Citation: Kusumiyati, Agus Arip Munawar, Diding Suhandy. Fast, simultaneous and contactless assessment of intact mango fruit by means of near infrared spectroscopy[J]. AIMS Agriculture and Food, 2021, 6(1): 172-184. doi: 10.3934/agrfood.2021011
[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 |
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
∫r0ρn−1f(u(ρ,t))dρ=∫r0χ{u(⋅,t)≥C∗}(ρ)⋅ρn−1f(u(ρ,t))dρ+∫r0χ{u(⋅,t)<C∗}(ρ)⋅ρn−1f(u(ρ,t))dρ. |
Combining these we have
hr=1nμ(t)r−r1−n∫r0χ{u(⋅,t)≥C∗}(ρ)⋅ρn−1f(u(ρ,t))dρ−r1−n∫r0χ{u(⋅,t)<C∗}(ρ)⋅ρn−1f(u(ρ,t))dρ≤1nμ(t)r−r1−n∫r0χ{u(⋅,t)<C∗}(ρ)⋅ρn−1f(u(ρ,t))dρ≤1nμ(t)r+C0r1−n∫r0χ{u(⋅,t)<C∗}(ρ)⋅ρn−1dρ≤1nμ(t)r+C0r1−n∫r0ρn−1dρ≤1n(μ(t)+C0)r, |
so we complete this proof.
To show the existence of a finite-time blow-up solution of (2.4), we need to prove that Uss is nonpositive by the following lemma. The proof follows the strategy in [48].
Lemma 2.4. Suppose that D,f and u0 satisfy (I1),(I2) and (I3) respectively. Then
ur(r,t)≤0 for all r∈(0,R),t∈(0,Tmax). | (2.9) |
Moreover,
Uss(s,t)≤0 for all r∈(0,R),t∈(0,Tmax). | (2.10) |
Proof. Without loss of generality we may assume that u0∈C2 ([0,∞)) and f∈C2([0,∞)). Applying the regularity theory in ([49,50]), we all know that u and ur belong to C0([0,R]×[0,T))∩C2,1((0,R)×(0,T)) and we fixed T∈(0,Tmax). From (2.4), we have for r∈(0,R) and t∈(0,T)
hrr+n−1rhr=μ(t)−f(u), | (2.11) |
and from (2.4) we obtain
urt=((D(u)ur)r+n−1rD(u)ur+uf(u)−uμ(t)−urhr)r=(D(u)ur)rr+a1(D(u)ur)r+a2urr+bur, |
for all r∈(0,R) and t∈(0,T), where
a1(r,t)=n−1r,a2(r,t)=−hr,b(r,t)=−n−1r2D(u)−μ(t)−hrr+f(u)+uf′(u), |
for all r∈(0,R) and t∈(0,T). Moreover, we have hr≤rn(μ(t)+C0) by (2.8) and from (2.11) such that
−hrr=n−1rhr−μ(t)+f(u)≤n−1nμ(t)+n−1nC0−μ(t)+f(u)≤f(u)+C0for all r∈(0,R) and t∈(0,T), |
then setting c1:=sup(r,t)∈(0,R)×(0,T)(2f(u)+uf′(u)+C0), we obtain
b(r,t)≤c1for all r∈(0,R) and t∈(0,T), |
and we introduce
c2:=sup(r,t)∈(0,R)×(0,T)((D(u))rr+a1(D(u))r)<∞, |
and set c3=2(c1+c2+1). Since ur(r,t)=0 for r∈{0,R},t∈(0,T) (because u is radially symmetric) and u0r≤0, the function y:[0,R]×[0,T]→R, (r,t)↦ur(r,t)−εec3t belongs to C0([0,R]×[0,T]) and fulfills
