
Environmental concern is a determinant in acquiring new green innovation. We aimed to investigate the relationship between environmental concern, consumer attitude, and behavioral intention to switch in the context of drone delivery. The motivation that comes from a green perspective is believed to create a behavior that is also keen on innovative green products. One of the examples is the implementation of drones in delivering parcels, which is believed to cut the carbon footprint. Our purpose was to analyze the direct impact of environmental concern on consumers' behavioral intentions regarding e-commerce drone delivery. Additionally, we aimed to examine the mediating role of consumers' attitudes toward innovation in the relationship between environmental concern and behavioral intention. We sought to provide insights into how environmental awareness and the adoption of innovative delivery technologies like drones can influence consumer behavior, contributing to more sustainable and eco-friendly e-commerce practices. Structured questionnaires were provided to e-commerce users, reliability and validity tests were confirmed, and structural equation modeling (SEM) was used to analyze the relationships among variables. The results of the SEM analysis proved that environmental concern and consumer attitudes have a positive impact on behavioral intention. Customer attitude mediates the relationship between environmental concern and behavioral intention. This research provides a deeper understanding of how environmental concerns influence consumer behavior towards drone delivery innovation within the e-commerce sector. The implications integrate environmental concerns with consumer behavior and innovation adoption, providing a comprehensive view that goes beyond traditional marketing and consumer research.
Citation: Veronica Veronica, Muhtosim Arief, Asnan Furinto, Lim Sanny. E-commerce user's intention to switch toward drone delivery innovation: The role of environmental concern and customers' attitude[J]. AIMS Environmental Science, 2024, 11(5): 847-865. doi: 10.3934/environsci.2024042
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Environmental concern is a determinant in acquiring new green innovation. We aimed to investigate the relationship between environmental concern, consumer attitude, and behavioral intention to switch in the context of drone delivery. The motivation that comes from a green perspective is believed to create a behavior that is also keen on innovative green products. One of the examples is the implementation of drones in delivering parcels, which is believed to cut the carbon footprint. Our purpose was to analyze the direct impact of environmental concern on consumers' behavioral intentions regarding e-commerce drone delivery. Additionally, we aimed to examine the mediating role of consumers' attitudes toward innovation in the relationship between environmental concern and behavioral intention. We sought to provide insights into how environmental awareness and the adoption of innovative delivery technologies like drones can influence consumer behavior, contributing to more sustainable and eco-friendly e-commerce practices. Structured questionnaires were provided to e-commerce users, reliability and validity tests were confirmed, and structural equation modeling (SEM) was used to analyze the relationships among variables. The results of the SEM analysis proved that environmental concern and consumer attitudes have a positive impact on behavioral intention. Customer attitude mediates the relationship between environmental concern and behavioral intention. This research provides a deeper understanding of how environmental concerns influence consumer behavior towards drone delivery innovation within the e-commerce sector. The implications integrate environmental concerns with consumer behavior and innovation adoption, providing a comprehensive view that goes beyond traditional marketing and consumer research.
In this paper, we consider the following initial-boundary value problem
{ut−Δut−Δu=|x|σ|u|p−1u,x∈Ω,t>0,u(x,t)=0,x∈∂Ω,t>0,u(x,0)=u0(x),x∈Ω | (1) |
and its corresponding steady-state problem
{−Δu=|x|σ|u|p−1u,x∈Ω,u=0,x∈∂Ω, | (2) |
where
1<p<{∞,n=1,2;n+2n−2,n≥3,σ>{−n,n=1,2;(p+1)(n−2)2−n,n≥3. | (3) |
(1) was called homogeneous (inhomogeneous) pseudo-parabolic equation when
The homogeneous problem, i.e.
Li and Du [12] studied the Cauchy problem of equation in (1) with
(1) If
(2) If
Φα:={ξ(x)∈BC(Rn):ξ(x)≥0,lim inf|x|↑∞|x|αξ(x)>0}, |
and
Φα:={ξ(x)∈BC(Rn):ξ(x)≥0,lim sup|x|↑∞|x|αξ(x)<∞}. |
Here
In view of the above introductions, we find that
(1) for Cauchy problem in
(2) for zero Dirichlet problem in a bounded domain
The difficulty of allowing
σ>(p+1)(n−2)2−n⏟<0 if n≥3 |
for
The main results of this paper can be summarized as follows: Let
(1) (the case
(2) (the case
(3) (arbitrary initial energy level) For any
(4) Moreover, under suitable assumptions, we show the exponential decay of global solutions and lifespan (i.e. the upper bound of blow-up time) of the blowing-up solutions.
The organizations of the remain part of this paper are as follows. In Section 2, we introduce the notations used in this paper and the main results of this paper; in Section 3, we give some preliminaries which will be used in the proofs; in Section 4, we give the proofs of the main results.
Throughout this paper we denote the norm of
‖ϕ‖Lγ={(∫Ω|ϕ(x)|γdx)1γ, if 1≤γ<∞;esssupx∈Ω|ϕ(x)|, if γ=∞. |
We denote the
Lp+1σ(Ω):={ϕ:ϕ is measurable on Ω and ‖u‖Lp+1σ<∞}, | (4) |
where
‖ϕ‖Lp+1σ:=(∫Ω|x|σ|ϕ(x)|p+1dx)1p+1,ϕ∈Lp+1σ(Ω). | (5) |
By standard arguments as the space
We denote the inner product of
(ϕ,φ)H10:=∫Ω(∇ϕ(x)⋅∇φ(x)+ϕ(x)φ(x))dx,ϕ,φ∈H10(Ω). | (6) |
The norm of
‖ϕ‖H10:=√(ϕ,ϕ)H10=√‖∇ϕ‖2L2+‖ϕ‖2L2,ϕ∈H10(Ω). | (7) |
An equivalent norm of
‖∇ϕ‖L2≤‖ϕ‖H10≤√λ1+1λ1‖∇ϕ‖L2,ϕ∈H10(Ω), | (8) |
where
λ1=infϕ∈H10(Ω)‖∇ϕ‖2L2‖ϕ‖2L2. | (9) |
Moreover, by Theorem 3.2, we have
for p and σ satisfying (4), H10(Ω)↪Lp+1σ(Ω) continuously and compactly. | (10) |
Then we let
Cpσ=supu∈H10(Ω)∖{0}‖ϕ‖Lp+1σ‖∇ϕ‖L2. | (11) |
We define two functionals
J(ϕ):=12‖∇ϕ‖2L2−1p+1‖ϕ‖p+1Lp+1σ | (12) |
and
I(ϕ):=‖∇ϕ‖2L2−‖ϕ‖p+1Lp+1σ. | (13) |
By (3) and (10), we know that
We denote the mountain-pass level
d:=infϕ∈NJ(ϕ), | (14) |
where
N:={ϕ∈H10(Ω)∖{0}:I(ϕ)=0}. | (15) |
By Theorem 3.3, we have
d=p−12(p+1)C−2(p+1)p−1pσ, | (16) |
where
For
Jρ={ϕ∈H10(Ω):J(ϕ)<ρ}. | (17) |
Then, we define the set
Nρ={ϕ∈N:‖∇ϕ‖2L2<2(p+1)ρp−1},ρ>d. | (18) |
For
λρ:=infϕ∈Nρ‖ϕ‖H10,Λρ:=supϕ∈Nρ‖ϕ‖H10 | (19) |
and two sets
Sρ:={ϕ∈H10(Ω):‖ϕ‖H10≤λρ,I(ϕ)>0},Sρ:={ϕ∈H10(Ω):‖ϕ‖H10≥Λρ,I(ϕ)<0}. | (20) |
Remark 1. There are two remarks on the above definitions.
