
Internet of Things (IoT) is a terminology used for a mixed connection of heterogeneous objects to the internet and to each other with the employment of recent technological and communication infrastructures. Its incorporation into engineering systems have gradually become very popular in recent times as it promises to transform and ease the life of end users. The use of IoT in smart energy systems (SES) facilitates an ample offer of variety of applications that transverses through a wide range of areas in energy systems. With the numerous benefits that includes unmatched fast communication between subsystems, the maximization of energy use, the decrease in environmental impacts and a boost in the dividends of renewable energies, IoT has grown into an emerging innovative technology to be integrated into smart energy systems. In this work, we have provided an overview of the link between SES, IoT and Internet of Energy (IoE). The main applications of IoT in smart energy systems consisting of smart industries, smart homes and buildings, and smart cities are explored and analyzed. The paper also explores the challenges limiting the employment of IoT technologies in SES and the possible remedies to these challenges. In addition, the future trends of this technology, its research direction and reasons why industry should adopt it are also addressed. The aim of this work is to furnish researchers in this field, decision and energy policy makers, energy economist and energy administrators with a possible literature outline on the roles and impacts of IoT technology in smart energy systems.
Citation: Efe Francis Orumwense, Khaled Abo-Al-Ez. Internet of Things for smart energy systems: A review on its applications, challenges and future trends[J]. AIMS Electronics and Electrical Engineering, 2023, 7(1): 50-74. doi: 10.3934/electreng.2023004
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Internet of Things (IoT) is a terminology used for a mixed connection of heterogeneous objects to the internet and to each other with the employment of recent technological and communication infrastructures. Its incorporation into engineering systems have gradually become very popular in recent times as it promises to transform and ease the life of end users. The use of IoT in smart energy systems (SES) facilitates an ample offer of variety of applications that transverses through a wide range of areas in energy systems. With the numerous benefits that includes unmatched fast communication between subsystems, the maximization of energy use, the decrease in environmental impacts and a boost in the dividends of renewable energies, IoT has grown into an emerging innovative technology to be integrated into smart energy systems. In this work, we have provided an overview of the link between SES, IoT and Internet of Energy (IoE). The main applications of IoT in smart energy systems consisting of smart industries, smart homes and buildings, and smart cities are explored and analyzed. The paper also explores the challenges limiting the employment of IoT technologies in SES and the possible remedies to these challenges. In addition, the future trends of this technology, its research direction and reasons why industry should adopt it are also addressed. The aim of this work is to furnish researchers in this field, decision and energy policy makers, energy economist and energy administrators with a possible literature outline on the roles and impacts of IoT technology in smart energy systems.
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.
\begin{equation} \hbox{$H_0^1( \Omega)\hookrightarrow L^{p+1}( \Omega)$ continuously and compactly.} \end{equation} | (37) |
Then we have, for any
\begin{equation*} \|u\|_{L^{p+1}_{\sigma}}^{p+1} = \int_ \Omega|x|^\sigma|u|^{p+1}dx\leq R^\sigma\|u\|_{L^{p+1}}^{p+1}\lesssim\|u\|_{H_0^1}^{p+1}, \end{equation*} |
which, together with (37), implies
Case 2.
\begin{equation} \hbox{$H_0^1( \Omega)\hookrightarrow L^{\frac{(p+1)r}{r-1}}( \Omega)$ continuously and compactly,} \end{equation} | (38) |
for any
\begin{align*} \|u\|_{L^{p+1}_{\sigma}}^{p+1}& = \int_ \Omega|x|^\sigma|u|^{p+1}dx\\&\leq \left(\int_{B(0,R)}|x|^{\sigma r}dx\right)^{\frac1r}\left(\int_ \Omega|u|^{\frac{(p+1)r}{r-1}}dx\right)^{\frac{r-1}{r}}\\ &\leq\left\{ \begin{array}{ll} \left(\frac{2}{\sigma r+1}R^{\sigma r+1}\right)^{\frac 1r}\|u\|_{L^{\frac{(p+1)r}{r-1}}}^{p+1}\lesssim\|u\|_{H_0^1}^{p+1}, & n = 1;\\ \left(\frac{2\pi}{\sigma r+2}R^{\sigma r+2}\right)^{\frac 1r}\|u\|_{L^{\frac{(p+1)r}{r-1}}}^{p+1}\lesssim\|u\|_{H_0^1}^{p+1}, & n = 2, \end{array} \right. \end{align*} |
which, together with (38), implies
Case 3.
\begin{equation*} -\frac{\sigma}{n} < \frac1r < 1-\frac{(p+1)(n-2)}{2n}. \end{equation*} |
By the second inequality of the above inequalities, we have
\begin{equation*} \frac{(p+1)r}{r-1} = \frac{p+1}{1-\frac1r} < \frac{p+1}{\frac{(p+1)(n-2)}{2n}} = \frac{2n}{n-2}. \end{equation*} |
So,
\begin{equation} \hbox{$H_0^1( \Omega)\hookrightarrow L^{\frac{(p+1)r}{r-1}}( \Omega)$ continuously and compactly.} \end{equation} | (39) |
Then by Hölder's inequality, for any
\begin{align*} \|u\|_{L^{p+1}_{\sigma}}^{p+1}& = \int_ \Omega|x|^\sigma|u|^{p+1}dx\\&\leq \left(\int_{B(0,R)}|x|^{\sigma r}dx\right)^{\frac1r}\left(\int_ \Omega|u|^{\frac{(p+1)r}{r-1}}dx\right)^{\frac{r-1}{r}}\\ &\leq\left(\frac{\omega_{n-1}}{\sigma r+n}R^{\sigma r+n}\right)^{\frac1r}\|u\|_{L^{\frac{(p+1)r}{r-1}}}^{p+1}\lesssim\|u\|_{H_0^1}^{p+1}, \end{align*} |
which, together with (39), implies
Theorem 3.3. Assume
\begin{equation*} d = \frac{p-1}{2(p+1)}C_{p\sigma}^{\frac{2(p+1)}{p-1}}, \end{equation*} |
where
Proof. Firstly, we show
\begin{equation} \inf\limits_{\phi\in N}J(\phi) = \min\limits_{\phi\in H_0^1( \Omega)\setminus\{0\}}J(s_\phi^*\phi), \end{equation} | (40) |
where
\begin{equation} s_\phi^*: = \left(\frac{\|\nabla \phi\|_{L^2}^2}{\|\phi\|_{L_\sigma^{p+1}}^{p+1}}\right)^{\frac{1}{p-1}}. \end{equation} | (41) |
By the definition of
On one hand, since
\begin{equation*} \min\limits_{\phi\in H_0^1( \Omega)\setminus\{0\}}J(s_\phi^*\phi)\leq\min\limits_{\phi\in N}J(s_\phi^*\phi) = \min\limits_{\phi\in N}J(\phi). \end{equation*} |
On the other hand, since
\begin{equation*} \inf\limits_{\phi\in N}J(\phi)\leq \inf\limits_{\phi\in H_0^1( \Omega)\setminus\{0\}}J(s_\phi^*\phi). \end{equation*} |
Then (40) follows from the above two inequalities.
