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Research article

Does land transfer have an impact on land use efficiency? A case study on rural China

  • Received: 07 May 2022 Revised: 01 June 2022 Accepted: 02 June 2022 Published: 13 June 2022
  • JEL Codes: G00, O13

  • Land use efficiency is one of the core elements for the high-quality development of food production and agricultural industry, and land transfer can optimize the allocation of land resources, adjust the structure of the agricultural industry, and drive the improvement of agricultural labor productivity, thereby promoting land use efficiency and realizing agricultural modernization. Taking 30 provinces (autonomous regions and municipalities) in Chinese mainland from 2005 to 2019 as the research objects, this paper adopts panel OLS and panel Tobit estimation to study the overall impact of land transfer on land use efficiency. In addition, this paper explores the heterogeneous impact of land transfer on land use efficiency through cluster analysis and panel regression. Finally, this paper further analyzes the influence mechanism of land transfer on land use efficiency through industrial structure and labor productivity. The following conclusions are drawn. Firstly, land transfer significantly promotes the improvement of provincial rural land use efficiency. Secondly, the land use efficiency of various provinces (autonomous regions and municipalities) in Chinese mainland has the characteristics of periodic changes. From the perspective of time and space, there are large differences in the land use efficiency of various provinces, autonomous regions, and municipalities, and there are regional heterogeneity effects of land transfer on provincial rural land use efficiency. Thirdly, the industrial structure can enhance the promotion effect of land transfer on land use efficiency, and land transfer can promote the improvement of land use efficiency by improving agricultural labor productivity.

    Citation: Jiehua Ma, Shuanglian Chen. Does land transfer have an impact on land use efficiency? A case study on rural China[J]. National Accounting Review, 2022, 4(2): 112-134. doi: 10.3934/NAR.2022007

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  • Land use efficiency is one of the core elements for the high-quality development of food production and agricultural industry, and land transfer can optimize the allocation of land resources, adjust the structure of the agricultural industry, and drive the improvement of agricultural labor productivity, thereby promoting land use efficiency and realizing agricultural modernization. Taking 30 provinces (autonomous regions and municipalities) in Chinese mainland from 2005 to 2019 as the research objects, this paper adopts panel OLS and panel Tobit estimation to study the overall impact of land transfer on land use efficiency. In addition, this paper explores the heterogeneous impact of land transfer on land use efficiency through cluster analysis and panel regression. Finally, this paper further analyzes the influence mechanism of land transfer on land use efficiency through industrial structure and labor productivity. The following conclusions are drawn. Firstly, land transfer significantly promotes the improvement of provincial rural land use efficiency. Secondly, the land use efficiency of various provinces (autonomous regions and municipalities) in Chinese mainland has the characteristics of periodic changes. From the perspective of time and space, there are large differences in the land use efficiency of various provinces, autonomous regions, and municipalities, and there are regional heterogeneity effects of land transfer on provincial rural land use efficiency. Thirdly, the industrial structure can enhance the promotion effect of land transfer on land use efficiency, and land transfer can promote the improvement of land use efficiency by improving agricultural labor productivity.



    This paper is concerned with the initial boundary value problem

    {utt(x,t)uxx(x,t)+μ1(t)ut(x,t)+μ2(t)ut(x,tτ(t))=0in Ω×]0,+[,u(0,t)=u(L,t)=0on ]0,+[,u(x,0)=u0(x),ut(x,0)=u1(x)on Ω,ut(x,tτ(0))=f0(x,tτ(0))in Ω×]0,τ(0)[, (1)

    where Ω=]0,L[, 0<τ(t) are a non-constant time delay, μ1(t),μ2(t) are non-constant weights and the initial data (u0,u1,f0) belong to a suitable function space.

    This problem has been first proposed and studied in Nicaise and Pignotti [22] in case of constant coefficients μ1,μ2 and constant time delay. Under suitable assumptions, the authors proved the exponential stability of the solution by introducing suitable energies and by using some observability inequalities. Some instability results are also given for the case of the some assumptions is not satisfied.

    With a weight depending on time, μ1(t),μ2(t) and constant time delay, this problem was studied in [2], where the existence of solution was made by Faedo-Galerkin method and a decay rate estimate for the energy was given by using the multiplier method.

    W. Liu in [19] studied the weak viscoelastic equation with an internal time varying delay term. By introducing suitable energy and Lyapunov functionals, he establishes a general decay rate estimate for the energy under suitable assumptions.

    F. Tahamtani and A. Peyravi [29] investigated the nonlinear viscoelastic wave equation with source term. Using the Potential well theory they showed that the solutions blow up in finite time under some restrictions on initial data and for arbitrary initial energy.

    Global existence and asymptotic behavior of solutions to the viscoelastic wave equation with a constant delay term was considered by M. Remil and A. Hakem in [28].

    Global existence and asymptotic stability for a coupled viscoelastic wave equation with time-varying delay was studied in [3] by combining the energy method with the Faedo-Galerkin's procedure.

    The stabilization problem by interior damping of the wave equation with boundary or internal time-varying delay was studied in [23] by introducing suitable Lyapunov functionals.

    Energy decay of solutions for the wave equation with a time varying delay term in the weakly nonlinear internal feedbacks was considered in [11].

    For problems with delay in different contexts we cite [9,10,30,32] with references therein. In absence of delay (μ2(t)=0), the problem (1) is exponentially stable provided that μ1(t) is constant, see, for instance [5,6,16,17,21] and reference therein.