{yt=(D(u)(y+εec3t))rr+a1(D(u)(y+εec3t))r+a2yr+b(y+εec3t)−c3εec3t=(D(u)y)rr+a1(D(u)y)r+a2yr+by+εec3t((D(u))rr+a1(D(u))r+b−c3)≤(D(u)y)rr+a1(D(u)y)r+a2yr+by+εec3t(c1+c2−c3),in (0,R)×(0,T),y<0,on {0,R}×(0,T),y(⋅,0)<0,in (0,R). | (2.12) |
By the estimate for y(⋅,0) in (2.12) and continuity of y, the time t0:=sup{t∈(0,T):y≤0 in [0,R]×(0,T)}∈(0,T] is defined. Suppose that t0<T, then there exists r0∈[0,R] such that y(r0,t0)=0 and y(r,t)≤0 for all r∈[0,R] and t∈[0,t0]; hence, yt(r0,t0)≥0. As D≥0 in [0,∞), not only y(⋅,t0) but also z:(0,R)→R,r⟼D(u(r,t0))y(r,t0) attains its maximum 0 at r0. Since the second equality in (2.12) asserts r0∈(0,R), we conclude zrr(r0)≤0,zr(r0)=0 and yr(r0,t0)=0. Hence, we could obtain the contradiction
0≤yt(r0,t0)≤zrr(r0)+a1(r0,t0)zr(r0)+a2(r0,t0)yr(r0,t0)+b(r0,t0)y(r0,t0)+εec3t0(c1+c2−c3)≤−c32εec3t0<0, |
since we have
c1+c2≤c32. |
So that t0=T, implying y≤0 in [0,R]×[0,T] and hence ur≤εec3t in [0,R]×[0,T]. Letting first ε↘0 and then T↗Tmax, this proves that ur≤0 in [0,R]×[0,Tmax), and we have Uss≤0 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 Tmax by counter evidence. Firstly, we introduce a moment-like functional as follows
ϕ(t):=∫s00s−γ(s0−s)U(s,t)ds,t∈[0,Tmax), | (3.1) |
with γ∈(−∞,1) and s0∈(0,Rn). As a preparation of the subsequent analysis of ϕ, we denote
Sϕ:={t∈(0,Tmax)|ϕ(t)≥nM−s0(1−γ)(2−γ)ωn⋅s2−γ0}. | (3.2) |
The following lemma provides a lower bound for U.
Lemma 3.1. Let γ∈(−∞,1) and s0∈(0,Rn), then
U(s02,t)≥1ωn⋅(nM−4s02γ(3−γ)). | (3.3) |
Proof. If (3.3) was false for some t∈Sϕ such that U(s02,t)<1ωn⋅(nM−4s02γ(3−γ)), then necessarily δ:=4s02γ(3−γ)<nM. By the monotonicity of U(⋅,t) we would obtain that U(s,t)<nM−δωn for all s∈(0,s02). Since U(s,t)<nMωn for all s∈(0,Rn), we have
ϕ(t)<nM−δωn⋅∫s020s−γ(s0−s)ds+nMωn⋅∫s0s02s−γ(s0−s)ds=nMωn⋅∫s00s−γ(s0−s)ds−δωn⋅∫s020s−γ(s0−s)ds |
=nMωn⋅s2−γ0(1−γ)(2−γ)−δωn⋅2γ(3−γ)s2−γ04(1−γ)(2−γ). |
In view of the definition of Sϕ, we find that nM−s0<nM−2γ(3−γ)δ4, which contradicts our definition of δ.
An upper bound for μ is established by the following lemma.
Lemma 3.2. Let γ∈(−∞,1) and s0>0 such that s0≤Rn6. Then the function μ(t) has property that
μ(t)≤C1+12s∫s0f(Us(σ,t))dσ for all s∈(0,s0) and any t∈Sϕ, | (3.4) |
where C1=χ2C0+C0+C23+C3=13(χ2C0+C0+χk1(γ1−γ2)2γ2(2ξk2γ2χk1γ1)γ1γ1−γ2)+χf1(2δωns0).