By (40), the definition of
\begin{align*} d& = \min\limits_{\phi\in H_0^1( \Omega)\setminus\{0\}}J(s_\phi^*\phi)\\ = &\frac{p-1}{2(p+1)}\min\limits_{\phi\in H_0^1( \Omega)\setminus\{0\}}\left(\frac{\|\nabla \phi\|_{L^2}}{\|\phi\|_{L_\sigma^{p+1}}}\right)^{\frac{2(p+1)}{p-1}}\\ = &\frac{p-1}{2(p+1)}C_{p\sigma}^{-\frac{2(p+1)}{p-1}}. \end{align*} |
Theorem 3.4. Assume (3) holds. Let
\begin{equation} \sqrt{\frac{2(p+1)d}{p-1}}\leq \lambda_\rho\leq \Lambda_\rho\leq \sqrt{\frac{2(p+1)( \lambda_1+1)\rho}{ \lambda_1(p-1)}}. \end{equation} | (42) |
Proof. Let
\begin{equation} \lambda_\rho\leq\Lambda_\rho. \end{equation} | (43) |
Since
\begin{align*} d& = \inf\limits_{\phi\in N}J(\phi)\\ & = \frac{p-1}{2(p+1)}\inf\limits_{\phi\in N}\|\nabla\phi\|_{L^2}^2\\ &\leq\frac{p-1}{2(p+1)}\inf\limits_{\phi\in N^\rho}\|\phi\|_{H_0^1}^2\\ & = \frac{p-1}{2(p+1)} \lambda_\rho^2, \end{align*} |
which implies
\begin{equation*} \lambda_\rho\geq\sqrt{\frac{2(p+1)d}{p-1}} \end{equation*} |
On the other hand, by (8) and (18), we have
\begin{equation*} \begin{split} \Lambda_\rho& = \sup\limits_{\phi\in N^\rho}\|\phi\|_{H_0^1}\\&\leq\sqrt{\frac{ \lambda_1+1}{ \lambda_1}}\sup\limits_{\phi\in N^\rho}\|\nabla\phi\|_{L^2}\\&\leq\sqrt{\frac{ \lambda_1+1}{ \lambda_1}}\sqrt{\frac{2(p+1)\rho}{p-1}}. \end{split} \end{equation*} |
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
\begin{equation*} \|\nabla\phi\|_{L^2}^2 < \|\phi\|_{L_\sigma^{p+1}}^{p+1}\leq C_{p\sigma}^{p+1}\|\nabla \phi\|_{L^2}^{p+1}, \end{equation*} |
which implies
\begin{equation} \|\nabla\phi\|_{L^2} > C_{p\sigma}^{-\frac{p+1}{p-1}}. \end{equation} | (44) |
Let
\begin{equation} I(u(\cdot,t)) < 0,\; \; \; t\in[0, \varepsilon]. \end{equation} | (45) |
Then by (24),
\begin{equation} J(u(\cdot,t)) < d\hbox{ for }t\in(0, \varepsilon]. \end{equation} | (46) |
We argument by contradiction. Since
\begin{equation} J(u(\cdot,t_0)) < d \end{equation} | (47) |
(note (25) and (46),
\begin{equation*} \|\nabla u(\cdot,t_0)\|_{L^2}\geq C_{p\sigma}^{-\frac{p+1}{p-1}} > 0, \end{equation*} |
which, together with
\begin{equation*} J(u(\cdot,t_0))\geq d, \end{equation*} |
which contradicts (47). So the conclusion holds.
Theorem 3.6. Assume (3) holds and
\begin{equation} \|\nabla u(\cdot,t)\|_{L^2}^2\geq\frac{2(p+1)}{p-1}d,\; \; \; 0\leq t < {T_{\max}}, \end{equation} | (48) |
where
Proof. Let
By the proof in Theorem 3.3,
\begin{align*} d& = \min\limits_{\phi \in H_0^1( \Omega)\setminus\{0\}}J(s^*_\phi\phi)\\ &\leq\min\limits_{\phi\in N^-}J(s_\phi^*\phi)\\ &\leq J(s_u^*u(\cdot,t))\\ & = \frac{(s_u^*)^2}{2}\|\nabla u(\cdot,t)\|_{L^2}^2-\frac{(s_u^*)^{p+1}}{p+1}\|u(\cdot,t)\|_{L_\sigma^{p+1}}^{p+1}\\ &\leq\left(\frac{(s_u^*)^2}{2}-\frac{(s_u^*)^{p+1}}{p+1}\right)\|\nabla u(\cdot,t)\|_{L^2}^2, \end{align*} |
where we have used
\begin{align*} d& = \min\limits_{\phi \in H_0^1( \Omega)\setminus\{0\}}J(s^*_\phi\phi)\\ &\leq\min\limits_{\phi\in N^-}J(s_\phi^*\phi)\\ &\leq J(s_u^*u(\cdot,t))\\ & = \frac{(s_u^*)^2}{2}\|\nabla u(\cdot,t)\|_{L^2}^2-\frac{(s_u^*)^{p+1}}{p+1}\|u(\cdot,t)\|_{L_\sigma^{p+1}}^{p+1}\\ &\leq\left(\frac{(s_u^*)^2}{2}-\frac{(s_u^*)^{p+1}}{p+1}\right)\|\nabla u(\cdot,t)\|_{L^2}^2, \end{align*} |
Then
\begin{equation*} \begin{split} d&\leq\max\limits_{0\leq s\leq 1}\left(\frac{s^2}{2}-\frac{s^{p+1}}{p+1}\right)\|\nabla u(\cdot,t)\|_{L^2}^2\\ & = \left(\frac{s^2}{2}-\frac{s^{p+1}}{p+1}\right)_{s = 1}\|\nabla u(\cdot,t)\|_{L^2}^2\\ & = \frac{p-1}{2(p+1)}\|\nabla u(\cdot,t)\|_{L^2}^2, \end{split} \end{equation*} |
and (48) follows from the above inequality.
