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,t−τ(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,t−τ(0))=f0(x,t−τ(0))in Ω×]0,τ(0)[, | (1) |
where
This problem has been first proposed and studied in Nicaise and Pignotti [22] in case of constant coefficients
With a weight depending on time,
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 (
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)
|μ′1(t)μ1(t)|≤M1,0<α0≤μ1(t),∀t≥0, | (2) |
where
(H2)
|μ2(t)|≤βμ1(t), | (3) |
|μ′2(t)|≤M2μ1(t), | (4) |
for some
We now state a lemma needed later.
Lemma 2.1 (Sobolev-Poincare's inequality). Let
‖Ψ‖q≤c∗‖Ψx‖2,forΨ∈H10(]0,L[). |
Lemma 2.2 ([13][16]). Let
∫+∞SE1+σ(t)dt≤1ωEσ(0)E(S), 0≤S<+∞. |
Then
E(t)=0 ∀t≥Eσ(0)ω|σ|, if−1<σ<0,E(t)≤E(0)(1+σ1+ωσt)1σ ∀t≥0, ifσ>0,E(t)≤E(0)e1−ωt ∀t≥0, ifσ=0. |
As in [23], we assume that
τ(t)∈W2,+∞([0,T]), for T>0 | (5) |
and there exist positive constants
0<τ0≤τ(t)≤τ1, ∀t>0 | (6) |
and
τ′(t)≤d<1, ∀t>0. | (7) |
We introduce the new variable
z(x,ρ,t)=ut(x,t−τ(t)ρ), x∈Ω,ρ∈]0,1[,t>0. | (8) |
Then
τ(t)zt(x,ρ,t)+(1−τ′(t)ρ)zρ(x,ρ,t)=0, x∈Ω, 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)+(1−τ′(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)=12‖ut‖2L2(Ω)+12‖ux‖2L2(Ω)+ξ(t)τ(t)2∫Ω∫10z2(x,ρ,t)dρdx, | (10) |
where
ξ(t)=ˉξμ1(t) | (11) |
is a non-increasing function of class
β√1−d<ˉξ<2−β√1−d. | (12) |
Our first result states that the energy is a non-increasing function.
Lemma 2.3. Let
E′(t)≤−μ1(t)(1−ˉξ2−β2√1−d)‖ut‖2L2(Ω)−μ1(t)(ˉξ(1−τ′(t))2−β√1−d2)‖z(x,1,t)‖2L2(Ω)≤0. | (13) |
Proof. Multiplying the first equation (9) by
12ddt(‖ut‖2L2(Ω)+‖ux‖2L2(Ω))+μ1(t)‖ut‖2L2(Ω)+μ2(t)∫Ωz(x,1,t)utdx. | (14) |
Now multiplying the second equation (9) by
τ(t)ξ(t)∫Ω∫10zt(x,ρ,t)z(x,ρ,t)dρdx=−ξ(t)2∫Ω∫10(1−τ′(t)ρ)∂∂ρ(z(x,ρ,t))2dρdx. |
Consequently,
ddt(ξ(t)τ(t)2∫Ω∫10z2(x,ρ,t)dρdx)=−ξ(t)2∫Ω∫10(1−τ′(t)ρ)∂∂ρ(z(x,ρ,t))2dρdx+ξ′(t)τ(t)2∫Ω∫10z2(x,ρ,t)dρdx=ξ(t)2∫Ω(z2(x,0,t)−z2(x,1,t))dx+ξ(t)τ′(t)2∫Ω∫10z2(x,1,t)dρdx+ξ′(t)τ(t)2∫Ω∫10z2(x,ρ,t)dρdx. | (15) |