Proof. First for any fixed t∈Sϕ, we may invoke Lemma 3.1 to see that
U(s02,t)≥nM−δωn, |
and thus, as U≤nMωn,
U(s0,t)−U(s02,t)s02≤nMωn−nM−δωns02=2δωns0. |
However, by concavity of U(⋅,t), as asserted by Lemma 2.4,
U(s0,t)−U(s02,t)s02≥Us(s0,t)≥Us(s,t) for all s∈(s0,Rn). |
Then let s0∈(0,Rn), we know that
μ(t)=1Rn∫s00f(Us(σ,t))dσ+1Rn∫Rns0f(Us(σ,t))dσ=1Rn∫s0f(Us(σ,t))dσ+1Rn∫s0sf(Us(σ,t))dσ+1Rn∫Rns0f(Us(σ,t))dσ,∀t∈(0,Tmax). | (3.5) |
Since γ1>γ2 and Young's inequality such that ξf2(u)≤ξk2(1+u)γ2≤χk12(1+u)γ1+C2≤χ2f1(u)+C2 with C2=χk1(γ1−γ2)2γ2(2ξk2γ2χk1γ1)γ1γ1−γ2 for u≥0, then for all s∈(0,Rn) and t∈(0,Tmax) we show that
χ2f1(Us(s,t))−C2≤f(Us(s,t))≤χf1(Us(s,t)). | (3.6) |
Accordingly, by the monotonicity of Us(⋅,t) along with (1.7) and (3.6), we have
∫s0sf(Us(σ,t))dσ≤∫s0sχf1(Us(σ,t))dσ≤∫s0sχf1(Us(s,t))dσ≤s0χf1(Us(s,t)),∀s∈(0,s0), t∈(0,Tmax). |
Since the condition of (2.7) implies that
∫s0f(Us(σ,t))dσ=∫s0χ{Us(⋅,t)≥1}(σ)⋅f(Us(σ,t)dσ+∫s0χ{Us(⋅,t)<1}(σ)⋅f(Us(σ,t)dσ≥∫s0χ{Us(⋅,t)≥1}(σ)⋅(χ2f1(Us(σ,t))−C2)dσ−C0s≥∫s0χ{Us(⋅,t)≥1}(σ)⋅χ2f1(Us(σ,t))dσ−(C0+C2)s=∫s0χ2f1(Us(σ,t))dσ−∫s0χ{Us(⋅,t)<1}(σ)⋅χ2f1(Us(σ,t))dσ−(C0+C2)s≥∫s0χ2f1(Us(s,t))dσ−χ2C0s−(C0+C2)s≥sχ2f1(Us(s,t))−(χ2C0+C0+C2)s. |
Therefore, we obtain
∫s0sf(Us(σ,t))dσ≤2s0s∫s0f(Us(σ,t))dσ+2(χ2C0+C0+C2)s0. |
Since (3.5) we have for all s∈(0,s0)
1Rn∫s0f(Us(σ,t))dσ+1Rn∫s0sf(Us(σ,t))dσ≤1Rn∫s0f(Us(σ,t))dσ+2s0Rns∫s0f(Us(σ,t))dσ+2(χ2C0+C0+C2)s0Rn, | (3.7) |
where s0≤Rn6 such that 1Rn≤16s0≤16s, 2s0Rns≤13s and s0Rn≤16 for all s∈(0,s0). Finally, we estimate the last summand of (3.5)
1Rn∫Rns0f(Us(σ,t))dσ≤1Rn∫Rns0χf1(Us(σ,t))dσ≤χf1(2δωns0)=C3. | (3.8) |
Together with (3.5), (3.7) and (3.8) imply (3.4).
Lemma 3.3. Assume that γ∈(−∞,1) satisfying
γ<2−2n, |
and s0∈(0,Rn6]. Then the function ϕ:[0,Tmax)→R defined by (3.1) belongs to C0([0,Tmax))∩C1((0,Tmax)) and satisfies
ϕ′(t)≥n2∫s00s2−2n−γ(s0−s)UssD(Us)ds+12∫s00s−γ(s0−s)Us⋅{∫s0f(Us(σ,t))dσ}ds−C1∫s00s1−γ(s0−s)Usds=:J1(t)+J2(t)+J3(t), | (3.9) |
for all t∈[0,Tmax), where C1 is defined in Lemma 3.2.
Proof. Combining (2.6) and (3.4) we have
Ut(s,t)=n2s2−2nD(Us)Uss−sμ(t)Us+Us⋅∫s0f(Us(σ,t))dσ≥n2s2−2nUssD(Us)+12Us⋅∫s0f(Us(σ,t))dσ−C1sUs. |
Notice ϕ(t) conforms ϕ(t)=∫s00s−γ(s0−s)U(s,t)ds. So (3.9) is a direct consequence.