Theorem 3.7. Assume (3) holds and
Proof. Firstly, we show
\begin{align*} \frac12\|\nabla u_0\|_{L^2}^2-\frac{1}{p+1}\|u_0\|_{L_\sigma^{p+1}}^{p+1}& = J(u_0)\\ & < \frac{ \lambda_1(p-1)}{2( \lambda_1+1)(p+1)}\|u_0\|_{H_0^1}^2\\ &\leq\frac{p-1}{2(p+1)}\|\nabla u_0\|_{L^2}^2, \end{align*} |
which implies
\begin{equation*} I(u_0) = \|\nabla u_0\|_{L^2}^2-\|u_0\|_{L_\sigma^{p+1}}^{p+1} < 0. \end{equation*} |
Secondly, we prove
\begin{equation} \begin{split} J(u_0)& < \frac{ \lambda_1(p-1)}{2( \lambda_1+1)(p+1)}\|u_0\|_{H_0^1}^2\\ & < \frac{ \lambda_1(p-1)}{2( \lambda_1+1)(p+1)}\|u(\cdot,t_0)\|_{H_0^1}^2\\ &\leq \frac{p-1}{2(p+1)}\|\nabla u(\cdot,t_0)\|_{L^2}^2. \end{split} \end{equation} | (49) |
On the other hand, by (24), (12), (13) and
\begin{equation*} J(u_0)\geq J(u(\cdot,t_0)) = \frac{p-1}{2(p+1)}\|\nabla u(\cdot,t_0)\|_{L^2}^2, \end{equation*} |
which contradicts (49). The proof is complete.
Proof of Theorem 2.4. Let
\begin{align*} J(u_0)&\geq J(u(\cdot,t))\geq\frac{p-1}{2(p+1)}\|\nabla u(\cdot,t)\|_{L^2}^2,\; \; \; 0\leq t < {T_{\max}}, \end{align*} |
which implies
\begin{equation} \|\nabla u(\cdot,t)\|_{L^2}\leq \sqrt{\frac{2(p+1)J(u_0)}{p-1}},\; \; \; 0\leq t < \infty. \end{equation} | (50) |
Next, we prove
\begin{align*} \frac{d}{dt}\left(\|u(\cdot,t)\|_{H_0^1}^2\right) = &-2I(u(\cdot,t)) = -2\left(\|\nabla u(\cdot,t)\|_{L^2}^2-\|u(\cdot,t)\|_{L_\sigma^{p+1}}^{p+1}\right)\\ \leq &-2\left(1-C_{p\sigma}^{p+1}\|\nabla u(\cdot,t)\|_{L^2}^{p-1}\right)\|\nabla u(\cdot,t)\|_{L^2}^2\\ \leq &-2\left(1-C_{p\sigma}^{p+1}\left(\sqrt{\frac{2(p+1)J(u_0)}{p-1}}\right)^{p-1}\right)\|\nabla u(\cdot,t)\|_{L^2}^2\\ = &-2\left(1-\left(\frac{J(u_0)}{d}\right)^{\frac{p-1}{2}}\right)\|\nabla u(\cdot,t)\|_{L^2}^2\\ \leq&-\frac{2 \lambda_1}{ \lambda_1+1}\left(1-\left(\frac{J(u_0)}{d}\right)^{\frac{p-1}{2}}\right)\|u(\cdot,t)\|_{H_0^1}^2, \end{align*} |
which leads to
\begin{equation*} \|u(\cdot,t)\|_{H_0^1}^2\leq\|u_0\|_{H_0^1}^2\exp\left[-\frac{2 \lambda_1}{ \lambda_1+1}\left(1-\left(\frac{J(u_0)}{d}\right)^{\frac{p-1}{2}}\right)t\right]. \end{equation*} |
The proof is complete.
Proof of Theorem 2.5. Let
Firstly, we consider the case
\begin{equation} \xi(t): = \left(\int_0^t\|u(\cdot,s)\|_{H_0^1}^2ds\right)^{\frac 12},\; \; \; \eta(t): = \left(\int_0^t\|u_s(\cdot,s)\|_{H_0^1}^2ds\right)^{\frac 12},\; \; \; \; 0\leq t < {T_{\max}}. \end{equation} | (51) |
For any
\begin{equation} F(t): = \xi^2(t)+(T^*-t)\|u_0\|_{H_0^1}^2+ \beta(t+ \alpha)^2,\; \; \; 0\leq t\leq T^*. \end{equation} | (52) |
Then
\begin{equation} F(0) = T^*\|u_0\|_{H_0^1}^2+ \beta \alpha^2 > 0, \end{equation} | (53) |
\begin{equation} \begin{split} F'(t)& = \|u(\cdot,t)\|_{H_0^1}^2-\|u_0\|_{H_0^1}^2+2 \beta(t+ \alpha)\\ & = 2\left(\frac12\int_0^t\frac{d}{ds}\|u(\cdot,s)\|_{H_0^1}^2ds+ \beta(t+ \alpha)\right),\; \; \; 0\leq t\leq T^*, \end{split} \end{equation} | (54) |
and (by (24), (12), (13), (48), (25))
\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&2(p+1)(d-J(u_0))+2(p+1)\eta^2(t)+2 \beta,\; \; \; 0\leq t\leq T^*. \end{split} \end{equation} | (55) |
Since
\begin{equation*} F'(t)\geq2 \beta(t+ \alpha). \end{equation*} |
Then
\begin{equation} F(t) = F(0)+\int_0^tF'(s)ds\geq T^*\|u_0\|_{H_0^1}^2+ \beta \alpha^2+2 \alpha \beta t+ \beta t^2,\; \; \; 0\leq t\leq T^*. \end{equation} | (56) |
By (6), Schwartz's inequality and Hölder's inequality, we have
\begin{align*} \frac12\int_0^t\frac{d}{ds}\|u(\cdot,s)\|_{H_0^1}^2ds& = \int_0^t(u(\cdot,s),u_s(\cdot,s))_{H_0^1}ds\\ &\leq\int_0^t\|u(\cdot,s)\|_{H_0^1}\|u_s(\cdot,s)\|_{H_0^1}ds\leq \xi(t)\eta(t),\; \; \; 0\leq t\leq T^*, \end{align*} |
which, together with the definition of
\begin{align*} &\ \ \ \left(F(t)-(T^*-t)\|u_0\|_{H_0^1}^2\right)\left(\eta^2(t)+ \beta\right)\\ & = \left(\xi^2(t)+ \beta(t+ \alpha)^2\right)\left(\eta^2(t)+ \beta\right)\\ & = \xi^2(t)\eta^2(t)+ \beta\xi^2(t)+ \beta(t+ \alpha)^2\eta^2(t)+ \beta^2(t+ \alpha)^2\\ &\geq\xi^2(t)\eta^2(t)+2\xi(t)\eta(t) \beta(t+ \alpha)+ \beta^2(t+ \alpha)^2\\ &\geq\left(\xi(t)\eta(t)+ \beta(t+ \alpha)\right)^2\\ &\geq\left(\frac12\int_0^t\frac{d}{ds}\|u(\cdot,s)\|_{H_0^1}^2ds+ \beta(t+ \alpha)\right)^2,\; \; \; 0\leq t\leq T^*. \end{align*} |
Then it follows from (54) and the above inequality that
\begin{equation} \begin{split} \left(F'(t)\right)^2& = 4\left(\frac12 \int\limits_0^t\frac{d}{ds}\|u(s)\|_{H_0^1}^2ds+ \beta(t+ \alpha)\right)^2\\ &\leq 4F(t)\left(\eta^2(t)+ \beta\right),\; \; \; \; \; \; 0\leq t\leq T^*. \end{split} \end{equation} | (57) |