(1) By the definitions of
(2) By Theorem 3.4, we have
√2(p+1)dp−1≤λρ≤Λρ≤√2(p+1)(λ1+1)ρλ1(p−1). | (21) |
Then the sets
‖sϕ‖H10≤√2(p+1)dp−1⇔s≤δ1:=√2(p+1)dp−1‖ϕ‖−1H10,I(sϕ)=s2‖∇ϕ‖2L2−sp+1‖ϕ‖p−1Lp+1σ>0⇔s<δ2:=(‖∇ϕ‖2L2‖ϕ‖p+1Lp+1σ)1p−1,‖sϕ‖H10≥√2(p+1)(λ1+1)ρλ1(p−1)⇔s≥δ3:=√2(p+1)(λ1+1)ρλ1(p−1)‖ϕ‖−1H10,I(sϕ)=s2‖∇ϕ‖2L2−sp+1‖ϕ‖p−1Lp+1σ<0⇔s>δ2. |
So,
{sϕ:0<s<min{δ1,δ2}}⊂Sρ,{sϕ:s>max{δ2,δ3}}⊂Sρ. |
In this paper we consider weak solutions to problem (1), local existence of which can be obtained by Galerkin's method (see for example [22,Chapter II,Sections 3 and 4]) and a standard limit process and the details are omitted.
Definition 2.1. Assume
∫Ω(utv+∇ut⋅∇v+∇u⋅∇v−|x|σ|u|p−1uv)dx=0 | (22) |
holds for any
u(⋅,0)=u0(⋅) in H10(Ω). | (23) |
Remark 2. There are some remarks on the above definition.
(1) Since
(2) Denote by
(3) Taking
‖u(⋅,t)‖2H10=‖u0‖2H10−2∫t0I(u(⋅,s))ds,0≤t≤T, | (24) |
where
(4) Taking
J(u(⋅,t))=J(u0)−∫t0‖us(⋅,s)‖2H10ds,0≤t≤T, | (25) |
where
Definition 2.2. Assume (3) holds. A function
∫Ω(∇u⋅∇v−|x|σ|u|p−1uv)dx=0 | (26) |
holds for any
Remark 3. There are some remarks to the above definition.
(1) By (10), we know all the terms in (26) are well-defined.
(2) If we denote by
Φ={ϕ∈H10(Ω):J′(ϕ)=0 in H−1(Ω)}⊂(N∪{0}), | (27) |
where
With the set
Definition 2.3. Assume (3) holds. A function
J(u)=infϕ∈Φ∖{0}J(ϕ). |
With the above preparations, now we can state the main results of this paper. Firstly, we consider the case
(1)
(2)
(3)
Theorem 2.4. Assume (3) holds and
‖∇u(⋅,t)‖L2≤√2(p+1)J(u0)p−1,0≤t<∞, | (28) |
where
V:={ϕ∈H10(Ω):J(ϕ)≤d,I(ϕ)>0}. | (29) |
In, in addition,
‖u(⋅,t)‖H10≤‖u0‖H10exp[−λ1λ1+1(1−(J(u0)d)p−12)t]. | (30) |
Remark 4. Since
J(u0)>p−12(p+1)‖∇u0‖2L2>0. |
So the equality (28) makes sense.
Theorem 2.5. Assume (3) holds and
limt↑Tmax∫t0‖u(⋅,s)‖2H10ds=∞, |
where
W:={ϕ∈H10(Ω):J(ϕ)≤d,I(ϕ)<0} | (31) |
and
Tmax≤4p‖u0‖2H10(p−1)2(p+1)(d−J(u0)). | (32) |
Remark 5. There are two remarks.
(1) If
(2) The sets
f(s)=J(sϕ)=s22‖∇ϕ‖2L2−sp+1p+1‖ϕ‖p+1Lp+1σ,g(s)=I(sϕ)=s2‖∇ϕ‖2L2−sp+1‖ϕ‖p+1Lp+1σ. |
Then (see Fig. 2)
(a)
maxs∈[0,∞)f(s)=f(s∗3)=p−12(p+1)(‖∇ϕ‖L2‖ϕ‖Lp+1σ)2(p+1)p−1≤d⏟By (14) since s∗3ϕ∈N, | (33) |
(b)
maxs∈[0,∞)g(s)=g(s∗1)=p−1p+1(2p+1)2p−1(‖∇ϕ‖L2‖ϕ‖Lp+1σ)2(p+1)p−1, |
(c)
f(s∗2)=g(s∗2)=p−12p(p+12p)2p−1(‖∇ϕ‖L2‖ϕ‖Lp+1σ)2(p+1)p−1, |
where
s∗1:=(2‖∇ϕ‖2L2(p+1)‖ϕ‖p+1Lp+1σ)1p−1<s∗2:=((p+1)‖∇ϕ‖2L22p‖ϕ‖p+1Lp+1σ)1p−1<s∗3:=(‖∇ϕ‖2L2‖ϕ‖p+1Lp+1σ)1p−1<s∗4:=((p+1)‖∇ϕ‖2L22‖ϕ‖p+1Lp+1σ)1p−1. |
So,
Theorem 2.6. Assume (3) holds and
G:={ϕ∈H10(Ω):J(ϕ)=d,I(ϕ)=0}. | (34) |
Remark 6. There are two remarks on the above theorem.