Lemma 3.4. Let s0∈(0,Rn6], and γ∈(−∞,1) satisfying γ<2−2n. Then J1(t) in (3.9) satisfies
J1(t)≥−I, | (3.10) |
where
I:={−n2dm(2−2n−γ)∫s00s1−2n−γ(s0−s),m<0,n2d(2−2n−γ)∫s00s1−2n−γ(s0−s)ln(Us+1),m=0,n2dm(2−2n−γ)∫s00s1−2n−γ(s0−s)(Us+1)m,m>0, | (3.11) |
for all t∈Sϕ.
Proof. Since D∈C2([0,∞)), suppose that
G(τ)=∫τ0D(δ)dδ, |
then
0<G(τ)≤d∫τ0(1+δ)m−1dδ≤{−dm,m<0,dln(τ+1),m=0,dm(τ+1)m,m>0. |
Here integrating by parts we obtain
J1(t)=n2∫s00s2−2n−γ(s0−s)dG(Us)=n2s2−2n−γ(s0−s)G(Us)|s00+n2∫s00s2−2n−γG(Us)ds−n2(2−2n−γ)∫s00s1−2n−γ(s0−s)G(Us)ds. |
Hence a direct calculation yields
J1(t)≥{n2dm(2−2n−γ)∫s00s1−2n−γ(s0−s),m<0,−n2d(2−2n−γ)∫s00s1−2n−γ(s0−s)ln(Us+1),m=0,−n2dm(2−2n−γ)∫s00s1−2n−γ(s0−s)(Us+1)m,m>0, |
for all t∈Sϕ. We conclude (3.10).
Lemma 3.5. Assume that γ∈(−∞,1) satisfying γ<2−2n and s0∈(0,Rn6]. Then we have
J2(t)+J3(t)≥k1χ4∫s00s1−γ(s0−s)U1+γ1sds−C4∫s00s1−γ(s0−s)Usds | (3.12) |
for all t∈Sϕ, where C4=C1+(χ2C0+C0+C2)2.
Proof. Since Lemma 3.2 we have
∫s0f(Us(σ,t))dσ≥s2χf1(Us(s,t))−(χ2C0+C0+C2)sfor all s∈(0,s0) and t∈(0,Tmax). |
Therefore,
J2(t)=12∫s00s−γ(s0−s)Us⋅{∫s0f(Us(σ,t))dσ}ds≥χ4∫s00s1−γ(s0−s)Usf1(Us(s,t))ds−(χ2C0+C0+C2)2∫s00s1−γ(s0−s)Usds≥k1χ4∫s00s1−γ(s0−s)U1+γ1sds−(χ2C0+C0+C2)2∫s00s1−γ(s0−s)Usds, |
where f1(Us(s,t))≥k1(1+Us)γ1≥k1(Us)γ1. Combining these inequalities we can deduce (3.12).
Lemma 3.6. Let γ1>max{0,m−1}. For any γ∈(−∞,1) satisfying
γ∈min{2−2n⋅1+γ1γ1, 2−2n⋅1+γ11+γ1−m}, | (3.13) |
and s0∈(0,Rn6], the function ϕ:[0,Tmax)→R defined in (3.1) satisfies
ϕ′(t)≥{Cψ(t)−Cs3−γ−2n⋅1+γ1γ10,m≤1,Cψ(t)−Cs3−γ−2n⋅1+γ11+γ1−m0,m>1, | (3.14) |
with C>0 for all t∈Sϕ, where ψ(t):=∫s00s1−γ(s0−s)U1+γ1sds.