    Time delay is the property of a physical system by which the response to an applied force is delayed in its effect, and the central question is that delays source can destabilize a system that is asymptotically stable in the absence of delays, see [7]. In fact, an arbitrarily small delay may destabilize a system that is uniformly asymptotically stable in the absence of delay unless additional control terms have been used, see for example [8,12,31]

    By energy method in [24] was studied the stabilization of the wave equation with boundary or internal distributed delay. By semigroup approach in [27] was proved the well-posedness and exponential stability for a wave equation with frictional damping and nonlocal time-delayed condition. Transmission problem with distributed delay was studied in [18] where was established the exponential stability of the solution by introducing a suitable Lyapunov functional.

    Here we consider a wave equation with non-constant delay and nonlinear weights, thus, the present paper is a generalization of the previous ones. The remaining part of this paper is organized as follows. In the section 2 we introduce some notations and prove the dissipative property of the full energy of the system. In the section 3, for an approach combining semigroup theory (see [21] and [4]) with the energy estimate method we prove the existence and uniqueness of solution. In section 4 we present the result of exponential stability.

    We will need the following hypotheses:

    (H1) μ1:R+]0,+[ is a non-increasing function of class C1(R+) satisfying

    |μ1(t)μ1(t)|M1,0<α0μ1(t),t0, (2)

    where α0 and M1 are constants such that M1>0.

    (H2) μ2:R+R is a function of class C1(R+), which is not necessarily positives or monotones, such that

    |μ2(t)|βμ1(t), (3)
    |μ2(t)|M2μ1(t), (4)

    for some 0<β<1d and M2>0.

    We now state a lemma needed later.

    Lemma 2.1 (Sobolev-Poincare's inequality). Let q be a number with 2q+. Then there is a constant c=c(]0,L[,q) such that

    ΨqcΨx2,forΨH10(]0,L[).

    Lemma 2.2 ([13][16]). Let E:R+R+ be a non increasing function and assume that there are two constants σ>1 and ω>0 such that

    +SE1+σ(t)dt1ωEσ(0)E(S), 0S<+.

    Then

    E(t)=0 tEσ(0)ω|σ|, if1<σ<0,E(t)E(0)(1+σ1+ωσt)1σ t0, ifσ>0,E(t)E(0)e1ωt t0, ifσ=0.

    As in [23], we assume that

    τ(t)W2,+([0,T]),  for T>0 (5)

    and there exist positive constants τ0,τ1 and d satisfying

    0<τ0τ(t)τ1, t>0 (6)

    and

    τ(t)d<1, t>0. (7)

    We introduce the new variable

    z(x,ρ,t)=ut(x,tτ(t)ρ), xΩ,ρ]0,1[,t>0. (8)

    Then

    τ(t)zt(x,ρ,t)+(1τ(t)ρ)zρ(x,ρ,t)=0, xΩ, ρ]0,1[, t>0

    and problem (1) takes the form

    {utt(x,t)uxx(x,t)+μ1(t)ut(x,t)+μ2(t)z(x,1,t)=0inΩ×]0,+[,τ(t)zt(x,ρ,t)+(1τ(t)ρ)zρ(x,ρ,t)=0inΩ×]0,1[×]0,+[,u(0,t)=u(L,t)=0on]0,+[,u(x,0)=u0(x),ut(x,0)=u1(x)onΩ,z(x,ρ,0)=ut(x,τ(0)ρ)=f0(x,τ(0)ρ)inΩ×]0,1[. (9)

    We define the energy of the solution of problem (9) by

    E(t)=12ut2L2(Ω)+12ux2L2(Ω)+ξ(t)τ(t)2Ω10z2(x,ρ,t)dρdx, (10)

    where

    ξ(t)=ˉξμ1(t) (11)

    is a non-increasing function of class C1(R+) and ˉξ be a positive constant such that

    β1d<ˉξ<2β1d. (12)

    Our first result states that the energy is a non-increasing function.

    Lemma 2.3. Let (u,z) be a solution to the problem (9). Then, the energy functional defined by (10) satisfies

    E(t)μ1(t)(1ˉξ2β21d)ut2L2(Ω)μ1(t)(ˉξ(1τ(t))2β1d2)z(x,1,t)2L2(Ω)0. (13)

    Proof. Multiplying the first equation (9) by ut(x,t), integrating on Ω and using integration by parts, we get

    12ddt(ut2L2(Ω)+ux2L2(Ω))+μ1(t)ut2L2(Ω)+μ2(t)Ωz(x,1,t)utdx. (14)

    Now multiplying the second equation (9) by ξ(t)z(x,ρ,t) and integrate on Ω×]0,1[, to obtain

    τ(t)ξ(t)Ω10zt(x,ρ,t)z(x,ρ,t)dρdx=ξ(t)2Ω10(1τ(t)ρ)ρ(z(x,ρ,t))2dρdx.

    Consequently,

    ddt(ξ(t)τ(t)2Ω10z2(x,ρ,t)dρdx)=ξ(t)2Ω10(1τ(t)ρ)ρ(z(x,ρ,t))2dρdx+ξ(t)τ(t)2Ω10z2(x,ρ,t)dρdx=ξ(t)2Ω(z2(x,0,t)z2(x,1,t))dx+ξ(t)τ(t)2Ω10z2(x,1,t)dρdx+ξ(t)τ(t)2Ω10z2(x,ρ,t)dρdx. (15)

    From (10), (14) and (15) we obtain

    E(t)=ξ(t)2ut2L2(Ω)ξ(t)2z(x,1,t)2L2(Ω)+ξ(t)τ(t)2z(x,1,t)2L2(Ω)+ξ(t)τ(t)2Ω10z2(x,ρ,t)dρdxμ1(t)ut2L2(Ω)μ2(t)Ωz(x,1,t)utdx. (16)

    Due to Young's inequality, we have

    μ2(t)Ωz(x,1,t)utdx|μ2(t)|21dut2L2(Ω)+|μ2(t)|1d2z(x,1,t)2L2(Ω). (17)

    Inserting (17) into (16), we obtain

    E(t)(μ1(t)ξ(t)2|μ2(t)|21d)ut2L2(Ω)(ξ(t)2ξ(t)τ(t)2|μ2(t)|1d2)z(x,1,t)2L2(Ω)+ξ(t)τ(t)2Ω10z2(x,ρ,t)dρdxμ1(t)(1ˉξ2β21d)ut2L2(Ω)μ1(t)(ˉξ(1τ(t))2β1d2)z(x,1,t)2L2(Ω)0.