(1) Unlike Remark 5, it is not easy to show
(2) To prove the above Theorem, we only need to show
Theorem 2.7. Assume (3) holds and let
Secondly, we consider the case
Theorem 2.8. Assume (3) holds and the initial value
(i): If
(ii): If
Here
Next, we show the solution of the problem (1) can blow up at arbitrary initial energy level (Theorem 2.10). To this end, we firstly introduce the following theorem.
Theorem 2.9. Assume (3) holds and
Tmax≤8p‖u0‖2H10(p−1)2(λ1(p−1)λ1+1‖u0‖2H10−2(p+1)J(u0)) | (35) |
and
limt↑Tmax∫t0‖u(⋅,s)‖2H10ds=∞, |
where
ˆW:={ϕ∈H10(Ω):J(ϕ)<λ1(p−1)2(λ1+1)(p+1)‖ϕ‖2H10}. | (36) |
and
By using the above theorem, we get the following theorem.
Theorem 2.10. For any
The following lemma can be found in [11].
Lemma 3.1. Suppose that
F″(t)F(t)−(1+γ)(F′(t))2≥0 |
for some constant
T≤F(0)γF′(0)<∞ |
and
Theorem 3.2. Assume
Proof. Since
We divide the proof into three cases. We will use the notation
Case 1.
H10(Ω)↪Lp+1(Ω) continuously and compactly. | (37) |
Then we have, for any
‖u‖p+1Lp+1σ=∫Ω|x|σ|u|p+1dx≤Rσ‖u‖p+1Lp+1≲‖u‖p+1H10, |
which, together with (37), implies
Case 2.
H10(Ω)↪L(p+1)rr−1(Ω) continuously and compactly, | (38) |
for any
‖u‖p+1Lp+1σ=∫Ω|x|σ|u|p+1dx≤(∫B(0,R)|x|σrdx)1r(∫Ω|u|(p+1)rr−1dx)r−1r≤{(2σr+1Rσr+1)1r‖u‖p+1L(p+1)rr−1≲‖u‖p+1H10,n=1;(2πσr+2Rσr+2)1r‖u‖p+1L(p+1)rr−1≲‖u‖p+1H10,n=2, |
which, together with (38), implies
Case 3.
−σn<1r<1−(p+1)(n−2)2n. |
By the second inequality of the above inequalities, we have
(p+1)rr−1=p+11−1r<p+1(p+1)(n−2)2n=2nn−2. |
So,
H10(Ω)↪L(p+1)rr−1(Ω) continuously and compactly. | (39) |
Then by Hölder's inequality, for any
‖u‖p+1Lp+1σ=∫Ω|x|σ|u|p+1dx≤(∫B(0,R)|x|σrdx)1r(∫Ω|u|(p+1)rr−1dx)r−1r≤(ωn−1σr+nRσr+n)1r‖u‖p+1L(p+1)rr−1≲‖u‖p+1H10, |
which, together with (39), implies
Theorem 3.3. Assume
d=p−12(p+1)C2(p+1)p−1pσ, |
where
Proof. Firstly, we show
infϕ∈NJ(ϕ)=minϕ∈H10(Ω)∖{0}J(s∗ϕϕ), | (40) |
where
s∗ϕ:=(‖∇ϕ‖2L2‖ϕ‖p+1Lp+1σ)1p−1. | (41) |
By the definition of
On one hand, since
minϕ∈H10(Ω)∖{0}J(s∗ϕϕ)≤minϕ∈NJ(s∗ϕϕ)=minϕ∈NJ(ϕ). |
On the other hand, since
infϕ∈NJ(ϕ)≤infϕ∈H10(Ω)∖{0}J(s∗ϕϕ). |
Then (40) follows from the above two inequalities.
By (40), the definition of
d=minϕ∈H10(Ω)∖{0}J(s∗ϕϕ)=p−12(p+1)minϕ∈H10(Ω)∖{0}(‖∇ϕ‖L2‖ϕ‖Lp+1σ)2(p+1)p−1=p−12(p+1)C−2(p+1)p−1pσ. |
Theorem 3.4. Assume (3) holds. Let
√2(p+1)dp−1≤λρ≤Λρ≤√2(p+1)(λ1+1)ρλ1(p−1). | (42) |
Proof. Let
λρ≤Λρ. | (43) |
Since
d=infϕ∈NJ(ϕ)=p−12(p+1)infϕ∈N‖∇ϕ‖2L2≤p−12(p+1)infϕ∈Nρ‖ϕ‖2H10=p−12(p+1)λ2ρ, |
which implies
λρ≥√2(p+1)dp−1 |
On the other hand, by (8) and (18), we have
Λρ=supϕ∈Nρ‖ϕ‖H10≤√λ1+1λ1supϕ∈Nρ‖∇ϕ‖L2≤√λ1+1λ1√2(p+1)ρp−1. |
Combining the above two inequalities with (43), we get (42), the proof is complete.
Theorem 3.5. Assume (3) holds and
Proof. We only prove the invariance of
For any
‖∇ϕ‖2L2<‖ϕ‖p+1Lp+1σ≤Cp+1pσ‖∇ϕ‖p+1L2, |
which implies
‖∇ϕ‖L2>C−p+1p−1pσ. | (44) |
Let
I(u(⋅,t))<0,t∈[0,ε]. | (45) |
Then by (24),
J(u(⋅,t))<d for t∈(0,ε]. | (46) |
We argument by contradiction. Since
J(u(⋅,t0))<d | (47) |
(note (25) and (46),
‖∇u(⋅,t0)‖L2≥C−p+1p−1pσ>0, |
which, together with
J(u(⋅,t0))≥d, |
which contradicts (47). So the conclusion holds.
Theorem 3.6. Assume (3) holds and
‖∇u(⋅,t)‖2L2≥2(p+1)p−1d,0≤t<Tmax, | (48) |
where
Proof. Let
By the proof in Theorem 3.3,
d=minϕ∈H10(Ω)∖{0}J(s∗ϕϕ)≤minϕ∈N−J(s∗ϕϕ)≤J(s∗uu(⋅,t))=(s∗u)22‖∇u(⋅,t)‖2L2−(s∗u)p+1p+1‖u(⋅,t)‖p+1Lp+1σ≤((s∗u)22−(s∗u)p+1p+1)‖∇u(⋅,t)‖2L2, |
where we have used
d=minϕ∈H10(Ω)∖{0}J(s∗ϕϕ)≤minϕ∈N−J(s∗ϕϕ)≤J(s∗uu(⋅,t))=(s∗u)22‖∇u(⋅,t)‖2L2−(s∗u)p+1p+1‖u(⋅,t)‖p+1Lp+1σ≤((s∗u)22−(s∗u)p+1p+1)‖∇u(⋅,t)‖2L2, |
Then
d≤max0≤s≤1(s22−sp+1p+1)‖∇u(⋅,t)‖2L2=(s22−sp+1p+1)s=1‖∇u(⋅,t)‖2L2=p−12(p+1)‖∇u(⋅,t)‖2L2, |
and (48) follows from the above inequality.