In view of (55), (56), and (57), we have
\begin{align*} F(t)F''(t)-\frac{p+1}{2}(F'(t))^2\geq F(t)\left(2(p+1)(d-J(u_0))-2p \beta\right),\; \; \; 0\leq t\leq T^*. \end{align*} |
If we take
\begin{equation} 0 < \beta\leq\frac{p+1}{p}(d-J(u_0)), \end{equation} | (58) |
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} | (59) |
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{4p\|u_0\|_{H_0^1}^2}{(p-1)^2(p+1)(d-J(u_0))}. \end{equation*} |
By the arbitrariness of
\begin{equation*} {T_{\max}}\leq\frac{4p\|u_0\|_{H_0^1}^2}{(p-1)^2(p+1)(d-J(u_0))}. \end{equation*} |
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
\begin{equation*} d = \inf\limits_{\phi\in N}J(\phi) = \frac{p-1}{2(p+1)}\inf\limits_{\phi\in N}\|\nabla\phi\|_{L^2}^2. \end{equation*} |
Then a minimizing sequence
\begin{equation} \lim\limits_{k\uparrow\infty}J(\phi_k) = \frac{p-1}{2(p+1)}\lim\limits_{k\uparrow\infty}\|\nabla\phi_k\|_{L^2}^2 = d, \end{equation} | (60) |
which implies
(1)
(2)
Now, in view of
\begin{equation} \lim\limits_{k\uparrow\infty}J(\phi_k) = \frac{p-1}{2(p+1)}\lim\limits_{k\uparrow\infty}\|\nabla\phi_k\|_{L^2}^2 = d, \end{equation} | (60) |
We claim
\begin{equation} \|\nabla\varphi\|_{L^2}^2 = \|\varphi\|_{L_\sigma^{p+1}}^{p+1}\hbox{ i.e. }I(\varphi) = 0. \end{equation} | (62) |
In fact, if the claim is not true, then by (61),
\begin{equation*} \|\nabla\varphi\|_{L^2}^2 < \|\varphi\|_{L_\sigma^{p+1}}^{p+1}. \end{equation*} |
By the proof of Theorem 3.3, we know that
\begin{equation} J(s^*_\varphi\varphi)\geq d, \end{equation} | (63) |
where
\begin{equation} J(s^*_\varphi\varphi)\geq d, \end{equation} | (63) |
On the other hand, since
\begin{equation*} \begin{split} J(s^*_\varphi\varphi)& = \frac{p-1}{2(p+1)}(s^*_\varphi)^2\|\nabla \varphi\|_{L^2}^2\\ & < \frac{p-1}{2(p+1)}\|\nabla \varphi\|_{L^2}^2\\ &\leq\frac{p-1}{2(p+1)}\liminf\limits_{k\uparrow\infty}\|\nabla \phi_k\|_{L^2}^2\\& = d, \end{split} \end{equation*} |
which contradicts to (63). So the claim is true, i.e.
\begin{equation*} \lim\limits_{k\uparrow\infty}\|\nabla\phi_k\|_{L^2}^2 = \|\varphi\|_{L_\sigma^{p+1}}^{p+1}, \end{equation*} |
which, together with
Second, we prove
\begin{equation*} \lim\limits_{k\uparrow\infty}\|\nabla\phi_k\|_{L^2}^2 = \|\varphi\|_{L_\sigma^{p+1}}^{p+1}, \end{equation*} |
Then
\begin{equation*} A: = \{\tau(s)(\varphi+sv):s\in(- \varepsilon, \varepsilon)\} \end{equation*} |
is a curve on
\begin{equation*} A: = \{\tau(s)(\varphi+sv):s\in(- \varepsilon, \varepsilon)\} \end{equation*} |
where
\begin{align*} \xi&: = 2\int_ \Omega\nabla(\varphi+sv)\cdot\nabla vdx\|\varphi+sv\|_{L_\sigma^{p+1}}^{p+1},\\ \eta&: = (p+1)\int_ \Omega|x|^\sigma|\varphi+sv|^{p-1}(\varphi+sv)vdx\|\nabla(\varphi+sv)\|_{L^2}^2. \end{align*} |
Since (62), we get
\begin{equation} \tau'(0) = \frac{1}{(p-1)\|\varphi\|_{L_\sigma^{p+1}}^{p+1}}\left(2\int_ \Omega\nabla\varphi\nabla vdx-(p+1)\int_ \Omega|x|^\sigma|\varphi|^{p-1}\varphi vdx\right). \end{equation} | (65) |
Let
\begin{equation*} \varrho(s): = J(\tau(s)(\varphi+sv)) = \frac{\tau^2(s)}{2}\|\nabla(\varphi+sv)\|_{L^2}^2-\frac{\tau^{p+1}(s)}{p+1}\|\varphi+sv\|_{L^{p+1}_{\sigma}}^{p+1},\; \; \; s\in(- \varepsilon, \varepsilon). \end{equation*} |
Since
\begin{align*} 0 = &\varrho'(0) = \left.\tau(s)\tau'(s)\|\nabla(\varphi+sv)\|_{L^2}^2+\tau^2(s)\int_ \Omega\nabla(\varphi+sv)\cdot\nabla vdx\right|_{s = 0}\\ &\left.-\tau^p(s)\tau'(s)\|\varphi+sv\|_{L_\sigma^{p+1}}^{p+1}-\tau^{p+1}(s)\int_ \Omega|x|^\sigma|\varphi+sv|^{p-1}(\varphi+sv)vdx\right|_{s = 0}\\ = &\int_ \Omega\nabla\varphi\cdot\nabla vdx-\int_ \Omega|x|^\sigma|\varphi|^{p-1}\varphi vdx. \end{align*} |
So,
Finally, in view of Definition 2.3 and
\begin{equation} d = \inf\limits_{\phi\in\Phi\setminus\{0\}}J(\phi). \end{equation} | (66) |
In fact, by the above proof and (27), we have
\begin{equation*} d = \inf\limits_{\phi\in N}J(\phi) \end{equation*} |
and
Proof of Theorem 2.8. Let
\begin{equation*} \omega(u_0) = \cap_{t\geq0} \overline{\{u(\cdot,s):s\geq t\}}^{H_0^1( \Omega)} \end{equation*} |
the
(i) Assume
\begin{equation*} v(x,t) = \left\{ \begin{array}{ll} u(x,t), & \hbox{ if }0\leq t\leq t_0; \\ 0, & \hbox{ if }t > t_0 \end{array} \right. \end{equation*} |
is a global weak solution of problem (1), and the proof is complete.
