5.4)
    \begin{align} \mathcal{D}_i = 0, \Sigma = 0,\ & on\ D\times\{t > 0\}, \end{align} (5.5)
    \begin{align} \Sigma(x_1, x_2, x_3, 0) = 0, \ &in \ \Omega \end{align} (5.6)
    \begin{align} |\mathit{\boldsymbol{u}}|, |\Sigma| = O(1), |\mathcal{D}_i|, |\nabla\Sigma|, |\pi| = o(x_3^{-1}),&\ as\ x_3\rightarrow \infty. \end{align} (5.7)

    We also define \Phi_1(z, t) as that in Eq (3.8). Similar to Eq (3.10), we have

    \begin{align} \Phi_1(z, t) & = \frac{b}{2}\int_{0}^{t}\int_{\Omega_z}e^{-\omega\eta}(\xi-z)\Big[|\mathit{\boldsymbol{u}}|+|\mathit{\boldsymbol{u}}^*|\Big]\mathcal{D}_i\mathcal{D}_idx d\eta \\ &+\int_{0}^{t}\int_{\Omega_z}e^{-\omega\eta}(\xi-z)(1+\gamma_2 T^*)\mathcal{D}_i\mathcal{D}_idx d\eta \\ &+\frac{b}{2}\int_{0}^{t}\int_{\Omega_z}e^{-\omega\eta}(\xi-z)\Big[|\mathit{\boldsymbol{u}}|-|\mathit{\boldsymbol{u}}^*|\Big]^2\Big[|\mathit{\boldsymbol{u}}|+|\mathit{\boldsymbol{u}}^*|\Big]dx d\eta \\ &+\gamma_2\int_{0}^{t}\int_{\Omega_z}e^{-\omega\eta}(\xi-z)\mathcal{D}_i\Sigma u_i^*dx d\eta -\int_{0}^{t}\int_{\Omega_z}e^{-\omega\eta}(\xi-z)\mathcal{D}_ig_i\Sigma dx d\eta \\ &+\widetilde{\gamma}\int_{0}^{t}\int_{\Omega_z}e^{-\omega\eta}(\xi-z)Tu_i\mathcal{D}_i dx d\eta. \end{align} (5.8)

    Using the Hölder inequality, Young's inequality and Lemmas 2.5 and 2.6, we obtain

    \begin{align} & \widetilde{\gamma}\int_{0}^{t}\int_{\Omega_z}e^{-\omega\eta}Tu_i\mathcal{D}_i dx d\eta\\ & \geq-\frac{1}{2}T_M^2\widetilde{\gamma}^2\int_0^t\int_{\Omega_z}e^{-\omega\eta}u_{i}u_{i}dxd\eta -\frac{1}{2}\int_0^t\int_{\Omega_z}e^{-\omega\eta}(1+\gamma_2 T^*)\mathcal{D}_i\mathcal{D}_idxd\eta \\ &\geq-\frac{k_4(t)}{2a_2}T_M^2\widetilde{\gamma}^2e^{-\frac{z}{k_3}} -\frac{1}{2}\int_0^t\int_{\Omega_z}e^{-\omega\eta}(1+\gamma_2 T^*)\mathcal{D}_i\mathcal{D}_idxd\eta, \end{align} (5.9)

    Combining Eqs (3.12), (3.13), (5.8) and (5.9), we obtain

    \begin{align}& -\frac{\partial}{\partial z}\Phi_1(z, t)\\ &\geq\frac{b}{2}\int_{0}^{t}\int_{\Omega_z}e^{-\omega\eta}\Big[|\mathit{\boldsymbol{u}}|+|\mathit{\boldsymbol{u}}^*|\Big]\mathcal{D}_i\mathcal{D}_idx d\eta +\frac{1}{2}\int_{0}^{t}\int_{\Omega_z}e^{-\omega\eta}(1+\gamma_2 T^*)\mathcal{D}_i\mathcal{D}_idx d\eta \\ &-\frac{4\gamma_2^2\sqrt{k_5(t)k_2}}{b}\int_{0}^{t}\int_{\Omega_z}e^{-\omega\eta}\Sigma_{,i}\Sigma_{,i}dx d\eta -\frac{1}{2\gamma}\int_{0}^{t}\int_{\Omega_z}e^{-\omega\eta}\Sigma^2dx d\eta \\ &-\frac{k_4(t)}{2a_2}T_M^2\widetilde{\gamma}^2e^{-\frac{z}{k_3}}. \end{align} (5.10)