    Lemma 2.4. Let (u,z) be a solution to the problem (9). Then the energy functional defined by (10) satisfies

    ut(x,t)2L2(Ω)<1σE(t),

    where σ=a0(1ˉξ2β21d).

    Proof. From Lemma 2.3, we have that

    E(t)μ1(t)(1ˉξ2+β21d)ut2L2(Ω)+μ1(t)(ˉξ(1τ(t))2+β1d2)z(x,1,t)2L2(Ω)0

    and from (H1), we obtain

    0a0(1ˉξ2+β21d)ut2L2(Ω)μ1(t)(1ˉξ2+β21d)ut2L2(Ω)E(t)

    and the lemma is proved.

    For the semigroup setup we U=(u,ut,z)T and rewrite (9) as

    {Ut=A(t)U,U(0)=U0=(u0,u1,f0(,,τ(0)))T, (18)

    where the operator A(t) is defined by

    AU=(v,uxxμ1(t)vμ2(t)z(x,1,t),1τ(t)ρτ(t)zρ(x,ρ,t))T. (19)

    We introduce the phase space

    H=H10(Ω)×L2(Ω)×L2(Ω×]0,1[)

    and the domain of A is defined by

    D(A(t))={(u,v,z)TH/v=z(,0)  in Ω}, (20)

    where

    H=H2(Ω)H10(Ω)×H10(Ω)×L2(Ω;H10(]0,1[)).

    Notice that the domain of the operator A(t) is independent of the time t, i.e.,

    D(A(t))=D(A(0)),t>0. (21)

    H is a Hilbert space provided with the inner product

    U,ˉUH=Ωuxˉuxdx+Ωvˉvdx+ξ(t)τ(t)Ω10zˉzdρdx, (22)

    for U=(u,v,z)T and ˉU=(ˉu,ˉv,ˉz)T.

    Using this time-dependent inner product and the next theorem we will get a result of existence and uniqueness.

    Theorem 3.1. Assume that

    (i) Y=D(A(0)) is dense subset of H,

    (ii) (21) holds,

    (iii) for all t[0,T], A(t) generates a strongly continuous semigroup on H and the family A(t)={A(t)/t[0,T]} is stable with stability constants C and m independent of t (i.e., the semigroup (St(s))s0 generated by A(t) satisfies St(s)uHCemsuH, for all uH and s0),

    (iv) tA(t) belongs to L([0,T],B(Y,H)), which is the space of equivalent classes of essentially bounded, strongly measurable functions from [0,T] into the set B(Y,H) of bounded operators from Y into H.

    Then, problem (18) has a solution UC([0,T],Y)C1([0,T],H) for any initial datum in Y.

    Our goal is then to check the above assumptions for problem (18).

    First, we prove D(A(0)) is dense in H.

    The proof is the same as the one Lemma 2.2 of [25], we give it for the sake of completeness.

    Let (f,g,h)T be orthogonal to all elements of DA(0), namely

    0=(u,v,z)T,(f,g,h)TH=Ωuxfxdx+Ωvgdx+ξ(t)τ(t)Ω10zhdρdx,

    for all (u,v,z)TD(A(0)).

    We first take u=v=0 and zD(Ω×]0,1[). As (0,0,z)TD(A(0)), we get

    Ω10zhdρdx=0.

    Since D(Ω×]0,1[) is dense in L2(Ω×]0,1[), we deduce that h=0. In the same manner, by taking u=z=0 e vD(Ω) we see that g=0.

    The above orthogonality condition is then reduced to

    0=Ωuxfxdx,(u,v,z)TD(A(0)).

    By restricting ourselves to v=0 and z=0, we obtain

    0=Ωuxfxdx,(u,0,0)TD(A(0)).

    Since D(Ω) is dense in H10(Ω) (equipped with the inner product ,H10(Ω), we deduce that f=0.

    We consequently

    D(A(0) is dense in H. (23)

    Secondly, we notice that

    ΦtΦsec2τ0|ts|,t,s[0,T], (24)

    where Φ=(u,v,z)T and c is a positive constant and is the norm associated the inner product (22). For all t,s[0,T], we have

    Φ2tΦ2secτ0|ts|=(1ec2τ0|ts|)(ux2L2(Ω)+v2L2(Ω))+(ξ(t)τ(t)ξ(s)τ(s)ecτ0|ts|)Ω10z2(x,ρ,t)dρdx.

    It is clear that 1ecτ0|ts|0. Now we will prove ξ(t)τ(t)ξ(s)τ(s)ecτ0|ts|0 for some c>0. To do this, we have

    τ(t)=τ(s)+τ(r)(ts),

    where r]s,t[.

    Hence ξ is a non increasing function and ξ>0, we get

    ξ(t)τ(t)ξ(s)τ(s)+ξ(s)τ(r)(ts),

    which implies

    ξ(t)τ(t)ξ(s)τ(s)1+|τ(r)|τ(s)|ts|.

    Using (5) and τ is bounded, we deduce that

    ξ(t)τ(t)ξ(s)τ(s)1+cτ0|ts|ecτ0|ts|,

    which proves (24) and therefore (iii) follows.

