In December 2019, the severe respiratory syndrome coronavirus-2 was discovered in China. The virus spread rapidly and, by March 2020, the World Health Organization (WHO) declared COVID-19 to be a global pandemic. Scientists expected the African continent to be among the worst affected by the sanitary emergency in terms of prevalence, incidence and mortality. This prediction was refuted by evidence, considering that Africa reported the least number of cases and deaths compared to Europe, Asia and America. The first case in Africa was registered in Egypt on February 14, 2020. By the end of 2021, the continent recorded a cumulative of 7,110,817 cases and 155,505 deaths. Nonetheless, estimates are likely to be distorted due to the lack of available data about the impact of COVID-19 and the limited documentary capacity of most African countries. There are several theories to explain why, contrary to the expected trend, Africa had the fewest COVID-19 incidences compared to other continents. Africa is characterized by a young population, which is notoriously less susceptible to COVID-19, with an average age of 19.7 years. In addition, most of the Africans (59%) live in rural areas, with few opportunities to travel or get in contact with outsiders. Moreover, governments enforced outstanding measures to contain the spread of the virus and safeguard the national economy, such as strengthening their documentary capacity and enforcing effective social safety nets. However, most of these policies have aggravated entrenched patterns of discrimination, making certain populations uniquely vulnerable. Indeed, mobility restrictions and border closures severely affected people with mobile livelihoods. In Morocco, the emergency measures compromised the resilience capacity of sub-Saharan migrants, particularly women and girls. To study the phenomenon of African migration to Morocco, we conducted fieldwork research from October to December 2021, interrupted by the closure of the kingdom's borders, and continued remotely thanks to key informants.
Citation: Daniela Santus, Sara Ansaloni. Mobility issues and multidimensional inequalities: exploring the limits of the National Strategy for Immigration and Asylum during the COVID-19 pandemic in Morocco[J]. AIMS Geosciences, 2023, 9(1): 191-218. doi: 10.3934/geosci.2023011
[1] | Karl Hajjar, Lénaïc Chizat . On the symmetries in the dynamics of wide two-layer neural networks. Electronic Research Archive, 2023, 31(4): 2175-2212. doi: 10.3934/era.2023112 |
[2] | Eray Önler . Feature fusion based artificial neural network model for disease detection of bean leaves. Electronic Research Archive, 2023, 31(5): 2409-2427. doi: 10.3934/era.2023122 |
[3] | Dong-hyeon Kim, Se-woon Choe, Sung-Uk Zhang . Recognition of adherent polychaetes on oysters and scallops using Microsoft Azure Custom Vision. Electronic Research Archive, 2023, 31(3): 1691-1709. doi: 10.3934/era.2023088 |
[4] | Ziqing Yang, Ruiping Niu, Miaomiao Chen, Hongen Jia, Shengli Li . Adaptive fractional physical information neural network based on PQI scheme for solving time-fractional partial differential equations. Electronic Research Archive, 2024, 32(4): 2699-2727. doi: 10.3934/era.2024122 |
[5] | Ilyоs Abdullaev, Natalia Prodanova, Mohammed Altaf Ahmed, E. Laxmi Lydia, Bhanu Shrestha, Gyanendra Prasad Joshi, Woong Cho . Leveraging metaheuristics with artificial intelligence for customer churn prediction in telecom industries. Electronic Research Archive, 2023, 31(8): 4443-4458. doi: 10.3934/era.2023227 |
[6] | Kai Huang, Chang Jiang, Pei Li, Ali Shan, Jian Wan, Wenhu Qin . A systematic framework for urban smart transportation towards traffic management and parking. Electronic Research Archive, 2022, 30(11): 4191-4208. doi: 10.3934/era.2022212 |
[7] | Alejandro Ballesteros-Coll, Koldo Portal-Porras, Unai Fernandez-Gamiz, Iñigo Aramendia, Daniel Teso-Fz-Betoño . Generative adversarial network for inverse design of airfoils with flow control devices. Electronic Research Archive, 2025, 33(5): 3271-3284. doi: 10.3934/era.2025144 |
[8] | Ruyu Yan, Jiafei Jin, Kun Han . Reinforcement learning for deep portfolio optimization. Electronic Research Archive, 2024, 32(9): 5176-5200. doi: 10.3934/era.2024239 |
[9] | Xin Liu, Yuan Zhang, Kai Zhang, Qixiu Cheng, Jiping Xing, Zhiyuan Liu . A scalable learning approach for user equilibrium traffic assignment problem using graph convolutional networks. Electronic Research Archive, 2025, 33(5): 3246-3270. doi: 10.3934/era.2025143 |
[10] | Mohd. Rehan Ghazi, N. S. Raghava . Securing cloud-enabled smart cities by detecting intrusion using spark-based stacking ensemble of machine learning algorithms. Electronic Research Archive, 2024, 32(2): 1268-1307. doi: 10.3934/era.2024060 |
In December 2019, the severe respiratory syndrome coronavirus-2 was discovered in China. The virus spread rapidly and, by March 2020, the World Health Organization (WHO) declared COVID-19 to be a global pandemic. Scientists expected the African continent to be among the worst affected by the sanitary emergency in terms of prevalence, incidence and mortality. This prediction was refuted by evidence, considering that Africa reported the least number of cases and deaths compared to Europe, Asia and America. The first case in Africa was registered in Egypt on February 14, 2020. By the end of 2021, the continent recorded a cumulative of 7,110,817 cases and 155,505 deaths. Nonetheless, estimates are likely to be distorted due to the lack of available data about the impact of COVID-19 and the limited documentary capacity of most African countries. There are several theories to explain why, contrary to the expected trend, Africa had the fewest COVID-19 incidences compared to other continents. Africa is characterized by a young population, which is notoriously less susceptible to COVID-19, with an average age of 19.7 years. In addition, most of the Africans (59%) live in rural areas, with few opportunities to travel or get in contact with outsiders. Moreover, governments enforced outstanding measures to contain the spread of the virus and safeguard the national economy, such as strengthening their documentary capacity and enforcing effective social safety nets. However, most of these policies have aggravated entrenched patterns of discrimination, making certain populations uniquely vulnerable. Indeed, mobility restrictions and border closures severely affected people with mobile livelihoods. In Morocco, the emergency measures compromised the resilience capacity of sub-Saharan migrants, particularly women and girls. To study the phenomenon of African migration to Morocco, we conducted fieldwork research from October to December 2021, interrupted by the closure of the kingdom's borders, and continued remotely thanks to key informants.
Chemotaxis is the property of cells to move in an oriented manner in response to an increasing concentration of chemo-attractant or decreasing concentration of chemo-repellent, where the former is referred to as attractive chemotaxis and the later to repulsive chemotaxis. To begin with, it is important to study the quasilinear Keller-Segel system as follows
{ut=∇⋅(D(u)∇u)−χ∇⋅(ϕ(u)∇v),x∈Ω,t>0,τvt=Δv−αv+βu,x∈Ω,t>0, | (1.1) |
subject to homogeneous Neumann boundary conditions, where the functions D(u) and ϕ(u) denote the strength of diffusion and chemoattractant, respectively, and the function u=u(x,t) idealizes the density of cell, v=x(x,t) represents the concentration of the chemoattractant. Here the attractive (repulsive) chemotaxis corresponds to χ>0 (χ<0), and |χ|∈R∖{0} measures the strength of chemotactic response. The parameters τ∈{0,1}, and α,β>0 denote the production and degradation rates of the chemical. The above system describes the chemotactic interaction between cells and one chemical signal (either attractive or repulsive), and it has been investigated quite extensively on the existence of global bounded solutions or the occurrence of blow-up in finite time in the past four decades. In particular, the system (1.1) is the prototypical Keller-Segel model [1] when D(u)=1,ϕ(u)=u. In the case τ=1, there are many works to show that the solution is bounded [2,3,4,5], and blow-up in finite time [6,7,8,9,10,11]. If the cell's movement is much slower than the chemical signal diffusing, the second equation of (1.1) is reduced to 0=Δv−M+u, where M:=1|Ω|∫Ωu(x,t)dx and the simplified system has many significant results [12,13,14,15].
For further information concerning nonlinear signal production, when the chemical signal function is denoted by g(u), authors derived for more general nonlinear diffusive system as follows
{ut=∇⋅(D(u)∇u)−∇⋅(ϕ(u)∇v),x∈Ω,t>0,0=Δv−M+g(u),x∈Ω,t>0, | (1.2) |
where M:=1|Ω|∫Ωg(u(x,t))dx. Recently, when D(u)=u−p,ϕ(u)=u and g(u)=ul, it has been shown that all solutions are global and uniformly bounded if p+l<2n, whereas p+l>2n implies that the solution blows up in finite time [16]. What's more, there are many significant works [17,18,19] associated with this system.
Subsequently, the attraction-repulsion system has been introduced in ([20,21]) as follows
{ut=Δu−χ∇⋅(u∇v)+ξ∇⋅(u∇w),x∈Ω,t>0,τ1vt=Δv+αu−βv,x∈Ω,t>0,τ2wt=Δw+δu−γw,x∈Ω,t>0, | (1.3) |
subject to homogeneous Neumann boundary conditions, where χ,ξ,α,β,δ,γ>0 are constants, and the functions u(x,t),v(x,t) and w(x,t) denote the cell density, the concentration of the chemoattractant and chemorepellent, respectively. The above attraction-repulsion chemotaxis system has been studied actively in recent years, and there are many significant works to be shown as follows.
For example, if τ1=τ2=0, Perthame [22] investigated a hyperbolic Keller-Segel system with attraction and repulsion when n=1. Subsequently, Tao and Wang [23] proved that the solution of (1.3) is globally bounded provided ξγ−χα>0 when n≥2, and the solution would blow up in finite time provided ξγ−χα<0,α=β when n=2. Then, there is a blow-up solution when χα−ξγ>0,δ≥β or χαδ−ξγβ>0,δ<β for n=2 [24]. Moreover, Viglialoro [25] studied the explicit lower bound of blow-up time when n=2. In another hand, if τ1=1,τ2=0, Jin and Wang [26] showed that the solution is bounded when n=2 with ξγ−χα≥0, and Zhong et al. [27] obtained the global existence of weak solution when ξγ−χα≥0 for n=3. Furthermore, if τ1=τ2=1, Liu and Wang [28] obtained the global existence of solutions, and Jin et al. [29,30,31] also showed a uniform-in-time bound for solutions. In addition, there are plenty of available results of the attraction-repulsion system with logistic terms [32,33,34,35,36,37,38,39,40], and for further information concerning (1.3) based on the nonlinear signal production, it was used to model the aggregation patterns formed by some bacterial chemotaxis in [41,42,43].
We turn our eyes into a multi-dimensional attraction-repulsion system
{ut=Δu−χ∇⋅(ϕ(u)∇v)+ξ∇⋅(ψ(u)∇w),x∈Ω,t>0,τ1vt=Δv−μ1(t)+f(u),x∈Ω,t>0,τ2wt=Δw−μ2(t)+g(u),x∈Ω,t>0, | (1.4) |
where Ω∈Rn(n≥2) is a bounded domain with smooth boundary, μ1(t)=1|Ω|∫Ωf(u)dx,μ2(t)=1|Ω|∫Ωg(u)dx and τ1,τ2∈{0,1}. Later on, the system (1.4) has attracted great attention of many mathematicians. In particular, when ϕ(u)=ψ(u)=u,f(u)=uk and g(u)=ul, Liu and Li [44] proved that all solutions are bounded if k<2n, while blow-up occurs for k>l and k>2n in the case τ1=τ2=0.
Inspired by the above literature, we are devoted to deal with the quasilinear attraction-repulsion chemotaxis system
{ut=∇⋅(D(u)∇u)−χ∇⋅(u∇v)+ξ∇⋅(u∇w),x∈Ω,t>0,0=Δv−μ1(t)+f1(u),x∈Ω,t>0,0=Δw−μ2(t)+f2(u),x∈Ω,t>0,∂u∂ν=∂v∂ν=∂w∂ν=0,x∈∂Ω,t>0,u(x,0)=u0(x),x∈Ω, | (1.5) |
in a bounded domain Ω⊂Rn,n≥2 with smooth boundary, where ∂∂ν denotes outward normal derivatives on ∂Ω. The function u(x,t) denotes the cell density, v(x,t) represents the concentration of an attractive signal (chemo-attractant), and w(x,t) is the concentration of a repulsive signal (chemo-repellent). The parameters satisfy χ,ξ≥0, which denote the strength of the attraction and repulsion, respectively. Here μ1(t)=1|Ω|∫Ωf1(u(x,t))dx, μ2(t)=1|Ω|∫Ωf2(u(x,t))dx, and f1,f2 are nonnegative Hölder continuous functions.
In the end, we propose the following assumptions on D,f1,f2 and u0 for the system (1.5).
(I1) The nonlinear diffusivity D is positive function satisfying
D∈C2([0,∞)). | (1.6) |
(I2) The function fi is nonnegative and nondecreasing and satisfies
fi∈⋃θ∈(0,1)Cθloc([0,∞))∩C1((0,∞)) | (1.7) |
with i∈{1,2}.
(I3) The initial datum
u0∈⋃θ∈(0,1)Cθ(¯Ω) is nonnegative and radially decreasing,∂u0∂ν=0 on ∂Ω. | (1.8) |
The goal of the article is twofold. On the one hand, we need to find out the mutual effect of the nonlinear diffusivity D(u) and the nonlinear signal production fi(u)(i=1,2). On the other hand, we need to make a substantial step towards the dynamic of blowing up in finite time. Hence, we draw our main results concerning (1.5) read as follows.
Theorem 1.1. Let n≥2, R>0 and Ω=BR(0)⊂Rn be a ball, and suppose that the function D satisfies (1.6) and f1,f2 are assumed to fulfill (1.7) as well as
D(u)≤d(1+u)m−1, f1(u)≥k1(1+u)γ1, f2(u)≤k2(1+u)γ2 for all u≥0, |
with m∈R, k1,k2,γ1,γ2,d>0 and
γ1>γ2 and 1+γ1−m>2n. | (1.9) |
For any M>0 there exist ε=ε(γ1,M,R)∈(0,M) and r∗=r∗(γ1,M,R)∈(0,R) such that if u0 satisfies (1.8) with
∫Ωu0=M and ∫Br∗(0)u0≥M−ε, |
then the corresponding solution of the system (1.5) blows up in finite time.