Theorem 3.7. Assume (3) holds and
Proof. Firstly, we show
12‖∇u0‖2L2−1p+1‖u0‖p+1Lp+1σ=J(u0)<λ1(p−1)2(λ1+1)(p+1)‖u0‖2H10≤p−12(p+1)‖∇u0‖2L2, |
which implies
I(u0)=‖∇u0‖2L2−‖u0‖p+1Lp+1σ<0. |
Secondly, we prove
J(u0)<λ1(p−1)2(λ1+1)(p+1)‖u0‖2H10<λ1(p−1)2(λ1+1)(p+1)‖u(⋅,t0)‖2H10≤p−12(p+1)‖∇u(⋅,t0)‖2L2. | (49) |
On the other hand, by (24), (12), (13) and
J(u0)≥J(u(⋅,t0))=p−12(p+1)‖∇u(⋅,t0)‖2L2, |
which contradicts (49). The proof is complete.
Proof of Theorem 2.4. Let
J(u0)≥J(u(⋅,t))≥p−12(p+1)‖∇u(⋅,t)‖2L2,0≤t<Tmax, |
which implies
‖∇u(⋅,t)‖L2≤√2(p+1)J(u0)p−1,0≤t<∞. | (50) |
Next, we prove
ddt(‖u(⋅,t)‖2H10)=−2I(u(⋅,t))=−2(‖∇u(⋅,t)‖2L2−‖u(⋅,t)‖p+1Lp+1σ)≤−2(1−Cp+1pσ‖∇u(⋅,t)‖p−1L2)‖∇u(⋅,t)‖2L2≤−2(1−Cp+1pσ(√2(p+1)J(u0)p−1)p−1)‖∇u(⋅,t)‖2L2=−2(1−(J(u0)d)p−12)‖∇u(⋅,t)‖2L2≤−2λ1λ1+1(1−(J(u0)d)p−12)‖u(⋅,t)‖2H10, |
which leads to
‖u(⋅,t)‖2H10≤‖u0‖2H10exp[−2λ1λ1+1(1−(J(u0)d)p−12)t]. |
The proof is complete.
Proof of Theorem 2.5. Let
Firstly, we consider the case
ξ(t):=(∫t0‖u(⋅,s)‖2H10ds)12,η(t):=(∫t0‖us(⋅,s)‖2H10ds)12,0≤t<Tmax. | (51) |
For any
F(t):=ξ2(t)+(T∗−t)‖u0‖2H10+β(t+α)2,0≤t≤T∗. | (52) |
Then
F(0)=T∗‖u0‖2H10+βα2>0, | (53) |
F′(t)=‖u(⋅,t)‖2H10−‖u0‖2H10+2β(t+α)=2(12∫t0dds‖u(⋅,s)‖2H10ds+β(t+α)),0≤t≤T∗, | (54) |
and (by (24), (12), (13), (48), (25))
F″(t)=−2I(u(⋅,t))+2β=(p−1)‖∇u(⋅,t)‖2L2−2(p+1)J(u(⋅,t))+2β≥2(p+1)(d−J(u0))+2(p+1)η2(t)+2β,0≤t≤T∗. | (55) |
Since
F′(t)≥2β(t+α). |
Then
F(t)=F(0)+∫t0F′(s)ds≥T∗‖u0‖2H10+βα2+2αβt+βt2,0≤t≤T∗. | (56) |
By (6), Schwartz's inequality and Hölder's inequality, we have
12∫t0dds‖u(⋅,s)‖2H10ds=∫t0(u(⋅,s),us(⋅,s))H10ds≤∫t0‖u(⋅,s)‖H10‖us(⋅,s)‖H10ds≤ξ(t)η(t),0≤t≤T∗, |
which, together with the definition of
(F(t)−(T∗−t)‖u0‖2H10)(η2(t)+β)=(ξ2(t)+β(t+α)2)(η2(t)+β)=ξ2(t)η2(t)+βξ2(t)+β(t+α)2η2(t)+β2(t+α)2≥ξ2(t)η2(t)+2ξ(t)η(t)β(t+α)+β2(t+α)2≥(ξ(t)η(t)+β(t+α))2≥(12∫t0dds‖u(⋅,s)‖2H10ds+β(t+α))2,0≤t≤T∗. |
Then it follows from (54) and the above inequality that
(F′(t))2=4(12t∫0dds‖u(s)‖2H10ds+β(t+α))2≤4F(t)(η2(t)+β),0≤t≤T∗. | (57) |
In view of (55), (56), and (57), we have
F(t)F″(t)−p+12(F′(t))2≥F(t)(2(p+1)(d−J(u0))−2pβ),0≤t≤T∗. |
If we take
0<β≤p+1p(d−J(u0)), | (58) |
then
T∗≤F(0)(p+12−1)F′(0)=T∗‖u0‖2H10+βα2(p−1)αβ. |
Then for
α∈(‖u0‖2H10(p−1)β,∞), | (59) |
we get
T∗≤βα2(p−1)αβ−‖u0‖2H10. |
Minimizing the above inequality for
T∗≤βα2(p−1)αβ−‖u0‖2H10|α=2‖u0‖2H10(p−1)β=4‖u0‖2H10(p−1)2β. |
Minimizing the above inequality for
T∗≤4p‖u0‖2H10(p−1)2(p+1)(d−J(u0)). |
By the arbitrariness of
Tmax≤4p‖u0‖2H10(p−1)2(p+1)(d−J(u0)). |
Secondly, we consider the case
Proof of Theorems 2.6 and 2.7. Since Theorem 2.6 follows from Theorem 2.7 directly, we only need to prove Theorem 2.7.