We claim that
\begin{equation} I(u(\cdot,t)) > 0,\; \; \; 0\leq t < {T_{\max}}. \end{equation} | (67) |
Since
\begin{equation} I(u(\cdot,t)) > 0,\; \; \; 0\leq t < t_0 \end{equation} | (68) |
and
\begin{equation} I(u(\cdot,t_0)) = 0, \end{equation} | (69) |
which together with the definition of
\begin{equation} \|u(\cdot,t_0)\|_{H_0^1}\geq \lambda_{\rho}. \end{equation} | (70) |
On the other hand, it follows from (24), (68) and
\begin{equation*} \|u(\cdot,t)\|_{H_0^1} < \|u_0\|_{H_0^1}\leq \lambda_\rho, \end{equation*} |
which contradicts (70). So (67) is true. Then by (24) again, we get
\begin{equation*} \|u(\cdot,t)\|_{H_0^1}\leq\|u_0\|_{H_0^1},\; \; \; 0\leq t < {T_{\max}}, \end{equation*} |
which implies
By (24) and (67),
\begin{equation*} \lim\limits_{t\uparrow\infty}\|u(\cdot,t)\|_{H_0^1} = c. \end{equation*} |
Taking
\begin{equation*} \int_0^\infty I(u(\cdot,s))ds\leq\frac12\left(\|u_0\|_{H_0^1}^2-c\right) < \infty. \end{equation*} |
Note that
\begin{equation} \lim\limits_{n\uparrow\infty}I(u(\cdot,t_n)) = 0. \end{equation} | (71) |
Let
\begin{equation} u(\cdot,t_n)\rightarrow\omega \hbox{ in }H_0^1( \Omega)\hbox{ as }n\uparrow\infty. \end{equation} | (72) |
Then by (71), we get
\begin{equation} I(\omega) = \lim\limits_{n\uparrow\infty}I(u(\cdot,t_n)) = 0. \end{equation} | (73) |
As the above, one can easily see
\begin{equation*} \|\omega\|_{H_0^1} < \lambda_\rho\leq \lambda_{J(u_0)}, \; \; \; \underbrace{J(\omega) < J(u_0)}_{\Rightarrow\omega\in J^{J(u_0)}}, \end{equation*} |
which implies
\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} = 0. \end{equation*} |
(ⅱ) Assume
\begin{equation} I(u(\cdot,t)) < 0,\; \; \; 0\leq t < {T_{\max}}. \end{equation} | (74) |
Since
\begin{equation} I(u(\cdot,t)) < 0,\; \; \; 0\leq t < t_0 \end{equation} | (75) |
and
\begin{equation} I(u(\cdot,t_0)) = 0. \end{equation} | (76) |
Since (75), by (44) and
\begin{equation*} \|\nabla u(\cdot,t_0)\|_{L^2}\geq C_{p\sigma}^{-\frac{p+1}{p-1}}, \end{equation*} |
which, together with the definition of
\begin{equation} \|u(\cdot,t_0)\|_{H_0^1}\leq \Lambda_{\rho}. \end{equation} | (77) |
On the other hand, it follows from (24), (75) and
\begin{equation*} \|u(\cdot,t)\|_{H_0^1} > \|u_0\|_{H_0^1}\geq\Lambda_\rho, \end{equation*} |
which contradicts (77). So (74) is true.
Suppose by contradiction that
\begin{equation*} \lim\limits_{t\uparrow\infty}\|u(\cdot,t)\|_{H_0^1} = \tilde c, \end{equation*} |
Taking
\begin{equation*} -\int_0^\infty I(u(\cdot,s))ds\leq\frac12\left( \tilde c-\|u_0\|_{H_0^1}^2\right) < \infty. \end{equation*} |
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] | International Energy Agency (IEA), 2022. Global Energy & CO2 Status Report 2022. |
[2] | IEA, 2019. Global Energy and CO2 Report 2019. Available from: https://iea.org/reports/global-energy-co2-status-report-2019 |
[3] |
Jensen M (1993) The Modern Industrial Revolution, Exit, and the Failure of Internal Control Systems. J Financ 48: 831–880. https://doi.org/10.1111/j.1540-6261.1993.tb04022.x doi: 10.1111/j.1540-6261.1993.tb04022.x
![]() |
[4] |
Saleem Y, Crespi N, Rehmani MH, et al. (2017) Internet of things-aided Smart Grid: technologies, architectures, applications, prototypes, and future research directions. IEEE Access 7: 62962-63003. https://doi.org/10.1109/ACCESS.2019.2913984. doi: 10.1109/ACCESS.2019.2913984
![]() |
[5] |
Chelloug SA, El-Zawawy MA (2017) Middleware for Internet of Things: Survey and Challenges. Intell Autom Soft Comput 3: 70–95. https://doi.org/10.1080/10798587.2017.1290328 doi: 10.1080/10798587.2017.1290328
![]() |
[6] | Shrouf F, Ordieres J, Miragliotta G (2014) Smart factories in Industry 4.0: A review of the concept and of energy management approached in production based on the Internet of Things paradigm. Proceedings of the 2014 IEEE International Conference on Industrial Engineering and Engineering Management; 697–701. https://doi.org/10.1109/IEEM.2014.7058728 |
[7] | Wang Q, Wang YG (2018) Research on Power Internet of Things Architecture for Smart Grid Demand. 2018 2nd IEEE Conference on Energy Internet and Energy System Integration (EI2). https://doi.org/10.1109/EI2.2018.8582132 |
[8] | Vu TL, Le NT, Jang YM, et al. (2018) An Overview of Internet of Energy (IoE) Based Building Energy Management System. 2018 International Conference on Information and Communication Technology Convergence (ICTC), 852-855. https://doi.org/10.1109/ICTC.2018.8539513 |
[9] | Bin X, Qing C, Jun M, et al. (2019) Research on a Kind of Ubiquitous Power Internet of Things System for Strong Smart Power Grid. 2019 IEEE Innovative Smart Grid Technologies - Asia (ISGT Asia), 2805-2808. https://doi.org/10.1109/ISGT-Asia.2019.8881652 |
[10] | Wang W, Zhou Z (2020) Exploring Novel Internet-of-things Based on Free Space Optical Communications for Smart Grids. 2020 IEEE 4th Conference on Energy Internet and Energy System Integration (EI2), 4277-4281. https://doi.org/10.1109/EI250167.2020.9347166 |
[11] | Xiang W, Wang Y, Gao X, et al. (2021) Design and Implementation of Internet of Things System Based on Customer Electricity Behavior Analysis. 2021 IEEE 5th Conference on Energy Internet and Energy System Integration (EI2), 3411-3415. https://doi.org/10.1109/EI252483.2021.9713593 |