Theorem 1.2. Let n≥2, Ω⊂Rn be a smooth bounded domain, and suppose that the function D satisfies (1.6) and f1,f2 are assumed to fulfill (1.7) as well as
D(u)≥d(1+u)m−1, f1(u)≤k1(1+u)γ1, f2(u)=k2(1+u)γ2 for all u≥0, |
with m∈R, k1,k2,γ1,γ2,d>0 and
γ2<1+γ1<2n+m. | (1.10) |
Then for each u0∈⋃θ∈(0,1)Cθ(¯Ω), u0≥0 with ∂u0∂ν=0 on ∂Ω, and the system (1.5) admits a unique global classical solution (u,v,w) with
u,v,w∈C2,1(¯Ω×(0,∞))∩C0(¯Ω×[0,∞)). |
Furthermore, u,v and w are all non-negative and bounded.
The structure of this paper reads as follows: In section 2, we will show the local-in-time existence of a classical solution to the system (1.5) and some lemmas that we will use later. In section 3, we will prove Theorem 1.1 by establishing a superlinear differential inequality. In section 4, we will solve the boundedness of u in L∞ and prove Theorem 1.2.
Firstly, we state one result concerning local-in-time existence of a classical solution to the system (1.5). Then, we denote some new variables to transfer the original equations in (1.5) to a new system according to the ideas in [19,20,21,22,23,24,25]. In addition, in order to prove the main result, we will state some lemmas which will be needed later.
Lemma 2.1. Let Ω⊂Rn with n≥2 be a bounded domain with smooth boundary. Assume that D fulfills (1.6), f1,f2 satisfy (1.7) and u0∈⋃θ∈(0,1)Cθ(¯Ω) with ∂u0∂ν=0 on ∂Ω as well as u0≥0, then there exist Tmax∈(0,∞] and a classical solution (u,v,w) to (1.5) uniquely determined by
{u∈C0(¯Ω×[0,Tmax))∩C2,1(¯Ω×(0,Tmax)),v∈∩q>nL∞((0,Tmax);W1,q(Ω))∩C2,0(¯Ω×(0,Tmax)),w∈∩q>nL∞((0,Tmax);W1,q(Ω))∩C2,0(¯Ω×(0,Tmax)). |
In addition, the function u≥0 in Ω×(0,Tmax) and if Tmax<∞ then
limt↗Tmaxsup‖u(⋅,t)‖L∞(Ω)=∞. | (2.1) |
Moreover,
∫Ωv(⋅,t)=0,∫Ωw(⋅,t)=0 for all t∈(0,Tmax). | (2.2) |
Finally, the solution (u,v,w) is radially symmetric with respect to |x| if u0 satisfies (1.8).
Proof. The proof of this lemma needs to be divided into four steps. Firstly, the method to solve the local time existence of the classical solution to the problem (1.5) is based on a standard fixed point theorem. Next, we will use the standard extension theorem to obtain (2.1). Then, we are going to use integration by parts to deduce (2.2). Finally, we would use the comparison principle to conclude that the solution is radially symmetric. For the details, we refer to [45,46,47,48].
For the convenience of analysis and in order to prove Theorem 1.1, we set h=χv−ξw, then the system (1.5) is rewritten as
{ut=∇⋅(D(u)∇u)−∇⋅(u∇h),x∈Ω,t>0,0=Δh−μ(t)+f(u),x∈Ω,t>0,∂u∂ν=∂h∂ν=0,x∈∂Ω,t>0,u(x,0)=u0(x),x∈Ω, | (2.3) |
where μ(t)=χμ1(t)−ξμ2(t)=1|Ω|∫Ωf(u(x,t))dx and f(u)=χf1(u)−ξf2(u).
For the same reason, we will convert the system (2.3) into a scalar equation. Let us assume Ω=BR(0) with some R>0 is a ball and the initial data u0=u0(r) with r=|x|∈[0,R] satisfies (1.8). In the radial framework, the system (2.3) can be transformed into the following form
{rn−1ut=(rn−1D(u)ur)r−(rn−1uhr)r,r∈(0,R),t>0,0=(rn−1hr)r−rn−1μ(t)+rn−1f(u),r∈(0,R),t>0,ur=hr=0, r=R,t>0,u(r,0)=u0(r),r∈(0,R). | (2.4) |
Lemma 2.2. Let us introduce the function
U(s,t)=n∫s1n0ρn−1u(ρ,t)dρ,s=rn∈[0,Rn], t∈(0,Tmax), |
then
Us(t)=u(s1n,t), Uss(t)=1ns1n−1ur(s1n,t), | (2.5) |
and
Ut(s,t)=n2s2−2nD(Us)Uss−sμ(t)Us+Us⋅∫s0f(Us(σ,t))dσ. | (2.6) |
Proof. Firstly, integrating the second equation of (2.4) over (0,r), we have
rn−1hr(r,t)=rnnμ(t)−∫r0ρn−1f(u(ρ,t))dρ, |
so
s1−1nhr(s1n,t)=snμ(t)−1n∫s0f(u(σ1n,t))dσ,∀s∈(0,Rn), t∈(0,Tmax). |
Then, a direct calculation yields
Us(s,t)=u(s1n,t),∀s∈(0,Rn), t∈(0,Tmax), |
and
Uss(s,t)=1ns1n−1ur(s1n,t),∀s∈(0,Rn), t∈(0,Tmax), |
as well as
Ut(s,t)=n∫s1n0ρn−1ut(ρ,t)dρ=n2s2−2nD(Us)Uss−ns1−1nUshr=n2s2−2nD(Us)Uss−sμ(t)Us+Us⋅∫s0f(Us(σ,t))dσ |
for all s∈(0,Rn) and t∈(0,Tmax).
Furthermore, by a direct calculation and (1.7), we know that the functions U and f satisfy the following results
{Us(s,t)=u(s1n,t)>0,s∈(0,Rn),t∈(0,Tmax),U(0,t)=0,U(Rn,t)=nωn∫Ωu(⋅,t)=nMωn,t∈[0,Tmax),|f(s)|,f1(s),f2(s)≤C0,0≤s≤A,C0=C0(A)>0, | (2.7) |
where ωn=n|B1(0)| and A is a positive constant.
Lemma 2.3. Suppose that (1.7), (1.8) and (2.7) hold, then we have
hr(r,t)=1nμ(t)r−r1−n∫r0ρn−1f(u(ρ,t))dρfor all r∈(0,R),t∈(0,Tmax). |
In particular,
hr(r,t)≤1n(μ(t)+C0)r. | (2.8) |
Proof. By integration the second equation in (2.4), we obtain that
rn−1hr=μ(t)⋅∫r0ρn−1dρ−∫r0ρn−1f(u(ρ,t))dρ for all r∈(0,R),t∈(0,Tmax). |
According to (1.9), we can easily get that f(u)≥0 if u≥C∗=max{0,(k2ξk1χ)1γ1−γ2−1}, and split
\begin{equation} \nonumber \int^{r}_{0}\rho^{n-1}f(u(\rho,t))d\rho = \int^{r}_{0}\boldsymbol{\chi}_{\{u(\cdot,t)\geq C^{\ast}\}}(\rho)\cdot\rho^{n-1}f(u(\rho,t))d\rho+\int^{r}_{0}\boldsymbol{\chi}_{\{u(\cdot,t) < C^{\ast}\}}(\rho)\cdot\rho^{n-1}f(u(\rho,t))d\rho. \end{equation} |
Combining these we have
\begin{align} h_{r}& = \frac{1}{n}\mu(t)r-r^{1-n}\int^{r}_{0}\boldsymbol{\chi}_{\{u(\cdot,t)\geq C^{\ast}\}}(\rho)\cdot\rho^{n-1}f(u(\rho,t))d\rho-r^{1-n}\\ &\int^{r}_{0} \boldsymbol{\chi}_{\{u(\cdot,t) < C^{\ast}\}}(\rho)\cdot\rho^{n-1}f(u(\rho,t))d\rho\\ &\leq\frac{1}{n}\mu(t)r-r^{1-n}\int^{r}_{0}\boldsymbol{\chi}_{\{u(\cdot,t) < C^{\ast}\}}(\rho)\cdot\rho^{n-1}f(u(\rho,t))d\rho\\ &\leq\frac{1}{n}\mu(t)r+C_{0}r^{1-n}\int^{r}_{0}\boldsymbol{\chi}_{\{u(\cdot,t) < C^{\ast}\}}(\rho)\cdot\rho^{n-1}d\rho\\ &\leq\frac{1}{n}\mu(t)r+C_{0}r^{1-n}\int^{r}_{0}\rho^{n-1}d\rho\\ &\leq\frac{1}{n}(\mu(t)+C_{0})r, \end{align} |
so we complete this proof.
To show the existence of a finite-time blow-up solution of (2.4), we need to prove that U_{ss} is nonpositive by the following lemma. The proof follows the strategy in [48].
Lemma 2.4. Suppose that D, f and u_{0} satisfy (I_{1}), (I_{2}) and (I_{3}) respectively. Then
\begin{equation} u_{r}(r,t)\leq0 \ for \ all \ r\in(0,R), t\in(0,T_{max}). \end{equation} | (2.9) |
Moreover,
\begin{equation} U_{ss}(s,t)\leq0\ for \ all \ r\in(0,R), t\in(0,T_{max}). \end{equation} | (2.10) |
Proof. Without loss of generality we may assume that \nonumber u_{0}\in C^{2} ([0, \infty)) and f\in C^{2}([0, \infty)) . Applying the regularity theory in ([49,50]), we all know that u and u_{r} belong to C^{0}([0, R]\times[0, T))\cap C^{2, 1}((0, R)\times(0, T)) and we fixed T\in(0, T_{max}) . From (2.4), we have for r\in(0, R) and t\in(0, T)
\begin{equation} h_{rr}+\frac{n-1}{r}h_{r} = \mu(t)-f(u), \end{equation} | (2.11) |
and from (2.4) we obtain
\begin{align} u_{rt}& = \Big((D(u)u_{r})_{r}+\frac{n-1}{r}D(u)u_{r}+uf(u)-u\mu(t)-u_{r}h_{r}\Big)_{r}\\ & = (D(u)u_{r})_{rr}+a_{1}(D(u)u_{r})_{r}+a_{2}u_{rr}+bu_{r}, \end{align} |
for all r\in(0, R) and t\in(0, T) , where
\begin{equation} \nonumber a_{1}(r,t) = \frac{n-1}{r}, \quad a_{2}(r,t) = -h_{r}, \quad b(r,t) = -\frac{n-1}{r^{2}}D(u)-\mu(t)-h_{rr}+f(u)+uf'(u), \end{equation} |
for all r\in(0, R) and t\in(0, T) . Moreover, we have h_{r}\leq\frac{r}{n}(\mu(t)+C_{0}) by (2.8) and from (2.11) such that
\begin{equation} \nonumber-h_{rr} = \frac{n-1}{r}h_{r}-\mu(t)+f(u)\leq\frac{n-1}{n}\mu(t)+\frac{n-1}{n}C_{0}-\mu(t)+f(u)\leq f(u)+C_{0} \quad {\rm for\ all\ } r\in(0,R)\ {\rm and\ } t\in(0,T), \end{equation} |
then setting c_{1}: = \sup_{(r, t)\in(0, R)\times(0, T)}(2f(u)+uf'(u)+C_{0}) , we obtain
\begin{equation} \nonumber b(r,t)\leq c_{1} \quad {\rm for\ all\ } r\in(0,R)\ {\rm and\ } t\in(0,T), \end{equation} |
and we introduce
\begin{equation} \nonumber c_{2}: = \sup\limits_{(r,t)\in(0,R)\times(0,T)}((D(u))_{rr}+a_{1}(D(u))_{r}) < \infty, \end{equation} |
and set c_{3} = 2(c_{1}+c_{2}+1) . Since u_{r}(r, t) = 0 for r\in\{0, R\}, t\in(0, T) (because u is radially symmetric) and u_{0_{r}}\leq 0 , the function y:[0, R]\times[0, T]\rightarrow \mathbb{R} , (r, t)\mapsto u_{r}(r, t)-\varepsilon e^{c_{3}t} belongs to C^{0}([0, R]\times[0, T]) and fulfills
\begin{equation} \left\{ \begin{split} y_{t}& = (D(u)(y+\varepsilon e^{c_{3}t}))_{rr}+a_{1}(D(u)(y+\varepsilon e^{c_{3}t}))_{r}+a_{2}y_{r}+b(y+\varepsilon e^{c_{3}t})-c_{3}\varepsilon e^{c_{3}t}\\ & = (D(u)y)_{rr}+a_{1}(D(u)y)_{r}+a_{2}y_{r}+by+\varepsilon e^{c_{3}t}((D(u))_{rr}+a_{1}(D(u))_{r}+b-c_{3})\\ &\leq(D(u)y)_{rr}+a_{1}(D(u)y)_{r}+a_{2}y_{r}+by+\varepsilon e^{c_{3}t}(c_{1}+c_{2}-c_{3}), \quad {\rm in\ } (0,R)\times(0,T),\\ y& < 0, \quad {\rm on\ } \{0,R\} \times(0,T),\\ y&(\cdot,0) < 0, \quad {\rm in\ } (0,R). \end{split} \right. \end{equation} | (2.12) |
By the estimate for y(\cdot, 0) in (2.12) and continuity of y , the time t_{0}: = \sup\{t\in(0, T):y\leq 0\ { \rm in} \ [0, R]\times (0, T)\} \in (0, T] is defined. Suppose that t_{0} < T , then there exists r_{0}\in [0, R] such that y(r_{0}, t_{0}) = 0 and y(r, t)\leq 0 for all r\in[0, R] and t\in[0, t_{0}] ; hence, y_{t}(r_{0}, t_{0})\geq 0 . As D\geq 0 in [0, \infty) , not only y(\cdot, t_{0}) but also z:(0, R)\rightarrow \mathbb{R}, r\longmapsto D(u(r, t_{0}))y(r, t_{0}) attains its maximum 0 at r_{0} . Since the second equality in (2.12) asserts r_{0}\in(0, R) , we conclude z_{rr}(r_{0})\leq 0, z_{r}(r_{0}) = 0 and y_{r}(r_{0}, t_{0}) = 0 . Hence, we could obtain the contradiction
\begin{align} 0&\leq y_{t}(r_{0},t_{0})\\ &\leq z_{rr}(r_{0})+a_{1}(r_{0},t_{0})z_{r}(r_{0})+a_{2}(r_{0},t_{0})y_{r}(r_{0},t_{0})+b(r_{0},t_{0})y(r_{0},t_{0})+\varepsilon e^{c_{3}t_{0}}\big(c_{1}+c_{2}-c_{3}\big)\\ &\leq-\frac{c_{3}}{2}\varepsilon e^{c_{3}t_{0}} < 0, \end{align} |
since we have
\begin{equation} \nonumber c_{1}+c_{2}\leq \frac{c_{3}}{2}. \end{equation} |
So that t_{0} = T, implying y\leq 0 in [0, R]\times [0, T] and hence u_{r}\leq\varepsilon e^{c_{3}t} in [0, R]\times [0, T] . Letting first \varepsilon\searrow 0 and then T\nearrow T_{max} , this proves that u_{r}\leq 0 in [0, R]\times [0, T_{max}) , and we have U_{ss}\leq0 because of (2.5).
