Drivers of changes in natural resources consumption of Central African countries

Yvette Baninla; Qian Zhang; Xiaoqi Zheng; Yonglong Lu; Yvette Baninla; Qian Zhang; Xiaoqi Zheng; Yonglong Lu

doi:10.3934/ctr.2022005

Clean Technologies and Recycling

2022, Volume 2, Issue 2: 80-102. doi: 10.3934/ctr.2022005

Previous Article Next Article

Research article

Drivers of changes in natural resources consumption of Central African countries

1.
Department of Geology, Mining and Environmental Science, University of Bamenda, P.O Box 39 Bambili, NW Region, Cameroon
2.
Graduate School of Humanities and Social Sciences, Hiroshima University, Hiroshima 739-8511, Japan
3.
Robert M. Buchan Department of Mining, Queen's University, Kingston, Ontario K7L 3N6, Canada
4.
College of Economics, Nanjing University of Post and Telecommunications, Nanjing 210023, China
5.
State Key Laboratory of Urban and Regional Ecology, Research Center for Eco-Environmental Sciences, Chinese Academy of Sciences, Beijing 100085, China

Received: 13 December 2021 Revised: 14 March 2022 Accepted: 24 March 2022 Published: 29 April 2022

Consumption of nine different natural resources has kept an increasing trend in Central African countries from 1970 to 2018. This study therefore, investigates the changes and major determinants that have driven the patterns of resource use in six Central African countries over almost fifty years. We used the logarithmic mean Divisia index (LMDI) method to quantitatively analyze different effects of technology, affluence and population associated with domestic material consumption (DMC) of Cameroon, Chad, Central African Republic, Equatorial Guinea, Democratic Republic of the Congo and Gabon from 1970 to 2018. We further subdivided the affluence effect into energy productivity (GDP/energy) and per capita energy use (energy/cap) and conducted a four-factor LMDI analysis of Cameroon as a case study. The results highlight that decreased affluence during certain periods has slowed down DMC growth in four of six Central African countries except for Cameroon and Equatorial Guinea, while significant technology offset in Equatorial Guinea reduces DMC growth by 28%. Population remains the main positive driving factor of DMC growth, with the highest share in the Democratic Republic of the Congo. The case of Cameroon shows that technological intensity and energy intensity play different roles in changing DMC. This study confirms that the rising population and economic growth, combined with a gradual improvement in technology in the region are insufficient to reduce natural resource use. A stringent management plan of natural resources for Central African countries should focus on technological improvement while remaining balanced with the future demand for socioeconomic development in the coming decades.

Keywords:

Citation: Yvette Baninla, Qian Zhang, Xiaoqi Zheng, Yonglong Lu. Drivers of changes in natural resources consumption of Central African countries[J]. Clean Technologies and Recycling, 2022, 2(2): 80-102. doi: 10.3934/ctr.2022005

Related Papers:

[1]	Jiang Li, Xiaohui Liu, Chunjin Wei . The impact of fear factor and self-defence on the dynamics of predator-prey model with digestion delay. Mathematical Biosciences and Engineering, 2021, 18(5): 5478-5504. doi: 10.3934/mbe.2021277
[2]	Jiajun Zhang, Tianshou Zhou . Stationary moments, distribution conjugation and phenotypic regions in stochastic gene transcription. Mathematical Biosciences and Engineering, 2019, 16(5): 6134-6166. doi: 10.3934/mbe.2019307
[3]	Jian Zu, Wendi Wang, Bo Zu . Evolutionary dynamics of prey-predator systems with Holling type II functional response. Mathematical Biosciences and Engineering, 2007, 4(2): 221-237. doi: 10.3934/mbe.2007.4.221
[4]	A. Swierniak, M. Krzeslak, D. Borys, M. Kimmel . The role of interventions in the cancer evolution–an evolutionary games approach. Mathematical Biosciences and Engineering, 2019, 16(1): 265-291. doi: 10.3934/mbe.2019014
[5]	Andrei Korobeinikov, Conor Dempsey . A continuous phenotype space model of RNA virus evolution within a host. Mathematical Biosciences and Engineering, 2014, 11(4): 919-927. doi: 10.3934/mbe.2014.11.919
[6]	Thanh Nam Nguyen, Jean Clairambault, Thierry Jaffredo, Benoît Perthame, Delphine Salort . Adaptive dynamics of hematopoietic stem cells and their supporting stroma: a model and mathematical analysis. Mathematical Biosciences and Engineering, 2019, 16(5): 4818-4845. doi: 10.3934/mbe.2019243
[7]	Farinaz Forouzannia, Sivabal Sivaloganathan, Mohammad Kohandel . A mathematical study of the impact of cell plasticity on tumour control probability. Mathematical Biosciences and Engineering, 2020, 17(5): 5250-5266. doi: 10.3934/mbe.2020284
[8]	Kazunori Sato . Effects of cyclic allele dominance rules and spatial structure on the dynamics of cyclic competition models. Mathematical Biosciences and Engineering, 2020, 17(2): 1479-1494. doi: 10.3934/mbe.2020076
[9]	Thomas Imbert, Jean-Christophe Poggiale, Mathias Gauduchon . Intra-specific diversity and adaptation modify regime shifts dynamics under environmental change. Mathematical Biosciences and Engineering, 2024, 21(12): 7783-7804. doi: 10.3934/mbe.2024342
[10]	Dongmei Wu, Hao Wang, Sanling Yuan . Stochastic sensitivity analysis of noise-induced transitions in a predator-prey model with environmental toxins. Mathematical Biosciences and Engineering, 2019, 16(4): 2141-2153. doi: 10.3934/mbe.2019104

Abstract

1. Introduction

Let $x \in(r_1, r_2), t \in [1, T), T > 0, u_1 = u_1(t, x)$ and $u_2 = u_2(t, x)$ . We consider a new kind of coupled system of Emden-Fowler type wave equations in viscoelasticities with strong nonlinear terms

$\begin{equation} \left\{ \begin{array}{ll} t^2\partial_{tt}u_i- u_{ixx} +\int_{1}^{t}\mu_i\left(s\right) u_{ixx}\left(t-s\right) \, ds = f_i(u_1, u_2) \quad \hbox{ in }[1, T) \times(r_1, r_2), \\\\ u_i(1, x) = u_{i0}(x) \in H^2(r_1, r_2) \cap H_0^{1}(r_1, r_2), \\\\ \partial_t u_{i}(1, x) = u_{i1}(x) \in H_0^{1}(r_1, r_2), \\\\ u_i(t, r_1) = u_i(t, r_2) = 0 \quad \hbox{ in }[1, T), \end{array} \right. \end{equation}$

(1.1)

where $i = 1, 2$ and

$\begin{eqnarray} \left\{ \begin{array}{ll} f_1(\xi_1, \xi_2) = |\xi_1+\xi_2|^{2(\rho +1)}(\xi_1+\xi_2)+|\xi_1|^\rho \xi_1|\xi_2|^{\rho+2} \\ f_2(\xi_1, \xi_2) = |\xi_1+\xi_2|^{2(\rho +1)}(\xi_1+\xi_2)+|\xi_2|^\rho \xi_2|\xi_1|^{\rho+2}, \end{array} \right. \end{eqnarray}$

(1.2)

for $\rho > -1, r_i$ are real numbers and the scalar functions $\mu_i$ (so-called relaxation kernels) are assumed only to be nonincreasing $\mu_i \in C^1(\mathbb{R}^+, \mathbb{R}^+)$ satisfying

$\begin{eqnarray} \mu_i(0) \gt 0, \ 1-\int_{0}^{\infty}e^{-s/2}\mu_i(s) \, ds = l \gt 0. \end{eqnarray}$

(1.3)

Many issues in physics and engineering pose problems that deal with coupled evolution equations. For example, in diffusion theory and some mechanical applications, such evolution equations are in the form of a system of nonlinear hyperbolic equations. An important example of such systems goes back to ^[13], which introduced a three-dimensional system of space similar to our system, without dissipations. (see ^{[1,11,12,16,17]}).