[12] |
Amin SM, Wollenberg BF (2005) Toward a smart grid: power delivery for the 21st century. IEEE power and energy magazine 3: 34–41. https://doi.org/10.1109/MPAE.2005.1507024 doi: 10.1109/MPAE.2005.1507024
![]() |
[13] |
Orumwense EF, Abo-Al-Ez KM (2019) A systematic review to aligning research paths: Energy cyber-physical systems. Cogent Eng 6: 1700738. https://doi.org/10.1080/23311916.2019.1700738 doi: 10.1080/23311916.2019.1700738
![]() |
[14] |
Zeng Z, Ding T, Xu Y, et al. (2020) Reliability Evaluation for Integrated Power-Gas Systems with Power-to-Gas and Gas Storagesin. IEEE T Power Syst 35: 571-583. https://doi.org/10.1109/TPWRS.2019.2935771 doi: 10.1109/TPWRS.2019.2935771
![]() |
[15] |
Gahleitner G (2013) Hydrogen from renewable electricity: An international review of power-to-gas pilot plants for stationary applications. Int J Hydrogen Energy 38: 2039–2061. https://doi.org/10.1016/j.ijhydene.2012.12.010 doi: 10.1016/j.ijhydene.2012.12.010
![]() |
[16] |
Cui S, Wang Y, Xiao J (2019) Peer-to-Peer Energy Sharing Among Smart Energy Buildings by Distributed Transaction. IEEE T Smart Grid 10: 6491-6501. https://doi.org/10.1109/TSG.2019.2906059 doi: 10.1109/TSG.2019.2906059
![]() |
[17] | Vision of Smart Energy – Research, development and Demonstration, Smart Energy Networks. |
[18] | IEEE smart grid domains - IEEE smart grid. (2020) https://smartgrid.ieee.org/domains. |
[19] |
Mohassel R, Fung A, Mohammadi F, et al. (2014) A survey of Advanced Metering Infrastructure. International Journal of Electrical Power and Energy Systems 63: 473-484. https://doi.org/10.1016/j.ijepes.2014.06.025 doi: 10.1016/j.ijepes.2014.06.025
![]() |
[20] |
Pereira R, Figueiredo J, Melicio R, et al. (2015) Consumer energy management system with integration of smart meters. Energy Rep 1: 22–29. https://doi.org/10.1016/j.egyr.2014.10.001 doi: 10.1016/j.egyr.2014.10.001
![]() |
[21] | Sayed K, Gabbar HA (2017) SCADA and smart energy grid control automation. Smart Energy Grid Engineering, 481–514. https://doi.org/10.1016/B978-0-12-805343-0.00018-8 |
[22] |
Mohanty SP, Choppali U, Kougianos K (2016) Everything you wanted to know about smart cities. IEEE Consum Electron Mag 5: 60–70. https://doi.org/10.1109/MCE.2016.2556879 doi: 10.1109/MCE.2016.2556879
![]() |
[23] |
Bouzid AM, Guerrero JM, Cheriti A, et al. (2015) A survey on control of electric power distributed generation systems for microgrid applications. Renew Sustain Energy Rev 44: 751–766. https://doi.org/10.1016/j.rser.2015.01.016 doi: 10.1016/j.rser.2015.01.016
![]() |
[24] | Haseeb K, Almogren A, Islam N, et al. (2019) An Energy-Efficient and Secure Routing Protocol for Intrusion Avoidance in IoT-Based WSN. Energies 12, 4174. https://doi.org/10.3390/en12214174 |
[25] | Zouinkhi A, Ayadi H, Val T, et al. (2019) Auto-management of energy in IoT networks. Int J Commun Syst 33: e4168. https://doi.org/10.1002/dac.4168. |
[26] | Hö ller J, Tsiatsis V, Mulligan C, et al. (2014) From Machine-to-Machine to the Internet of Things: Introduction to a New Age of Intelligence; Elsevier: Amsterdam, The Netherlands. |
[27] | Hersent O, Boswarthick D, Elloumi O (2011) The internet of things: Key Applications and Protocols. John Wiley & Sons. https://doi.org/10.1002/9781119958352 |
[28] | Tulemissova G (2016) The Impact of the IoT and IoE Technologies on Changes of Knowledge Management Strategy. ECIC2016-Proceedings of the 8th European Conference on Intellectual Capital: ECIC2016, 300. Academic Conferences and publishing limited. |
[29] |
Zhou K, Yang S, Shao Z (2016) Energy Internet: The business perspective. Appl Energy 178: 212–222. https://doi.org/10.1016/j.apenergy.2016.06.052 doi: 10.1016/j.apenergy.2016.06.052
![]() |
[30] |
Tahanan M, Van Ackooij W, Frangioni A, et al. (2015) Large-scale Unit Commitment under uncertainty. 4OR 13: 115–171, https://doi.org/10.1007/s10288-014-0279-y doi: 10.1007/s10288-014-0279-y
![]() |
[31] |
Anvari-Moghaddam A, Monsef H, Rahimi-Kian A (2015) Cost-effective and comfort-aware residential energy management under different pricing schemes and weather conditions. Energy Build 86: 782–793, https://doi.org/10.1016/j.enbuild.2014.10.017 doi: 10.1016/j.enbuild.2014.10.017
![]() |
[32] |
Mahmud K, Town GE, Morsalin S, et al. (2017) Integration of electric vehicles and management in the internet of energy. Renew Sustain Energy Rev 82: 4179–4203, https://doi.org/10.1016/j.rser.2017.11.004 doi: 10.1016/j.rser.2017.11.004
![]() |
[33] | Orumwense EF, Abo-Al-Ez K (2019) An Energy Efficient Cognitive Radio based Smart Grid Communication Architecture. Proceedings of the 17th IEEE Industrial and Commercial Use of Energy, Cape Town, South Africa. https://doi.org/10.2139/ssrn.3638151 |
[34] |
Patil K, Lahudkar PSL (2015) Survey of MAC Layer Issues and Application layer Protocols for Machine-to-Machine Communications. IEEE Internet Things J 2: 175–186. https://doi.org/10.1109/JIOT.2015.2394438 doi: 10.1109/JIOT.2015.2394438
![]() |
[35] |
Li Z, Shahidehpour M, Aminifar F (2017) Cybersecurity in Distributed Power Systems. Proc IEEE, 105: 1367–1388, https://doi.org/10.1109/JPROC.2017.2687865 doi: 10.1109/JPROC.2017.2687865