In this section our aim is to establish a function and to select appropriate parameters such that the function satisfies ODI, which means finiteness of T_{max} by counter evidence. Firstly, we introduce a moment-like functional as follows
\begin{equation} \phi(t): = \int^{s_{0}}_{0}s^{-\gamma}(s_{0}-s)U(s,t)ds, \quad t\in[0,T_{max}), \end{equation} | (3.1) |
with \gamma\in(-\infty, 1) and s_{0}\in(0, R^{n}) . As a preparation of the subsequent analysis of \phi , we denote
\begin{equation} S_{\phi}: = \bigg\{t\in(0,T_{max})|\phi(t)\geq\frac{nM-s_{0}}{(1-\gamma)(2-\gamma)\omega_{n}}\cdot s_{0}^{2-\gamma}\bigg\}. \end{equation} | (3.2) |
The following lemma provides a lower bound for U .
Lemma 3.1. Let \gamma\in(-\infty, 1) and s_{0}\in(0, R^{n}) , then
\begin{equation} U(\frac{s_{0}}{2},t)\geq\frac{1}{\omega_{n}}\cdot(nM-\frac{4s_{0}}{2^{\gamma}(3-\gamma)}). \end{equation} | (3.3) |
Proof. If (3.3) was false for some t\in S_{\phi} such that U(\frac{s_{0}}{2}, t) < \frac{1}{\omega_{n}}\cdot(nM-\frac{4s_{0}}{2^{\gamma}(3-\gamma)}) , then necessarily \delta: = \frac{4s_{0}}{2^{\gamma}(3-\gamma)} < nM . By the monotonicity of U(\cdot, t) we would obtain that U(s, t) < \frac{nM-\delta}{\omega_{n}} for all s\in(0, \frac{s_{0}}{2}) . Since U(s, t) < \frac{nM}{\omega_{n}} for all s\in(0, R^{n}) , we have
\begin{align} \phi(t)& < \frac{nM-\delta}{\omega_{n}}\cdot\int^{\frac{s_{0}}{2}}_{0}s^{-\gamma}(s_{0}-s)ds+\frac{nM}{\omega_{n}}\cdot\int^{s_{0}}_{\frac{s_{0}}{2}}s^{-\gamma}(s_{0}-s)ds\\ & = \frac{nM}{\omega_{n}}\cdot\int^{s_{0}}_{0}s^{-\gamma}(s_{0}-s)ds-\frac{\delta}{\omega_{n}}\cdot\int^{\frac{s_{0}}{2}}_{0}s^{-\gamma}(s_{0}-s)ds \end{align} |
\begin{equation} \nonumber = \frac{nM}{\omega_{n}}\cdot\frac{s_{0}^{2-\gamma}}{(1-\gamma)(2-\gamma)}-\frac{\delta}{\omega_{n}}\cdot\frac{2^{\gamma}(3-\gamma)s_{0}^{2-\gamma}}{4(1-\gamma)(2-\gamma)}. \quad \end{equation} |
In view of the definition of S_{\phi} , we find that nM-s_{0} < nM-\frac{2^{\gamma}(3-\gamma)\delta}{4} , which contradicts our definition of \delta .
An upper bound for \mu is established by the following lemma.
Lemma 3.2. Let \gamma\in(-\infty, 1) and s_{0} > 0 such that s_{0}\leq\frac{R^{n}}{6} . Then the function \mu(t) has property that
\begin{equation} \mu(t)\leq C_{1}+\frac{1}{2s}\int^{s}_{0}f(U_{s}(\sigma,t))d\sigma \quad \ for \ all\ s\in(0,s_{0}) \ and \ any\ t\in S_{\phi}, \end{equation} | (3.4) |
where C_{1} = \frac{\frac{\chi}{2}C_{0}+C_{0}+C_{2}}{3}+C_{3} = \frac{1}{3}\bigg(\frac{\chi}{2}C_{0}+C_{0}+\frac{\chi k_{1}(\gamma_{1}-\gamma_{2})}{2\gamma_{2}}(\frac{2\xi k_{2}\gamma_{2}}{\chi k_{1}\gamma_{1}})^{\frac{\gamma_{1}}{\gamma_{1}-\gamma_{2}}}\bigg)+\chi f_{1}\bigg(\frac{2\delta}{\omega_{n}s_{0}}\bigg) .
Proof. First for any fixed t\in S_{\phi} , we may invoke Lemma 3.1 to see that
\begin{equation} \nonumber U(\frac{s_{0}}{2},t)\geq\frac{nM-\delta}{\omega_{n}}, \end{equation} |
and thus, as U\leq\frac{nM}{\omega_{n}} ,
\begin{equation} \nonumber\frac{U(s_{0},t)-U(\frac{s_{0}}{2},t)}{\frac{s_{0}}{2}}\leq\frac{\frac{nM}{\omega_{n}}-\frac{nM-\delta}{\omega_{n}}}{\frac{s_{0}}{2}} = \frac{2\delta}{\omega_{n}s_{0}}. \end{equation} |
However, by concavity of U(\cdot, t) , as asserted by Lemma 2.4,
\begin{equation} \nonumber\frac{U(s_{0},t)-U(\frac{s_{0}}{2},t)}{\frac{s_{0}}{2}}\geq U_{s}(s_{0},t)\geq U_{s}(s,t) \quad {\rm \ for \ all} \ s\in(s_{0},R^{n}). \end{equation} |
Then let s_{0}\in(0, R^{n}) , we know that
\begin{align} \mu(t)& = \frac{1}{R^{n}}\int^{s_{0}}_{0}f(U_{s}(\sigma,t))d\sigma+\frac{1}{R^{n}}\int^{R^{n}}_{s_{0}}f(U_{s}(\sigma,t))d\sigma \\ & = \frac{1}{R^{n}}\int^{s}_{0}f(U_{s}(\sigma,t))d\sigma+\frac{1}{R^{n}}\int^{s_{0}}_{s}f(U_{s}(\sigma,t))d\sigma+\frac{1}{R^{n}}\int^{R^{n}}_{s_{0}}f(U_{s}(\sigma,t))d\sigma, \forall t\in(0,T_{max}). \end{align} | (3.5) |
Since \gamma_{1} > \gamma_{2} and Young's inequality such that \xi f_{2}(u)\leq\xi k_{2}(1+u)^{\gamma_{2}}\leq\frac{\chi k_{1}}{2}(1+u)^{\gamma_{1}}+C_{2}\leq\frac{\chi}{2}f_{1}(u)+C_{2} with C_{2} = \frac{\chi k_{1}(\gamma_{1}-\gamma_{2})}{2\gamma_{2}}(\frac{2\xi k_{2}\gamma_{2}}{\chi k_{1}\gamma_{1}})^{\frac{\gamma_{1}}{\gamma_{1}-\gamma_{2}}} for u\geq0 , then for all s\in(0, R^{n}) and t\in(0, T_{max}) we show that
\begin{equation} \frac{\chi}{2}f_{1}(U_{s}(s,t))-C_{2}\leq f(U_{s}(s,t))\leq\chi f_{1}(U_{s}(s,t)). \end{equation} | (3.6) |
Accordingly, by the monotonicity of U_{s}(\cdot, t) along with (1.7) and (3.6), we have
\begin{align} \int^{s_{0}}_{s}f(U_{s}(\sigma,t))d\sigma&\leq\int^{s_{0}}_{s}\chi f_{1}(U_{s}(\sigma,t))d\sigma\\ &\leq\int^{s_{0}}_{s}\chi f_{1}(U_{s}(s,t))d\sigma\\ &\leq s_{0}\chi f_{1}(U_{s}(s,t)), \quad \forall s\in(0,s_{0}) ,\ t\in(0,T_{max}). \end{align} |
Since the condition of (2.7) implies that
\begin{align} \int^{s}_{0}f(U_{s}(\sigma,t))d\sigma& = \int^{s}_{0}\boldsymbol{\chi}_{\{U_{s}(\cdot,t)\geq1\}}(\sigma)\cdot f(U_{s}(\sigma,t)d\sigma+\int^{s}_{0}\boldsymbol{\chi}_{\{U_{s}(\cdot,t) < 1\}}(\sigma)\cdot f(U_{s}(\sigma,t)d\sigma\\ &\geq\int^{s}_{0}\boldsymbol{\chi}_{\{U_{s}(\cdot,t)\geq1\}}(\sigma)\cdot\bigg(\frac{\chi}{2}f_{1}(U_{s}(\sigma,t))-C_{2}\bigg)d\sigma-C_{0}s\\ \quad &\geq\int^{s}_{0}\boldsymbol{\chi}_{\{U_{s}(\cdot,t)\geq1\}}(\sigma)\cdot\frac{\chi}{2}f_{1}(U_{s}(\sigma,t))d\sigma-(C_{0}+C_{2})s\\ & = \int^{s}_{0}\frac{\chi}{2}f_{1}(U_{s}(\sigma,t))d\sigma-\int^{s}_{0} \boldsymbol{\chi}_{\{U_{s}(\cdot,t) < 1\}}(\sigma)\cdot\frac{\chi}{2}f_{1}(U_{s}(\sigma,t))d\sigma-(C_{0}+C_{2})s\\ &\geq\int^{s}_{0}\frac{\chi}{2}f_{1}(U_{s}(s,t))d\sigma-\frac{\chi}{2}C_{0}s-(C_{0}+C_{2})s\\ &\geq\frac{s\chi}{2}f_{1}(U_{s}(s,t))-(\frac{\chi}{2}C_{0}+C_{0}+C_{2})s. \end{align} |
Therefore, we obtain
\begin{equation} \nonumber\int^{s_{0}}_{s}f(U_{s}(\sigma,t))d\sigma\leq\frac{2s_{0}}{s}\int^{s}_{0}f(U_{s}(\sigma,t))d\sigma+2(\frac{\chi}{2}C_{0}+C_{0}+C_{2})s_{0}. \end{equation} |
Since (3.5) we have for all s\in(0, s_{0})
\begin{align} &\frac{1}{R^{n}}\int^{s}_{0}f(U_{s}(\sigma,t))d\sigma+\frac{1}{R^{n}}\int^{s_{0}}_{s}f(U_{s}(\sigma,t))d\sigma\\ &\leq\frac{1}{R^{n}}\int^{s}_{0}f(U_{s}(\sigma,t))d\sigma+\frac{2s_{0}}{R^{n}s}\int^{s}_{0}f(U_{s}(\sigma,t))d\sigma+\frac{2(\frac{\chi}{2}C_{0}+C_{0}+C_{2})s_{0}}{R^{n}}, \end{align} | (3.7) |
where s_{0}\leq\frac{R^{n}}{6} such that \frac{1}{R^{n}}\leq\frac{1}{6s_{0}}\leq\frac{1}{6s}, \frac{2s_{0}}{R^{n}s}\leq\frac{1}{3s} and \frac{s_{0}}{R^{n}}\leq\frac{1}{6} for all s\in(0, s_{0}) . Finally, we estimate the last summand of (3.5)
\begin{equation} \frac{1}{R^{n}}\int^{R^{n}}_{s_{0}}f(U_{s}(\sigma,t))d\sigma\leq\frac{1}{R^{n}}\int^{R^{n}}_{s_{0}}\chi f_{1}(U_{s}(\sigma,t))d\sigma\leq\chi f_{1}\bigg(\frac{2\delta}{\omega_{n}s_{0}}\bigg) = C_{3}. \end{equation} | (3.8) |
Together with (3.5), (3.7) and (3.8) imply (3.4).
Lemma 3.3. Assume that \gamma\in(-\infty, 1) satisfying
\begin{equation} \nonumber\gamma < 2-\frac{2}{n}, \end{equation} |
and s_{0}\in(0, \frac{R^{n}}{6}] . Then the function \phi:[0, T_{max})\rightarrow \mathbb{R} defined by (3.1) belongs to C^{0}([0, T_{max}))\cap C^{1}((0, T_{max})) and satisfies
\begin{align} \phi'(t)&\geq n^{2}\int^{s_{0}}_{0}s^{2-\frac{2}{n}-\gamma}(s_{0}-s)U_{ss}D(U_{s})ds\\ & \quad +\frac{1}{2}\int^{s_{0}}_{0}s^{-\gamma}(s_{0}-s)U_{s}\cdot\bigg\{\int^{s}_{0}f(U_{s}(\sigma,t))d\sigma\bigg\}ds-C_{1}\int^{s_{0}}_{0}s^{1-\gamma}(s_{0}-s)U_{s}ds\\ & = :J_{1}(t)+J_{2}(t)+J_{3}(t), \end{align} | (3.9) |
for all t\in[0, T_{max}) , where C_{1} is defined in Lemma 3.2.
Proof. Combining (2.6) and (3.4) we have
\begin{align} U_{t}(s,t)& = n^{2}s^{2-\frac{2}{n}}D(U_{s})U_{ss}-s\mu(t)U_{s}+U_{s}\cdot\int^{s}_{0}f(U_{s}(\sigma,t))d\sigma\\ &\geq n^{2}s^{2-\frac{2}{n}}U_{ss}D(U_{s})+\frac{1}{2}U_{s}\cdot\int^{s}_{0}f(U_{s}(\sigma,t))d\sigma-C_{1}sU_{s}. \end{align} |
Notice \phi(t) conforms \phi(t) = \int^{s_{0}}_{0}s^{-\gamma}(s_{0}-s)U(s, t)ds . So (3.9) is a direct consequence.
Lemma 3.4. Let s_{0}\in (0, \frac{R^{n}}{6}] , and \gamma\in(-\infty, 1) satisfying \gamma < 2-\frac{2}{n}. Then J_{1}(t) in (3.9) satisfies
\begin{equation} J_{1}(t)\geq-I, \end{equation} | (3.10) |
where
\begin{equation} I: = \left\{ \begin{split} &-\frac{n^{2}d}{m}(2-\frac{2}{n}-\gamma)\int^{s_{0}}_{0}s^{1-\frac{2}{n}-\gamma}(s_{0}-s), &\qquad &m < 0, \\ &n^{2}d(2-\frac{2}{n}-\gamma)\int^{s_{0}}_{0}s^{1-\frac{2}{n}-\gamma}(s_{0}-s)\ln(U_{s}+1),&\qquad &m = 0,\\ &\frac{n^{2}d}{m}(2-\frac{2}{n}-\gamma)\int^{s_{0}}_{0}s^{1-\frac{2}{n}-\gamma}(s_{0}-s)(U_{s}+1)^{m},&\qquad &m > 0, \end{split} \right. \end{equation} | (3.11) |
for all t\in S_{\phi} .