    Inserting Eqs (3.18) and (5.10) into Eq (3.19), choosing \omega > \frac{4T_M^2}{\gamma^2} and the boundary data satisfies

    \begin{align} \frac{8(\gamma^*)^2\sqrt{k_5(t)k_2}}{b}\leq\frac{1}{4}, \end{align} (5.11)

    we have

    \begin{align} & -\frac{\partial}{\partial z}\Phi(z, t)\\ &\geq\frac{b}{\gamma^*}T_M^2\int_{0}^{t}\int_{\Omega_z}e^{-\omega\eta}\Big[|\mathit{\boldsymbol{u}}|+|\mathit{\boldsymbol{u}}^*|\Big]\mathcal{D}_i\mathcal{D}_idx d\eta +\frac{1}{(\gamma^*)^2}T_M^2\int_{0}^{t}\int_{\Omega_z}e^{-\omega\eta}(1+\gamma_2 T^*)\mathcal{D}_i\mathcal{D}_idx d\eta \\ &+\frac{1}{2}e^{-\omega t}\int_{\Omega_z}\Sigma^2dx+\int_{0}^{t}\int_{\Omega_z}e^{-\omega\eta}\Big[ \frac{1}{4}\omega\Sigma^2+\frac{1}{4}\Sigma_{,i}\Sigma_{,i}\Big]dxd\eta \\ &-\frac{k_4(t)}{2a_2\gamma^*}T_M^4\widetilde{\gamma}^2e^{-\frac{z}{k_3}}. \end{align} (5.12)

    Integrating Eq (5.12) from z to \infty , we obtain

    \begin{align} \Phi(z, t)&\geq\frac{b}{\gamma^*}T_M^2\\ &\int_{0}^{t}\int_{\Omega_z}e^{-\omega\eta}(\xi-z)\Big[|\mathit{\boldsymbol{u}}|+|\mathit{\boldsymbol{u}}^*|\Big]\mathcal{D}_i\mathcal{D}_idx d\eta +\frac{1}{(\gamma^*)^2}T_M^2\int_{0}^{t}\int_{\Omega_z}e^{-\omega\eta}(1+\gamma_2 T^*)\mathcal{D}_i\mathcal{D}_idx d\eta \\ &+\frac{1}{2}e^{-\omega t}\int_{\Omega_z}(\xi-z)\Sigma^2dx+\int_{0}^{t}\int_{\Omega_z}e^{-\omega\eta}(\xi-z)\Big[ \frac{1}{4}\omega\Sigma^2+\frac{1}{4}\Sigma_{,i}\Sigma_{,i}\Big]dxd\eta \\ &-\frac{k_4(t)}{2a_2\gamma^*}k_3T_M^4\widetilde{\gamma}^2e^{-\frac{z}{k_3}}. \end{align} (5.13)

    Similar to the calculation in Eqs (3.33) and (3.37), we can get

    \begin{align} \Phi(z, t)&\leq n_6'(t)\Big[-\frac{\partial}{\partial z}\Phi(z, t)\Big]+n_7'(t)\widetilde{b}^2e^{-\frac{z}{k_3}}, \end{align} (5.14)

    for n_6'(t), n_7'(t) > 0 .

    After similar analysis as in the previous section, we can get the following theorem from Eq (5.14).

    Theorem 5.1. Let (u_i, T, p) and (u_i^*, T^*, p^*) be solutions of the Eqs (1.1)–(1.8) in \Omega , corresponding to b_1 and b_2 , respectively. If \int_Df_3dA = 0 , Equation (5.11) holds and f_{\alpha, \alpha}-\gamma_1f_3 = 0, H\in L^\infty(\Omega\times\{t > 0\}) , then

    (u_i, T)\rightarrow (u_i^*, T^*),\ as\ b_1\rightarrow b_2.