Firstly, we show
d=infϕ∈NJ(ϕ)=p−12(p+1)infϕ∈N‖∇ϕ‖2L2. |
Then a minimizing sequence
limk↑∞J(ϕk)=p−12(p+1)limk↑∞‖∇ϕk‖2L2=d, | (60) |
which implies
(1)
(2)
Now, in view of
limk↑∞J(ϕk)=p−12(p+1)limk↑∞‖∇ϕk‖2L2=d, | (60) |
We claim
‖∇φ‖2L2=‖φ‖p+1Lp+1σ i.e. I(φ)=0. | (62) |
In fact, if the claim is not true, then by (61),
‖∇φ‖2L2<‖φ‖p+1Lp+1σ. |
By the proof of Theorem 3.3, we know that
J(s∗φφ)≥d, | (63) |
where
J(s∗φφ)≥d, | (63) |
On the other hand, since
J(s∗φφ)=p−12(p+1)(s∗φ)2‖∇φ‖2L2<p−12(p+1)‖∇φ‖2L2≤p−12(p+1)lim infk↑∞‖∇ϕk‖2L2=d, |
which contradicts to (63). So the claim is true, i.e.
limk↑∞‖∇ϕk‖2L2=‖φ‖p+1Lp+1σ, |
which, together with
Second, we prove
limk↑∞‖∇ϕk‖2L2=‖φ‖p+1Lp+1σ, |
Then
A:={τ(s)(φ+sv):s∈(−ε,ε)} |
is a curve on
A:={τ(s)(φ+sv):s∈(−ε,ε)} |
where
ξ:=2∫Ω∇(φ+sv)⋅∇vdx‖φ+sv‖p+1Lp+1σ,η:=(p+1)∫Ω|x|σ|φ+sv|p−1(φ+sv)vdx‖∇(φ+sv)‖2L2. |
Since (62), we get
τ′(0)=1(p−1)‖φ‖p+1Lp+1σ(2∫Ω∇φ∇vdx−(p+1)∫Ω|x|σ|φ|p−1φvdx). | (65) |
Let
ϱ(s):=J(τ(s)(φ+sv))=τ2(s)2‖∇(φ+sv)‖2L2−τp+1(s)p+1‖φ+sv‖p+1Lp+1σ,s∈(−ε,ε). |
Since
0=ϱ′(0)=τ(s)τ′(s)‖∇(φ+sv)‖2L2+τ2(s)∫Ω∇(φ+sv)⋅∇vdx|s=0−τp(s)τ′(s)‖φ+sv‖p+1Lp+1σ−τp+1(s)∫Ω|x|σ|φ+sv|p−1(φ+sv)vdx|s=0=∫Ω∇φ⋅∇vdx−∫Ω|x|σ|φ|p−1φvdx. |
So,
Finally, in view of Definition 2.3 and
d=infϕ∈Φ∖{0}J(ϕ). | (66) |
In fact, by the above proof and (27), we have
d=infϕ∈NJ(ϕ) |
and
Proof of Theorem 2.8. Let
ω(u0)=∩t≥0¯{u(⋅,s):s≥t}H10(Ω) |
the
(i) Assume
v(x,t)={u(x,t), if 0≤t≤t0;0, if t>t0 |
is a global weak solution of problem (1), and the proof is complete.
We claim that
I(u(⋅,t))>0,0≤t<Tmax. | (67) |
Since
I(u(⋅,t))>0,0≤t<t0 | (68) |
and
I(u(⋅,t0))=0, | (69) |
which together with the definition of
‖u(⋅,t0)‖H10≥λρ. | (70) |
On the other hand, it follows from (24), (68) and
‖u(⋅,t)‖H10<‖u0‖H10≤λρ, |
which contradicts (70). So (67) is true. Then by (24) again, we get
‖u(⋅,t)‖H10≤‖u0‖H10,0≤t<Tmax, |
which implies
By (24) and (67),
limt↑∞‖u(⋅,t)‖H10=c. |
Taking
∫∞0I(u(⋅,s))ds≤12(‖u0‖2H10−c)<∞. |
Note that
limn↑∞I(u(⋅,tn))=0. | (71) |
Let
u(⋅,tn)→ω in H10(Ω) as n↑∞. | (72) |
Then by (71), we get
I(ω)=limn↑∞I(u(⋅,tn))=0. | (73) |
As the above, one can easily see
‖ω‖H10<λρ≤λJ(u0),J(ω)<J(u0)⏟⇒ω∈JJ(u0), |
which implies
limt↑∞‖u(⋅,t)‖H10=limn↑∞‖u(⋅,tn)‖H10=‖ω‖H10=0. |
(ⅱ) Assume
I(u(⋅,t))<0,0≤t<Tmax. | (74) |
Since
I(u(⋅,t))<0,0≤t<t0 | (75) |
and
I(u(⋅,t0))=0. | (76) |
Since (75), by (44) and
‖∇u(⋅,t0)‖L2≥C−p+1p−1pσ, |
which, together with the definition of
‖u(⋅,t0)‖H10≤Λρ. | (77) |
On the other hand, it follows from (24), (75) and
‖u(⋅,t)‖H10>‖u0‖H10≥Λρ, |
which contradicts (77). So (74) is true.