The Emden-Fowler equation has an impact on many astrophysics evolution phenomena. It has been poorly studied by scientists until now, essentially of the qualitative point of view.

In 1862, Land ^[15] proposed the well known Lane-Emden equation

$\begin{equation} \partial_t\Big(t^2\partial_t u\Big) +t^2u^p = 0, \end{equation}$

(1.4)

where $p = 1.5$ and $2.5$ . When $p = 1$ , Eq (1.4) has a solution $u = \sin t/t$ , and when $p = 5$ , the explicit solution is given by $u = 1/\sqrt{1+t^2/3}$ (^[2,3,14]).

The generalization of such equation is given by

$\partial_t\Big(t^{\rho}\partial_t u\Big) +t^{\sigma}u^{\gamma} = 0, \quad t \geq0.$

It was considered by Fowler ^[4,5,6,7] in a series of four papers during 1914–1931.

Next, the generalized Emden-Fowler equation

$\partial_{tt}u+a(t) \vert u \vert^{\gamma} \operatorname{sgn}u = 0, \quad t\geq0,$

was studied by Atkinson et al.

Recently, M. R. Li in ^[8] considered and studied the blow-up phenomena of solutions to the Emden-Fowler type semilinear wave equation

$t^2\partial_{tt}u -u_{xx} = u^p \quad \hbox{ in }[1, T) \times (r_{1}, r_{2}).$

The present research aims to extend the study of Emden-Fowler type wave equation to the case when the viscoelastic term is injected in domain $[r_1, r_2]$ where there is no result about this topic as equations and as coupled systems. Thus, a wider class of phenomena can be modeled.

The main results here are to exhibit the role of viscoelasticities, which makes our system (1.1) dissipative. When the energy is negative, the blow up of solutions in $L^2$ at finite time given by

$\ln T^{\ast} = \frac{2}{\rho+1} \Bigg( \sum\limits_{i = 1}^2 \int_{r_1}^{r_2} \vert u_{i0} \vert^2 \, dx \Bigg) \Bigg( \sum\limits_{i = 1}^2 \int_{r_1}^{r_2} \left( 2 u_{i0}u_{i1} - \vert u_{i0} \vert^2 \right) \, dx \Bigg)^{-1},$

will be the main result in Theorem 3.1.

The plan of this article is as follows. We present some notations and assumptions needed for our results in section 2. Section 3 is devoted to the blow up result of solutions.

2. Preliminaries and position of problem

Under some suitable transformations, we can get the local existence of solutions to Eq (1.1). Taking the first transform

$\tau = \ln t, \qquad v_i(\tau, x) = u_i(t, x),$

for $i = 1, 2$ , we have

$u_{ixx} = v_{ixx}, \quad \partial_tu_i = t^{-1}\partial_\tau v_{i}, \quad t^2 \partial_{tt}u_i = -\partial_\tau v_{i}+\partial_{\tau\tau}v_{i} .$

Then problem (1.1) takes the form

$\begin{eqnarray} \left\{ \begin{array}{ll} \partial_{\tau\tau}v_{i }- v_{ixx} +\int_{0}^{\tau}\mu_i(s) v_{ixx}(\tau-s) \, ds = \partial_{\tau}v_{i}+f_i(v_1, v_2)\quad \hbox{ in }[0, \ln T) \times(r_1, r_2), \\\\ v_i(0, x) = u_{i0}(x) \quad \hbox{ in }(r_1, r_2), \\\\ \partial_{\tau}v_{i}(0, x) = u_{i1}(x) \quad \hbox{ in }(r_1, r_2), \\\\ v_i(\tau, r_1) = v_i(\tau, r_2) = 0\quad \hbox{ in }[0, \ln T). \end{array} \right. \end{eqnarray}$

(2.1)

To make the second transformation, let

$\begin{equation} v_i(\tau, x) = e^{\tau/2}w_i(\tau, x). \nonumber \end{equation}$

Since we have

$\begin{eqnarray} &&\partial_{\tau}v_{i}(\tau, x) = e^{\tau/2}\partial_{\tau}w_{i}(\tau, x)+\frac{1}{2}e^{\tau/2}w_i(\tau, x), \\ &&\partial_{\tau\tau}v_{i}(\tau, x) = e^{\tau/2}\partial_{\tau\tau}w_{i}(\tau, x)+e^{\tau/2}\partial_{\tau}w_{i}(\tau, x)+\frac{1}{4}e^{\tau/2}w_i(\tau, x), \end{eqnarray}$

then Eq in (2.1) can be rewritten as

$\begin{equation} e^{\tau/2}\partial_{\tau\tau}w_{i }-e^{\tau/2}w_{ixx}+\int_{0}^{\tau}e^{(\tau-s)/2}\mu_i(s)w_{ixx}(\tau-s) \, ds = \frac{1}{4}e^{\tau/2}w_i+f_i(e^{\tau/2}w_1, e^{\tau/2}w_2), \nonumber \end{equation}$

and converted to

$\begin{equation} \partial_{tt}w_{i }- w_{ixx} + \int_{0}^{t}e^{-s/2}\mu_i(s) w_{ixx}(t-s) \, ds = \frac{1}{4}w_i+e^{-t/2}f_i(e^{t/2}w_1, e^{t/2}w_2), \end{equation}$

(2.2)

with the corresponding initial and boundary conditions. Throughout this paper, we shall write $w_{i} = w_i(t, x)$ where no confusion occurs. The following technical Lemma will play an important role.

Lemma 2.1. For any $y \in C^{1}\left(0, \ln T; H_{0}^{1}(r_1, r_2)\right)$ and $i = 1, 2$ , we have

$\begin{eqnarray} &&\int_{r_1}^{r_2} \int_{0}^{t} e^{-s/2}\mu_i(s) y_{xx}(t-s) \partial_{t}y(t) \, ds \, dx \\ \\ & = &\frac{1}{2}\partial_{t} \Bigg( \int_{0}^{t} e^{-(t-p)/2}\mu_i(t-p)\int_{r_1}^{r_2} \vert y_x(p)-y_x(t)\vert^2 \, dx \, dp \Bigg) \\ \\ &&-\frac{1}{2}\partial_{t}\Bigg( \int_{0}^{t}e^{-s/2}\mu_i(s) \, ds \int_{r_1}^{r_2} \left\vert y_{x}(t)\right\vert^{2} dx \Bigg) \\\\ &&+\frac{1}{4}\int_{0}^{t}e^{-(t-p)/2}\mu_i(t-p) \int_{r_1}^{r_2} \vert y_x(p)-y_x(t) \vert^2 \, dx \, dp \\ \\ &&-\frac{1}{2}\int_{0}^{t} e^{-(t-p)/2}\partial_t\mu_i(t-p) \int_{r_1}^{r_2} \vert y_x(p)-y_x(t) \vert^2 \, dx \, dp \\\\ &&+\frac{1}{2}e^{-t/2}\mu_i(t)\int_{r_1}^{r_2} \vert y_x(t) \vert^2 \, dx. \end{eqnarray}$