Proof. From (3.10) and (3.12) we have
ϕ′(t)≥k1χ4ψ(t)−I−C4∫s00s1−γ(s0−s)Usds, | (3.15) |
for all t∈Sϕ and I is given by (3.11). In the case m<0,
−n2dm(2−2n−γ)∫s00s1−2n−γ(s0−s)ds≤−n2dm(2−2n−γ)s0∫s00s1−2n−γds=−n2dms3−2n−γ0. |
If m=0, we use the fact that ln(1+x)x<1 for any x>0 and Hölder's inequality to estimate
n2d(2−2n−γ)∫s00s1−2n−γ(s0−s)ln(Us+1)ds=n2d(2−2n−γ)∫s00[s1−γ(s0−s)U1+γ1s]11+γ1⋅s1−2n−γ−1−γ1+γ1(s0−s)1−11+γ1ln(1+Us)Usds≤n2d(2−2n−γ){∫s00s1−γ(s0−s)U1+γ1sds}11+γ1⋅{∫s00(s1−2n−γ−1−γ1+γ1(s0−s)γ11+γ1)1+γ1γ1ds}γ11+γ1≤n2d(2−2n−γ)sγ11+γ10{∫s00s(1−2n−γ)γ1−2nγ1ds}γ11+γ1ψ11+γ1(t)=C5ψ11+γ1(t)s(3−2n−γ)γ1−2n1+γ10, |
for all t∈Sϕ with C5:=n2d(2−2n−γ)⋅(12−γ−2n⋅1+γ1γ1)γ11+γ1>0 by (3.13). In the case m>0, by using the elementary inequality (a+b)α≤2α(aα+bα) for all a,b>0 and every α>0, we obtain
n2dm(2−2n−γ)∫s00s1−2n−γ(s0−s)(Us+1)mds≤2mn2dm(2−2n−γ)∫s00s1−2n−γ(s0−s)Umsds+2mn2dm(2−2n−γ)∫s00s1−2n−γ(s0−s)ds, | (3.16) |
for all t∈Sϕ, and we first estimate the second term on the right of (3.16)
2mn2dm(2−2n−γ)∫s00s1−2n−γ(s0−s)ds≤2mn2dms3−2n−γ0. |
Since γ1>m−1 and by Hölder's inequality we deduce that
2mn2dm(2−2n−γ)∫s00s1−2n−γ(s0−s)Umsds=2mn2dm(2−2n−γ)∫s00s(1−γ)⋅m1+γ1(s0−s)m1+γ1Ums⋅s1−2n−γ−(1−γ)⋅m1+γ1(s0−s)1−m1+γ1ds≤2mn2dm(2−2n−γ){∫s00[s(1−γ)⋅m1+γ1(s0−s)m1+γ1Ums]1+γ1mds}m1+γ1×{∫s00[s1−2n−γ−(1−γ)⋅m1+γ1(s0−s)1−m1+γ1]1+γ11+γ1−mds}1+γ1−m1+γ1≤2mn2dm(2−2n−γ)ψm1+γ1(t)s1+γ1−m1+γ10⋅{∫s00s(1+γ1−m)(1−γ)−2n(1+γ1)1+γ1−mds}1+γ1−m1+γ1≤C6ψm1+γ1(t)s(1+γ1−m)(3−γ)−2n(1+γ1)1+γ10, |
for all t∈Sϕ with C6=2mn2dm(2−2n−γ)(12−γ−2n⋅1+γ11+γ1−m)1+γ1−m1+γ1>0 where γ<2−2n⋅1+γ11+γ1−m from (3.13).