    Now we calculate A(t)U,Ut for a fixed t. Take U=(u,v,z)TD(A(t)). Then

    A(t)U,Ut=Ωvxuxdx+Ω(uxxμ1(t)vμ2(t)z(,1))vdxξ(t)Ω10(1τ(t)ρ)zρ(x,ρ)z(x,ρ)dρdx.

    Integrating by parts, we obtain

    A(t)U,Ut=μ1(t)v2L2(Ω)μ2(t)Ωz(,1)vdxΩ10(1τ(t)ρ)ρz2(x,ρ)dρdx.

    Since

    (1τ(t)ρ)ρz2(x,ρ)=ρ((1τ(t)ρ)z2(x,ρ))+τ(t)z2(x,ρ),

    we have

    10(1τ(t)ρ)ρz2(x,ρ)dρ=(1τ(t))z2(x,1)z2(x,0)+τ(t)10z2(x,ρ)dρ.

    So we get

    A(t)U,Ut=μ1(t)v2L2(Ω)μ2(t)Ωz(x,1)vdx+ξ(t)2z(x,0)2L2(Ω)ξ(t)(1τ(t))2z(x,1)2L2(Ω)ξ(t)τ(t)2Ω10z2(x,ρ)dρdx.

    Therefore, from (16) and (17), we deduce

    A(t)U,Utμ1(t)(1ˉξ2β21d)v2L2(Ω)μ1(t)(ˉξ(1τ(t))2β1d2)z(x,1,t)2L2(Ω)+ξ(t)|τ(t)|2τ(t)τ(t)Ω10z2(x,ρ)dρdx.

    Then, we have

    A(t)U,Utμ1(t)(1ˉξ2β21d)v2L2(Ω)μ1(t)(ˉξ(1τ(t))2β1d2)z(x,1,t)2L2(Ω)+κ(t)U,Ut,

    where

    κ(t)=1+τ(t)22τ(t).

    From the (13), we obtain

    A(t)U,Utκ(t)U,Ut0, (25)

    which means that the operator ˜A=A(t)κ(t)I is dissipative.

    Moreover, κ(t)=τ(t)τ(t)2τ(t)1+τ(t)2τ(t)1+τ(t)22τ(t)2 is bounded on [0,T] for all T>0 (by (5) and (12)) and we have

    ddtA(t)U=(0,0,τ(t)τ(t)ρτ(t)(τ(t)ρ1)τ(t)2zρ)T,

    with τ(t)τ(t)ρτ(t)(τ(t)ρ1)τ(t)2 bounded on [0,T] by (5) and (12). Thus

    ddt˜A(t)L([0,T],B(D(A(0)),H)), (26)

    the space of equivalence classes of essentially bounded, strongly measurable functions from [0,T] into B(D(A(0)),H).

    Now, we will show that λIA(t) is surjective for fixed t>0 and λ>0. For this purpose, let F=(f1,f2,f3)TH, we seek U=(u,v,z)TD(A(t)) solution of

    (λIA(t))U=F,

    that is verifying following system of equations

    {λuv=f1,λvuxx+μ1(t)vμ2(t)z(,1)=f2,λz+1τ(t)ρτ(t)zρ=f3. (27)

    Suppose that we have found u with the appropriated regularity. Then

    v=λuf1. (28)

    It is clear that vH10(Ω). Furthermore, by (27) we can find z. From (20), we have

    z(x,0)=v(x),  for xΩ. (29)

    Following the same approach as in [22], we obtain, by using equation for z in (27),

    z(x,ρ)=v(x)eϑ(ρ,t)+τ(t)eϑ(ρ,t)ρ0f3(x,s)eϑ(s,t)ds,

    if τ(t)=0, where ϑ(,t)=λτ(t), and

    z(x,ρ)=v(x)eζ(ρ,t)+eζ(ρ,t)ρ0τ(t)f3(x,s)1sτ(s)eζ(s,t)ds,

    otherwise, where ζ(,t)=λτ(t)τ(t)ln(1τ(t)).

    From (28), we obtain

    z(x,ρ)=λu(x)eϑ(ρ,t)f1(x,ρ)eϑ(ρ,t)+τ(t)eϑ(ρ,t)ρ0f3(x,s)eϑ(s,t)ds, (30)

    if τ(t)=0, and

    z(x,ρ)=λu(x)eζ(ρ,t)f1(x,ρ)eζ(ρ,t)+eζ(ρ,t)ρ0τ(t)f3(x,s)1sτ(s)eζ(s,t)ds, (31)

    otherwise.

    In particular, if τ(t)=0 and from (30), we have

    z(x,1)=λu(x)eϑ(1,t)f1(x,1)eϑ(1,t)+τ(t)eϑ(1,t)10f3(x,s)eϑ(s,t)ds, (32)

    and if τ(t)0 and from (31), we have

    z(x,1)=λu(x)eζ(1,t)f1(x,1)eζ(1,t)+eζ(1,t)10τ(t)f3(x,s)1sτ(s)eζ(s,t)ds. (33)

    By using (27) and (28), the function u satisfies

    λ2uuxx+μ1(t)v+μ2(t)z(,1)=f2+λf1. (34)

    Solving the equation (34) is equivalent to finding uH2(Ω)H10(Ω) such that

    Ω(λ2uη+uxηx+μ1(t)vη+μ2(t)z(,1)η)dx=Ω(f2+λf1)ηdx, (35)

    for all ηH10(Ω).