Proof. It's not hard to see

$\begin{eqnarray} &&\int_{r_1}^{r_2} \int_{0}^{t} e^{-s/2}\mu_i(s) y_{xx}(t-s) \partial_{t}y(t) \, ds \, dx \\ \\ & = &-\int_{0}^{t}e^{-s/2}\mu_i(s) \int_{r_1}^{r_2} y_x(t-s) \partial_{t}y_{x}(t) \, dx \, ds \\\\ & = &-\int_{0}^{t}e^{-(t-p)/2} \mu_i(t-p) \int_{r_1}^{r_2} y_x(p) \partial_{t}y_{x}(t) \, dx \, dp \\\\ & = &-\int_{0}^{t}e^{-(t-p)/2} \mu_i(t-p) \int_{r_1}^{r_2} \Big( y_x(p) - y_x(t) \Big) \partial_{t}y_{x}(t) \, dx \, dp \\\\ &&-\int_{0}^{t}e^{-s/2} \mu_i(s) \, ds \int_{r_1}^{r_2} y_x(t) \partial_{t}y_{x}(t) \, dx. \end{eqnarray}$

Consequently, we obtain

$\begin{eqnarray} &&\int_{r_1}^{r_2} \int_{0}^{t} e^{-s/2}\mu_i(s) y_{xx}(t-s) \partial_{t}y(t) \, ds \, dx \\ \\ & = &\frac{1}{2} \int_{0}^{t}e^{-(t-p)/2} \mu_i(t-p) \partial_{t}\Bigg( \int_{r_1}^{r_2}\left\vert y_{x}(p)- y_{x}(t) \right\vert^{2} \, dx \Bigg) \, dp \\ \\ &&-\frac{1}{2} \int_{0}^{t}e^{-s/2} \mu_i(s) \, ds \partial_{t}\Bigg( \int_{r_1}^{r_2}\left\vert y_{x}(t) \right\vert^{2} \, dx \Bigg), \end{eqnarray}$

which implies

$\begin{eqnarray} &&\int_{r_1}^{r_2} \int_{0}^{t} e^{-s/2}\mu_i(s) y_{xx}(t-s) \partial_{t}y(t) \, ds \, dx \\ \\ & = &\frac{1}{2}\partial_{t}\Bigg( \int_{0}^{t} e^{-(t-p)/2}\mu_i(t-p)\int_{r_1}^{r_2} \vert y_x(p)-y_x(t)\vert^2 \, dx \, dp \Bigg) \\ \\ &&+\frac{1}{4}\int_{0}^{t}e^{-(t-p)/2}\mu_i(t-p) \int_{r_1}^{r_2} \vert y_x(p)-y_x(t) \vert^2 \, dx \, dp \\ \\ &&-\frac{1}{2}\int_{0}^{t} e^{-(t-p)/2}\partial_t\mu_i(t-p) \int_{r_1}^{r_2} \vert y_x(p)-y_x(t) \vert^2 \, dx \, dp \\\\ &&-\frac{1}{2}\partial_{t} \Bigg( \int_{0}^{t}e^{-s/2}\mu_i(s) \, ds \int_{r_1}^{r_2} \left\vert y_{x}(t)\right\vert^{2} dx \Bigg) \\\\ &&+\frac{1}{2}e^{-t/2}\mu_i(t)\int_{r_1}^{r_2} \vert y_x(t) \vert^2 \, dx. \end{eqnarray}$

This completes the proof.

The modified energy associated to problem (2.2) is introduced as

$\begin{eqnarray} &&2E_{w}(t) = \sum\limits_{i = 1}^{2} \int_{r_1}^{r_2} \vert \partial_{t}w_{i}\vert^2 \, dx +\sum\limits_{i = 1}^{2} \Bigg( 1-\int_{0}^{t}e^{-s/2}\mu_i(s) \, ds \Bigg) \int_{r_1}^{r_2} \vert w_{ix} \vert^2 \, dx \\\\ &&+\sum\limits_{i = 1}^{2} \int_{0}^{t}e^{-(t-p)/2}\mu_i(t-p)\int_{r_1}^{r_2} \left\vert w_{ix}(p)- w_{ix}(t) \right\vert^{2} \, dx \, dp \\\\ &&-\frac{1}{4}\sum\limits_{i = 1}^{2}\int_{r_1}^{r_2} \vert w_i\vert^2 \, dx -\frac{1}{\rho +2}e^{(\rho +1)t}\int_{r_1}^{r_2} \Bigg( \vert w_1+w_2 \vert^{2(\rho +2)} + 2 \vert w_1w_2 \vert^{\rho+2} \Bigg) \, dx, \end{eqnarray}$

(2.3)

and

$\begin{eqnarray} &&2E_{w}(0) = \sum\limits_{i = 1}^{2} \int_{r_1}^{r_2}\Big( u_{i1}-\frac{1}{2}u_{i0} \Big)^2 \, dx +\sum\limits_{i = 1}^{2}\int_{r_1}^{r_2} \vert u_{i0x}\vert^2 \, dx -\frac{1}{4}\sum\limits_{i = 1}^{2}\int_{r_1}^{r_2} \vert u_{i0} \vert^2 \, dx\\\\ &&-\frac{1}{\rho +2}\int_{r_1}^{r_2} \Bigg( \vert u_{10}+u_{20} \vert^{2(\rho +2)} + 2 \vert u_{10}u_{20} \vert^{\rho+2} \Bigg) \, dx. \end{eqnarray}$

For this energy, Lemma 2.3 leads to

$\begin{eqnarray} \partial_{t}E_{w}(t)\leq 0. \end{eqnarray}$

As in ^[10], one can easily verify that

$e^{-t/2} \sum\limits_{i = 1}^2 \partial_{t}w_{i} f_{i} ( e^{t/2}w_{1}, e^{t/2}u_{2} ) = \frac{1}{2(\rho +2)}e^{(\rho +1)t} \partial_{t} \Big( \vert w_1+w_2 \vert^{2(\rho +2)} + 2 \vert w_1w_2 \vert^{\rho+2} \Big),$

and

$\begin{eqnarray} e^{-t/2} \sum\limits_{i = 1}^2 w_i f_i(e^{t/2}w_1, e^{t/2}w_2) \, dx = e^{(\rho +1)t} \Big( \vert w_1+w_2 \vert^{2(\rho +2)} + 2 \vert w_1w_2 \vert^{\rho+2} \Big). \end{eqnarray}$

(2.4)

Next, we introduce the Dirichlet-Poincaré's inequality in one spatial variable.

Lemma 2.2. For any $v \in H_0^{1}(r_1, r_2)$ , we have

$\begin{eqnarray} \int_{r_1}^{r_2} \vert v \vert^2 \, dx \leq \frac{(r_2-r_1)^2}{2}\int_{r_1}^{r_2} \vert v_{x} \vert^2 \, dx. \end{eqnarray}$

Proof. From (2.1), we have $w(t, r_1) = w(t, r_2) = 0$ . By the Fundamental Theorem of Calculus

$\begin{eqnarray} w(s) = \int_{r_1}^{s}w_xdx. \end{eqnarray}$

(2.5)

Therefore

$\begin{eqnarray} |w(s)|\leq\int_{r_1}^{s}|w_x|dx. \end{eqnarray}$

(2.6)

Recall the Cauchy-Schwar's inequality

$\int fgdx\leq \Big(\int f^2dx\Big)^{1/2}\Big(\int g^2dx\Big)^{1/2}.$

Apply this with $f = 1, g = |w_x|$ to get

$\begin{eqnarray} |w(s)|&\leq&\Big(\int_{r_1}^{s}|w_x|^2dx\Big)^{1/2}(s-r_1)^{1/2}\\ &\leq &\Big(\int_{r_1}^{r_2}|w_x|^2dx\Big)^{1/2}(r_2-r_1)^{1/2}. \end{eqnarray}$

Squaring both sides gives

$\begin{eqnarray} |w(s)|^2&\leq& \Big(\int_{r_1}^{r_2}|w_x|^2dx\Big) (r_2-r_1), \end{eqnarray}$

and finally we integrate over $[r_1, r_2]$ to give the required.

Now, in order to deal with nonlinear terms (1.2) which are considered as a sources of dissipativity in (2.2), we need the next important Lemma. This Lemma shows that the energy functional is decreasing.