From (10), (14) and (15) we obtain
E′(t)=ξ(t)2‖ut‖2L2(Ω)−ξ(t)2‖z(x,1,t)‖2L2(Ω)+ξ(t)τ′(t)2‖z(x,1,t)‖2L2(Ω)+ξ′(t)τ(t)2∫Ω∫10z2(x,ρ,t)dρdx−μ1(t)‖ut‖2L2(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)|2√1−d‖ut‖2L2(Ω)+|μ2(t)|√1−d2‖z(x,1,t)‖2L2(Ω). | (17) |
Inserting (17) into (16), we obtain
E′(t)≤−(μ1(t)−ξ(t)2−|μ2(t)|2√1−d)‖ut‖2L2(Ω)−(ξ(t)2−ξ(t)τ′(t)2−|μ2(t)|√1−d2)‖z(x,1,t)‖2L2(Ω)+ξ′(t)τ(t)2∫Ω∫10z2(x,ρ,t)dρdx≤−μ1(t)(1−ˉξ2−β2√1−d)‖ut‖2L2(Ω)−μ1(t)(ˉξ(1−τ′(t))2−β√1−d2)‖z(x,1,t)‖2L2(Ω)≤0. |
Lemma 2.4. Let
‖ut(x,t)‖2L2(Ω)<−1σE′(t), |
where
Proof. From Lemma 2.3, we have that
−E′(t)≥μ1(t)(1−ˉξ2+β2√1−d)‖ut‖2L2(Ω)+μ1(t)(ˉξ(1−τ′(t))2+β√1−d2)‖z(x,1,t)‖2L2(Ω)≥0 |
and from (H1), we obtain
0≤a0(1−ˉξ2+β2√1−d)‖ut‖2L2(Ω)≤μ1(t)(1−ˉξ2+β2√1−d)‖ut‖2L2(Ω)≤−E′(t) |
and the lemma is proved.
For the semigroup setup we
{Ut=A(t)U,U(0)=U0=(u0,u1,f0(⋅,−,τ(0)))T, | (18) |
where the operator
AU=(v,uxx−μ1(t)v−μ2(t)z(x,1,t),−1−τ′(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
D(A(t))={(u,v,z)T∈H/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
D(A(t))=D(A(0)),∀t>0. | (21) |
⟨U,ˉU⟩H=∫Ωuxˉuxdx+∫Ωvˉvdx+ξ(t)τ(t)∫Ω∫10zˉzdρdx, | (22) |
for
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)
(ii) (21) holds,
(iii) for all
(iv)
Then, problem (18) has a solution
Our goal is then to check the above assumptions for problem (18).
First, we prove
The proof is the same as the one Lemma
Let
0=⟨(u,v,z)T,(f,g,h)T⟩H=∫Ωuxfxdx+∫Ωvgdx+ξ(t)τ(t)∫Ω∫10zhdρdx, |
for all
We first take
∫Ω∫10zhdρdx=0. |
Since
The above orthogonality condition is then reduced to
0=∫Ωuxfxdx,∀(u,v,z)T∈D(A(0)). |
By restricting ourselves to
0=∫Ωuxfxdx,∀(u,0,0)T∈D(A(0)). |
Since
We consequently
D(A(0) is dense in H. | (23) |
Secondly, we notice that
‖Φ‖t‖Φ‖s≤ec2τ0|t−s|,∀t,s∈[0,T], | (24) |
where
‖Φ‖2t−‖Φ‖2secτ0|t−s|=(1−ec2τ0|t−s|)(‖ux‖2L2(Ω)+‖v‖2L2(Ω))+(ξ(t)τ(t)−ξ(s)τ(s)ecτ0|t−s|)∫Ω∫10z2(x,ρ,t)dρdx. |
It is clear that
τ(t)=τ(s)+τ′(r)(t−s), |