Next, we can estimate the third expression on the right of (3.15) as follows
C4∫s00s1−γ(s0−s)Usds=C4∫s00[s1−γ(s0−s)U1+γ1s]11+γ1⋅s1−γ−1−γ1+γ1(s0−s)1−11+γ1ds≤C4{∫s00s1−γ(s0−s)U1+γ1sds}11+γ1⋅{∫s00[s1−γ−1−γ1+γ1(s0−s)1−11+γ1]1+γ1γ1ds}γ11+γ1≤C4ψ11+γ1(t)sγ11+γ10{∫s00s1−γds}γ11+γ1=C7ψ11+γ1(t)s(3−γ)γ11+γ10, |
where C7=C4(12−γ)γ11+γ1 for all t∈Sϕ. By (3.15) and collecting the estimates above we have
ϕ′(t)≥{k1χ4ψ(t)+n2dms3−2n−γ0−C7ψ11+γ1(t)s(3−γ)γ11+γ10,m<0, t∈Sϕ,k1χ4ψ(t)−C5ψ11+γ1(t)s(3−2n−γ)γ1−2n1+γ10−C7ψ11+γ1(t)s(3−γ)γ11+γ10,m=0, t∈Sϕ,k1χ4ψ(t)−2mn2dms3−2n−γ0−C6ψm1+γ1(t)s(1+γ1−m)(3−γ)−2n(1+γ1)1+γ10−C7ψ11+γ1(t)s(3−γ)γ11+γ10,m>0, t∈Sϕ. |
If m=0, by Young's inequality we can find positive constants C8,C9 such that
C5ψ11+γ1(t)s(3−2n−γ)γ1−2n1+γ10≤k1χ16ψ(t)+C8s3−γ−2n⋅1+γ1γ10,∀t∈Sϕ, |
while as m>0 we have
C6ψm1+γ1(t)s(1+γ1−m)(3−γ)−2n(1+γ1)1+γ10≤k1χ16ψ(t)+C9s3−γ−2n⋅1+γ11+γ1−m0,∀t∈Sϕ. |
On the other hand, we use Young's inequality again
C7ψ11+γ1(t)s(3−γ)γ11+γ10≤k1χ16ψ(t)+C10s3−γ0,∀t∈Sϕ. |
In the case m<0, because of s0∈(0,Rn6], we have
s3−2n−γ0=s3−γ−2n⋅1+γ1γ10⋅s2nγ10≤(Rn6)2nγ1s3−γ−2n⋅1+γ1γ10, |
when m>0 we have
s3−2n−γ0=s3−γ−2n⋅1+γ11+γ1−m0⋅s2mn(1+γ1−m)0≤(Rn6)2mn(1+γ1−m)s3−γ−2n⋅1+γ11+γ1−m0. |
All in all, we have
ϕ′(t)≥{k1χ8ψ(t)−C11s3−γ−2n⋅1+γ1γ10−C10s3−γ0,m≤0,k1χ8ψ(t)−C12s3−γ−2n⋅1+γ11+γ1−m0−C10s3−γ0,m>0, | (3.17) |
for all t∈Sϕ with C11=C8−n2dm(Rn6)2nγ1 and C12=C9+2mn2dm(Rn6)2mn(1+γ1−m). When 0<m≤1, we have 1+γ11+γ1−m≤1+γ1γ1 such that s3−γ−2n⋅1+γ11+γ1−m0=s3−γ−2n⋅1+γ1γ10s2n(1+γ1γ1−1+γ11+γ1−m)0≤(Rn6)2(1−m)(1+γ1)nγ1(1+γ1−m)s3−γ−2n⋅1+γ1γ10. In the case m≤1
s3−γ0=s2n⋅1+γ1γ10⋅s3−γ−2n⋅1+γ1γ10≤(Rn6)2n⋅1+γ1γ1⋅s3−γ−2n⋅1+γ1γ10, |
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] | Wendel A, Underwood J, Walsh K (2018) Maturity estimation of mangoes using hyperspectral imaging from a ground based mobile platform. Comput Electron Agric 155: 298-313. |
[2] | Cortés V, Ortiz C, Aleixos N, et al. (2016) A new internal quality index for mango and its prediction by external visible and near-infrared reflection spectroscopy. Postharvest Biol Technol 118: 148-158. |
[3] | Nagle M, Mahayothee B, Rungpichayapichet P, et al. (2010) Effect of irrigation on near-infrared (NIR) based prediction of mango maturity. Sci Hortic 125: 771-774. |
[4] | Hayati R, Munawar AA, Fachruddin F (2020) Enhanced near infrared spectral data to improve prediction accuracy in determining quality parameters of intact mango. Data Brief 30: 105577. |
[5] | Munawar AA, Kusumiyati, Hafidh, et al. (2019) The application of near infrared technology as a rapid and non-destructive method to determine vitamin C content of intact mango fruit. INMATEH-Agric Eng 58: 285-292. |
[6] | Arendse E, Fawole OA, Magwaza LS, et al. (2018) Non-destructive prediction of internal and external quality attributes of fruit with thick rind: A review. J Food Eng 217: 11-23. |
[7] | Pasquini C (2018) Near infrared spectroscopy: A mature analytical technique with new perspectives-A review. Anal Chim Acta 1026: 8-36. |
[8] | Munawar AA, Kusumiyati, Wahyuni D (2019) Near infrared spectroscopic data for rapid and simultaneous prediction of quality attributes in intact mango fruits. Data Brief 27: 104789. |
[9] | Santos UJ dos, Demattê JA de M, Menezes RSC, et al. (2020) Predicting carbon and nitrogen by visible near-infrared (Vis-NIR) and mid-infrared (MIR) spectroscopy in soils of Northeast Brazil. Geoderma Reg 23: e00333. |