Lemma 2.3. Suppose that $v\in C^{1}(0, \ln T; H_0^{1}(r_1, r_2)) \cap C^2(0, \ln T; L^2(r_1, r_2))$ is a solution of the semi-linear wave equation (2.2). Then for $t \geq 0$ , we have

$\begin{eqnarray} E_{w}(t) \leq E_{w}(0) -\frac{\rho+1}{2(\rho+2)} \int_{0}^{t} e^{(\rho +1)s} \int_{r_1}^{r_2} \Bigg( \vert w_1+w_2 \vert^{2(\rho +2)} + 2 \vert w_1w_2 \vert^{\rho+2} \Bigg) \, dx \, ds.\\ \end{eqnarray}$

(2.7)

Proof. Taking the $L^2$ product of $(2.2)_i$ with $\partial_{t}w_{i}$ yields

$\begin{eqnarray} &&\int_{r_1}^{r_2}\partial_{tt}w_{i}\partial_{t}w_{i} \, dx -\int_{r_1}^{r_2}w_{ixx}\partial_{t}w_{i} \, dx +\int_{r_1}^{r_2}\int_{0}^{t}e^{-s/2}\mu_i(s) w_{ixx}(t-s) \, ds \partial_{t}w_{i} \, dx\\ && = \frac{1}{4}\int_{r_1}^{r_2}w_i \partial_{t}w_{i} \, dx +e^{-t/2}\int_{r_1}^{r_2}f_i(e^{t/2}w_1, e^{t/2}w_2)\partial_{t}w_{i} \, dx, \end{eqnarray}$

for $i = 1, 2$ . Adding each other, we have

$\begin{eqnarray} &&\frac{1}{2} \sum\limits_{i = 1}^2 \partial_{t}\int_{r_1}^{r_2} \Big[ \vert \partial_{t}w_{i}\vert^2 + \vert w_{ix}\vert^2 -\frac{1}{4}\vert w_{i} \vert^2 \Big] \, dx \\ &&+\sum\limits_{i = 1}^2 \int_{r_1}^{r_2}\int_{0}^{t} e^{-s/2}\mu_i(s) w_{ixx}(t-s) \partial_{t}w_{i}(t) \, ds \, dx \\ && = e^{-t/2} \sum\limits_{i = 1}^2 \int_{r_1}^{r_2} \partial_{t}w_{i} f_{i} ( e^{t/2}w_{1}, e^{t/2}w_{2} ) \, dx. \end{eqnarray}$

Thus, by Lemma 2.1 and (2.4), we obtain

$\begin{eqnarray} &&\frac{1}{2} \sum\limits_{i = 1}^2 \partial_{t}\int_{r_1}^{r_2} \Big[ \vert \partial_{t}w_{i}\vert^2 + \vert w_{ix}\vert^2 -\frac{1}{4}\vert w_{i} \vert^2 \Big] \, dx \\ &&+\frac{1}{2}\sum\limits_{i = 1}^2\partial_{t} \Bigg( \int_{0}^{t} e^{-(t-p)/2}\mu_i(t-p) \int_{r_1}^{r_2} \vert w_{ix}(p)-w_{ix}(t)\vert^2 \, dx \, dp \Bigg) \\ \\ &&-\frac{1}{2}\sum\limits_{i = 1}^2\partial_{t} \Bigg( \int_{0}^{t}e^{-s/2}\mu_i(s) \, ds \int_{r_1}^{r_2} \left\vert w_{ix}(t)\right\vert^{2} dx \Bigg) \\\\ &&+\frac{1}{4}\sum\limits_{i = 1}^2\int_{0}^{t}e^{-(t-p)/2}\mu_i(t-p) \int_{r_1}^{r_2} \vert w_{ix}(p)-w_{ix}(t) \vert^2 \, dx \, dp \\ \\ &&-\frac{1}{2}\sum\limits_{i = 1}^2\int_{0}^{t} e^{-(t-p)/2}\partial_t\mu_i(t-p) \int_{r_1}^{r_2}\vert w_{ix}(p)-w_{ix}(t) \vert^2 \, dx \, dp \\\\ &&+\frac{1}{2}\sum\limits_{i = 1}^2e^{-t/2}\mu_i(t)\int_{r_1}^{r_2} \vert w_{ix}(t) \vert^2 \, dx \\ && = \frac{1}{2(\rho +2)} \partial_{t} \Bigg( e^{(\rho +1)t} \int_{r_1}^{r_2} \Big[ \vert w_1+w_2 \vert^{2(\rho +2)} + 2 \vert w_1w_2 \vert^{\rho+2} \Big] \Bigg) \, dx \\ &&-\frac{\rho+1}{2(\rho +2)} e^{(\rho +1)t} \int_{r_1}^{r_2} \Big[ \vert w_1+w_2 \vert^{2(\rho +2)} + 2 \vert w_1w_2 \vert^{\rho+2} \Big] \, dx . \end{eqnarray}$

Dropping the positive terms from the left-hand side, we have

$\partial_{t}E_{w}(t) \leq -\frac{\rho+1}{2(\rho +2)} e^{(\rho +1)t} \int_{r_1}^{r_2} \Big[ \vert w_1+w_2 \vert^{2(\rho +2)} + 2 \vert w_1w_2 \vert^{\rho+2} \Big] \, dx,$

which gives the conclusion by integrating both sides with respect to $t$ .

Remark 2.4. Concerning the local existence, we can follow the steps of results in ^[9] as equations, with some modifications imposed by the existence of the memories terms, where we replace the operator $\partial_{tt}-\Delta$ by $\partial_{tt}-\Delta \Big(1-\int_{0}^{t}\mu_i (t-s) \, ds \Big)$ with some conditions on the exponent $\rho$ . The local existence results for one equation still valid for a coupled system in the same type.

3. Blow up result

We prove that $(u_1, u_2)$ blows up in $L^2$ at finite time $T^{\ast}$ in the following Theorem.

Theorem 3.1. Let $\rho > -1$ and $\left(r_{2} - r_{1} \right)^2 < 8$ . Suppose that

$(u_1, u_2) \in \Big(C^{1}(0, T;H_0^{1}(r_1, r_2)) \cap C^2(0, T;L^2(r_1, r_2))\Big)^2,$

is a weak solution of (1.1) with

$e(0): = \sum\limits_{i = 1}^2 \int_{r_1}^{r_2}u_{i0}\left( u_{i1}- \frac{1}{2}u_{i0} \right) \, dx \gt 0 \quad \mathit{\text{and}} \quad E_w(0) \leq 0.$

Assume that $l$ satisfies

$\frac{2 \left( 4 \rho + 9 \right) + (\rho+1)(\rho+2)(r_2-r_1)^2}{2 \left( 4\rho^2+16\rho+17 \right)} \leq l .$

Then there exists $T^{\ast}$ such that

$\sum\limits_{i = 1}^2 \int_{r_1}^{r_2} \vert u_i \vert^2 \, dx \to +\infty \quad as\ t\to T^{\ast},$

where

$\ln T^{\ast} = \frac{2}{\rho+1}\Big( \sum\limits_{i = 1}^2 \int_{r_1}^{r_2} \vert u_{i0}\vert^2 \, dx \Big) \Big( \sum\limits_{i = 1}^2\int_{r_1}^{r_2} \left( 2 u_{i0} u_{i1} - \vert u_{i0}\vert^2 \right) \, dx \Big)^{-1}.$

Remark 3.2. Let

$g(x) \equiv \frac{2 \left( 4x + 9 \right) + (x+1)(x+2)(r_2-r_1)^2}{2 \left( 4x^2+16x+17 \right)}.$

Now that $(r_{2} - r_{1})^2 < 8$ holds, we have

$\partial_{x}g(x) = \frac{4x^2 +18x +19}{2 \left( 4x^2+16x+17 \right)^2}\left( (r_2-r_1)^2 - 8 \right) \lt 0.$

Thus by the monotonicity and $\rho > -1$ , we obtain

$\frac{(r_2-r_1)^2}{8} = \lim\limits_{t \to +\infty} g(t) \lt g(x) \lt g(-1) = 1.$

Hence the assumptions of Theorem make sense.