where
Hence
ξ(t)τ(t)≤ξ(s)τ(s)+ξ(s)τ′(r)(t−s), |
which implies
ξ(t)τ(t)ξ(s)τ(s)≤1+|τ′(r)|τ(s)|t−s|. |
Using (5) and
ξ(t)τ(t)ξ(s)τ(s)≤1+cτ0|t−s|≤ecτ0|t−s|, |
which proves (24) and therefore
Now we calculate
⟨A(t)U,U⟩t=∫Ωvxuxdx+∫Ω(uxx−μ1(t)v−μ2(t)z(⋅,1))vdx−ξ(t)∫Ω∫10(1−τ′(t)ρ)zρ(x,ρ)z(x,ρ)dρdx. |
Integrating by parts, we obtain
⟨A(t)U,U⟩t=−μ1(t)‖v‖2L2(Ω)−μ2(t)∫Ωz(⋅,1)vdx−∫Ω∫10(1−τ′(t)ρ)∂∂ρz2(x,ρ)dρdx. |
Since
(1−τ′(t)ρ)∂∂ρz2(x,ρ)=∂∂ρ((1−τ′(t)ρ)z2(x,ρ))+τ′(t)z2(x,ρ), |
we have
∫10(1−τ′(t)ρ)∂∂ρz2(x,ρ)dρ=(1−τ′(t))z2(x,1)−z2(x,0)+τ′(t)∫10z2(x,ρ)dρ. |
So we get
⟨A(t)U,U⟩t=−μ1(t)‖v‖2L2(Ω)−μ2(t)∫Ωz(x,1)vdx+ξ(t)2‖z(x,0)‖2L2(Ω)−ξ(t)(1−τ′(t))2‖z(x,1)‖2L2(Ω)−ξ(t)τ′(t)2∫Ω∫10z2(x,ρ)dρdx. |
Therefore, from (16) and (17), we deduce
⟨A(t)U,U⟩t≤−μ1(t)(1−ˉξ2−β2√1−d)‖v‖2L2(Ω)−μ1(t)(ˉξ(1−τ′(t))2−β√1−d2)‖z(x,1,t)‖2L2(Ω)+ξ(t)|τ′(t)|2τ(t)τ(t)∫Ω∫10z2(x,ρ)dρdx. |
Then, we have
⟨A(t)U,U⟩t≤−μ1(t)(1−ˉξ2−β2√1−d)‖v‖2L2(Ω)−μ1(t)(ˉξ(1−τ′(t))2−β√1−d2)‖z(x,1,t)‖2L2(Ω)+κ(t)⟨U,U⟩t, |
where
κ(t)=√1+τ′(t)22τ(t). |
From the (13), we obtain
⟨A(t)U,U⟩t−κ(t)⟨U,U⟩t≤0, | (25) |
which means that the operator
Moreover,
ddtA(t)U=(0,0,τ″(t)τ(t)ρ−τ′(t)(τ′(t)ρ−1)τ(t)2zρ)T, |
with
ddt˜A(t)∈L∞∗([0,T],B(D(A(0)),H)), | (26) |
the space of equivalence classes of essentially bounded, strongly measurable functions from
Now, we will show that
(λI−A(t))U=F, |
that is verifying following system of equations
{λu−v=f1,λv−uxx+μ1(t)v−μ2(t)z(⋅,1)=f2,λz+1−τ′(t)ρτ(t)zρ=f3. | (27) |
Suppose that we have found
v=λu−f1. | (28) |
It is clear that
z(x,0)=v(x), for x∈Ω. | (29) |
Following the same approach as in [22], we obtain, by using equation for
z(x,ρ)=v(x)e−ϑ(ρ,t)+τ(t)e−ϑ(ρ,t)∫ρ0f3(x,s)eϑ(s,t)ds, |
if
z(x,ρ)=v(x)eζ(ρ,t)+eζ(ρ,t)∫ρ0τ(t)f3(x,s)1−sτ′(s)e−ζ(s,t)ds, |
otherwise, where
From (28), we obtain
z(x,ρ)=λu(x)e−ϑ(ρ,t)−f1(x,ρ)e−ϑ(ρ,t)+τ(t)e−ϑ(ρ,t)∫ρ0f3(x,s)eϑ(s,t)ds, | (30) |
if
z(x,ρ)=λu(x)eζ(ρ,t)−f1(x,ρ)eζ(ρ,t)+eζ(ρ,t)∫ρ0τ(t)f3(x,s)1−sτ′(s)e−ζ(s,t)ds, | (31) |
otherwise.