    Consequently, the equation (35) is equivalent to the problem

    Υ(u,η)=L(η), (36)

    where the bilinear form

    Υ:H10(Ω)×H10(Ω)R

    and the linear form

    L:H10(Ω)R

    are defined by

    Υ(u,η)=Ω(λ2uη+uxηx)dx+Ωλu(μ1(t)+μ2(t)N1)ηdx

    and

    L(η)=Ω(μ1(t)f1η+μ2(t)N2)ηdx+Ω(f2+λf1)ηdx,

    where

    N1={eϑ(1,t),ifτ(t)=0,eζ(1,t),ifτ(t)0

    and

    N2={f1(x,1)eϑ(1,t)+τ(t)eϑ(1,t)10f3(x,s)eϑ(s,t)ds,ifτ(t)=0,f1(x,1)ezeta(1,t)+ezeta(1,t)10τ(t)f3(x,s)1sτ(t)eζ(s,t)ds,ifτ(t)0.

    It is easy to verify that Υ is continuous and coercive, and L is continuous. So applying the Lax-Milgram theorem, we deduce that for all ηH10(Ω) the problem (36) admits a unique solution

    uH10(Ω).

    Applying the classical elliptic regularity, it follows from (35) that

    uH2(Ω).

    Therefore, the operator λIA(t) is surjective for any λ>0 and t>0. Again as κ(t)>0, this prove that

    λI˜A(t)=(λ+κ(t))IA(t) is surjective, (37)

    for any λ>0 and t>0.

    Then, (24), (25) and (37) imply that the family ˜A={˜A(t)/t[0,T]} is a stable family of generators in H with stability constants independent of t, by Proposition 1.1 from [14]. Therefore, the assumptions (i)(iv) of Theorem 3.1 are verified by (21), (24), (25), (26), (37) and (23), and thus, the problem

    {˜Ut=˜A(t)˜U,˜U(0)=U0=(u0,u1,f0(,,τ(0)))T (38)

    has a unique solution ˜UC([0,+[,D(A(0)))C1([0,+[,H) for U0D(A(0)). The requested solution of (18) is then given by

    U(t)=et0κ(s)ds˜U(t)

    because

    Ut(t)=κ(t)et0κ(s)ds˜U(t)+et0κ(s)ds˜Ut(t)=et0κ(s)ds(κ(t)+˜A(t))˜U(t)=A(t)et0κ(s)ds˜U(t)=A(t)U(t),

    which concludes the proof.

    The existence and uniqueness are obtained by the following result.

    Theorem 3.2 (Global solution). For any initial datum U0H there exists a unique solution U satisfying

    UC([0,+[,H)

    for problem (18).

    Moreover, if U0D(A(0)), then

    UC([0,+[,D(A(0)))C1([0,+[,H).

    Proof. A general theory for equations of type (18) has been developed using semigroup theory [14], [15] and [26]. The simplest way to prove existence and uniqueness results in to show that the triplet {(A,H,Y)}, with A={A(t)/t[0,T]}, for some fixed T>0 and Y=A(0), forms a CD-systems (or constant domain system, see [14] and [15]). More precisely, the following theorem gives the existence and uniqueness results and is proved in Theorem 1.9 of [14] (see also Theorem 2.13 of [15] or [1]).

    In this section we shall investigate the asymptotic behavior of problem (1). The stability result will be obtained using Lemma 2.2.

    Theorem 4.1 (Stability Result). Let (u0,u1,f0(,,τ(0)))H10(Ω)×L2(Ω)×L2(Ω×]0,1[). Assume that the hypotheses (H1), (H2) and (5)-(7) hold. Then problem (1) admits a unique solution

    uC([0,+[,H10(Ω))C1([0,+[,L2(Ω)),
    zC([0,+[,L2(Ω)×]0,1[).

    Proof. From now on, we denote by c various positive constants which may be different at different occurrences.

    Given 0S<T< we start by multiplying the first equation of (9) by uEq and then integrating over (S,T)×Ω, we obtain

    TSEqΩu(uttuxx+μ1(t)ut+μ2(t)z(x,1,t))dxdt=0.

    Notice that

    uttu=(utu)tu2t,

    using integration by parts and the boundary conditions we know that

    0=[Eq(t)Ωuutdx]TSTSqEq1(t)E(t)ΩuutdxdtTSEq(t)ut2L2(Ω)dt+TSEq(t)ux2L2(Ω)dt+TSEq(t)Ωμ1(t)uutdxdt+TSEq(t)Ωμ2(t)uz(x,1,t)dxdt. (39)

    Similarly, we multiply the second equation of (9) by Eqξ(t)e2ρτ(t)z(x,ρ,t) and then integrate over Ω×(0,1)×(S,T) to see that

    0=TSΩ10Eq(t)ξ(t)e2ρτ(t)z(τ(t)zt+(1ρτ(t))zρ)dρdxdt
    =12Ω10TSEq(t)ξ(t)e2ρτ(t)tz2dtdρdx+12TSEq(t)ξ(t)Ω10e2ρτ(t)(1ρτ(t))ρz2dρdxdt.

    Using integration by parts and the boundary conditions we know that

    0=[ξ(t)τ(t)2Eq(t)Ω10e2ρτ(t)z2dρdx]TS12TSqEq1(t)E(t)ξ(t)τ(t)Ω10e2ρτ(t)z2dρdxdt12TSqEq(t)ξ(t)τ(t)Ω10e2ρτ(t)z2dρdxdt+12TSEq(t)ξ(t)Ω[e2ρτ(t)(1τ(t))z2(x,1,t)z2(x,0,t)]dxdt+TSEq(t)ξ(t)τ(t)Ω10e2ρτ(t)z2dρdxdt. (40)

    Since μ1 is a non-increasing function of class C1(R), its derivatives is non-positive, which implies that ξ(t)0. This result this

    TSqEq(t)ξ(t)τ(t)Ω10e2ρτ(t)z2dρdxdt0. (41)