Remark 3.3. In the case of $r_{1} = 0$ and $r_{2} = 1$ , we introduce the example of $\mu_{i}(t)$ satisfying (1.3) and assumptions of Theorem 3.1. Let

$\rho \gt -1 \quad {and} \quad \mu_{i}(t) = e^{-kt} \quad {for} \quad k \gt \frac{9}{14}.$

Then we have

$\begin{eqnarray} l = 1-\int_{0}^{\infty}e^{- \left( k+1/2 \right) t} \, dt = \frac{2k-1}{2k+1} \in \left( \frac{1}{8}, 1 \right). \end{eqnarray}$

The condition $g(\rho) \leq l$ is equivalent to

$k \geq \frac{9}{14} + \frac{8\rho+18}{7 \left( \rho+1 \right) \left( \rho+2 \right)}.$

We need to state and prove the next intermediate Lemma.

Lemma 3.4. Under the assumptions in Theorem 3.1, we have

${e^{(\rho +1)t}\int_{r_1}^{r_2} \Bigg( \vert w_1+w_2 \vert^{2(\rho +2)} + 2 \vert w_1w_2 \vert^{\rho+2} \Bigg) \, dx} \\ \geq \left( \rho +2 \right) \sum\limits_{i = 1}^{2} \int_{r_1}^{r_2} \Big( \vert \partial_{t}w_{i}\vert^2 + l \vert w_{ix} \vert^2 -\frac{1}{4}\vert w_i\vert^2 \Big) \, dx \\ + \left( \rho +2 \right) \sum\limits_{i = 1}^{2} \int_{r_1}^{r_2} \int_{0}^{t} e^{-(t-p)/2}\mu_i(t-p) \left\vert w_{ix}(p)- w_{ix}(t) \right\vert^{2} \, dp \, dx.$

Proof. Let

$L(t) \equiv e^{(\rho +1)t}\int_{r_1}^{r_2} \Bigg( \vert w_1+w_2 \vert^{2(\rho +2)} + 2 \vert w_1w_2 \vert^{\rho+2} \Bigg) \, dx.$

We have

$\begin{eqnarray} L(t) & = & \left( \rho +2 \right) \sum\limits_{i = 1}^{2} \int_{r_1}^{r_2} \Bigg[ \vert \partial_{t}w_{i}\vert^2 + \Bigg( 1-\int_{0}^{t}e^{-p/2}\mu_i(p) \, dp \Bigg) \vert w_{ix} \vert^2 -\frac{1}{4}\vert w_i\vert^2 \Bigg] \, dx \\ &&+ \left( \rho +2 \right) \sum\limits_{i = 1}^{2} \int_{r_1}^{r_2} \int_{0}^{t}e^{-(t-p)/2}\mu_i(t-p) \left\vert w_{ix}(p)- w_{ix}(t) \right\vert^{2} \, dp \, dx \\ &&-2 \left( \rho +2 \right) E_{w}(t), \end{eqnarray}$

by (2.3). Since

$E_{w}(t) \leq 0,$

holds by (2.7), we have

$\begin{eqnarray} L(t) &\geq& \left( \rho +2 \right) \sum\limits_{i = 1}^{2} \int_{r_1}^{r_2} \Big( \vert \partial_{t}w_{i}\vert^2 + l \vert w_{ix} \vert^2 -\frac{1}{4}\vert w_i\vert^2 \Big) \, dx \\ &&+ \left( \rho +2 \right) \sum\limits_{i = 1}^{2} \int_{r_1}^{r_2} \int_{0}^{t}e^{-(t-p)/2}\mu_i(t-p) \left\vert w_{ix}(p)- w_{ix}(t) \right\vert^{2} \, dp \, dx, \end{eqnarray}$

by (1.3) and Lemma 2.3.

We are now ready to prove Theorem 3.1

Proof. (of Theorem 3.1)

Let

$A(t) : = \sum\limits_{i = 1}^2\int_{r_1}^{r_2}\vert w_i(t, x) \vert^2 \, dx.$

Then we have

$\partial_{t}A(t) = 2\sum\limits_{i = 1}^2\int_{r_1}^{r_2}w_i(t, x)\partial_{t} w_{i}(t, x) \, dx,$

and

$\begin{eqnarray} \partial_{tt}A(t) & = &2\sum\limits_{i = 1}^2\int_{r_1}^{r_2} w_i(t, x) \partial_{tt}w_{i}(t, x) \, dx +2\sum\limits_{i = 1}^2\int_{r_1}^{r_2} \vert\partial_{t} w_{i}(t, x) \vert^2 \, dx \\\\ & = &2\sum\limits_{i = 1}^2\int_{r_1}^{r_2} \Bigg( w_i w_{ixx}- w_i \int_{0}^{t}e^{-p/2}\mu_i(p)w_{ixx}(t-p) \, dp +\frac{1}{4} \vert w_i \vert^2 + \vert\partial_{t} w_{i} \vert^2 \Bigg) \, dx \\\\ &&+2\sum\limits_{i = 1}^2 \int_{r_1}^{r_2} e^{-t/2} w_i f_i(e^{t/2}w_1, e^{t/2}w_2) \, dx\\ & = &2\sum\limits_{i = 1}^2\int_{r_1}^{r_2}\Big( -\vert w_{ix} \vert^2 +\frac{1}{4}\vert w_{i} \vert^2 +\vert \partial_{t}w_{i} \vert^2 \Big) \, dx\\ &&+2\sum\limits_{i = 1}^2\int_{r_1}^{r_2} \int_{0}^{t}e^{-p/2}\mu_i(p)w_{ix}(t) w_{ix}(t-p) \, dp \, dx +2L(t). \label{A-second} \end{eqnarray}$

By Lemma 2.4 and similar computation to Lemma 2.1 with Young's inequality

$ab \leq \frac{a^2}{2 \theta} + \frac{\theta b^2}{2},$

for $a, b \geq 0$ and $\theta > 0$ , we have

$\begin{eqnarray} &&2\sum\limits_{i = 1}^2\int_{r_1}^{r_2} \int_{0}^{t}e^{-p/2}\mu_i(p)w_{ix}(t) w_{ix}(t-p) \, dp \, dx \\ &&\geq -2\sum\limits_{i = 1}^2\int_{r_1}^{r_2} \int_{0}^{t}e^{-p/2}\mu_i(p) \vert w_{ix}(t) \vert \vert w_{ix}(t-p) - w_{ix}(t) \vert \, dp \, dx \\ &&+2 \sum\limits_{i = 1}^2\int_{r_1}^{r_2} \int_{0}^{t}e^{-p/2}\mu_i(p) \vert w_{ix}(t) \vert^2 \, dp \, dx \\ &&\geq \left( 2 - \frac{1}{\theta} \right) \sum\limits_{i = 1}^2\int_{r_1}^{r_2} \int_{0}^{t}e^{-p/2}\mu_i(p) \vert w_{ix}(t) \vert^2 \, dp \, dx \\ &&-\theta \sum\limits_{i = 1}^2\int_{r_1}^{r_2} \int_{0}^{t}e^{-(t-p)/2}\mu_i(t-p) \vert w_{ix}(p) - w_{ix}(t) \vert^2 \, dp \, dx, \end{eqnarray}$

where $\theta$ is a positive constant to be chosen later. This estimate implies that