In particular, if
z(x,1)=λu(x)e−ϑ(1,t)−f1(x,1)e−ϑ(1,t)+τ(t)e−ϑ(1,t)∫10f3(x,s)eϑ(s,t)ds, | (32) |
and if
z(x,1)=λu(x)eζ(1,t)−f1(x,1)eζ(1,t)+eζ(1,t)∫10τ(t)f3(x,s)1−sτ′(s)e−ζ(s,t)ds. | (33) |
By using (27) and (28), the function
λ2u−uxx+μ1(t)v+μ2(t)z(⋅,1)=f2+λf1. | (34) |
Solving the equation (34) is equivalent to finding
∫Ω(λ2uη+uxηx+μ1(t)vη+μ2(t)z(⋅,1)η)dx=∫Ω(f2+λf1)ηdx, | (35) |
for all
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={e−ϑ(1,t),ifτ′(t)=0,eζ(1,t),ifτ′(t)≠0 |
and
N2={−f1(x,1)e−ϑ(1,t)+τ(t)e−ϑ(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)1−sτ′(t)e−ζ(s,t)ds,ifτ′(t)≠0. |
It is easy to verify that
u∈H10(Ω). |
Applying the classical elliptic regularity, it follows from (35) that
u∈H2(Ω). |
Therefore, the operator
λI−˜A(t)=(λ+κ(t))I−A(t) is surjective, | (37) |
for any
Then, (24), (25) and (37) imply that the family
{˜Ut=˜A(t)˜U,˜U(0)=U0=(u0,u1,f0(⋅,−,τ(0)))T | (38) |
has a unique solution
U(t)=e∫t0κ(s)ds˜U(t) |
because
Ut(t)=κ(t)e∫t0κ(s)ds˜U(t)+e∫t0κ(s)ds˜Ut(t)=e∫t0κ(s)ds(κ(t)+˜A(t))˜U(t)=A(t)e∫t0κ(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
U∈C([0,+∞[,H) |
for problem (18).
Moreover, if
U∈C([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
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
u∈C([0,+∞[,H10(Ω))∩C1([0,+∞[,L2(Ω)), |
z∈C([0,+∞[,L2(Ω)×]0,1[). |
Proof. From now on, we denote by
Given
∫TSEq∫Ωu(utt−uxx+μ1(t)ut+μ2(t)z(x,1,t))dxdt=0. |
Notice that
uttu=(utu)t−u2t, |
using integration by parts and the boundary conditions we know that
0=[Eq(t)∫Ωuutdx]TS−∫TSqEq−1(t)E′(t)∫Ωuutdxdt−∫TSEq(t)‖ut‖2L2(Ω)dt+∫TSEq(t)‖ux‖2L2(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
0=∫TS∫Ω∫10Eq(t)ξ(t)e−2ρτ(t)z(τ(t)zt+(1−ρτ′(t))zρ)dρdxdt |
=12∫Ω∫10∫TSEq(t)ξ(t)e−2ρτ(t)∂∂tz2dtdρdx+12∫TSEq(t)ξ(t)∫Ω∫10e−2ρτ(t)(1−ρτ′(t))∂∂ρz2dρdxdt. |
Using integration by parts and the boundary conditions we know that