    Moreover, as

    12TSEq(t)ξ(t)Ωe2ρτ(t)(1τ(t))z2(x,1,t)dxdt0, (42)

    then, from (40), (41) and (42), we have that

    TSEq(t)ξ(t)τ(t)Ω10e2ρτ(t)z2dρdxdt[ξ(t)τ(t)2Eq(t)Ω10e2ρτ(t)z2dρdx]TS+12TSqEq1(t)E(t)ξ(t)τ(t)Ω10e2ρτ(t)z2dρdxdt12TSEq(t)ξ(t)Ωz2(x,0,t)dxdt. (43)

    Using the definition of E, (39) and (43), we get

    γ0TSEq+1dt[Eq(t)Ωuutdx]TS[ξ(t)τ(t)2Eq(t)Ω10e2ρτ(t)z2dρdx]TS+qTSEq1(t)E(t)Ωuutdxdt+qTSξ(t)τ(t)2Eq1(t)E(t)Ω10e2ρτ(t)z2dρdxdt
    +2TSEq(t)ut2L2(Ω)dtTSEq(t)Ωμ1(t)uutdxdtTSEq(t)Ωμ2(t)uz(x,1,t)dxdt+12TSξ(t)Eq(t)e2ρτ(t)Ωz2(x,0,t)dxdt, (44)

    where γ0=2min{1,e2τ1}.

    Using the Young and Sobolev-Poincaré inequalities and Lemma 2.3, we find that

    [Eq(t)Ωuutdx]TSEq(S)Ωu(x,S)ut(x,S)dxEq(T)Ωu(x,T)ut(x,T)dxcEq+1(S).

    Now, we known that

    [ξ(t)τ(t)2Eq(t)Ω10e2ρτ(t)z2dρdx]TSξ(S)τ(S)2Eq(S)Ω10e2ρτ(S)z2(x,ρ,S)dρdxcEq(S)ξ(S)τ(S)Ω10z2(x,ρ,S)dρdxcEq+1(S).

    By (13), we have

    TSEq1(t)E(t)ΩuutdxdtcTS(E(t))Eq(t)dtcEq+1(S).

    Similarly,

    TSEq1(t)E(t)ξ(t)τ(t)2Ω10e2ρτ(t)z2dρdxdtcEq+1(S).

    From Lemma 2.4, we deduce that

    TSEq(t)ut2L2(Ω)dtcTSEq(t)E(t)dtcEq+1(S).

    Now, we get that

    |TSEq(t)Ωμ1(t)uutdxdt|μ1(0)|TSEq(t)Ωuutdxdt|c(ε1)TSEq(t)Ωu2tdxdt+ε1TSEq(t)Ωu2xdxdtc(ε1)TSEq(t)(E(t))dt+ε1TSEq(t)E(t)dtc(ε1)Eq+1(S)+ε1TSEq+1(t)dt (45)

    and from (H2) we obtain that

    |TSEq(t)Ωμ2(t)uz(x,1,t)dxdt|βμ1(0)|TSEq(t)Ωφz(x,1,t)dxdt|c(ε2)Eq+1(S)+ε2TSEq+1(t)dt. (46)

    Finally,

    12TSEq(t)ξ(t)Ωz2(x,0,t)dxdtˉξμ1(0)2TSEq(t)ut2L2(Ω)dtcTSEq(t)(E(t))dtcEq+1(S).

    Choosing ε1 and ε2 small enough, we deduce from (45) and (46) that

    TSEq+1dt1γEq+1(S).

    Since E(S)E(0) for S0, we have that

    TSEq+1dt1γE(0)Eq(S).

    We choose q=0, we conclude from Lemma 2.2 that

    E(t)E(0)e1γt.

    This ends the proof of Theorem 4.1.