by using Lemma 3.4. Hence we choose $\theta = 2 (\rho+2)$ to obtain

$\begin{eqnarray} \partial_{tt}A(t) &\geq&2\sum\limits_{i = 1}^2\int_{r_1}^{r_2} \Big( -\vert w_{ix} \vert^2 +\frac{1}{4}\vert w_{i} \vert^2 +\vert \partial_{t}w_{i} \vert^2 \Big) \, dx \\ &&- \frac{1}{2 (\rho+2)}\sum\limits_{i = 1}^2\int_{r_1}^{r_2} \int_{0}^{t}e^{-p/2}\mu_i(p) \vert w_{ix}(t) \vert^2 \, dp \, dx \\ &&+2 \left( \rho +2 \right) \sum\limits_{i = 1}^{2} \int_{r_1}^{r_2} \Big( \vert \partial_{t}w_{i}\vert^2 + l \vert w_{ix} \vert^2 -\frac{1}{4}\vert w_i\vert^2 \Big) \, dx \\ &\geq& 2 \left( \rho +3 \right) \sum\limits_{i = 1}^2\int_{r_1}^{r_2} \vert \partial_{t}w_{i} \vert^2 \, dx \\ && +\Bigg( 2 \left( \rho +2 \right)l -2 - \frac{1-l}{2 (\rho+2)} \Bigg) \sum\limits_{i = 1}^2\int_{r_1}^{r_2} \vert w_{ix} \vert^2 \, dx \\ &&-\frac{(r_2-r_1)^2}{4}\left( \rho +1 \right)\sum\limits_{i = 1}^2\int_{r_1}^{r_2} \vert w_{ix} \vert^2 \, dx \\ &\geq& 2 \left( \rho +3 \right) \sum\limits_{i = 1}^2\int_{r_1}^{r_2} \vert\partial_{t} w_{i} \vert^2 \, dx +\frac{4 \rho^2 +16 \rho + 17}{2 \left( \rho +2 \right)} \left( l - g(\rho) \right) \sum\limits_{i = 1}^2\int_{r_1}^{r_2} \vert w_{ix} \vert^2 \, dx, \end{eqnarray}$

by Lemma 2.2, we have

$\begin{equation} \partial_{tt}A(t) \geq 2(\rho+3) \sum\limits_{i = 1}^2 \int_{r_1}^{r_2} \vert \partial_{t}w_{t} \vert^2 \, dx, \end{equation}$

(3.1)

where $g(\rho)$ is defined in Remark 3.2. Now under the assumption $e(0) > 0$ , (3.1) yields

$\partial_{t}A(t) \geq \partial_{t}A(0) \gt 0,$

and

$\begin{eqnarray} A(t) \geq A(0) + \partial_{t}A(0)t \geq A(0) \gt 0. \label{A} \end{eqnarray}$

Here, thanks to $e(0) > 0$ , $A(0) > 0$ follows. Hence we have just showed that $A(t)$ blows up. To complete the proof, we'll prove that the blow-up time $T_{1}^{\ast}$ is finite. As in ^[8], let us now set

$\begin{eqnarray} J(t) : = A(t)^{-k}, \qquad 2k = \rho+1 \gt 0. \label{J(t)} \end{eqnarray}$

We have only to show that $J(t)$ reaches $0$ in finite time. Then we have

$\partial_{t}J(t) = -kA(t)^{-k-1}\partial_{t}A(t) \lt 0,$

and

$\begin{eqnarray} \partial_{tt}J(t) & = &-kA(t)^{-k-2} \Big[ A(t) \partial_{tt}A(t)-(k+1)\partial_{t} A(t)^2 \Big] \\ &\leq& -kA(t)^{-k-1}\Big[\partial_{tt}A(t) -2(\rho+3) \sum\limits_{i = 1}^2 \int_{r_1}^{r_2}\left\vert\partial_{t} w_{i} \right\vert^2 \, dx \Big] \\ &\leq& 0, \end{eqnarray}$

(3.2)

by using Cauchy-Schwarz and Hölder's inequalities. Integrating (3.2) twice, we have

$0 \lt J(t) \leq J(0) + \partial_{t}J(0) t.$

Noting that $J'(0) < 0$ , we take

$T_{1}^{\ast} = -\frac{J(0)}{J'(0)} = \frac{1}{\rho+1}\frac{\sum_{i = 1}^2\int_{r_1}^{r_2}\vert w_{i0} \vert^2 \, dx} {\sum_{i = 1}^2\int_{r_1}^{r_2}w_{i0} w_{i1} \, dx} = \frac{e(0)^{-1}}{\rho+1} \sum\limits_{i = 1}^2 \int_{r_1}^{r_2} \vert u_{i0} \vert^2 \, dx \gt 0,$

so that $J(t) \to 0$ as $t \to T_{1}^{\ast}$ . Thus for a solution $w_{i}$ of (2.2), we obtain

$A(t) = \sum\limits_{i = 1}^2 \int_{r_1}^{r_2} \left\vert w_{i}(t, x) \right\vert^2 \, dx \to +\infty,$

as $t \to T_{1}^{\ast}$ . Since

$\sum\limits_{i = 1}^2 \int_{r_1}^{r_2} \vert u_{i}(t, x) \vert^2 \, dx = e^{\tau} A(\tau) = t A(\ln t),$

holds for all $t \in [1, \exp{T_{1}}^{\ast})$ by denoting $w_{i} = w_{i}(\tau, x)$ , the conclusion follows right away together with $T^{\ast} = \exp{T_{1}^{\ast}}$ .

Acknowledgement

The authors would like to thank the anonymous referees and the handling editor for their careful reading and for relevant remarks/suggestions to improve the paper.

Conflict of interest

The authors agree with the contents of the manuscript, and there is no conflict of interest among the authors.