Proof. Since D\in C^{2}([0, \infty)) , suppose that
\begin{equation} \nonumber G(\tau) = \int^{\tau}_{0}D(\delta)d\delta, \end{equation} |
then
\begin{equation} 0 < G(\tau)\leq d\int^{\tau}_{0}(1+\delta)^{m-1}d\delta\leq\left\{ \begin{split} &\frac{-d}{m}, &\qquad m < 0, \\ &d\ln(\tau+1),&\qquad m = 0,\\ &\frac{d}{m}(\tau+1)^{m},&\qquad m > 0. \end{split} \right. \nonumber \end{equation} |
Here integrating by parts we obtain
\begin{align} J_{1}(t)& = n^{2}\int^{s_{0}}_{0}s^{2-\frac{2}{n}-\gamma}(s_{0}-s)dG(U_{s})\\ & = n^{2}s^{2-\frac{2}{n}-\gamma}(s_{0}-s)G(U_{s})|^{s_{0}}_{0}+n^{2}\int^{s_{0}}_{0}s^{2-\frac{2}{n}-\gamma}G(U_{s})ds\\ & \quad -n^{2}(2-\frac{2}{n}-\gamma)\int^{s_{0}}_{0}s^{1-\frac{2}{n}-\gamma}(s_{0}-s)G(U_{s})ds. \end{align} |
Hence a direct calculation yields
\begin{equation} J_{1}(t)\geq\left\{ \begin{split} &\frac{n^{2}d}{m}(2-\frac{2}{n}-\gamma)\int^{s_{0}}_{0}s^{1-\frac{2}{n}-\gamma}(s_{0}-s), &\qquad m < 0, \\ &-n^{2}d(2-\frac{2}{n}-\gamma)\int^{s_{0}}_{0}s^{1-\frac{2}{n}-\gamma}(s_{0}-s)\ln(U_{s}+1),&\qquad m = 0,\\ &-\frac{n^{2}d}{m}(2-\frac{2}{n}-\gamma)\int^{s_{0}}_{0}s^{1-\frac{2}{n}-\gamma}(s_{0}-s)(U_{s}+1)^{m},&\qquad m > 0,\\ \end{split} \right. \nonumber \end{equation} |
for all t\in S_{\phi} . We conclude (3.10).
Lemma 3.5. Assume that \gamma\in(-\infty, 1) satisfying \gamma < 2-\frac{2}{n} and s_{0}\in(0, \frac{R^{n}}{6}] . Then we have
\begin{equation} J_{2}(t)+J_{3}(t)\geq\frac{k_{1}\chi }{4}\int^{s_{0}}_{0}s^{1-\gamma}(s_{0}-s)U_{s}^{1+\gamma_{1}}ds-C_{4}\int^{s_{0}}_{0}s^{1-\gamma}(s_{0}-s)U_{s}ds \end{equation} | (3.12) |
for all t\in S_{\phi} , where C_{4} = C_{1}+\frac{(\frac{\chi}{2}C_{0}+C_{0}+C_{2})}{2} .
Proof. Since Lemma 3.2 we have
\begin{equation} \nonumber\int^{s}_{0}f(U_{s}(\sigma,t))d\sigma\geq \frac{s}{2}\chi f_{1}(U_{s}(s,t))-(\frac{\chi}{2}C_{0}+C_{0}+C_{2})s \quad {\rm for\ all\ } s\in(0,s_{0})\ {\rm and\ } t\in(0,T_{max}). \end{equation} |
Therefore,
\begin{align} J_{2}(t)& = \frac{1}{2}\int^{s_{0}}_{0}s^{-\gamma}(s_{0}-s)U_{s}\cdot\bigg\{\int^{s}_{0}f(U_{s}(\sigma,t))d\sigma\bigg\}ds\\ &\geq\frac{\chi}{4}\int^{s_{0}}_{0}s^{1-\gamma}(s_{0}-s)U_{s}f_{1}(U_{s}(s,t))ds-\frac{(\frac{\chi}{2}C_{0}+C_{0}+C_{2})}{2}\int^{s_{0}}_{0}s^{1-\gamma}(s_{0}-s)U_{s}ds\\ &\geq\frac{k_{1}\chi }{4}\int^{s_{0}}_{0}s^{1-\gamma}(s_{0}-s)U_{s}^{1+\gamma_{1}}ds-\frac{(\frac{\chi}{2}C_{0}+C_{0}+C_{2})}{2}\int^{s_{0}}_{0}s^{1-\gamma}(s_{0}-s)U_{s}ds, \end{align} |
where f_{1}(U_{s}(s, t))\geq k_{1}(1+U_{s})^{\gamma_{1}}\geq k_{1}(U_{s})^{\gamma_{1}} . Combining these inequalities we can deduce (3.12).
Lemma 3.6. Let \gamma_{1} > \max\big\{0, m-1\big\} . For any \gamma\in(-\infty, 1) satisfying
\begin{equation} \gamma\in\min\bigg\{2-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{\gamma_{1}}, \ 2-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{1+\gamma_{1}-m}\bigg\}, \end{equation} | (3.13) |
and s_{0}\in(0, \frac{R^{n}}{6}], the function \phi:[0, T_{max})\rightarrow \mathbb{R} defined in (3.1) satisfies
\begin{equation} \phi'(t)\geq\left\{ \begin{split} &C\psi(t)-Cs_{0}^{3-\gamma-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{\gamma_{1}}}, &m\leq1, \\ &C\psi(t)-Cs_{0}^{3-\gamma-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{1+\gamma_{1}-m}},&m > 1, \ \end{split} \right. \end{equation} | (3.14) |
with C > 0 for all t\in S_{\phi} , where \psi(t): = \int^{s_{0}}_{0}s^{1-\gamma}(s_{0}-s)U_{s}^{1+\gamma_{1}}ds .
Proof. From (3.10) and (3.12) we have
\begin{equation} \phi'(t)\geq\frac{k_{1}\chi }{4}\psi(t)-I-C_{4}\int^{s_{0}}_{0}s^{1-\gamma}(s_{0}-s)U_{s}ds, \end{equation} | (3.15) |
for all t\in S_{\phi} and I is given by (3.11). In the case m < 0 ,
\begin{align} -\frac{n^{2}d}{m}(2-\frac{2}{n}-\gamma)\int^{s_{0}}_{0}s^{1-\frac{2}{n}-\gamma}(s_{0}-s)ds&\leq-\frac{n^{2}d}{m}(2-\frac{2}{n}-\gamma)s_{0}\int^{s_{0}}_{0}s^{1-\frac{2}{n}-\gamma}ds\\ & = -\frac{n^{2}d}{m}s_{0}^{3-\frac{2}{n}-\gamma}. \end{align} |
If m = 0 , we use the fact that \frac{\ln(1+x)}{x} < 1 for any x > 0 and Hölder's inequality to estimate
\begin{align} &n^{2}d(2-\frac{2}{n}-\gamma)\int^{s_{0}}_{0}s^{1-\frac{2}{n}-\gamma}(s_{0}-s)\ln(U_{s}+1)ds\\ & = n^{2}d(2-\frac{2}{n}-\gamma)\int^{s_{0}}_{0}[s^{1-\gamma}(s_{0}-s)U_{s}^{1+\gamma_{1}}]^{\frac{1}{1+\gamma_{1}}}\\ & \cdot s^{1-\frac{2}{n}-\gamma-\frac{1-\gamma}{1+\gamma_{1}}}(s_{0}-s)^{1-\frac{1}{1+\gamma_{1}}}\frac{\ln(1+U_{s})}{U_{s}}ds\\ &\leq n^{2}d(2-\frac{2}{n}-\gamma)\bigg\{\int^{s_{0}}_{0}s^{1-\gamma}(s_{0}-s)U_{s}^{1+\gamma_{1}}ds\bigg\}^{\frac{1}{1+\gamma_{1}}}\cdot \\ &\bigg\{\int^{s_{0}}_{0}\Big(s^{1-\frac{2}{n}-\gamma-\frac{1-\gamma}{1+\gamma_{1}}}(s_{0}-s)^{\frac{\gamma_{1}}{1+\gamma_{1}}}\Big)^{\frac{1+\gamma_{1}}{\gamma_{1}}}ds\bigg\}^{\frac{\gamma_{1}}{1+\gamma_{1}}}\\ &\leq n^{2}d(2-\frac{2}{n}-\gamma)s_{0}^{\frac{\gamma_{1}}{1+\gamma_{1}}}\bigg\{\int^{s_{0}}_{0}s^{\frac{(1-\frac{2}{n}-\gamma)\gamma_{1}-\frac{2}{n}}{\gamma_{1}}}ds\bigg\}^{\frac{\gamma_{1}}{1+\gamma_{1}}}\psi^{\frac{1}{1+\gamma_{1}}}(t)\\ & = C_{5}\psi^{\frac{1}{1+\gamma_{1}}}(t)s_{0}^{\frac{(3-\frac{2}{n}-\gamma)\gamma_{1}-\frac{2}{n}}{1+\gamma_{1}}}, \end{align} |
for all t\in S_{\phi} with C_{5}: = n^{2}d(2-\frac{2}{n}-\gamma)\cdot\Big(\frac{1}{2-\gamma-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{\gamma_{1}}}\Big)^{\frac{\gamma_{1}}{1+\gamma_{1}}} > 0 by (3.13). In the case m > 0 , by using the elementary inequality (a+b)^{\alpha}\leq2^{\alpha}(a^{\alpha}+b^{\alpha}) for all a, b > 0 and every \alpha > 0 , we obtain
\begin{align} &\frac{n^{2}d}{m}(2-\frac{2}{n}-\gamma)\int^{s_{0}}_{0}s^{1-\frac{2}{n}-\gamma}(s_{0}-s)(U_{s}+1)^{m}ds\\ &\leq2^{m}\frac{n^{2}d}{m}(2-\frac{2}{n}-\gamma)\int^{s_{0}}_{0}s^{1-\frac{2}{n}-\gamma}(s_{0}-s)U_{s}^{m}ds+2^{m}\frac{n^{2}d}{m}(2-\frac{2}{n}-\gamma)\int^{s_{0}}_{0}s^{1-\frac{2}{n}-\gamma}(s_{0}-s)ds, \end{align} | (3.16) |
for all t\in S_{\phi} , and we first estimate the second term on the right of (3.16)
\begin{equation} \nonumber2^{m}\frac{n^{2}d}{m}(2-\frac{2}{n}-\gamma)\int^{s_{0}}_{0}s^{1-\frac{2}{n}-\gamma}(s_{0}-s)ds\leq2^{m}\frac{n^{2}d}{m}s_{0}^{3-\frac{2}{n}-\gamma}. \end{equation} |
Since \gamma_{1} > m-1 and by Hölder's inequality we deduce that
\begin{align} 2^{m}\frac{n^{2}d}{m}&(2-\frac{2}{n}-\gamma)\int^{s_{0}}_{0}s^{1-\frac{2}{n}-\gamma}(s_{0}-s)U_{s}^{m}ds \quad \\ & = 2^{m}\frac{n^{2}d}{m}(2-\frac{2}{n}-\gamma)\int^{s_{0}}_{0}s^{(1-\gamma)\cdot\frac{m}{1+\gamma_{1}}}(s_{0}-s)^{\frac{m}{1+\gamma_{1}}}U_{s}^{m}\cdot s^{1-\frac{2}{n}-\gamma-(1-\gamma)\cdot\frac{m}{1+\gamma_{1}}}(s_{0}-s)^{1-\frac{m}{1+\gamma_{1}}}ds\\ &\leq2^{m}\frac{n^{2}d}{m}(2-\frac{2}{n}-\gamma)\bigg\{\int^{s_{0}}_{0}[s^{(1-\gamma)\cdot\frac{m}{1+\gamma_{1}}}(s_{0}-s)^{\frac{m}{1+\gamma_{1}}}U_{s}^{m}]^{\frac{1+\gamma_{1}}{m}}ds\bigg\}^{\frac{m}{1+\gamma_{1}}}\\ & \quad \times\bigg\{\int^{s_{0}}_{0}[s^{1-\frac{2}{n}-\gamma-(1-\gamma)\cdot\frac{m}{1+\gamma_{1}}}(s_{0}-s)^{1-\frac{m}{1+\gamma_{1}}}]^{\frac{1+\gamma_{1}}{1+\gamma_{1}-m}}ds\bigg\}^{\frac{1+\gamma_{1}-m}{1+\gamma_{1}}}\\ &\leq2^{m}\frac{n^{2}d}{m}(2-\frac{2}{n}-\gamma)\psi^{\frac{m}{1+\gamma_{1}}}(t)s_{0}^{\frac{1+\gamma_{1}-m}{1+\gamma_{1}}}\cdot\bigg\{\int^{s_{0}}_{0}s^{\frac{(1+\gamma_{1}-m)(1-\gamma)-\frac{2}{n}(1+\gamma_{1}) }{1+\gamma_{1}-m}}ds\bigg\}^{\frac{1+\gamma_{1}-m}{1+\gamma_{1}}}\\ &\leq C_{6}\psi^{\frac{m}{1+\gamma_{1}}}(t)s_{0}^{\frac{(1+\gamma_{1}-m)(3-\gamma)-\frac{2}{n}(1+\gamma_{1})}{1+\gamma_{1}}}, \end{align} |
for all t\in S_{\phi} with C_{6} = 2^{m}\frac{n^{2}d}{m}(2-\frac{2}{n}-\gamma)\Big(\frac{1}{2-\gamma-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{1+\gamma_{1}-m}}\Big)^{\frac{1+\gamma_{1}-m}{1+\gamma_{1}}} > 0 where \gamma < 2-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{1+\gamma_{1}-m} from (3.13).