[10] | Deng Y, Wang Y, Zhong G, et al. (2018) Simultaneous quantitative analysis of protein, carbohydrate and fat in nutritionally complete formulas of medical foods by near-infrared spectroscopy. Infrared Phys Technol 93: 124-129. |
[11] | Nordey T, Joas J, Davrieux F, et al. (2016) Robust NIRS models for non-destructive prediction of mango internal quality. Sci Hortic 216: 51-57. |
[12] | Cozzolino D (2014) An overview of the use of infrared spectroscopy and chemometrics in authenticity and traceability of cereals. Food Res Int 60: 262-265. |
[13] | Cen H, He Y (2007) Theory and application of near infrared reflectance spectroscopy in determination of food quality. Trends Food Sci Technol 18: 72-83. |
[14] | Le BT (2020) Application of deep learning and near infrared spectroscopy in cereal analysis. Vib Spectrosc 106: 103009. |
[15] | Kutsanedzie FYH, Chen Q, Hassan MM, et al. (2018) Near infrared system coupled chemometric algorithms for enumeration of total fungi count in cocoa beans neat solution. Food Chem 240: 231-238. |
[16] | Huang JH, Zhou RR, He D, et al. (2020) Rapid identification of Lilium species and polysaccharide contents based on near infrared spectroscopy and weighted partial least square method. Int J Biol Macromol 154: 182-187. |
[17] | Mandrile L, Barbosa-Pereira L, Sorensen KM, et al. (2019) Authentication of cocoa bean shells by near- and mid-infrared spectroscopy and inductively coupled plasma-optical emission spectroscopy. Food Chem 292: 47-57. |
[18] | Teye E, Huang XY, Lei W, et al. (2014) Feasibility study on the use of Fourier transform near-infrared spectroscopy together with chemometrics to discriminate and quantify adulteration in cocoa beans. Food Res Int 55: 288-293. |
[19] | Yisak H, Redi-Abshiro M, Chandravanshi BS (2018) Selective determination of caffeine and trigonelline in aqueous extract of green coffee beans by FT-MIR-ATR spectroscopy. Vib Spectrosc 97: 33-38. |
[20] | Santos JR, Lopo M, Rangel AOSS, et al. (2016) Exploiting near infrared spectroscopy as an analytical tool for on-line monitoring of acidity during coffee roasting. Food Control 60: 408-415. |
[21] | Dias RCE, Valderrama P, Março PH, et al. (2018) Data on roasted coffee with specific defects analyzed by infrared-photoacoustic spectroscopy and chemometrics. Data Brief 20: 242-249. |
[22] | Giovenzana V, Beghi R, Romaniello R, et al. (2018) Use of visible and near infrared spectroscopy with a view to on-line evaluation of oil content during olive processing. Biosyst Eng 172: 102-109. |
[23] | Lu J, Xiang B, Liu H, et al. (2008) Application of two-dimensional near-infrared correlation spectroscopy to the discrimination of Chinese herbal medicine of different geographic regions. Spectrochim Acta-Part A Mol Biomol Spectrosc 69: 580-586. |
[24] | Hao JW, Chen ND, Chen CW, et al. (2018) Rapid quantification of polysaccharide and the main onosaccharides in Dendrobium huoshanense by near-infrared attenuated total reflectance spectroscopy. J Pharm Biomed Anal 151: 331-318. |
[25] | Liu C, Yang SX, Deng L (2015) Determination of internal qualities of Newhall navel oranges based on NIR spectroscopy using machine learning. J Food Eng 161: 16-23. |
[26] | Ncama K, Opara UL, Tesfay SZ, et al. (2017) Application of Vis/NIR spectroscopy for predicting sweetness and flavour parameters of 'Valencia' orange (Citrus sinensis) and 'Star Ruby' grapefruit (Citrus x paradisi Macfad). J Food Eng 193: 86-94. |