References

[1]	UNEP, Decoupling natural resource use and environmental impacts from economic growth. United Nations Environment Program, 2011. Available from: https://www.resourcepanel.org/reports/decoupling-natural-resource-use-and-environmental-impacts-economic-growth.
[2]	Nilsson M, Griggs D, Visbeck M (2016) Policy: map the interactions between Sustainable Development Goals. Nature 534: 320–322. https://doi.org/10.1038/534320a doi: 10.1038/534320a
[3]	Van Soest HL, Van Vuuren DP, Hilaire J, et al. (2019) Analysing interactions among sustainable development goals with integrated assessment models. Glob Transit 1: 210–225. https://doi.org/10.1016/j.glt.2019.10.004 doi: 10.1016/j.glt.2019.10.004
[4]	Zhang Q, Liu S, Wang T, et al. (2019) Urbanization impacts on greenhouse gas (GHG) emissions of the water infrastructure in China: Trade-offs among sustainable development goals (SDGs). J Cleaner Prod 232: 474–486. https://doi.org/10.1016/j.jclepro.2019.05.333 doi: 10.1016/j.jclepro.2019.05.333
[5]	Wallace KJ, Kim MK, Rogers A, et al. (2020) Classifying human wellbeing values for planning the conservation and use of natural resources. J Environ Manage 256: 109955. https://doi.org/10.1016/j.jenvman.2019.109955 doi: 10.1016/j.jenvman.2019.109955
[6]	International Resource Panel (2011) Decoupling Natural Resource Use and Environmental Impacts from Economic Growth, UNEP/Earthprint.
[7]	Simonis UE (2013) Decoupling natural resource use and environmental impacts from economic growth. Int J Soc Econ 40: 385–386. https://doi.org/10.1108/03068291311305044 doi: 10.1108/03068291311305044
[8]	Haberl H, Fischer‐Kowalski M, Krausmann F, et al. (2011) A socio‐metabolic transition towards sustainability? Challenges for another Great Transformation. Sustain Dev 19: 1–14. https://doi.org/10.1002/sd.410 doi: 10.1002/sd.410
[9]	Moutinho V, Madaleno M, Inglesi-Lotz R, et al. (2018) Factors affecting CO₂ emissions in top countries on renewable energies: a LMDI decomposition application. Renewable Sustainable Energy Rev 90: 605–622. https://doi.org/10.1016/j.rser.2018.02.009 doi: 10.1016/j.rser.2018.02.009
[10]	Lin B, Agyeman SD (2019) Assessing Ghana's carbon dioxide emissions through energy consumption structure towards a sustainable development path. J Cleaner Prod 238: 117941. https://doi.org/10.1016/j.jclepro.2019.117941 doi: 10.1016/j.jclepro.2019.117941
[11]	Ang BW, Zhang FQ (2000) A survey of index decomposition analysis in energy and environmental studies. Energy 25 1149–1176. https://doi.org/10.1016/S0360-5442(00)00039-6 doi: 10.1016/S0360-5442(00)00039-6
[12]	Su B, Ang BW (2012) Structural decomposition analysis applied to energy and emissions: some methodological developments. Energy Econ 34: 177–188. https://doi.org/10.1016/j.eneco.2011.10.009 doi: 10.1016/j.eneco.2011.10.009
[13]	Ang BW, Liu FL (2001) A new energy decomposition method: perfect in decomposition and consistent in aggregation. Energy 26: 537–548. https://doi.org/10.1016/S0360-5442(01)00022-6 doi: 10.1016/S0360-5442(01)00022-6
[14]	Yang J, Cai W, Ma M, et al. (2020) Driving forces of China's CO₂ emissions from energy consumption based on Kaya-LMDI methods. Sci Total Environ 711: 134569. https://doi.org/10.1016/j.scitotenv.2019.134569 doi: 10.1016/j.scitotenv.2019.134569
[15]	Wang L, Wang Y, He H, et al. (2020) Driving force analysis of the nitrogen oxides intensity related to electricity sector in China based on the LMDI method. J Cleaner Prod 242: 118364. https://doi.org/10.1016/j.jclepro.2019.118364 doi: 10.1016/j.jclepro.2019.118364
[16]	Steckel JC, Hilaire J, Jakob M, et al. (2019) Coal and carbonization in sub-Saharan Africa. Nat Clim Change 10: 83–88. https://doi.org/10.1038/s41558-019-0649-8 doi: 10.1038/s41558-019-0649-8
[17]	Pothen F, Schymura M (2015). Bigger cakes with fewer ingredients? A comparison of material use of the world economy. Ecol Econ 109: 109–121. https://doi.org/10.1016/j.ecolecon.2014.10.009 doi: 10.1016/j.ecolecon.2014.10.009
[18]	Steinberger JK, Krausmann F, Getzner M et al. (2013) Development and dematerialization: an international study. PLoS One 8: e70385. https://doi.org/10.1371/journal.pone.0070385 doi: 10.1371/journal.pone.0070385
[19]	Wiedmann TO, Schandl H, Lenzen M, et al. (2015) The material footprint of nations. P Natl Acad Sci USA 112: 6271–6276. https://doi.org/10.1073/pnas.1220362110 doi: 10.1073/pnas.1220362110
[20]	Weinzettel J, Kovanda J (2011) Structural decomposition analysis of raw material consumption: the case of the Czech Republic. J Ind Ecol 15: 893–907. https://doi.org/10.1111/j.1530-9290.2011.00378.x doi: 10.1111/j.1530-9290.2011.00378.x
[21]	Azami S, Hajiloui MM (2022) How does the decomposition approach explain changes in Iran's energy consumption? What are the driving factors? Clean Responsible Consum 4: 100054. https://doi.org/10.1016/j.clrc.2022.100054 doi: 10.1016/j.clrc.2022.100054
[22]	Zhang J, Wang H, Ma L, et al. (2021) Structural path decomposition analysis of resource utilization in China, 1997–2017. J Cleaner Prod 322: 129006. https://doi.org/10.1016/j.jclepro.2021.129006 doi: 10.1016/j.jclepro.2021.129006
[23]	Krausmann F, Wiedenhofer D, Haberl H (2020) Growing stocks of buildings, infrastructures and machinery as key challenge for compliance with climate targets. Global Environ Chang 61: 102034. https://doi.org/10.1016/j.gloenvcha.2020.102034 doi: 10.1016/j.gloenvcha.2020.102034
[24]	Eyre N, Killip G (2019) Shifting the Focus: Energy Demand in a Net-Zero Carbon UK, 1 Ed., Oxford: Centre for Research into Energy Demand Solutions.
[25]	Gonzalez Hernandez A (2018) Site-level resource efficiency analysis[PhD's thesis]. University of Cambridge, United Kingdom.
[26]	Baninla Y, Lu Y, Zhang Q, et al. (2020) Material use and resource efficiency of African sub-regions. J Cleaner Prod 247: 119092. https://doi.org/10.1016/j.jclepro.2019.119092 doi: 10.1016/j.jclepro.2019.119092
[27]	OECD Statistics Database, OECD Statistics Database Domestic Material Consumption and Material Footprint. OECD, 2020. Available from: https://stats.oecd.org/Index.aspx?DataSetCode=MATERIAL_RESOURCES.
[28]	Ward JD, Sutton PC, Werner AD, et al. (2016) Is decoupling GDP growth from environmental impact possible? PLoS One 11: e0164733. https://doi.org/10.1371/journal.pone.0164733 doi: 10.1371/journal.pone.0164733
[29]	Bithas K, Kalimeris P (2018) Unmasking decoupling: redefining the resource intensity of the economy. Sci Total Environ 619: 338–351. https://doi.org/10.1016/j.scitotenv.2017.11.061 doi: 10.1016/j.scitotenv.2017.11.061
[30]	Pao HT, Chen CC (2019) Decoupling strategies: CO₂ emissions, energy resources, and economic growth in the Group of Twenty. J Cleaner Prod 206: 907–919. https://doi.org/10.1016/j.jclepro.2018.09.190 doi: 10.1016/j.jclepro.2018.09.190
[31]	Sanyé-Mengual E, Secchi M, Corrado S, et al. (2019) Assessing the decoupling of economic growth from environmental impacts in the European Union: A consumption-based approach. J Cleaner Prod 236: 117535. https://doi.org/10.1016/j.jclepro.2019.07.010 doi: 10.1016/j.jclepro.2019.07.010
[32]	Liu Z, Xin L (2019) Dynamic analysis of spatial convergence of green total factor productivity in China's primary provinces along its Belt and Road Initiative. Chin J Popul Resour Environ 17: 101–112. https://doi.org/10.1080/10042857.2019.1611342 doi: 10.1080/10042857.2019.1611342
[33]	Ang BW (2005) The LMDI approach to decomposition analysis: a practical guide. Energ Policy 33: 867–871. https://doi.org/10.1016/j.enpol.2003.10.010 doi: 10.1016/j.enpol.2003.10.010