Suppose by contradiction that
limt↑∞‖u(⋅,t)‖H10=˜c, |
Taking
−∫∞0I(u(⋅,s))ds≤12(˜c−‖u0‖2H10)<∞. |
Note
\begin{equation} \lim\limits_{n\uparrow\infty}I(u(\cdot,t_n)) = 0. \end{equation} | (78) |
Let
\begin{equation} u(\cdot,t_n)\rightarrow\omega \hbox{ in }H_0^1( \Omega)\hbox{ as }n\uparrow\infty. \end{equation} | (79) |
Since
\begin{equation*} \lim\limits_{t\uparrow\infty}\|u(\cdot,t)\|_{H_0^1} = \lim\limits_{n\uparrow\infty}\|u(\cdot,t_n)\|_{H_0^1} = \|\omega\|_{H_0^1}. \end{equation*} |
Then by (78), we get
\begin{equation} I(\omega) = \lim\limits_{n\uparrow\infty}I(u(\cdot,t_n)) = 0. \end{equation} | (80) |
By (24), (25) and (74), one can easily see
\begin{equation*} \|\omega\|_{H_0^1} > \|u_0\|_{H_0^1}\geq\Lambda_\rho\geq\Lambda_{J(u_0)},\; \; \; \underbrace{J(\omega) < J(u_0)}_{\Rightarrow\omega\in J^{J(u_0)}}, \end{equation*} |
which implies
Proof of Theorem 2.9. Let
\begin{equation} \|\nabla u(\cdot,t)\|_{L^2}^2\geq \frac{ \lambda_1}{ \lambda_1+1}\|u(\cdot,t)\|_{H_0^1}^2\geq\frac{ \lambda_1}{ \lambda_1+1}\|u_0\|_{H_0^1}^2,\; \; \; 0\leq t < {T_{\max}}. \end{equation} | (81) |
The remain proofs are similar to the proof of Theorem 2.9. For any
\begin{equation} \begin{split} F''(t) = &-2I(u(\cdot,t))+2 \beta\\ = &(p-1)\|\nabla u(\cdot,t)\|_{L^2}^2-2(p+1)J(u(\cdot,t))+2 \beta\\ \geq&\frac{ \lambda_1(p-1)}{ \lambda_1+1}\|u_0\|_{H_0^1}^2-2(p+1)J(u_0)+2(p+1)\eta^2(t)+2 \beta,\; \; \; 0\leq t\leq T^*. \end{split} \end{equation} | (82) |
We also have (56) and (57). Then it follows from (56), (57) and (82) that
\begin{align*} &F(t)F''(t)-\frac{p+1}{2}(F'(t))^2\\ \geq& F(t)\left(\frac{ \lambda_1(p-1)}{ \lambda_1+1}\|u_0\|_{H_0^1}^2-2(p+1)J(u_0)-2p \beta\right),\; \; \; 0\leq t\leq T^*. \end{align*} |
If we take
\begin{equation} 0 < \beta\leq\frac{1}{2p}\left(\frac{ \lambda_1(p-1)}{ \lambda_1+1}\|u_0\|_{H_0^1}^2-2(p+1)J(u_0)\right), \end{equation} | (83) |
then
\begin{equation*} T^*\leq\frac{F(0)}{\left(\frac{p+1}{2}-1\right) F'(0)} = \frac{T^*\|u_0\|_{H_0^1}^2+ \beta \alpha^2}{(p-1) \alpha \beta}. \end{equation*} |
Then for
\begin{equation} \alpha\in\left(\frac{\|u_0\|_{H_0^1}^2}{(p-1) \beta},\infty\right), \end{equation} | (84) |
we get
\begin{equation*} T^*\leq \frac{ \beta \alpha^2}{(p-1) \alpha \beta-\|u_0\|_{H_0^1}^2}. \end{equation*} |
Minimizing the above inequality for
\begin{equation*} T^*\leq\left.\frac{ \beta \alpha^2}{(p-1) \alpha \beta-\|u_0\|_{H_0^1}^2}\right|_{ \alpha = \frac{2\|u_0\|_{H_0^1}^2}{(p-1) \beta}} = \frac{4\|u_0\|_{H_0^1}^2}{(p-1)^2 \beta}. \end{equation*} |
Minimizing the above inequality for
\begin{equation*} T^*\leq\frac{8p\|u_0\|_{H_0^1}^2}{(p-1)^2\left(\frac{ \lambda_1(p-1)}{ \lambda_1+1}\|u_0\|_{H_0^1}^2-2(p+1)J(u_0)\right)}. \end{equation*} |
By the arbitrariness of
\begin{equation*} {T_{\max}}\leq\frac{8p\|u_0\|_{H_0^1}^2}{(p-1)^2\left(\frac{ \lambda_1(p-1)}{ \lambda_1+1}\|u_0\|_{H_0^1}^2-2(p+1)J(u_0)\right)}. \end{equation*} |
Proof of Theorem 2.10. For any
\begin{equation} \| \alpha\psi\|_{H_0^1}^2 > \frac{2( \lambda_1+1)(p+1)}{ \lambda_1(p-1)}M. \end{equation} | (85) |
For such
\begin{equation} J(s_3^*\phi)\geq M-J( \alpha \psi), \end{equation} | (86) |
where (see Remark 5)
\begin{equation} J(s_3^*\phi)\geq M-J( \alpha \psi), \end{equation} | (86) |
which can be done since
\begin{equation*} J(s_3^*\phi) = \frac{p-1}{2(p+1)}\left(\frac{\|\nabla \phi\|_{L^2}}{\|\phi\|_{L_\sigma^{p+1}}}\right)^{\frac{2(p+1)}{p-1}} \end{equation*} |
and
By Remark 5 again,
\begin{equation} J(\{s\phi:0\leq s < \infty\}) = (-\infty, J(s_3^*\phi)]. \end{equation} | (87) |
By (87) and (86), we can choose
\begin{equation*} J(u_0) = J(v)+J( \alpha\psi) = M \end{equation*} |
and (note (85))
\begin{align*} J(u_0)& = M < \frac{ \lambda_1(p-1)}{2( \lambda_1+1)(p+1)}\| \alpha\psi\|_{H_0^1}^2\\ &\leq\frac{ \lambda_1(p-1)}{2( \lambda_1+1)(p+1)}\left(\| \alpha\psi\|_{H_0^1}^2+\|v\|_{H_0^1}^2\right)\\ & = \frac{ \lambda_1(p-1)}{2( \lambda_1+1)(p+1)}\|u_0\|_{H_0^1}^2. \end{align*} |
Let
[1] | Statista (2023) E-commerce in Indonesia-statistics & Facts. Available from: https://www.statista.com/topics/5742/e-commerce-in-indonesia/#topicOverview. |
[2] | AJ Marketing (2023) Indonesia ecommerce market: Data, trends, top stories. Available from: https://www.indonesia-investments.com/business/business-columns/indonesia-ecommerce-market-data-trends-top-stories/item9630. |
[3] |
Allen J, Piecyk M, Piotrowska M, et al. (2018) Understanding the impact of e-commerce on last-mile light goods vehicle activity in urban areas: The case of London. Transport Res D Tr E 61: 325–338. https://doi.org/10.1016/j.trd.2017.07.020 doi: 10.1016/j.trd.2017.07.020
![]() |
[4] |
Mangiaracina R, Perego A, Seghezzi A, et al. (2019) Innovative solutions to increase last-mile delivery efficiency in B2C e-commerce: A literature review. Int J Phys Distr Log 49: 901–920. https://doi.org/10.1108/IJPDLM-02-2019-0048 doi: 10.1108/IJPDLM-02-2019-0048
![]() |
[5] |
Anvari R (2023) Green, closed loop, and reverse supply chain: A literature review. J Bus Manag 1: 33–57. https://doi.org/10.47747/jbm.v1i1.956 doi: 10.47747/jbm.v1i1.956
![]() |
[6] |
Chen C, Pan S (2016) Using the crowd of taxis to last mile delivery in E-commerce: A methodological research. Serv Orientat Holonic Multi-Ag Manufact 61–70. https://doi.org/10.1007/978-3-319-30337-6_6 doi: 10.1007/978-3-319-30337-6_6