Next, we can estimate the third expression on the right of (3.15) as follows
\begin{align} &C_{4}\int^{s_{0}}_{0}s^{1-\gamma}(s_{0}-s)U_{s}ds\\ & = C_{4}\int^{s_{0}}_{0}[s^{1-\gamma}(s_{0}-s)U_{s}^{1+\gamma_{1}}]^{\frac{1}{1+\gamma_{1}}}\cdot s^{1-\gamma-\frac{1-\gamma}{1+\gamma_{1}}}(s_{0}-s)^{1-\frac{1}{1+\gamma_{1}}}ds\\ &\leq C_{4}\bigg\{\int^{s_{0}}_{0}s^{1-\gamma}(s_{0}-s)U_{s}^{1+\gamma_{1}}ds\bigg\}^{\frac{1}{1+\gamma_{1}}}\cdot\bigg\{\int^{s_{0}}_{0}[s^{1-\gamma-\frac{1-\gamma}{1+\gamma_{1}}}(s_{0}-s)^{1-\frac{1}{1+\gamma_{1}}}]^{\frac{1+\gamma_{1}}{\gamma_{1}}}ds\bigg\}^{\frac{\gamma_{1}}{1+\gamma_{1}}}\\ &\leq C_{4}\psi^{\frac{1}{1+\gamma_{1}}}(t)s_{0}^{\frac{\gamma_{1}}{1+\gamma_{1}}}\bigg\{\int^{s_{0}}_{0}s^{1-\gamma}ds\bigg\}^{\frac{\gamma_{1}}{1+\gamma_{1}}}\\ & = C_{7}\psi^{\frac{1}{1+\gamma_{1}}}(t)s_{0}^{\frac{(3-\gamma)\gamma_{1}}{1+\gamma_{1}}}, \end{align} |
where C_{7} = C_{4}\big(\frac{1}{2-\gamma}\big)^{\frac{\gamma_{1}}{1+\gamma_{1}}} for all t\in S_{\phi} . By (3.15) and collecting the estimates above we have
\begin{equation} \phi'(t)\geq\left\{ \begin{split} &\frac{k_{1}\chi }{4}\psi(t)+\frac{n^{2}d}{m}s_{0}^{3-\frac{2}{n}-\gamma}-C_{7}\psi^{\frac{1}{1+\gamma_{1}}}(t)s_{0}^{\frac{(3-\gamma)\gamma_{1}}{1+\gamma_{1}}}, &m < 0,\ &t\in S_{\phi}, \\ &\frac{k_{1}\chi }{4}\psi(t)-C_{5}\psi^{\frac{1}{1+\gamma_{1}}}(t)s_{0}^{\frac{(3-\frac{2}{n}-\gamma)\gamma_{1}-\frac{2}{n}}{1+\gamma_{1}}}-C_{7}\psi^{\frac{1}{1+\gamma_{1}}}(t)s_{0}^{\frac{(3-\gamma)\gamma_{1}}{1+\gamma_{1}}},&m = 0,\ &t\in S_{\phi},\\ &\frac{k_{1}\chi }{4}\psi(t)-2^{m}\frac{n^{2}d}{m}s_{0}^{3-\frac{2}{n}-\gamma}-C_{6}\psi^{\frac{m}{1+\gamma_{1}}}(t)s_{0}^{\frac{(1+\gamma_{1}-m)(3-\gamma)-\frac{2}{n}(1+\gamma_{1})}{1+\gamma_{1}}}-C_{7}\psi^{\frac{1}{1+\gamma_{1}}}(t)s_{0}^{\frac{(3-\gamma)\gamma_{1}}{1+\gamma_{1}}},&m > 0,\ &t\in S_{\phi}. \end{split} \right. \nonumber \end{equation} |
If m = 0 , by Young's inequality we can find positive constants C_{8}, C_{9} such that
\begin{equation} \nonumber C_{5}\psi^{\frac{1}{1+\gamma_{1}}}(t)s_{0}^{\frac{(3-\frac{2}{n}-\gamma)\gamma_{1}-\frac{2}{n}}{1+\gamma_{1}}}\leq \frac{k_{1}\chi }{16}\psi(t)+C_{8}s_{0}^{3-\gamma-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{\gamma_{1}}}, \quad \forall t\in S_{\phi}, \end{equation} |
while as m > 0 we have
\begin{equation} \nonumber C_{6}\psi^{\frac{m}{1+\gamma_{1}}}(t)s_{0}^{\frac{(1+\gamma_{1}-m)(3-\gamma)-\frac{2}{n}(1+\gamma_{1})}{1+\gamma_{1}}}\leq\frac{k_{1}\chi }{16}\psi(t)+C_{9}s_{0}^{3-\gamma-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{1+\gamma_{1}-m}}, \quad \forall t\in S_{\phi}. \end{equation} |
On the other hand, we use Young's inequality again
\begin{equation} \nonumber C_{7}\psi^{\frac{1}{1+\gamma_{1}}}(t)s_{0}^{\frac{(3-\gamma)\gamma_{1}}{1+\gamma_{1}}}\leq\frac{k_{1}\chi }{16}\psi(t)+C_{10}s_{0}^{3-\gamma}, \quad \forall t\in S_{\phi}. \end{equation} |
In the case m < 0 , because of s_{0}\in(0, \frac{R^{n}}{6}] , we have
\begin{equation} \nonumber s_{0}^{3-\frac{2}{n}-\gamma} = s_{0}^{3-\gamma-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{\gamma_{1}}}\cdot s_{0}^{\frac{2}{n\gamma_{1}}}\leq\big(\frac{R^{n}}{6}\big)^{\frac{2}{n\gamma_{1}}}s_{0}^{3-\gamma-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{\gamma_{1}}}, \end{equation} |
when m > 0 we have
\begin{equation} \nonumber s_{0}^{3-\frac{2}{n}-\gamma} = s_{0}^{3-\gamma-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{1+\gamma_{1}-m}}\cdot s_{0}^{\frac{2m}{n(1+\gamma_{1}-m)}}\leq\big(\frac{R^{n}}{6}\big)^{\frac{2m}{n(1+\gamma_{1}-m)}}s_{0}^{3-\gamma-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{1+\gamma_{1}-m}}. \end{equation} |
All in all, we have
\begin{equation} \phi'(t)\geq\left\{ \begin{split} &\frac{k_{1}\chi }{8}\psi(t)-C_{11}s_{0}^{3-\gamma-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{\gamma_{1}}}-C_{10}s_{0}^{3-\gamma}, &m\leq0, \\ &\frac{k_{1}\chi }{8}\psi(t)-C_{12}s_{0}^{3-\gamma-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{1+\gamma_{1}-m}}-C_{10}s_{0}^{3-\gamma},& m > 0,\\ \end{split} \right. \end{equation} | (3.17) |
for all t\in S_{\phi} with C_{11} = C_{8}-\frac{n^{2}d}{m}\big(\frac{R^{n}}{6}\big)^{\frac{2}{n\gamma_{1}}} and C_{12} = C_{9}+\frac{2^{m}n^{2}d}{m}\big(\frac{R^{n}}{6}\big)^{\frac{2m}{n(1+\gamma_{1}-m)}} . When 0 < m\leq1 , we have \frac{1+\gamma_{1}}{1+\gamma_{1}-m}\leq\frac{1+\gamma_{1}}{\gamma_{1}} such that s_{0}^{3-\gamma-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{1+\gamma_{1}-m}} = s_{0}^{3-\gamma-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{\gamma_{1}}}s_{0}^{\frac{2}{n}(\frac{1+\gamma_{1}}{\gamma_{1}}-\frac{1+\gamma_{1}}{1+\gamma_{1}-m})} \leq\big(\frac{R^{n}}{6}\big)^{\frac{2(1-m)(1+\gamma_{1})}{n\gamma_{1}(1+\gamma_{1}-m)}}s_{0}^{3-\gamma-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{\gamma_{1}}} . In the case m\leq1
\begin{equation} s_{0}^{3-\gamma} = s_{0}^{\frac{2}{n}\cdot\frac{1+\gamma_{1}}{\gamma_{1}}}\cdot s_{0}^{3-\gamma-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{\gamma_{1}}}\leq\big(\frac{R^{n}}{6}\big)^{\frac{2}{n}\cdot\frac{1+\gamma_{1}}{\gamma_{1}}}\cdot s_{0}^{3-\gamma-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{\gamma_{1}}}, \quad \nonumber \end{equation} |
and if m > 1 we have
\begin{equation} s_{0}^{3-\gamma} = s_{0}^{\frac{2}{n}\cdot\frac{1+\gamma_{1}}{1+\gamma_{1}-m}}\cdot s_{0}^{3-\gamma-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{1+\gamma_{1}-m}}\leq\big(\frac{R^{n}}{6}\big)^{\frac{2}{n}\cdot\frac{1+\gamma_{1}}{(1+\gamma_{1}-m)}}\cdot s_{0}^{3-\gamma-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{1+\gamma_{1}-m}}. \nonumber \end{equation} |
Thus (3.17) turns into (3.14).
Next, we need to build a connection between \phi(t) and \psi(t) . Let us define
\begin{equation} S_{\psi}: = \bigg\{t\in(0,T_{max})|\psi(t)\geq s_{0}^{3-\gamma}\bigg\}. \end{equation} | (3.18) |
Lemma 3.7. Let \gamma\in(-\infty, 1) satisfying \gamma > 1-\gamma_{1} and (3.13) . Then for any choice of s_{0}\in(0, \frac{R^{n}}{6}] , the following inequality
\begin{equation} \phi'(t)\geq\left\{ \begin{split} &Cs_{0}^{-\gamma_{1}(3-\gamma)}\phi^{1+\gamma_{1}}(t)-Cs_{0}^{3-\gamma-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{\gamma_{1}}}, &m\leq1, \\ &Cs_{0}^{-\gamma_{1}(3-\gamma)}\phi^{1+\gamma_{1}}(t)-Cs_{0}^{3-\gamma-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{1+\gamma_{1}-m}},&m > 1, \ \end{split} \right. \end{equation} | (3.19) |
holds for all t\in S_{\phi}\cap S_{\psi} with C > 0 .
Proof. We first split
\begin{align} U(s,t) = \int^{s}_{0}U_{s}(\sigma,t)d\sigma& = \int^{s}_{0}\boldsymbol{\chi}_{\{U_{s}(\cdot,t) < 1\}}(\sigma)\cdot U_{s}(\sigma,t)d\sigma+\int^{s}_{0}\boldsymbol{\chi}_{\{U_{s}(\cdot,t)\geq1\}}(\sigma)\cdot U_{s}(\sigma,t)d\sigma\\ &\leq s+\int^{s}_{0} \boldsymbol{\chi}_{\{U_{s}(\cdot,t)\geq1\}}(\sigma)\cdot\big\{\sigma^{1-\gamma}(s_{0}-\sigma)U_{s}^{1+\gamma_{1}}\big\}^{\frac{1}{1+\gamma_{1}}}\cdot\sigma^{-\frac{1-\gamma}{1+\gamma_{1}}}(s_{0}-\sigma)^{-\frac{1}{1+\gamma_{1}}}d\sigma\\ &\leq s+(s_{0}-s)^{-\frac{1}{1+\gamma_{1}}}\psi^{\frac{1}{1+\gamma_{1}}}(t)\cdot\Big\{\int^{s}_{0}\sigma^{-\frac{1-\gamma}{1+\gamma_{1}}\cdot\frac{1+\gamma_{1}}{\gamma_{1}}}d\sigma\Big\}^{\frac{\gamma_{1}}{1+\gamma_{1}}}\\ & = s+\Big(\frac{\gamma_{1}}{\gamma+\gamma_{1}-1}\Big)^{\frac{\gamma_{1}}{1+\gamma_{1}}}(s_{0}-s)^{-\frac{1}{1+\gamma_{1}}}s^{\frac{\gamma+\gamma_{1}-1}{1+\gamma_{1}}}\psi^{\frac{1}{1+\gamma_{1}}}(t), \end{align} | (3.20) |
for all s\in(0, s_{0}) and t\in(0, T_{max}) where \frac{\gamma_{1}}{\gamma+\gamma_{1}-1} > 0 . According to the definition of S_{\psi} , we can find
\begin{align} \frac{s}{(s_{0}-s)^{-\frac{1}{1+\gamma_{1}}}s^{\frac{\gamma+\gamma_{1}-1}{1+\gamma_{1}}}\psi^{\frac{1}{1+\gamma_{1}}}(t)}& = s^{\frac{2-\gamma}{1+\gamma_{1}}}(s_{0}-s)^{\frac{1}{1+\gamma_{1}}}\psi^{-\frac{1}{1+\gamma_{1}}}(t)\\ &\leq s_{0}^{\frac{2-\gamma}{1+\gamma_{1}}}\cdot s_{0}^{\frac{1}{1+\gamma_{1}}}\cdot(s_{0}^{3-\gamma})^{-\frac{1}{1+\gamma_{1}}} = 1, \end{align} | (3.21) |
for all s\in(0, s_{0}) and t\in S_{\psi} . Combining (3.20) and (3.21) we have
\begin{equation} \nonumber U(s,t)\leq C_{1}s^{\frac{\gamma+\gamma_{1}-1}{1+\gamma_{1}}}(s_{0}-s)^{-\frac{1}{1+\gamma_{1}}}\psi^{\frac{1}{1+\gamma_{1}}}(t), \end{equation} |
where C_{1} = 1+\Big(\frac{\gamma_{1}}{\gamma+\gamma_{1}-1}\Big)^{\frac{\gamma_{1}}{1+\gamma_{1}}} for all s\in(0, s_{0}) and t\in S_{\psi} . Invoking Hölder's inequality, we get
\begin{align} \phi(t)& = \int^{s_{0}}_{0}s^{-\gamma}(s_{0}-s)U(s,t)ds \\ &\leq C_{1}\int^{s_{0}}_{0}s^{-\gamma+\frac{\gamma+\gamma_{1}-1}{1+\gamma_{1}}}(s_{0}-s)^{1-\frac{1}{1+\gamma_{1}}}ds\cdot\psi^{\frac{1}{1+\gamma_{1}}}(t)\\ &\leq C_{1}s_{0}^{\frac{\gamma_{1}}{1+\gamma_{1}}}\int^{s_{0}}_{0}s^{-\gamma+\frac{\gamma+\gamma_{1}-1}{1+\gamma_{1}}}ds\cdot\psi^{\frac{1}{1+\gamma_{1}}}(t)\\ & = C_{2}s_{0}^{\frac{\gamma_{1}(3-\gamma)}{1+\gamma_{1}}}\cdot\psi^{\frac{1}{1+\gamma_{1}}}(t), \end{align} | (3.22) |
where C_{2} = C_{1}\frac{1+\gamma_{1}}{\gamma_{1}(2-\gamma)} for all s\in(0, s_{0}) and t\in S_{\psi} . Employing these conclusion we deduce (3.19).
These preparations above will enable us to establish a superlinear ODI for \phi as mentioned earlier, and we prove our main result on blow-up based on a contradictory argument.