    Specifically, either the inequality

    \begin{align} \frac{b}{\gamma^*}T_M^2&\int_{0}^{t}\int_{\Omega_z}e^{-\omega\eta}(\xi-z)\Big[|\mathit{\boldsymbol{u}}|+|\mathit{\boldsymbol{u}}^*|\Big]\mathcal{D}_i\mathcal{D}_idx d\eta \\ &+ \frac{1}{(\gamma^*)^2}T_M^2\int_{0}^{t}\int_{\Omega_z}e^{-\omega\eta}(\xi-z)(1+\gamma^* T^*)\mathcal{D}_i\mathcal{D}_idx d\eta \\ &+\frac{1}{2}e^{-\omega t}\int_{\Omega_z}(\xi-z)\Sigma^2dx+\int_{0}^{t}\int_{\Omega_z}e^{-\omega\eta}(\xi-z)\Big[ \frac{1}{4}\omega\Sigma^2+\frac{1}{4}\Sigma_{,i}\Sigma_{,i}\Big]dxd\eta \\ &\leq \frac{k_4(t)}{2a_2\gamma^*}k_3T_M^4\widetilde{\gamma}^2e^{-\frac{z}{k_3}}+ \widetilde{\gamma}^2n_7'(t)e^{-\frac{1}{n_6^*}z} +\widetilde{\gamma}^2\frac{n_7'(t)}{n_6^*}ze^{-\frac{1}{n_6^*}z} \end{align}

    holds, or the inequality

    \begin{align} \frac{b}{\gamma^*}T_M^2&\int_{0}^{t}\int_{\Omega_z}e^{-\omega\eta}(\xi-z)\Big[|\mathit{\boldsymbol{u}}|+|\mathit{\boldsymbol{u}}^*|\Big]\mathcal{D}_i\mathcal{D}_idx d\eta \\ &+ \frac{1}{(\gamma^*)^2}T_M^2\int_{0}^{t}\int_{\Omega_z}e^{-\omega\eta}(\xi-z)(1+\gamma^* T^*)\mathcal{D}_i\mathcal{D}_idx d\eta \\ &+\frac{1}{2}e^{-\omega t}\int_{\Omega_z}(\xi-z)\Sigma^2dx+\int_{0}^{t}\int_{\Omega_z}e^{-\omega\eta}(\xi-z)\Big[ \frac{1}{4}\omega\Sigma^2+\frac{1}{4}\Sigma_{,i}\Sigma_{,i}\Big]dxd\eta \\ &\leq \frac{k_4(t)}{2a_2\gamma^*}k_3T_M^4\widetilde{\gamma}^2e^{-\frac{z}{k_3}}+ \widetilde{\gamma}^2n_7'(t)e^{-\frac{1}{n_6^*}z} +\widetilde{\gamma}^2\frac{n_7'(t)}{n_6^*(\frac{1}{n_6^*}-\frac{1}{k_3})}b_3(t)[e^{-\frac{1}{k_3}z}-e^{-\frac{1}{n_6^*}z}] \end{align}

    holds.

    In this paper, using a priori estimates of the solutions, we show how to control the nonlinear term, and obtain the structural stability of the solution of the Forchheimer equation in a semi-infinite cylinder. Meanwhile, the spatial decay results of the solution are also obtained. The methods in this paper can bring some inspiration for the structural stability of other nonlinear partial differential equations.

    The authors express their heartfelt thanks to the editors and referees who have provided some important suggestions. This work is supported by the Tutor System Rroject of Guangzhou Huashang College (2021HSDS13) and the Key projects of universities in Guangdong Province (NATURAL SCIENCE) (2019KZDXM042).

    The authors declare there is no conflict of interest. Conceptualization, and validation, Z. Li.; formal analysis, Z W. Zhang; investigation, Y. Li. All authors have read and agreed to the published version of the manuscript.



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