![]() |
[7] |
Anvari R, Askari KOA (2024) Evaluating the efficiency of resource and energy consumption of farmland systems. SSRN Electron J. https://dx.doi.org/10.2139/ssrn.4761505 doi: 10.2139/ssrn.4761505
![]() |
[8] | Lavanya R (2016) The 2015 Paris agreement: Interplay between hard, soft and non-obligations. J Environ law 28: 337–358. https://www.jstor.org/stable/26168923 |
[9] |
Rogelj J, Elzen MD, Hö hne N, et al. (2016) Paris Agreement climate proposals need a boost to keep warming well below 2 ℃. Nature 534: 631–639. https://doi.org/10.1038/nature18307 doi: 10.1038/nature18307
![]() |
[10] |
Maitreyee D, Rangarajan K (2020) Impact of policy initiatives and collaborative synergy on sustainability and business growth of Indian SMEs. Indian Growth Dev Rev 13: 607–627. https://doi.org/10.1108/IGDR-09-2019-0095 doi: 10.1108/IGDR-09-2019-0095
![]() |
[11] |
Muangmee C, Pikiewicz ZD, Meekaewkunchorn N, et al. (2021) Green entrepreneurial orientation and green innovation in small and medium-sized enterprises (Smes). Soc Sci 10: 2021. https://doi.org/10.3390/socsci10040136 doi: 10.3390/socsci10040136
![]() |
[12] |
Raj A, Sah B (2019) Analyzing critical success factors for implementation of drones in the logistics sector using grey-DEMATEL based approach. Comput Ind Eng 138: 106118. https://doi.org/10.1016/j.cie.2019.106118 doi: 10.1016/j.cie.2019.106118
![]() |
[13] |
Kellermann R, Biehle T, Fischer L (2020) Drones for parcel and passenger transportation: A literature review. Transp Res Interdisc 4: 100088. https://doi.org/10.1016/j.trip.2019.100088 doi: 10.1016/j.trip.2019.100088
![]() |
[14] |
Cam LNT (2023) A rising trend in eco-friendly products: A health-conscious approach to green buying. Heliyon 9: e19845. https://doi.org/10.1016/j.heliyon.2023.e19845 doi: 10.1016/j.heliyon.2023.e19845
![]() |
[15] |
Mahesh S, Ramadurai G, Nagendra SMS (2019) Real-world emissions of gaseous pollutants from motorcycles on Indian urban arterials. Transport Res D Tr E 76: 72–84. https://doi.org/10.1016/j.trd.2019.09.010 doi: 10.1016/j.trd.2019.09.010
![]() |
[16] |
Mathew AO, Jha AN, Lingappa AK, et al. (2021) Attitude towards drone food delivery services—role of innovativeness, perceived risk, and green image. J Open Innov Technol Mark Complex 7: 144. https://doi.org/10.3390/joitmc7020144 doi: 10.3390/joitmc7020144
![]() |
[17] | Dileep MR, Navaneeth AV, Ullagaddi S, et al. (2020) A study and analysis on various types of agricultural drones and its applications, in 2020 Fifth International Conference on Research in Computational Intelligence and Communication Networks, 181–185. https://doi.org/10.1109/ICRCICN50933.2020.9296195 |
[18] |
Puri V, Nayyar A, Raja L (2017) Agriculture drones: A modern breakthrough in precision agriculture. J Stat Manag Syst 20: 507–518. https://doi.org/10.1080/09720510.2017.1395171 doi: 10.1080/09720510.2017.1395171
![]() |
[19] |
Nwaogu JM, Yang Y, Chan AP, et al. (2023) Application of drones in the architecture, engineering, and construction (AEC) industry. Automat Constr 150. https://doi.org/10.1016/j.autcon.2023.104827 doi: 10.1016/j.autcon.2023.104827
![]() |
[20] |
Goodchild A, Toy J (2018) Delivery by drone: An evaluation of unmanned aerial vehicle technology in reducing CO2 emissions in the delivery service industry. Transport Res D Tr E 61: 58–67. https://doi.org/10.1016/j.trd.2017.02.017 doi: 10.1016/j.trd.2017.02.017
![]() |
[21] | Clutch (2020) Drone delivery: Benefits and challenges. Available from: https://clutch.co/logistics/resources/drone-delivery-statistics-benefits-challenges. |
[22] |
Hwang J, Choe JY (2019) Exploring perceived risk in building successful drone food delivery services. Int J Contemp Hosp M 31: 3249–3269. https://doi.org/10.1108/IJCHM-07-2018-0558 doi: 10.1108/IJCHM-07-2018-0558
![]() |
[23] |
Yoo W, Yu E, Jung J (2018) Drone delivery: Factors affecting the public's attitude and intention to adopt. Telemat Inform 35: 1687–1700. https://doi.org/10.1016/j.tele.2018.04.014 doi: 10.1016/j.tele.2018.04.014
![]() |
[24] |
Hu ZH, Huang YL, Li YN, et al. (2024) Drone-based instant delivery hub-and-spoke network optimization. Drones 8. https://doi.org/10.3390/drones8060247 doi: 10.3390/drones8060247
![]() |
[25] |
Chiang WC, Li Y, Shang J, et al. (2019) Impact of drone delivery on sustainability and cost: Realizing the UAV potential through vehicle routing optimization. Appl Energy 242: 1164–1175. https://doi.org/10.1016/j.apenergy.2019.03.117 doi: 10.1016/j.apenergy.2019.03.117
![]() |
[26] |
Adnan N, Nordin SM, Rahman I, et al. (2017) A new era of sustainable transport: An experimental examination on forecasting adoption behavior of EVs among Malaysian consumer. Transport Res A-Pol 103: 279–295. https://doi.org/10.1016/j.tra.2017.06.010 doi: 10.1016/j.tra.2017.06.010
![]() |
[27] |
Baeshen Y, Soomro YA, Bhutto MY (2021) Determinants of green innovation to achieve sustainable business performance: Evidence from SMEs. Front Psychol 12: 767968. https://doi.org/10.3389/fpsyg.2021.767968 doi: 10.3389/fpsyg.2021.767968
![]() |
[28] |
Rodrigues TA, Patrikar J, Oliveira NL, et al. (2022) Drone flight data reveal energy and greenhouse gas emissions savings for very small package delivery. Patterns 3: 100569. https://doi.org/10.1016/j.patter.2022.100569 doi: 10.1016/j.patter.2022.100569