![]() |
[36] | Groopman J, Etlinger S (2015) Consumer Perceptions of Privacy in the Internet of Things: What Brands Can Learn from a Concerned Citizenry. Altimeter Group: San Francisco, CA, USA, 1–25. |
[37] |
Zafar R, Mahmood A, Razzaq S, et al. (2018) Prosumer based energy management and sharing in smart grid. Renew Sustain Energy Rev 82: 1675–1684, https://doi.org/10.1016/j.rser.2017.07.018 doi: 10.1016/j.rser.2017.07.018
![]() |
[38] |
Luna AC, Diaz NL, Graells M, et al. (2016) Cooperative energy management for a cluster of households prosumers. IEEE T Consum Electron 62: 235–242. https://doi.org/10.1109/TCE.2016.7613189 doi: 10.1109/TCE.2016.7613189
![]() |
[39] | Iannello F, Simeone O, Spagnolini U (2010) Energy Management Policies for Passive RFID Sensors with RF-Energy Harvesting. Proceedings of the 2010 IEEE International Conference on Communications, 1–6, https://doi.org/10.1109/ICC.2010.5502035 |
[40] | Ramamurthy A, Jain P (2017) The Internet of Things in the Power Sector: Opportunities in Asia and the Pacific. https://doi.org/10.22617/WPS178914-2 |
[41] | Sigfox, Inc. Utilities & Energy (2019) Available from: https://www.sigfox.com/en/utilities-energy/. |
[42] | Immelt JR (2015) The Future of Electricity Is Digital; Technical Report; General Electric: Boston, MA, USA, 2015. |
[43] | Huneria HK, Yadav P, Shaw RN, et al. (2021) AI and IOT-Based Model for Photovoltaic Power Generation. Innovations in Electrical and Electronic Engineering, 697-706. https://doi.org/10.1007/978-981-16-0749-3_55 |
[44] | Singh R, Akram SV, Gehlot A, et al. (2022) Energy System 4.0: Digitalization of the Energy Sector with Inclination towards Sustainability. Sensors 22: 6619. https://doi.org/10.3390/s22176619 |
[45] |
Ejaz W, Naeem M, Shahid A, et al. (2017) Efficient energy management for the internet of things in smart cities. IEEE Commun Mag 55: 84–91. https://doi.org/10.1109/MCOM.2017.1600218CM doi: 10.1109/MCOM.2017.1600218CM
![]() |
[46] | Mitchell S, Villa N, Stewart-Weeks M, et al. (2013) The Internet of Everything for Cities; Cisco: San Jose, CA, USA, 2013. |
[47] |
Idwan S, Mahmood I, Zubairi J, et al. (2020) Optimal Management of Solid Waste in Smart Cities using Internet of Things. Wireless Pers Commun 110: 485-501. https://doi.org/10.1007/s11277-019-06738-8 doi: 10.1007/s11277-019-06738-8
![]() |
[48] |
Vakiloroaya V, Samali B, Fakhar A, et al. (2014) A review of different strategies for HVAC energy saving. Energy Convers Manag 77: 738–754. https://doi.org/10.1016/j.enconman.2013.10.023 doi: 10.1016/j.enconman.2013.10.023
![]() |
[49] | Arasteh H, Hosseinnezhad V, Loia V, et al. (2016) IoT-based smart cities: A survey. Proceedings of the 2016 IEEE 16th International Conference on Environment and Electrical Engineering (EEEIC), 1–6. https://doi.org/10.1109/EEEIC.2016.7555867 |
[50] | Lee C, Zhang S (2016) Development of an Industrial Internet of Things Suite for Smart Factory towards Re-industrialization in Hong Kong. Proceedings of the 6th International Workshop of Advanced Manufacturing and Automation, 285-289. https://doi.org/10.2991/iwama-16.2016.54 |
[51] | Reinfurt L, Falkenthal M, Breitenbücher U, et al. (2017) Applying IoT Patterns to Smart Factory Systems. Proceedings of the 2017 Advanced Summer School on Service Oriented Computing (Summer SOC), 66. |
[52] |
Cheng J, Chen W, Tao F, et al. (2018) Industrial IoT in 5G Environment towards Smart Manufacturing. J Ind Inf Integr 10: 10–19. https://doi.org/10.1016/j.jii.2018.04.001 doi: 10.1016/j.jii.2018.04.001
![]() |
[53] | IoT application areas (2016) https://www.iot-analytics.com/top-10-iot-project-application-areas-q3-2016/. |
[54] |
Shi W, Xie X, Chu C, et al. (2015) Distributed Optimal Energy Management in Microgrids. IEEE T Smart Grid 6: 1137–1146. https://doi.org/10.1109/TSG.2014.2373150 doi: 10.1109/TSG.2014.2373150
![]() |
[55] |
Kamalinejad P, Mahapatra C, Sheng Z, et al. (2015) Wireless energy harvesting for the Internet of Things. IEEE Commun Mag 53: 102–108. https://doi.org/10.1109/MCOM.2015.7120024 doi: 10.1109/MCOM.2015.7120024
![]() |
[56] |
Song T, Li R, Mei B, et al. (2017) A privacy preserving communication protocol for IoT applications in smart homes. IEEE Internet Things J 4: 1844–1852. https://doi.org/10.1109/JIOT.2017.2707489. doi: 10.1109/JIOT.2017.2707489
![]() |
[57] | Fhom HS, Kuntze N, Rudolph C, et al. (2010) A user-centric privacy manager for future energy systems. Proceedings of the 2010 International Conference on Power System Technology, 1–7. https://doi.org/10.1109/POWERCON.2010.5666447. |
[58] | Pramudita R, Hariadi IF, Achmad AS (2017) Development of IoT Authentication Mechanisms for Microgrid Applications. 2017 International Symposium on Electronics and Smart Devices (ISESD), 12-17. https://doi.org/10.1109/ISESD.2017.8253297 |
[59] |
Trappe W, Howard R, Moore RS (2015) Low-Energy Security: Limits and Opportunities in the Internet of Things. IEEE Secur Priv 13: 14-21. https://doi.org/10.1109/MSP.2015.7 doi: 10.1109/MSP.2015.7
![]() |
[60] | IEEE Std 802.15.4-2015 (Revision of IEEE Std 802.15.4-2011) (2016) IEEE Standard for Low-Rate Wireless Networks. IEEE Stand, 1–708, https://doi.org/10.1109/IEEESTD.2016.7460875 |