    [1] Britos B, Hernandez MA, Robles M, et al. (2022) Land market distortions and aggregate agricultural productivity: Evidence from Guatemala. J Dev Econ 155. https://doi.org/10.1016/j.jdeveco.2021.102787 doi: 10.1016/j.jdeveco.2021.102787
    [2] Chen W, Shen Y, Wang Y, et al. (2018) The effect of industrial relocation on industrial land use efficiency in China: A spatial econometrics approach. J Clean Prod 205: 525–535. https://doi.org/10.1016/j.jclepro.2018.09.106 doi: 10.1016/j.jclepro.2018.09.106
    [3] Corbelle-Rico E, Sánchez-Fernández P, López-Iglesias E, et al. (2022) Putting land to work: An evaluation of the economic effects of recultivating abandoned farmland. Land Use Policy 112. https://doi.org/10.1016/j.landusepol.2021.105808 doi: 10.1016/j.landusepol.2021.105808
    [4] Du W, Li M (2021) The impact of land resource mismatch and land marketization on pollution emissions of industrial enterprises in China. J Environ Manage 299. https://doi.org/10.1016/j.jenvman.2021.113565 doi: 10.1016/j.jenvman.2021.113565
    [5] Fan X, Qiu S, Sun Y (2020) Land finance dependence and urban land marketization in China: The perspective of strategic choice of local governments on land transfer. Land Use Policy 99. https://doi.org/10.1016/j.landusepol.2020.105023 doi: 10.1016/j.landusepol.2020.105023
    [6] Fei R, Lin Z, Chunga J (2021) How land transfer affects agricultural land use efficiency: Evidence from China's agricultural sector. Land Use Policy 103. https://doi.org/10.1016/j.landusepol.2021.105300 doi: 10.1016/j.landusepol.2021.105300
    [7] Haas R, Ajanovic A, Ramsebner J, et al. (2021) Financing the future infrastructure of sustainable energy systems. Green Financ 3: 90–118. https://doi.org/10.3934/gf.2021006 doi: 10.3934/GF.2021006
    [8] Farouq IS, Sambo NU, Ahmad UA, et al. (2021) Does financial globalization uncertainty affect CO2 emissions? Empirical evidence from some selected SSA countries. Quant Financ Econ 5: 247–263. https://doi.org/10.3934/QFE.2021011 doi: 10.3934/QFE.2021011
    [9] Jiang H (2021) Spatial-temporal differences of industrial land use efficiency and its influencing factors for China's central region: Analyzed by SBM model. Environ Technol Inno 22. https://doi.org/10.1016/j.eti.2021.101489 doi: 10.1016/j.eti.2021.101489
    [10] Jiang X, Lu X, Liu Q, et al. (2021) The effects of land transfer marketization on the urban land use efficiency: An empirical study based on 285 cities in China. Ecol Indic 132. https://doi.org/10.1016/j.ecolind.2021.108296 doi: 10.1016/j.ecolind.2021.108296
    [11] Joseph AM (2021) Commodity-linked bonds as an innovative financing instrument for African countries to build back better. Quant Financ Econ 5: 516–541. https://doi.org/10.3934/QFE.2021023 doi: 10.3934/QFE.2021023
    [12] Kanwal, M., & Khan, H. (2021) Does carbon asset add value to clean energy market? Evidence from EU. Green Finance, 3(4), 495-507. https://doi.org/10.3934/gf.2021023 doi: 10.3934/GF.2021023
    [13] Koroso NH, Zevenbergen JA, Lengoiboni M (2020) Urban land use efficiency in Ethiopia: An assessment of urban land use sustainability in Addis Ababa. Land Use Policy 99. https://doi.org/10.1016/j.landusepol.2020.105081 doi: 10.1016/j.landusepol.2020.105081
    [14] Leng Z, Wang Y, Hou X (2021) Structural and Efficiency Effects of Land Transfers on Food Planting: A Comparative Perspective on North and South of China. Sustainability 13. https://doi.org/10.3390/su13063327 doi: 10.3390/su13063327
    [15] Li F, Yang C, Li Z, et al. (2021a) Does Geopolitics Have an Impact on Energy Trade? Empirical Research on Emerging Countries. Sustainability 13. https://doi.org/10.3390/su13095199 doi: 10.3390/su13095199
    [16] Li T, Li X, Albitar K (2021b) Threshold effects of financialization on enterprise R & D innovation: a comparison research on heterogeneity. Quant Financ Econ 5: 496–515. https://doi.org/10.3934/QFE.2021022 doi: 10.3934/QFE.2021022
    [17] Li T, Li X, Liao G (2021c) Business Cycles and Energy Intensity. Evidence from Emerging Economies. Borsa Istanb Rev 22: 560–570. https://doi.org/10.1016/j.bir.2021.07.005 doi: 10.1016/j.bir.2021.07.005
    [18] Li Z, Chen L, Dong H (2021d) What are bitcoin market reactions to its-related events? Int Rev Econ Financ 73: 1–10. https://doi.org/10.1016/j.iref.2020.12.020 doi: 10.1016/j.iref.2020.12.020
    [19] Li Z, Dong H, Floros C, et al. (2021e) Re-examining bitcoin volatility: a caviar-based approach. Emerg Mark Financ Tr 58: 1320–1338. https://doi.org/10.1080/1540496x.2021.1873127 doi: 10.1080/1540496x.2021.1873127
    [20] Li Z, Huang Z, Failler P (2022) Dynamic Correlation between Crude Oil Price and Investor Sentiment in China: Heterogeneous and Asymmetric Effect. Energies 15. https://doi.org/10.3390/en1503068 doi: 10.3390/en1503068
    [21] Li Z, Mo B (2021) Revisiting the Valuable Roles of Global Financial Assets for International Stock Markets: Quantile Coherence and Causality-in-Quantiles Approaches. Mathematics 9. https://doi.org/10.3390/math9151750 doi: 10.3390/math9151750
    [22] Li Z, Zou F, Mo B (2021f) Does mandatory CSR disclosure affect enterprise total factor productivity? Econ Res-Ekon Istraž. https://doi.org/10.1080/1331677X.2021.201959 doi: 10.1080/1331677X.2021.201959
    [23] Li Z, Zou F, Tan Y, et al. (2021g) Does Financial Excess Support Land Urbanization-An Empirical Study of Cities in China. Land 10. https://doi.org/10.3390/land10060635 doi: 10.3390/land10060635
    [24] Liao G, Hou P, Shen X, et al. (2021) The impact of economic policy uncertainty on stock returns: The role of corporate environmental responsibility engagement. Int J Financ Econ 26: 4386–4389. https://doi.org/10.1002/ijfe.2020 doi: 10.1002/ijfe.2020
    [25] Lin B, Wang X, Jin S, et al. (2022) Impacts of cooperative membership on rice productivity: Evidence from China. World Dev 150. https://doi.org/10.1016/j.worlddev.2021.105669 doi: 10.1016/j.worlddev.2021.105669