[34]	Wang W, Li M, Zhang M (2017) Study on the changes of the decoupling indicator between energy-related CO₂ emission and GDP in China. Energy 128: 11–18. https://doi.org/10.1016/j.energy.2017.04.004 doi: 10.1016/j.energy.2017.04.004
[35]	Chen J, Wang P, Cui L (2018) Decomposition and decoupling analysis of CO₂ emissions in OECD. Appl Energy 231: 937–950. https://doi.org/10.1016/j.apenergy.2018.09.179 doi: 10.1016/j.apenergy.2018.09.179
[36]	Du G, Sun C, Ouyang X, et al. (2018) A decomposition analysis of energy-related CO₂ emissions in Chinese six high-energy intensive industries. J Clean Prod 184: 1102–1112. https://doi.org/10.1016/j.jclepro.2018.02.304 doi: 10.1016/j.jclepro.2018.02.304
[37]	Zheng X, Lu Y, Yuan J, et al. (2020) Drivers of change in China's energy-related CO₂ emissions. P Natl Acad Sci USA 117: 29–36. https://doi.org/10.1073/pnas.1908513117 doi: 10.1073/pnas.1908513117
[38]	Shao S, Yang L, Gan C, et al. (2016) Using an extended LMDI model to explore techno-economic drivers of energy-related industrial CO₂ emission changes: A case study for Shanghai (China). Renewable Sustainable Energy Rev 55: 516–536. https://doi.org/10.1016/j.rser.2015.10.081 doi: 10.1016/j.rser.2015.10.081
[39]	Li H, Zhao Y, Qiao X, et al. (2017) Identifying the driving forces of national and regional CO₂ emissions in China: based on temporal and spatial decomposition analysis models. Energy Econ 68: 522–538. https://doi.org/10.1016/j.eneco.2017.10.024 doi: 10.1016/j.eneco.2017.10.024
[40]	Guan D, Meng J, Reiner DM, et al. (2018) Structural decline in China's CO₂ emissions through transitions in industry and energy systems. Nat Geosci 11: 551–555. https://doi.org/10.1038/s41561-018-0161-1 doi: 10.1038/s41561-018-0161-1
[41]	Wu Y, Tam VW, Shuai C, et al. (2019) Decoupling China's economic growth from carbon emissions: Empirical studies from 30 Chinese provinces (2001–2015). Sci Total Environ 656: 576–588. https://doi.org/10.1016/j.scitotenv.2018.11.384 doi: 10.1016/j.scitotenv.2018.11.384
[42]	Bekun FV, Alola AA, Gyamfi BA, et al. (2021) The environmental aspects of conventional and clean energy policy in sub-Saharan Africa: is N-shaped hypothesis valid? Environ Sci Pollut R 28: 66695–66708. https://doi.org/10.1007/s11356-021-14758-w doi: 10.1007/s11356-021-14758-w
[43]	Adedoyin FF, Nwulu N, Bekun FV (2021) Environmental degradation, energy consumption and sustainable development: accounting for the role of economic complexities with evidence from World Bank income clusters. Bus Strategy Environ 30: 2727–2740. https://doi.org/10.1002/bse.2774 doi: 10.1002/bse.2774
[44]	Kwakwa PA, Alhassan H, Aboagye S (2018) Environmental Kuznets curve hypothesis in a financial development and natural resource extraction context: evidence from Tunisia. Quant Finance Econ 2: 981–1000. https://doi.org/10.3934/QFE.2018.4.981 doi: 10.3934/QFE.2018.4.981
[45]	Gyamfi BA (2022) Consumption-based carbon emission and foreign direct investment in oil-producing Sub-Sahara African countries: the role of natural resources and urbanization. Environ Sci Pollut R 29: 13154–13166. https://doi.org/10.1007/s11356-021-16509-3 doi: 10.1007/s11356-021-16509-3
[46]	Kwakwa PA, Alhassan H, Adu G (2020) Effect of natural resources extraction on energy consumption and carbon dioxide emission in Ghana. Int J Energy Sect Manag 14: 20–39. https://doi.org/10.1108/IJESM-09-2018-0003 doi: 10.1108/IJESM-09-2018-0003
[47]	Wiedenhofer D, Fishman T, Plank B, et al. (2021) Prospects for a saturation of humanity's resource use? An analysis of material stocks and flows in nine world regions from 1900 to 2035. Global Environ Chang 71: 102410. https://doi.org/10.1016/j.gloenvcha.2021.102410 doi: 10.1016/j.gloenvcha.2021.102410
[48]	Haberl H, Wiedenhofer D, Virág D, et al. (2020) A systematic review of the evidence on decoupling of GDP, resource use and GHG emissions, part Ⅱ: synthesizing the insights. Environ Res Lett 15: 065003. https://doi.org/10.1088/1748-9326/ab842a doi: 10.1088/1748-9326/ab842a
[49]	Bolt J, Inklaar R, de Jong H, et al. (2018) Rebasing 'Maddison': new income comparisons and the shape of long-run economic development. Maddison Project Database, version 2018. Maddison Project Working Paper 10. Available from: https://www.rug.nl/ggdc/historicaldevelopment/maddison/releases/maddison-project-database-2018.
[50]	Johansen S, Juselius K (1990) Some structural hypotheses in a multivariate cointegration analysis of the purchasing power parity and the uncovered interest parity for UK. Discussion Papers 90-05, University of Copenhagen.
[51]	Omay T, Emirmahmutoglu F, Denaux ZS (2017) Nonlinear error correction based cointegration test in panel data. Econ Lett 157: 1–4. https://doi.org/10.1016/j.econlet.2017.05.017 doi: 10.1016/j.econlet.2017.05.017
[52]	Odaki M (2015) Cointegration rank tests based on vector autoregressive approximations under alternative hypotheses. Econ Lett 136: 187–189. https://doi.org/10.1016/j.econlet.2015.09.028 doi: 10.1016/j.econlet.2015.09.028
[53]	Aslan A, Kula F, Kalyoncu H (2010) Additional evidence of long-run purchasing power parity with black and official exchange rates. Appl Econ Lett 17: 1379–1382. https://doi.org/10.1080/13504850902967522 doi: 10.1080/13504850902967522
[54]	Kaya Y (1989) Impact of carbon dioxide emission control on GNP growth: interpretation of proposed scenarios. Intergovernmental Panel on Climate Change/Response Strategies Working Group.
[55]	Ang BW, Liu FL, Chew EP (2003) Perfect decomposition techniques in energy and environmental analysis. Energ Policy 31: 1561–1566. https://doi.org/10.1016/S0301-4215(02)00206-9 doi: 10.1016/S0301-4215(02)00206-9
[56]	Bianchi M, del Valle I, Tapia C (2021) Material productivity, socioeconomic drivers and economic structures: A panel study for European regions. Ecol Econ 183: 106948. https://doi.org/10.1016/j.ecolecon.2021.106948 doi: 10.1016/j.ecolecon.2021.106948
[57]	Weisz H, Krausmann F, Amann C, et al. (2006) The physical economy of the European Union: Cross-country comparison and determinants of material consumption. Ecol Econ 58: 676–698. https://doi.org/10.1016/j.ecolecon.2005.08.016 doi: 10.1016/j.ecolecon.2005.08.016
[58]	Kassouri Y, Alola AA, Savaş S (2021) The dynamics of material consumption in phases of the economic cycle for selected emerging countries. Resour Policy 70: 101918. https://doi.org/10.1016/j.resourpol.2020.101918 doi: 10.1016/j.resourpol.2020.101918
[59]	Karakaya E, Sarı E, Alataş S (2021) What drives material use in the EU? Evidence from club convergence and decomposition analysis on domestic material consumption and material footprint. Resour Policy 70: 101904. https://doi.org/10.1016/j.resourpol.2020.101904 doi: 10.1016/j.resourpol.2020.101904
[60]	Jia H, Li T, Wang A, et al. (2021) Decoupling analysis of economic growth and mineral resources consumption in China from 1992 to 2017: A comparison between tonnage and exergy perspective. Resour Policy 74: 102448. https://doi.org/10.1016/j.resourpol.2021.102448 doi: 10.1016/j.resourpol.2021.102448
[61]	Khan I, Zakari A, Ahmad M, et al. (2022) Linking energy transitions, energy consumption, and environmental sustainability in OECD countries. Gondwana Res 103: 445–457. https://doi.org/10.1016/j.gr.2021.10.026 doi: 10.1016/j.gr.2021.10.026

This article has been cited by:

1.	Hamish G. Spencer, Anthony B. Pleasants, Peter D. Gluckman, Graeme C. Wake, A model of optimal timing for a predictive adaptive response, 2021, 2040-1744, 1, 10.1017/S2040174420001361
2.	Graeme Wake, 2015, Chapter 27, 978-3-319-22128-1, 155, 10.1007/978-3-319-22129-8_27

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)