[61] | Al-Qaseemi SA, Almulhim HA, Almulhim MF, et al. (2016) IoT architecture challenges and issues: Lack of standardization. Proceedings of the 2016 Future Technologies Conference (FTC), 731–738. https://doi.org/10.1109/FTC.2016.7821686 |
[62] |
Stojmenovic I (2014) Machine-to-Machine Communications with In-Network Data Aggregation, Processing, and Actuation for Large-Scale Cyber-Physical Systems. IEEE Internet Things J 1: 122–128. https://doi.org/10.1109/JIOT.2014.2311693 doi: 10.1109/JIOT.2014.2311693
![]() |
[63] |
Lloret J, Tomas J, Canovas A, et al. (2016) An Integrated IoT Architecture for Smart Metering. IEomEE Commun Mag 54: 50–57. https://doi.org/10.1109/MCOM.2016.1600647CM doi: 10.1109/MCOM.2016.1600647CM
![]() |
[64] |
Breur T (2015) Big data and Internet of Things. J Mark Anal 3: 1–4. https://doi.org/10.1057/jma.2015.7 doi: 10.1057/jma.2015.7
![]() |
[65] | Shakerighadi B, Anvari-Moghaddam A, Vasquez JC, et al. (2018) Internet of things for modern energy systems: state-of-the-art, challenges, and open issues. Energies 11: 1252. 10.3390/en11051252 |
[66] |
Xu J, Yao J, Wang L, et al. (2017) Narrowband internet of things: evolutions, technologies and open issues. IEEE Internet of things journal 5: 1449-1462. https://doi.org/10.1109/JIOT.2017.2783374 doi: 10.1109/JIOT.2017.2783374
![]() |
[67] | Venkatesh N (2015) Ensuring Coexistence of IoT Wireless Protocols Using a Convergence Module to Avoid Contention, Embedded Innovator, 12th Edition, 2015. |
[68] | Bedi G, Venayagamoorthy GK, Singh R (2016) Navigating the challenges of Internet of Things (IoT) for power and energy systems. 2016 Clemson University Power Systems Conference (PSC). https://doi.org/10.1109/PSC.2016.7462853 |
[69] | Singh R, Akram SV, Gehlot A, et al. (2022) Energy System 4.0: Digitalization of the Energy Sector with Inclination towards Sustainability. Sensors 22: 6619. https://doi.org/10.3390/s22176619. |
[70] |
Rana MM, Xiang W, Wang E (2018) IoT-based state estimation for microgrids. IEEE Internet of things Journal 5: 1345-1346. https://doi.org/10.1109/JIOT.2018.2793162 doi: 10.1109/JIOT.2018.2793162
![]() |
[71] | Rana MM, Xiang W, Wang E, et al. (2017) IoT Infrastructure and Potential Application to Smart Grid Communications. IEEE Global communication conference (GLOBECOM 2017). https://doi.org/10.1109/GLOCOM.2017.8254511 |
[72] | Naqvi SAR, Hassan SA, Hussain F (2017) IoT Applications and Business Models. Springer Briefs in Electrical and Computer Engineering, 45–61. https://doi.org/10.1007/978-3-319-55405-1_4 |
[73] | Research and Markets (2020) Global Internet of Things (IoT) in Energy Market Size Expected to Grow from USD 20.2 billion in 2020 to USD 35.2 billion by 2025, at a CAGR of 11.8%. https://www.globenewswire.com/news-release/2020/05/28/2040020/28124/en/Global-Internet-of-Things-IoT-in-Energy-Market-Size-Expected-to-Grow-from-USD-20-2-billion-in-2020-to-USD-35-2-billion-by-2025-at-a-CAGR-of-11-8.html. |
[74] | IoT Analytics, January (2022) https://iot-analytics.com/product/list-of1600-enterprise-iot-projects-2022/. |
[75] | The Insight planners, July (2022) https://www.theinsightpartners.com/reports/south-africa-iot-market/. Accessed 1o August 2022. |
[76] | Growth Enabler (2017) Market pulse report, Internet of Things (IoT). 1–35. GrowthEnabler. https://growthenabler.com/flipbook/pdf/IOTReport.pdf. |
[77] |
Hawlitschek F, Notheisen B, Teubner T (2018) The limits of trust-free systems: A literature review on blockchain technology and trust in the sharing economy. Electron Commer Res Appl 29: 50–63. https://doi.org/10.1016/j.elerap.2018.03.005 doi: 10.1016/j.elerap.2018.03.005
![]() |
[78] |
Christidis K, Devetsikiotis M (2016) Blockchains and Smart Contracts for the Internet of Things. IEEE Access 4: 2292–2303. https://doi.org/10.1109/ACCESS.2016.2566339. doi: 10.1109/ACCESS.2016.2566339
![]() |
[79] | LG and Samsung to Show Off New Food Identifying Smart Fridges at CES Next Week: https://thespoon.tech/lg-and-samsung-to-show-off-new-food-identifying-smart-fridges-at-ces-next-week/ |
[80] | IoT Statistics. https://www.statista.com/statistics/471264/iot-number-of-connected-devices-worldwide/. |
[81] |
Abalansa S, El Mahrad B, Icely J, et al. (2021) Electronic Waste, an Environmental Problem Exported to Developing Countries: The GOOD, the BAD and the UGLY. Sustainability 13: 5302. https://doi.org/10.3390/su13095302. doi: 10.3390/su13095302
![]() |
[82] |
Zhu C, Leung VCM, Shu L, et al. (2015) Green Internet of Things for Smart World. IEEE Access 3: 2151–2162. https://doi.org/10.1109/ACCESS.2015.2497312. doi: 10.1109/ACCESS.2015.2497312
![]() |
[83] | Kabalci Y, Ali M (2019) Emerging LPWAN Technologies for Smart Environments: An Outlook. Proceedings of the 2019 1st Global Power, Energy and Communication Conference (GPECOM), 24–29. https://doi.org/10.1109/GPECOM.2019.8778626 |
[84] |
Bembe M, Abu-Mahfouz A, Masonta M, et al. (2019) A survey on low-power wide area networks for IoT applications. Telecommun Syst 71: 249–274. https://doi.org/10.1007/s11235-019-00557-9. doi: 10.1007/s11235-019-00557-9
![]() |
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