0=[ξ(t)τ(t)2Eq(t)∫Ω∫10e−2ρτ(t)z2dρdx]TS−12∫TSqEq−1(t)E′(t)ξ(t)τ(t)∫Ω∫10e−2ρτ(t)z2dρdxdt−12∫TSqEq(t)ξ′(t)τ(t)∫Ω∫10e−2ρτ(t)z2dρdxdt+12∫TSEq(t)ξ(t)∫Ω[e−2ρτ(t)(1−τ′(t))z2(x,1,t)−z2(x,0,t)]dxdt+∫TSEq(t)ξ(t)τ(t)∫Ω∫10e−2ρτ(t)z2dρdxdt. | (40) |
Since
∫TSqEq(t)ξ′(t)τ(t)∫Ω∫10e−2ρτ(t)z2dρdxdt≤0. | (41) |
Moreover, as
−12∫TSEq(t)ξ(t)∫Ωe−2ρτ(t)(1−τ′(t))z2(x,1,t)dxdt≤0, | (42) |
then, from (40), (41) and (42), we have that
∫TSEq(t)ξ(t)τ(t)∫Ω∫10e−2ρτ(t)z2dρdxdt≤−[ξ(t)τ(t)2Eq(t)∫Ω∫10e−2ρτ(t)z2dρdx]TS+12∫TSqEq−1(t)E′(t)ξ(t)τ(t)∫Ω∫10e−2ρτ(t)z2dρdxdt−12∫TSEq(t)ξ(t)∫Ωz2(x,0,t)dxdt. | (43) |
Using the definition of
γ0∫TSEq+1dt≤−[Eq(t)∫Ωuutdx]TS−[ξ(t)τ(t)2Eq(t)∫Ω∫10e−2ρτ(t)z2dρdx]TS+q∫TSEq−1(t)E′(t)∫Ωuutdxdt+q∫TSξ(t)τ(t)2Eq−1(t)E′(t)∫Ω∫10e−2ρτ(t)z2dρdxdt |
+2∫TSEq(t)‖ut‖2L2(Ω)dt−∫TSEq(t)∫Ωμ1(t)uutdxdt−∫TSEq(t)∫Ωμ2(t)uz(x,1,t)dxdt+12∫TSξ(t)Eq(t)e−2ρτ(t)∫Ωz2(x,0,t)dxdt, | (44) |
where
Using the Young and Sobolev-Poincaré inequalities and Lemma 2.3, we find that
−[Eq(t)∫Ωuutdx]TS≤Eq(S)∫Ωu(x,S)ut(x,S)dx−Eq(T)∫Ωu(x,T)ut(x,T)dx≤cEq+1(S). |
Now, we known that
−[ξ(t)τ(t)2Eq(t)∫Ω∫10e−2ρτ(t)z2dρdx]TS≤ξ(S)τ(S)2Eq(S)∫Ω∫10e−2ρτ(S)z2(x,ρ,S)dρdx≤cEq(S)ξ(S)τ(S)∫Ω∫10z2(x,ρ,S)dρdx≤cEq+1(S). |
By (13), we have
∫TSEq−1(t)E′(t)∫Ωuutdxdt≤c∫TS(−E′(t))Eq(t)dt≤cEq+1(S). |
Similarly,
∫TSEq−1(t)E′(t)ξ(t)τ(t)2∫Ω∫10e−2ρτ(t)z2dρdxdt≤cEq+1(S). |
From Lemma 2.4, we deduce that
∫TSEq(t)‖ut‖2L2(Ω)dt≤−c∫TSEq(t)E′(t)dt≤cEq+1(S). |
Now, we get that
|∫TSEq(t)∫Ωμ1(t)uutdxdt|≤μ1(0)|∫TSEq(t)∫Ωuutdxdt|≤c(ε1)∫TSEq(t)∫Ωu2tdxdt+ε1∫TSEq(t)∫Ωu2xdxdt≤c(ε1)∫TSEq(t)(−E′(t))dt+ε1∫TSEq(t)E(t)dt≤c(ε1)Eq+1(S)+ε1∫TSEq+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)+ε2∫TSEq+1(t)dt. | (46) |
Finally,
12∫TSEq(t)ξ(t)∫Ωz2(x,0,t)dxdt≤ˉξμ1(0)2∫TSEq(t)‖ut‖2L2(Ω)dt≤c∫TSEq(t)(−E′(t))dt≤cEq+1(S). |
Choosing
∫TSEq+1dt≤1γEq+1(S). |
Since
∫TSEq+1dt≤1γE(0)Eq(S). |
We choose
E(t)≤E(0)e1−γt. |
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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