Proof of Theorem 1.1. Step 1. Assume on the contrary that T_{max} = +\infty , and we define the function
\begin{equation} S: = \bigg\{T\in(0,+\infty)\Big|\phi(t) > \frac{nM-s_{0}}{\omega_{n}(1-\gamma)(2-\gamma)}\cdot s_{0}^{2-\gamma} {\rm\ for \ all \ } t\in[0,T]\bigg\}. \end{equation} | (3.23) |
Let us choose s_{0} > 0 such that
\begin{equation} s_{0}\leq \min\bigg\{\frac{R^{n}}{6},\frac{nM}{2},\frac{nM\gamma_{1}}{2(1-\gamma)\omega_{n}[(C_{3}+1)(1+\gamma_{1})-1]}\bigg\}, \end{equation} | (3.24) |
where M and \omega_{n} were defined in (2.7) and C_{3} = \big(\frac{\gamma_{1}}{\gamma+\gamma_{1}-1}\big)^{\frac{\gamma_{1}}{1+\gamma_{1}}} has been mentioned in (3.20). Then we pick 0 < \varepsilon(\gamma_{1}, M, R) = \varepsilon < \frac{s_{0}}{n} and s^{\star}(\gamma_{1}, M, R)\in(0, s_{0}) wtih r^{\star}(\gamma_{1}, M, R) = (s^{\star})^{\frac{1}{n}}\in(0, R) such that
\begin{equation} \nonumber U(s,0)\geq U(s^{\star},0) = \frac{n}{\omega_{n}}\int_{B_{r^{\star}(0)}}u_{0}dx\geq\frac{n}{\omega_{n}}(M-\varepsilon), \quad \forall s\in(s^{\star},R^{n}). \end{equation} |
Therefore it is possible to estimate
\begin{align} \phi(0)& = \int^{s_{0}}_{0}s^{-\gamma}(s_{0}-s)U(s,0)ds\\ &\geq\int^{s_{0}}_{0}s^{-\gamma}(s_{0}-s)U(s^{\star},0)ds\\ & > \frac{n}{\omega_{n}}(M-\frac{s_{0}}{n})\int^{s_{0}}_{0}s^{-\gamma}(s_{0}-s)ds\\ & = \frac{nM-s_{0}}{\omega_{n}(1-\gamma)(2-\gamma)}\cdot s_{0}^{2-\gamma}. \end{align} | (3.25) |
Then S is non-empty and denote T = \sup S\in(0, \infty] . Next, we need to prove (0, T)\subset S_{\phi}\cap S_{\psi}\neq\emptyset . Note that
\begin{equation} \phi(t) > \frac{nM-s_{0}}{\omega_{n}(1-\gamma)(2-\gamma)}\cdot s_{0}^{2-\gamma}, \quad \forall t\in(0,T), \end{equation} | (3.26) |
we obtain (0, T)\subset S_{\phi} . From (3.20) we have
\begin{align} \phi(t)&\leq \int^{s_{0}}_{0}s^{-\gamma}(s_{0}-s)\big[s+C_{3}s^{\frac{\gamma+\gamma_{1}-1}{1+\gamma_{1}}}(s_{0}-s)^{-\frac{1}{1+\gamma_{1}}}\psi^{\frac{1}{1+\gamma_{1}}}(t)\big]ds\\ &\leq s_{0}\int^{s_{0}}_{0}s^{1-\gamma}ds+C_{3}\int^{s_{0}}_{0}s^{-\gamma+\frac{\gamma+\gamma_{1}-1}{1+\gamma_{1}}}(s_{0}-s)^{\frac{\gamma_{1}}{1+\gamma_{1}}}\psi^{\frac{1}{1+\gamma_{1}}}(t)ds\\ & = \frac{s^{3-\gamma}_{0}}{2-\gamma}+\frac{C_{3}(1+\gamma_{1})}{\gamma_{1}(2-\gamma)}s_{0}^{\frac{\gamma_{1}(3-\gamma)}{1+\gamma_{1}}}\cdot \psi^{\frac{1}{1+\gamma_{1}}}(t). \end{align} |
It follows from (3.24) and (3.26) that
\begin{equation} \phi(t)\geq \frac{nM}{2(1-\gamma)(2-\gamma)\omega_{n}}\cdot s^{2-\gamma}_{0} \quad {\rm for\ all \ } t\in(0,T). \end{equation} | (3.27) |
Then
\begin{equation} \nonumber\frac{C_{3}(1+\gamma_{1})}{\gamma_{1}(2-\gamma)}s^{\frac{\gamma_{1}(3-\gamma)}{1+\gamma_{1}}}_{0}\cdot \psi^{\frac{1}{1+\gamma_{1}}}(t)\geq\frac{nM}{2(1-\gamma)(2-\gamma)\omega_{n}}\cdot s^{2-\gamma}_{0}-\frac{s^{3-\gamma}_{0}}{2-\gamma}. \end{equation} |
Note that (3.24) implies
\begin{equation} \nonumber \frac{nM\gamma_{1}}{2C_{3}(1-\gamma)\omega_{n}(1+\gamma_{1})s_{0}}-\frac{\gamma_{1}}{C_{3}(1+\gamma_{1})}\geq1, \end{equation} |
then we have
\begin{align} \psi(t)&\geq\bigg[\Big(\frac{nMs^{2-\gamma}_{0}}{2(1-\gamma)(2-\gamma)\omega_{n}} -\frac{s^{3-\gamma}_{0}}{2-\gamma}\Big)\cdot \frac{\gamma_{1}(2-\gamma)}{C_{3}(1+\gamma_{1})}s_{0}^{-\frac{\gamma_{1}(3-\gamma)}{1+\gamma_{1}}}\bigg]^{1+\gamma_{1}}\\ &\geq\bigg[\frac{nM\gamma_{1}}{2C_{3}(1-\gamma)\omega_{n}(1+\gamma_{1})s_{0}}-\frac{\gamma_{1}}{C_{3}(1+\gamma_{1})}\bigg]^{1+\gamma_{1}}\cdot s_{0}^{3-\gamma}\\ &\geq s_{0}^{3-\gamma}. \end{align} |
Therefore, (0, T)\subset S_{\phi}\cap S_{\psi}\neq\emptyset.
Step 2. Applying Lemma 3.7 we can find \gamma\in(-\infty, 1) and C_{1}, C_{2} > 0 such that for all s_{0}\in(0, \frac{R^{n}}{6}]
\begin{equation} \phi'(t)\geq\left\{ \begin{split} &C_{1}s_{0}^{-\gamma_{1}(3-\gamma)}\phi^{1+\gamma_{1}}(t)-C_{2}s_{0}^{3-\gamma-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{\gamma_{1}}}, &m\leq1, \\ &C_{1}s_{0}^{-\gamma_{1}(3-\gamma)}\phi^{1+\gamma_{1}}(t)-C_{2}s_{0}^{3-\gamma-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{1+\gamma_{1}-m}},&m > 1, \ \end{split} \right. \nonumber \end{equation} |
for all t\in S_{\phi}\cap S_{\psi} and with (3.22) we have
\begin{equation} \nonumber\psi(t)\geq C_{3}s_{0}^{-\gamma_{1}(3-\gamma)}\phi^{1+\gamma_{1}}(t), \quad \forall t\in S_{\psi}. \end{equation} |
To specify our choice of s_{0} , for given M > 0 we choose s_{0}\in(0, \frac{R^{n}}{6}] small enough such that
\begin{equation} s_{0}\leq\frac{nM}{2}, \end{equation} | (3.28) |
and also
\begin{equation} s_{0}^{\gamma_{1}} < \frac{TC_{1}\gamma_{1}}{4}\bigg(\frac{nM}{2\omega_{n}(1-\gamma)(2-\gamma)}\bigg)^{\gamma_{1}}, \end{equation} | (3.29) |
as well as
\begin{equation} s_{0}^{1+\gamma_{1}}\leq C_{3}\bigg(\frac{nM}{2\omega_{n}(1-\gamma)(2-\gamma)}\bigg)^{1+\gamma_{1}}. \end{equation} | (3.30) |
From (3.23), (3.28) and (3.30) we have
\begin{equation} \nonumber\psi(t)\geq C_{3}s_{0}^{-\gamma_{1}(3-\gamma)}\phi^{1+\gamma_{1}}(t) > C_{3}\bigg(\frac{nM-s_{0}}{\omega_{n}(1-\gamma)(2-\gamma)}\cdot \frac{1}{s_{0}}\bigg)^{1+\gamma_{1}}\cdot s_{0}^{3-\gamma}\geq s_{0}^{3-\gamma}, \quad \forall t\in S_{\psi}, \end{equation} |
which shows that S\subset S_{\phi}\cap S_{\psi} . Since 1+\gamma_{1}-m > \frac{2}{n} , we have (1+\gamma_{1})(1-\frac{2}{n(1+\gamma_{1}-m)}) > 0 if m > 1 so that we can choose s_{0} sufficiently small satisfying (3.28) – (3.30) such that
\begin{equation} \nonumber s_{0}^{(1+\gamma_{1})(1-\frac{2}{n(1+\gamma_{1}-m)})}\leq\frac{C_{1}}{2C_{2}}\bigg(\frac{nM}{2\omega_{n}(1-\gamma)(2-\gamma)}\bigg)^{1+\gamma_{1}}, \end{equation} |
while in the case m\leq1 , the condition \gamma_{1} > m-1+\frac{2}{n}\geq\frac{2}{n} which infers that (1+\gamma_{1})(1-\frac{2}{n\gamma_{1}}) > 0 and we select s_{0} small enough fulfilling (3.28) – (3.30) such that
\begin{equation} \nonumber s_{0}^{(1+\gamma_{1})(1-\frac{2}{n\gamma_{1}})}\leq\frac{C_{1}}{2C_{2}}\bigg(\frac{nM}{2\omega_{n}(1-\gamma)(2-\gamma)}\bigg)^{1+\gamma_{1}}. \end{equation} |
It is possible to obtain
\begin{equation} \nonumber \frac{\frac{C_{1}}{2}s_{0}^{-\gamma_{1}(3-\gamma)}\phi^{1+\gamma_{1}}(0)}{C_{2}s_{0}^{3-\gamma-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{1+\gamma_{1}-m}}}\geq\frac{C_{1}}{2C_{2}}\bigg(\frac{nM}{2\omega_{n}(1-\gamma)(2-\gamma)}\bigg)^{1+\gamma_{1}}\cdot s_{0}^{-(1+\gamma_{1})+\frac{2}{n}\cdot\frac{1+\gamma_{1}}{(1+\gamma_{1}-m)}}\geq1, \quad \forall m > 1, \end{equation} |
and we have
\begin{equation} \nonumber \frac{\frac{C_{1}}{2}s_{0}^{-\gamma_{1}(3-\gamma)}\phi^{1+\gamma_{1}}(0)}{C_{2}s_{0}^{3-\gamma-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{\gamma_{1}}}}\geq\frac{C_{1}}{2C_{2}}\bigg(\frac{nM}{2\omega_{n}(1-\gamma)(2-\gamma)}\bigg)^{1+\gamma_{1}}\cdot s_{0}^{-(1+\gamma_{1})+\frac{2}{n}\cdot\frac{1+\gamma_{1}}{\gamma_{1}}}\geq1, \quad \forall m\leq1. \end{equation} |
All in all, for any m\in\mathbb{R} , we apply an ODI comparison argument to obtain that
\begin{equation} \nonumber \phi'(t)\geq\frac{C_{1}}{2}s_{0}^{-\gamma_{1}(3-\gamma)}\phi^{1+\gamma_{1}}(t), \quad \forall t\in(0,T). \end{equation} |
By a direct calculation we obtain
\begin{equation} \nonumber -\frac{1}{\gamma_{1}}\bigg(\frac{1}{\phi^{\gamma_{1}}(t)}-\frac{1}{\phi^{\gamma_{1}}(0)}\bigg)\geq\frac{C_{1}}{2}s_{0}^{-\gamma_{1}(3-\gamma)}t, \quad \forall t\in(0,T). \end{equation} |
Hence, according to (3.25) and (3.29) we conclude
\begin{equation} \nonumber t < \frac{2}{C_{1}\gamma_{1}}s_{0}^{\gamma_{1}}\bigg(\frac{2\omega_{n}(1-\gamma)(2-\gamma)}{nM}\bigg)^{\gamma_{1}}\leq\frac{T}{2}, \end{equation} |
for all t\in(0, T) . As a consequence, we infer that T_{max} must be finite.
In this section, we are preparing to prove Theorem 1.2 by providing the L^{p} estimate of u and the Moser-type iteration.