    [26] Liu D, Zhu X, Wang Y (2021a) China's agricultural green total factor productivity based on carbon emission: An analysis of evolution trend and influencing factors. J Clean Prod 278. https://doi.org/10.1016/j.jclepro.2020.123692 doi: 10.1016/j.jclepro.2020.123692
    [27] Liu J, Wang H, Rahman S, et al. (2021b) Energy Efficiency, Energy Conservation and Determinants in the Agricultural Sector in Emerging Economies. Agriculture 11. https://doi.org/10.3390/agriculture11080773 doi: 10.3390/agriculture11080773
    [28] Liu R, Gao Z, Nian Y, et al. (2020) Does Social Relation or Economic Interest Affect the Choice Behavior of Land Lease Agreement in China? Evidence from the Largest Wheat-Producing Henan Province. Sustainability 12. https://doi.org/10.3390/su12104279 doi: 10.3390/su12104279
    [29] Liu Y, Wang C, Tang Z, et al. (2017) Farmland Rental and Productivity of Wheat and Maize: An Empirical Study in Gansu, China. Sustainability 9. https://doi.org/10.3390/su9101678 doi: 10.3390/su9101678
    [30] Lu H, Xie H, Yao G (2019) Impact of land fragmentation on marginal productivity of agricultural labor and non-agricultural labor supply: A case study of Jiangsu, China. Habitat Int 83: 65–72. https://doi.org/10.1016/j.habitatint.2018.11.004 doi: 10.1016/j.habitatint.2018.11.004
    [31] Lu X, Jiang X, Gong M (2020) How land transfer marketization influence on green total factor productivity from the approach of industrial structure? Evidence from China. Land Use Policy 95. https://doi.org/10.1016/j.landusepol.2020.104610 doi: 10.1016/j.landusepol.2020.104610
    [32] Ngango J, Hong S (2021) Impacts of land tenure security on yield and technical efficiency of maize farmers in Rwanda. Land Use Policy 107. https://doi.org/10.1016/j.landusepol.2021.105488 doi: 10.1016/j.landusepol.2021.105488
    [33] Paltasingh KR, Basantaray AK, Jena PK (2022) Land tenure security and farm efficiency in Indian agriculture: Revisiting an old debate. Land Use Policy 114. https://doi.org/10.1016/j.landusepol.2021.105955 doi: 10.1016/j.landusepol.2021.105955
    [34] Pang Y, Wang X (2020) Land-Use Efficiency in Shandong (China): Empirical Analysis Based on a Super-SBM Model. Sustainability 12. https://doi.org/10.3390/su122410618 doi: 10.3390/su122410618
    [35] Qiu T, Luo B, Li S, et al. (2019) Does the basic farmland preservation hinder land transfers in rural China? China Agr Econ Rev 12: 39–56. https://doi.org/10.1108/caer-10-2018-0212 doi: 10.1108/CAER-10-2018-0212
    [36] Ricker-Gilbert J, Chamberlin J, Kanyamuka J, et al. (2019) How do informal farmland rental markets affect smallholders' well‐being? Evidence from a matched tenant–landlord survey in Malawi. Agr Econ 50: 595–613. https://doi.org/10.1111/agec.12512 doi: 10.1111/agec.12512
    [37] Shi M, Paudel KP, Chen F (2021) Mechanization and efficiency in rice production in China. J Integr Agr 20: 1996–2008. https://doi.org/10.1016/s2095-3119(20)63439-6 doi: 10.1016/S2095-3119(20)63439-6
    [38] Udimal TB, Liu E, Luo M, et al. (2020) Examining the effect of land transfer on landlords' income in China: An application of the endogenous switching model. Heliyon 6. https://doi.org/10.1016/j.heliyon.2020.e05071 doi: 10.1016/j.heliyon.2020.e05071
    [39] Wang H, Lu S, Lu B, et al. (2021) Overt and covert: The relationship between the transfer of land development rights and carbon emissions. Land Use Policy 108. https://doi.org/10.1016/j.landusepol.2021.105665 doi: 10.1016/j.landusepol.2021.105665
    [40] Wang Y, Xin L, Li X, et al. (2016) Impact of Land Use Rights Transfer on Household Labor Productivity: A Study Applying Propensity Score Matching in Chongqing, China. Sustainability 9. https://doi.org/10.3390/su9010004 doi: 10.3390/su9010004
    [41] Xie H, Wang W, Yang Z, et al. (2016) Measuring the sustainable performance of industrial land utilization in major industrial zones of China. Technol Forecast Soc Change 112: 207–219. https://doi.org/10.1016/j.techfore.2016.06.016 doi: 10.1016/j.techfore.2016.06.016
    [42] Xie H, Zhang Y, Choi Y (2018) Measuring the Cultivated Land Use Efficiency of the Main Grain-Producing Areas in China under the Constraints of Carbon Emissions and Agricultural Nonpoint Source Pollution. Sustainability 10. https://doi.org/10.3390/su10061932 doi: 10.3390/su10061932
    [43] Yang C, Li T, Albitar K (2021) Does Energy Efficiency Affect Ambient PM2.5? The Moderating Role of Energy Investment. Front Env Sci. https://doi.org/10.3389/fenvs.2021.707751 doi: 10.3389/fenvs.2021.707751
    [44] Zeng S, Zhu F, Chen F, et al. (2018) Assessing the Impacts of Land Consolidation on Agricultural Technical Efficiency of Producers: A Survey from Jiangsu Province, China. Sustainability 10. https://doi.org/10.3390/su10072490 doi: 10.3390/su10072490
    [45] Zhang J, Mishra AK, Zhu P, et al. (2020) Land rental market and agricultural labor productivity in rural China: A mediation analysis. World Dev 135. https://doi.org/10.1016/j.worlddev.2020.105089 doi: 10.1016/j.worlddev.2020.105089
    [46] Zhao J, Zhu D, Cheng J, et al. (2021a) Does regional economic integration promote urban land use efficiency? Evidence from the Yangtze River Delta, China. Habitat Int 116. https://doi.org/10.1016/j.habitatint.2021.102404 doi: 10.1016/j.habitatint.2021.102404
    [47] Zhao Q, Bao HXH, Zhang Z (2021b) Off-farm employment and agricultural land use efficiency in China. Land Use Policy 101. https://doi.org/10.1016/j.landusepol.2020.105097 doi: 10.1016/j.landusepol.2020.105097
    [48] Zhu J, Huang Z, Li Z, Albitar K (2021) The Impact of Urbanization on Energy Intensity—An Empirical Study on OECD Countries. Green Financ 3: 508–526. https://doi.org/10.3934/gf.2021024 doi: 10.3934/GF.2021024
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