[27] | Srivichien S, Terdwongworakul A, Teerachaichayut S (2015) Quantitative prediction of nitrate level in intact pineapple using Vis-NIRS. J Food Eng 150: 29-34. |
[28] | van Gastelen S, Mollenhorst H, Antunes-Fernandes EC, et al. (2018) Predicting enteric methane emission of dairy cows with milk Fourier-transform infrared spectra and gas chromatography-based milk fatty acid profiles. J Dairy Sci 101: 5582-5598. |
[29] | Coppa M, Martin B, Agabriel C, et al. (2012) Authentication of cow feeding and geographic origin on milk using visible and near-infrared spectroscopy. J Dairy Sci 95: 5544-5551. |
[30] | Sricharoonratana M, Thompson AK, Teerachaichayut S (2021) Use of near infrared hyperspectral imaging as a nondestructive method of determining and classifying shelf life of cakes. LWT 136: 110369. |
[31] | Suktanarak S, Teerachaichayut S (2017) Non-destructive quality assessment of hens' eggs using hyperspectral images. J Food Eng 215: 97-103. |
[32] | Iskandar CD, Zainuddin, Munawar AA (2020) Rapid assessment of frozen beef quality using near infrared technology. Int J Sci Technol Res 9: 1-5. |
[33] | Samadi, Wajizah S, Munawar AA (2020) Near infrared spectroscopy (NIRS) data analysis for a rapid and simultaneous prediction of feed nutritive parameters. Data Brief 29: 105211. |
[34] | Jha SN, Jaiswal P, Narsaiah K, et al. (2012) Non-destructive prediction of sweetness of intact mango using near infrared spectroscopy. Sci Hortic 138: 171-175. |
[35] | Marques EJN, De Freitas ST, Pimentel MF, et al. (2016) Rapid and non-destructive determination of quality parameters in the "Tommy Atkins" mango using a novel handheld near infrared spectrometer. Food Chem 197: 1207-1214. |
[36] | dos Santos Neto JP, de Assis MWD, Casagrande IP, et al. (2017) Determination of 'Palmer' mango maturity indices using portable near infrared (VIS-NIR) spectrometer. Postharvest Biol Technol 130: 75-80. |
[37] | Munawar AA, Devianti, Satriyo P, et al. (2019) Rapid and simultaneous prediction of soil quality attributes using near infrared technology. Int J Sci Technol Res 8: 725-728. |
[38] | Sudarjat, Kusumiyati, Hasanuddin, et al. (2019) Rapid and non-destructive detection of insect infestations on intact mango by means of near infrared spectroscopy. In: IOP Conference Series: Earth and Environmental Science 365. |
[39] | Bureau S, Cozzolino D, Clark CJ (2019) Contributions of Fourier-transform mid infrared (FT-MIR) spectroscopy to the study of fruit and vegetables: A review. Postharvest Biol Technol 148: 1-14. |
[40] | Munawar AA, von Hörsten D, Wegener JK, et al. (2016) Rapid and non-destructive prediction of mango quality attributes using Fourier transform near infrared spectroscopy and chemometrics. Eng Agric Environ Food 9: 208-215. |
[41] | Munawar AA, Kusumiyati, Wahyuni D (2019) Near infrared spectroscopic data for rapid and simultaneous prediction of quality attributes in intact mango fruits. Data Brief 27: 104789. |
[42] | Munawar AA, von Hörsten D, Wegener JK, et al. (2016) Rapid and non-destructive prediction of mango quality attributes using Fourier transform near infrared spectroscopy and chemometrics. Eng Agric Environ Food 9: 208-215. |
[43] | Nicolaï BM, Beullens K, Bobelyn E, et al. (2007) Nondestructive measurement of fruit and vegetable quality by means of NIR spectroscopy: A review. Postharvest Biol Technol 46: 99-118. |
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 |