Lemma 4.1. Let (u, v, w) be a classical solution of the system (1.5) under the condition of Theorem 1.2 . Suppose that
\begin{equation} \gamma_{2} < 1+\gamma_{1} < \frac{2}{n}+m. \end{equation} | (4.1) |
Then for any p > \max\big\{1, 2-m, \gamma_{2}\big\} , there exists C = C(p) > 0 such that
\begin{equation} \int_{\Omega}(1+u)^{p}(x,t)dx\leq C \quad on \ (0,T_{max}). \end{equation} | (4.2) |
Proof. Notice f_{1}(u)\leq k_{1}(1+u)^{\gamma_{1}}, \ f_{2}(u) = k_{2}(1+u)^{\gamma_{2}} for all u\geq0 . Multiplying the first equation of (1.5) by p(1+u)^{p-1} and integrating by parts with the boundary conditions for u, v and w , we have
\begin{align} \frac{d}{dt}\int_{\Omega}&(1+u)^{p}dx+p(p-1)\int_{\Omega}(1+u)^{p-2}D(u)|\nabla u|^{2}dx\\ & = \chi p(p-1)\int_{\Omega}u(1+u)^{p-2}\nabla u\cdot\nabla vdx-\xi p(p-1)\int_{\Omega}u(1+u)^{p-2}\nabla u\cdot\nabla wdx\\ & = -\chi(p-1)\int_{\Omega}(1+u)^{p}\Delta vdx+\chi p\int_{\Omega}(1+u)^{p-1}\Delta vdx\\ & \quad +\xi(p-1)\int_{\Omega}(1+u)^{p}\Delta wdx-\xi p\int_{\Omega}(1+u)^{p-1}\Delta wdx\\ \quad &\leq\chi(p-1)\int_{\Omega}(1+u)^{p}f_{1}(u)dx+\chi p\int_{\Omega}(1+u)^{p-1}\mu_{1}(t)dx+\xi(p-1)\int_{\Omega}(1+u)^{p}\mu_{2}(t)dx\\ & \quad -\xi(p-1)\int_{\Omega}(1+u)^{p}f_{2}(u)dx+\xi p\int_{\Omega}(1+u)^{p-1}f_{2}(u)dx\\ &\leq k_{1}\chi(p-1)\int_{\Omega}(1+u)^{p+\gamma_{1}}dx+\chi p\int_{\Omega}(1+u)^{p-1}\mu_{1}(t)dx+\xi(p-1)\int_{\Omega}(1+u)^{p}\mu_{2}(t)dx\\ & \quad -k_{2}\xi(p-1)\int_{\Omega}(1+u)^{p+\gamma_{2}}dx+k_{2}\xi p\int_{\Omega}(1+u)^{p+\gamma_{2}-1}dx, \quad \forall t\in(0,T_{max}). \end{align} | (4.3) |
Firstly,
\begin{align} p(p-1)\int_{\Omega}(1+u)^{p-2}D(u)|\nabla u|^{2}dx&\geq dp(p-1)\int_{\Omega}(1+u)^{p+m-3}|\nabla u|^{2}dx\\ & = \frac{4dp(p-1)}{(p+m-1)^{2}}\int_{\Omega}|\nabla(1+u)^{\frac{p+m-1}{2}}|^{2}dx. \end{align} |
By Young's inequality and Hölder's inequality, we obtain
\begin{align} \chi p\int_{\Omega}(1+u)^{p-1}\mu_{1}(t)dx&\leq C_{1}\int_{\Omega}(1+u)^{p+\gamma_{1}}dx+C_{2}\mu_{1}^{\frac{p+\gamma_{1}}{1+\gamma_{1}}}(t)\\ & = C_{1}\int_{\Omega}(1+u)^{p+\gamma_{1}}dx+C_{2}\bigg(\frac{1}{|\Omega|}\int_{\Omega}f_{1}(u)dx\bigg)^{\frac{p+\gamma_{1}}{1+\gamma_{1}}}\\ &\leq C_{1}\int_{\Omega}(1+u)^{p+\gamma_{1}}dx+C_{3}\bigg(\int_{\Omega}(1+u)^{1+\gamma_{1}}dx\bigg)^{\frac{p+\gamma_{1}}{1+\gamma_{1}}}\\ &\leq C_{1}\int_{\Omega}(1+u)^{p+\gamma_{1}}dx+C_{3}\bigg\{\Big(\int_{\Omega}(1+u)^{p+\gamma_{1}}dx\Big)^{\frac{1+\gamma_{1}}{p+\gamma_{1}}}\cdot|\Omega|^{\frac{p-1}{p+\gamma_{1}}}\bigg\}^{\frac{p+\gamma_{1}}{1+\gamma_{1}}}\\ & = C_{1}\int_{\Omega}(1+u)^{p+\gamma_{1}}dx+C_{3}|\Omega|^{\frac{p-1}{1+\gamma_{1}}}\int_{\Omega}(1+u)^{p+\gamma_{1}}dx. \end{align} |
for all t\in(0, T_{max}) . Then by Hölder's inequality we obtain
\begin{align} \xi(p-1)\int_{\Omega}(1+u)^{p}\mu_{2}(t)dx& = \frac{k_{2}\xi(p-1)}{|\Omega|}\int_{\Omega}(1+u)^{\gamma_{2}}dx\int_{\Omega}(1+u)^{p}dx\\ &\leq\frac{k_{2}\xi(p-1)}{|\Omega|}\bigg\{\int_{\Omega}(1+u)^{p+\gamma_{2}}dx\bigg\}^{\frac{\gamma_{2}}{p+\gamma_{2}}}|\Omega|^{\frac{p}{p+\gamma_{2}}}\times\bigg\{\int_{\Omega}(1+u)^{p+\gamma_{2}}dx\bigg\}^{\frac{p}{p+\gamma_{2}}}|\Omega|^{\frac{\gamma_{2}}{p+\gamma_{2}}}\\ & = k_{2}\xi(p-1)\int_{\Omega}(1+u)^{p+\gamma_{2}}dx, \quad \forall t\in(0,T_{max}). \end{align} |
Furthermore, by using Young's inequality and (4.1) we have
\begin{equation} \nonumber k_{2}\xi p\int_{\Omega}(1+u)^{p+\gamma_{2}-1}dx\leq C_{4}\int_{\Omega}(1+u)^{p+\gamma_{1}}dx+C_{5}, \end{equation} |
for all t\in(0, T_{max}) . Therefore, combining these we conclude
\begin{equation} \nonumber\frac{d}{dt}\int_{\Omega}(1+u)^{p}dx+\frac{4dp(p-1)}{(p+m-1)^{2}}\int_{\Omega}|\nabla(1+u)^{\frac{p+m-1}{2}}|^{2}dx\leq C_{6}\int_{\Omega}(1+u)^{p+\gamma_{1}}dx+C_{5}, \quad \forall t\in(0,T_{max}), \end{equation} |
where C_{6} = C_{1}+C_{3}|\Omega|^{\frac{p-1}{1+\gamma_{1}}}+C_{4}+k_{1}\chi(p-1) . By means of Gagliardo-Nirenberg inequality we can find C_{7} such that
\begin{align} C_{6}\int_{\Omega}(1+u)^{p+\gamma_{1}}dx& = C_{6}\|(1+u)^{\frac{p+m-1}{2}}\|^{\frac{2(p+\gamma_{1})}{p+m-1}}_{L^{\frac{2(p+\gamma_{1})}{p+m-1}}(\Omega)}\\ &\leq C_{7}\|\nabla(1+u)^{\frac{p+m-1}{2}}\|^{\frac{2(p+\gamma_{1})}{p+m-1}\cdot a}_{L^{2}(\Omega)}\cdot \|(1+u)^{\frac{p+m-1}{2}}\|^{\frac{2(p+\gamma_{1})}{p+m-1}\cdot(1-a)}_{L^{\frac{2}{p+m-1}}(\Omega)}\\ & \quad +C_{7}\|(1+u)^{\frac{p+m-1}{2}}\|^{\frac{2(p+\gamma_{1})}{p+m-1}}_{L^{\frac{2}{p+m-1}}(\Omega)} \end{align} |
for all t\in(0, T_{max}) , where
\begin{equation} \nonumber a = \frac{\frac{p+m-1}{2}-\frac{p+m-1}{2(p+\gamma_{1})}}{\frac{p+m-1}{2}-(\frac{1}{2}-\frac{1}{n})}\in(0,1). \end{equation} |
Since 1-m+\gamma_{1} < \frac{2}{n} , we have \frac{2(p+\gamma_{1})}{p+m-1}\cdot a < 2 , and we use Young's inequality to see that for all t\in(0, T_{max})
\begin{equation} \nonumber C_{6}\int_{\Omega}(1+u)^{p+\gamma_{1}}dx\leq\frac{2dp(p-1)}{(p+m-1)^{2}}\|\nabla(1+u)^{\frac{p+m-1}{2}}\|^{2}_{L^{2}(\Omega)}+C_{8}. \end{equation} |
In quite a similar manner, we obtain C_{9} = C_{9}(p) > 0 fulfilling
\begin{equation} \nonumber\int_{\Omega}(1+u)^{p}dx\leq\frac{2dp(p-1)}{(p+m-1)^{2}}\|\nabla(1+u)^{\frac{p+m-1}{2}}\|^{2}_{L^{2}(\Omega)}+C_{9} \quad {\rm for\ all\ } t\in(0,T_{max}). \end{equation} |
Finally, combining these to (4.3) we obtain
\begin{equation} \nonumber\frac{d}{dt}\int_{\Omega}(1+u)^{p}dx+\int_{\Omega}(1+u)^{p}dx\leq C_{5}+C_{8}+C_{9} \quad {\rm for\ all\ } t\in(0,T_{max}). \end{equation} |
Thus,
\begin{equation} \nonumber\int_{\Omega}(1+u)^{p}dx\leq\max\Big\{\int_{\Omega}(1+u_{0})^{p}dx,C_{5}+C_{8}+C_{9}\Big\} \quad {\rm for\ all\ } t\in(0,T_{max}). \end{equation} |
We have done the proof.
Under the condition of Lemma 4.1 we can use the above information to prove Theorem 1.2.
Proof of Theorem 1.2. From Lemma 4.1, we let p > \max\big\{\gamma_{1}n, \gamma_{2}n, 1\big\} . By the elliptic L^{p} -estimate to the two elliptic equations in (1.5), we get that for all t\in(0, T_{max}) there exists some C_{10}(p) > 0 such that
\begin{equation} \|v(\cdot,t)\| _{w^{2,\frac{p}{\gamma_{1}}}(\Omega)}\leq C_{10}(p), \quad \|w(\cdot,t)\| _{w^{2,\frac{p}{\gamma_{2}}}(\Omega)}\leq C_{10}(p), \end{equation} | (4.4) |
and hence, by the Sobolev embedding theorem, we get
\begin{equation} \|v(\cdot,t)\|_{C^{1}(\overline{\Omega})}\leq C_{10}(p), \quad \|w(\cdot,t)\|_{C^{1}(\overline{\Omega})}\leq C_{10}(p). \end{equation} | (4.5) |
Now the Moser iteration technique ([3,51]) ensures that \|u(\cdot, t)\|_{L^{\infty}(\Omega)}\leq C for any t\in(0, T_{max}) .
This concludes by Lemma 2.1 that T_{max} = \infty .
The paper is supported by the Research and Innovation Team of China West Normal University (CXTD2020–5).
The authors declare that there is no conflict of interest.
[1] | El Ghazouani D (2019) A Growing Destination for Sub-Saharan Africans, Morocco Wrestles with Immigrant Integration, Migration Policy Institute. Available from: https://www.migrationpolicy.org/article/growing-destination-sub-saharan-africans-morocco. |
[2] | Martini LS (2020) Marocco: se la pandemia aumenta le disuguaglianze, INSPI. Available from: https://www.ispionline.it/it/pubblicazione/marocco-se-la-pandemia-aumenta-le-disuguaglianze-27574. |
[3] | Al-Qays I, Jebril T (2020) Morocco: ECOWAS. Good intentions are not enough, Mipa Institute. Available from: https://mipa.institute/7323. |
[4] |
Abourabi Y. (2022) Governing African Migration in Morocco: The Challenge of Positive Desecuritisation. Governing Migration for Development from the Global Souths 14: 29–59. https://doi.org/10.1163/9789004522770_003 doi: 10.1163/9789004522770_003
![]() |
[5] | Messari N (2020) Moroccan Foreign Policy Under Mohammed Ⅵ: Balancing Diversity and Respect, Istituto Affari Internazionali. Available from: https://www.iai.it/it/pubblicazioni/moroccan-foreign-policy-under-mohammed-vi-balancing-diversity-and-respect. |
[6] | Benachour S (2020) What a life in the host country: narratives of immigrants in Morocco amidst the pandemic, Compas, 2020. Available from: https://www.compas.ox.ac.uk/2020/what-a-life-in-the-host-country-narratives-of-immigrants-in-morocco-amidst-the-pandemic/. |
[7] | De Bel-Air F (2016) Migration Profile: Morocco. Migration Policy Centre. Available from: https://cadmus.eui.eu/handle/1814/41124. |
[8] | Association Marocaine d'Etudes et de Recherche en Migrations, L'immigration subsaharienne au Maroc: analyse socio–économique, AMERM, 2008. Available from: https://amerm.org/. |
[9] | United Nations High Commissioner for Refugees, Morocco Factsheet: March 2016, UNHCR, 2016. Available from: https://www.unhcr.org/567162f79.html. |
[10] | Arab C, Charef M, Simon G (2015) Maroc, In: Simon G (ed.), Dictionnaire géo-historique des migrations internationales, Paris: Armand Colin, 276–282. https://doi.org/10.4000/remi.7893 |
[11] | Wayel S (2015) Labour Market situation of sub-Saharan migrants in Morocco: the case of call centers, In: Khrouz N, Lanza N (Eds.), Migrants au Maroc: Cosmopoliticsme, presence d'étrangers et transformations sociales, Rabat : Centre Jacques-Berge. https://doi.org/10.4000/books.cjb.889 |
[12] | Jabrane M, Idali M, Madi R (2021) The economic activities of sub-Saharan immigrants: informal sector and low wages, SHS web Conferences, 119. http://doi.org/10.1051/shsconf/202111906001 |
[13] |
Pickerill E (2011) Informal and entrepreneurial strategies among sub-Saharan migrants in Morocco. J North Afr Stud 16: 395–413. https://doi.org/10.1080/13629387.2010.484217 doi: 10.1080/13629387.2010.484217
![]() |
[14] | OECD (2020) The Covid-19 crisis in Morocco, Segretary-General OECD. Available from: https://www.oecd.org/countries/morocco/. |
[15] | ILO, ADWA (2021) Rapid Labour Force Survey on the Impact of Covid-19 in Morocco, Economic Research Forum. Available from: https://www.ilo.org/africa/countries-covered/morocco/WCMS_791952/lang--en/index.htm. |
[16] | El Rhaz L, Bouzineb Y (2021) Le secteur informel au Maroc: pricipales caractéristiques et tendances d'évolution, Division des études générales, DPP-HCP. Available from: https://www.hcp.ma/Les-Brefs-du-Plan-N-16-02-Mars-2021_a2668.html. |
[17] | Lopez-Acevedo G, Betcherman G, Khellaf A, et al. (2021) Morocco's Job Landscape. Identifying Constraints to an Inclusive Labour Market, International Development in Focus, Washington DC: World Bank. Available from: https://openknowledge.worldbank.org/handle/10986/35075. |
[18] | Guadagno L (2020) Migrants and the COVID-19 pandemic: An initial analysis. Migration Research Series, 60, International Organization for Migration (IOM), Geneva. |
[19] | Newland K (2020) Will International Migration Governance Survive the COVID-19 Pandemic? Washington DC: Migration Policy Institute. |
[20] |
Igoye A (2020) Migration and Immigration: Uganda and the COVID-19 Pandemic. Public Integr 22: 406–408. https://doi.org/10.1080/10999922.2020.1753383. doi: 10.1080/10999922.2020.1753383
![]() |
[21] |
Collins FL (2021) Migration ethics in pandemic times. Dialogues Hum Geogr 11: 78–82. https://doi.org/10.1177/2043820620975964 doi: 10.1177/2043820620975964
![]() |
[22] |
Delmas A, Gouery D (2020) Bordering the world as a response to emerging infectious disease. The case of SARS CoV-2. Borders Globalization Rev 2: 12–20. https://doi.org/10.18357/bigr21202019760 doi: 10.18357/bigr21202019760
![]() |
[23] |
Moumni O (2021) Covid-19: Between Panic, Racism and Social Change. Engl Stud Afr 64: 242–254. https://doi.org/10.1080/00138398.2021.1972602 doi: 10.1080/00138398.2021.1972602
![]() |
[24] |
Gravlee CC (2020) Systemic racism, chronic health inequities, and COVID-19: A syndemic in the making? Am J Hum Biol 32: e23482. https://doi.org/10.1002/ajhb.23482 doi: 10.1002/ajhb.23482
![]() |
[25] | Eligon J (2020) For urban poor, the coronavirus complicates health risks, New York Times. Available from: https://www.nytimes.com/2020/03/07/us/coronavirus-minorities.html. |
[26] | Eligon J, Burch ADS. (2020) Questions of bias in Covid-19 treatment add to the morning for black families, New York Times. Available from: https://www.nytimes.com/2020/05/10/us/coronavirus-african-americans-bias.html. |
[27] | Schwirtz M, Cook LR (2020) These N.Y.C. neighbourhoods have the highest rates of virus deaths, New York Times. Available from: https://www.nytimes.com/2020/05/18/nyregion/coronavirus-deaths-nyc. |
[28] | Vesoulis A (2020) Coronavirus may disproportionately hurt the poor – And that's bad for everyone, Time. Available from: https://time.com/5800930/how-coronavirus-will-hurt-the-poor/. |
[29] | Gawthrop E (2020) The Color of Coronavirus: Covid-19 Deaths by Race and Ethnicity in U.S., AMP Research Lab. Available from: https://www.apmresearchlab.org/covid/deaths-by-race. |