![]() |
[29] |
Balderjahn I (1988) Personality variables and environmental attitudes as predictors of ecologically responsible consumption patterns. J Bus Res 17: 51–56. https://doi.org/10.1016/0148-2963(88)90022-7 doi: 10.1016/0148-2963(88)90022-7
![]() |
[30] |
Crosby LA, Taylor JR (1982) Consumer satisfaction with Michigan's container deposit law: An ecological perspective. J Mark 46: 47–60. https://doi.org/10.2307/1251159 doi: 10.2307/1251159
![]() |
[31] |
Thieme J, Royne MB, Jha S, et al. (2015) Factors affecting the relationship between environmental concern and behaviors. Mark Intell Plan 33: 675–690. https://doi.org/10.1108/MIP-08-2014-0149 doi: 10.1108/MIP-08-2014-0149
![]() |
[32] |
Ozaki R, Sevastyanova K (2011) Going hybrid: An analysis of consumer purchase motivations. Energy Policy 39: 2217–2227. https://doi.org/10.1016/j.enpol.2010.04.024 doi: 10.1016/j.enpol.2010.04.024
![]() |
[33] |
Teoh CW, Gaur SS (2019) Environmental concern: An issue for poor or rich. Manag Environ Qual 30: 227–242. https://doi.org/10.1108/MEQ-02-2018-0046 doi: 10.1108/MEQ-02-2018-0046
![]() |
[34] |
Bamberg S (2003) How does environmental concern influence specific environmentally related behaviors? A new answer to an old question. J Environ Psychol 23: 21–32. https://doi.org/10.1016/S0272-4944(02)00078-6 doi: 10.1016/S0272-4944(02)00078-6
![]() |
[35] |
Stafford MR, Stafford TF, Collier JE (2006) The dimensionality of environmental concern: Validation of component measures. Interdiscip Environ Rev 8: 43–61. https://doi.org/10.1504/IER.2006.053946 doi: 10.1504/IER.2006.053946
![]() |
[36] |
Ajzen I (1991) The theory of planned behavior. Organ Behav Hum Dec 50: 179–211. https://doi.org/10.1016/0749-5978(91)90020-T doi: 10.1016/0749-5978(91)90020-T
![]() |
[37] | Schiffman LG, Kanuk LL (2004) Consumer behavior, 8th International Edition, Prentice Hall. |
[38] | Kotler P, (2009) Marketing management, Jakarta: Erlangga. |
[39] | Hawkins DI, Mothersbaugh DL (2013) Consumer behavior: Building marketing strategy, New York: McGraw-Hill Irwin. |
[40] |
Crosby LA, Gill JD, Taylor JR (1981) Consumer voter behavior in the passage of the Michigan container law. J Mark 45: 19–32. https://doi.org/10.1177/002224298104500203 doi: 10.1177/002224298104500203
![]() |
[41] |
Bettencourt LA, Hughner RS, Kuntze RJ, et al. (1999) Lifestyle of the tight and frugal: Theory and measurement. J Consum Res 26: 85–99. https://doi.org/10.1086/209552 doi: 10.1086/209552
![]() |
[42] | Schiffman LG, Wisenblit J (2015) Consumer behavior, England: Pearson Education Limited, 2015. |
[43] |
Agarwal J, Malhotra NK (2005) An integrated model of attitude and affect: Theoretical foundation and an empirical investigation. J Bus Res 58: 483–493. https://doi.org/10.1016/S0148-2963(03)00138-3 doi: 10.1016/S0148-2963(03)00138-3
![]() |
[44] |
Reaños MAT, Meier D, Curtis J, et al. (2023) The role of energy, financial attitudes and environmental concerns on perceived retrofitting benefits and barriers: Evidence from Irish home owners. Energy Buildings 297: 113448. https://doi.org/10.1016/j.enbuild.2023.113448 doi: 10.1016/j.enbuild.2023.113448
![]() |
[45] |
De Groot J, Steg L (2007) General beliefs and the theory of planned behavior: The role of environmental concerns in the TPB. J Appl Soc Psychol 37: 1817–1836. https://doi.org/10.1111/j.1559-1816.2007.00239.x doi: 10.1111/j.1559-1816.2007.00239.x
![]() |
[46] |
Lau JL, Hashim AH (2020) Mediation analysis of the relationship between environmental concern and intention to adopt green concepts. Smart Sustain Built 9: 539–556. https://doi.org/10.1108/SASBE-09-2018-0046 doi: 10.1108/SASBE-09-2018-0046
![]() |
[47] |
Li D, Zhao L, Ma S, et al. (2019) What influences an individual's pro-environmental behavior? A literature review. Resour Conserv Recy 146: 28–34. https://doi.org/10.1016/j.resconrec.2019.03.024 doi: 10.1016/j.resconrec.2019.03.024
![]() |
[48] |
Jaiswal D, Kaushal V, Kant R, et al. (2021) Consumer adoption intention for electric vehicles: Insights and evidence from Indian sustainable transportation. Technol Forecast Soc 173: 121089. https://doi.org/10.1016/j.techfore.2021.121089 doi: 10.1016/j.techfore.2021.121089
![]() |
[49] |
Yue B, Sheng G, She S, et al. (2020) Impact of consumer environmental responsibility on green consumption behavior in China: The role of environmental concern and price sensitivity. Sustainability 12. https://doi.org/10.3390/su12052074 doi: 10.3390/su12052074
![]() |
[50] | SIRCLO (2021) Navigating Indonesia's e-commerce: Omnichannel as the future of retail. Available from: https://www.sirclo.com/research-reports/navigating-indonesia-s-e-commerce-omnichannel-as-the-future-of-retail. |
[51] | Sekaran U, Bougie R (2016) Research methods for business: A skill-building approach, 7 Eds., Chichester, West Sussex, United Kingdom: John Wiley & Sons. |
[52] |
Bentler PM, Chou CP (1987) Practical issues in structural modeling. Sociol Method Res 16: 78–117. https://doi.org/10.1177/0049124187016001004 doi: 10.1177/0049124187016001004
![]() |
[53] | Nunnally JC, Bernstein IH (1967) Psychometric theory, New York: McGraw-Hill. |
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