[30] | Bassett MT, Chen JT, Krieger N (2020) The unequal toll of COVID-19 mortality by age in the United States: Quantifying racial/ethnic disparities. HCPDS Working Paper 19. |
[31] | HCP (2020) Rapports sociaux dans le contexte de la pandémie Covid-19. 2ème Panel sur l'impact du coronavirus sur la situation économique, sociale et psychologique des ménages. Available from: https://www.hcp.ma/Rapports-sociaux-dans-le-contexte-de-la-pandemie-COVID-19a2577.html. |
[32] | HCP (2021) La Migration Forcée au Maroc, Résultats de l'enquête nationale de 2021. Available from: https://www.hcp.ma/Note-sur-les-resultats-de-l-enquete-nationale-sur-la-migration-forcee-de-2021a2715.html. |
[33] | HCP, World Bank Group (2017) Pauvreté et prospérité partagée au Maroc du troisième millénaire: 2001–2014. Availbale from : https://www.hcp.ma/Pauvrete-et-prosperite-partagee-au-Maroc-du-troisieme-millenaire-2001-2014_a2055.html. |
[34] | Ennahkil Listening Center (2020) Projet «Contribution à la lutte contre la violence basée sur le genre impactée par la crise du COVID-19 dans la région Marrakech-Safi»: Réalisation d'une étude analytique sur l'impact de la crise du COVID-19 sur la violence basée sur le genre dans la région Marrakech-Safi, Ennahkil, USAID, Marrakech. |
[35] | FLDF (2020) Rapport sur la violence faite aux femmes pendant le confinement et l'état d'urgence sanitaire, Rabat. |
[36] | Fondation Orient et Occident (2020) Survey on the impact of Covid-19 on the educational outcomes of refugees. Available from: http://www.orient-occident.org/survey-on-the-impact-of-covid-19-on-the-educational-outcomes-of-refugees-in-french-only/. |
[37] | MRA (2019) Virtual violence, real harm: Promoting state responsibility for technology-facilitated gender-based violence against women in Morocco, Action Research Report, Rabat. Available from: https://mrawomen.ma/#. |
[38] | MRA (2020) The Impact of Covid-19 on Violence against Women in Morocco, Action Research Report, Rabat. Available from: https://mrawomen.ma/#. |
[39] | Mixed Migration Centre (2022) Understanding the Mixed Migration Landscape in Morocco, MMC. Available from: https://mixedmigration.org/resource/understanding-the-mixed-migration-landscape-in-morocco/. |
[40] |
Keygnaert I, Dialmy A, Manço A, et al. (2014) Sexual Violence and sub-Saharan migrants in Morocco: a community-based participatory assessment using respondents driven sampling. Global Health 10: 32. https://doi.org/10.1186/1744-8603-10-32 doi: 10.1186/1744-8603-10-32
![]() |
[41] | Migrants Refugees (2020) Migration Profile: Morocco. Available from: https://migrants-refugees.va/country-profile/morocco/. |
[42] | Lahlou M (2018) Migration dynamics in play in Morocco: Trafficking and political relationships and their implications at the regional level. Menara Working Paper, No. 26. |
[43] | Médecins Sans Frontières (2020) Violence, Vulnerability and Migration: Trapped at the Gates of Europe. Available from: https://www.msf.org/violence-vulnerability-and-migration-trapped-gateseurope. |
[44] | Association Marocaine des Droits Humains (2020) Morocco: Issues related to immigration detention, Global Detention Project. Available from: https://www.globaldetentionproject.org/countries/africa/morocco. |
[45] |
Bitari W (2020) Sub-Saharan Migrant Integration in Morocco, Oujda Case Study. Repères Perspect Econ 4: 86–102. https://doi.org/10.34874/IMIST.PRSM/RPE/23801 doi: 10.34874/IMIST.PRSM/RPE/23801
![]() |
[46] | Appiah-Nyamekye J, Abderebbi M (2019) Jobs loom large in Moroccans' attitudes toward in- and out-migration. Afrobarometer, No. 285. |
[47] |
Cherti M, Collyer M (2015) Immigration and Pensée d'Etat: Moroccan migration policy changes as transformation of 'geopolitical culture'. J North Afr Stud 20: 590–604. https://doi.org/10.1080/13629387.2015.1065043 doi: 10.1080/13629387.2015.1065043
![]() |
[48] |
Berriane M, de Hass H, Natter K (2015) Introduction: revisiting Moroccan migrations. J North Afr Stud 20: 503–521. https://doi.org/10.1080/13629387.2015.1065036 doi: 10.1080/13629387.2015.1065036
![]() |
[49] | Dennison J, Drazanova L (2018) Public attitudes on migration: rethinking how people perceive migration, ICMPD. Available from: https://www.researchgate.net/publication/346975152_Public_attitudes_on_migration_rethinking_how_people_perceive_migration. |
[50] |
El Otmani Dehbi Z, Sedrati H, Chaqsare S, et al. (2021) Moroccan Digital Health Response to the Covid-19 Crisis, Front Public Health 9: 690462. https://doi.org/10.3389/fpubh.2021.690462 doi: 10.3389/fpubh.2021.690462
![]() |
[51] | Medias 24 (2021) Les étrangers sans carte de séjour peuvent à présent se faire vacciner au Maroc. Available from: https://medias24.com/2021/11/21/les-etrangers-sans-carte-de-sejour-peuvent-a-present-se-fairevacciner-au-maroc/. |
[52] | Kessaba K, Halmi M (2021) Morocco Social Protection Response to Covid-19 and beyond: towards a Sustainable Social Protection Floor. International Policy Center for Inclusive Growth, 19: 29–31. |
[53] | Paul-Delvaux L, Crépon B, Devoto F, et al. (2021) Covid-19 in Morocco: Labor Market and Policy Responses, Harvard Kennedy School (EpoD). Available from: https://www.hks.harvard.edu/centers/cid/about-cid/news-announcements/MoroccoLaborMarket. |
[54] | Ennaji M (2021) Women and Gender Relations during the Pandemic in Morocco. Gender Women's Stud 4: 3. |
[55] | World Bank (2020) Morocco: Stepping up to the Covid-19 Pandemic Outbreak. Available from: https://www.worldbank.org/en/news/feature/2020/06/16/morocco-stepping-up-to-the-covid-19-pandemic-outbreak. |
[56] | El-Ouardighi S (2020) Mehdi Alioua: '20.000 migrants au Maroc risquent une catastrophe humanitaire', Medias24. Available from: https://medias24.com/2020/04/21/mehdi-alioua-20-000-migrants-au-maroc-risquent-unecatastrophe-humanitaire/. |
[57] | Fargues F, Rango M, Börgnas E, et al. (2020) Migration in West and North Africa and across the Mediterranean, Geneva: International Organization for Migration (IOM). Available from: https://publications.iom.int/books/migration-west-and-north-africa-and-across-mediterranean. |
[58] | Amnesty International (2021) Amnesty International Report 2020/2021. The State of the World's Human Rights. Available from: https://www.amnesty.org/en/documents/pol10/3202/2021/en/. |
[59] | Benachour S (2020) What a life in the host country: narratives of immigrants in Morocco amidst the pandemic, Compas. Available from: https://www.compas.ox.ac.uk/2020/what-a-life-in-the-host-countrynarratives-of-immigrants-in-morocco-amidst-the-pandemic/. |
[60] | Mobilizing for Rights Associates (2020) The Impact of Covid-19 on Violence against Women in Morocco. Available from: https://mrawomen.ma/#. |
[61] | Al-Karam (2015) La situation des Enfants de rue au Maroc. Available from: https://www.associationalkaram.org/situation.html. |
[62] | Babahaji L (2020) The Current State of Migrant Health in Morocco: Pre-and Peri-COVID-19 Pandemic, Independent Study Project (ISP), Collection 3353. |
[63] | Human Rights Watch (2022) Morocco/Spain: Horrific Migrant Deaths at Melilla Border. Available from: https://www.hrw.org/news/2022/06/29/morocco/spain-horrific-migrant-deaths-melilla-border. |
[64] | Leimazi S (2017) Morocco. The Invisible People who Should Take their Place on the Media Stage, Ethical Journalism Network, 2017. Available from: https://ethicaljournalismnetwork.org/media-mediterranean-migration-morocco. |
[65] | Royaume du Maroc (2019) La contribution de la société civile à l'effort de développement demeure 'faible'. Available from: https://www.maroc.ma/fr/actualites/la-contribution-de-la-societe-civile-leffort-de-devel%20oppe-ment-demeure-faible. |
[66] | Dimitrovova B (2009) Reshaping Civil Society in Morocco. Boundary Setting, Integration and Consolidation, CEPS Working Document, 323. Available from: https://ssrn.com/abstract = 1604037. |
[67] |
Cavatorta F (2006) Civil Society, Islamism and Democratization: the case of Morocco. J Mod Afr Stud 44: 203–222. https://doi.org/10.1017/S0022278X06001601 doi: 10.1017/S0022278X06001601
![]() |
[68] | ICEF (2019) Morocco priorities vocational training and strengthens ties with China. Available from: https://monitor.icef.com/2019/10/morocco-prioritises-vocational-training-and-strengthens-ties-with-china/. |
[69] | Belaid YN (2021) Participatory Governance in Moroccan education: What role for civil society organizations (CSOs)? J Res Humanit Soc Sci 9: 35–45. |
[70] | ETF (2022) Vocational Education and Training in Morocco and its relevance to the labour market. Available from: https://www.etf.europa.eu/en/publications-and-resources/publications/vocational-education-and-training-system-morocco-and-its. |
[71] | Naciri R (2009) Civic Society organizations in North Africa: Algeria, Morocco and Tunisia, Trust Africa, African Giving Knowledge Base. Available from: https://policycommons.net/artifacts/1848607/civil-society-organizations-in-north-africa/2594949/. |
[72] | Conseil Supérieur de l'enseignement (2008) État et Perspectives du Système d'Education et de Formation. Rapport Annuel 2008, Vol. 2: Rapport Analytique. Availble from: https://planipolis.iiep.unesco.org/fr/2008/etat-et-perspectives-du-syst%C3%A8me-d%C3%A9ducation-et-de-formation-volume-2-rapport-analytique-5164. |
[73] | HCP (2010) Etude sur les Associations marocaines de développement : diagnostic, analyse et perspectives ». Rapport final de la phase Ⅲ, Cabinet Conseil, MDSFS, Direction du Développement Social. Available from: http://www.abhatoo.net.ma/maalama-textuelle/developpement-economique-et-social/developpement-social/etat-politique/societe-civile/etude-sur-les-associations-marocaines-de-developpement-diagnostic-analyse-et-perspectives-rapport-iii-synthese-et-recommandations. |
[74] | Schoenen E (2016) Migrant Education in Morocco: Cross-Cultural Competence Favored Over Integrative Reform. An analysis of the Moroccan government's migrant integration efforts through education, ISP. Available fron: https://digitalcollections.sit.edu/isp_collection/2481. |
[75] |
Amthor RF, Roxas K (2016) Multicultural Education and Newcomer Youth: Re-Imagining a More Inclusive Vision for Immigrant and Refugee. J Am Educ Stud Assoc 52: 155–176. https://doi.org/10.1080/00131946.2016.1142992 doi: 10.1080/00131946.2016.1142992
![]() |
[76] | HCP (2021) Education, culture, jeunesse et loisir: Program 1: Education et Culture. Available from: https://marocainsdumonde.gov.ma/en/programmatic-achievements/. |
[77] |
Arcila-Calderón C, Blanco-Herrero D, Matsiola M, et al. (2023) Framing Migration in Southern European Media: Perceptions of Spanish, Italian, and Greek Specialized Journalists. J Pract 17: 24–47. https://doi.org/10.1080/17512786.2021.2014347 doi: 10.1080/17512786.2021.2014347
![]() |
[78] |
Castelli Gattinara P, Froio C (2019) Getting 'right' into the news: grassroots far-right mobilization and media coverage in Italy and France. Comp Eur Polit 17: 738–758. https://doi.org/10.1057/s41295-018-0123-4 doi: 10.1057/s41295-018-0123-4
![]() |
[79] | Berry M, Garcia-Blanco I, Moore K (2016) Press coverage of the refugee and migrant crisis in the EU: a content analysis of five European countries, Geneva: United Nations High Commissioner for Refugees. Available from: http://www.unhcr.org/56bb369c9.html. |
[80] |
Gemi E, Ulasiuk I, Triandafyllidou A (2013) Migrants and Media Newsmaking Practices. J Pract 7: 266-281. https://doi.org/10.1080/17512786.2012.740248 doi: 10.1080/17512786.2012.740248
![]() |
[81] |
Galantino MG (2022) The migration–terrorism nexus: An analysis of German and Italian press coverage of the 'refugee crisis'. Eur J Criminol 19: 259–281. https://doi.org/10.1177/1477370819896213 doi: 10.1177/1477370819896213
![]() |
[82] |
Geddes A, Pettrachin A (2020) Italian migration policy and politics: Exacerbating paradoxes. Contemp Ital Polit 12: 227–242. https://doi.org/10.1080/23248823.2020.1744918 doi: 10.1080/23248823.2020.1744918
![]() |
1. | W. Jung, C.A. Morales, Training neural networks from an ergodic perspective, 2023, 0233-1934, 1, 10.1080/02331934.2023.2239852 | |
2. | Steffen Dereich, Arnulf Jentzen, Sebastian Kassing, On the Existence of Minimizers in Shallow Residual ReLU Neural Network Optimization Landscapes, 2024, 62, 0036-1429, 2640, 10.1137/23M1556241 | |
3. | N Karthikeyan, K Madheswari, Hrithik Umesh, N Rajkumar, C Viji, Emotion Recognition with a Hybrid VGG-ResNet Deep Learning Model: A Novel Approach for Robust Emotion Classification, 2024, 3, 2953-4860, 960, 10.56294/sctconf2024960 |