Intelligent manufacturing (IM), sometimes referred to as smart manufacturing (SM), is the use of real-time data analysis, machine learning, and artificial intelligence (AI) in the production process to achieve the aforementioned efficiencies. Human-machine interaction technology has recently been a hot issue in smart manufacturing. The unique interactivity of virtual reality (VR) innovations makes it possible to create a virtual world and allow users to communicate with that environment, providing users with an interface to be immersed in the digital world of the smart factory. And virtual reality technology aims to stimulate the imagination and creativity of creators to the maximum extent possible for reconstructing the natural world in a virtual environment, generating new emotions, and transcending time and space in the familiar and unfamiliar virtual world. Recent years have seen a great leap in the development of intelligent manufacturing and virtual reality technologies, yet little research has been done to combine the two popular trends. To fill this gap, this paper specifically employs Preferred Reporting Items for Systematic Reviews and Meta-analysis (PRISMA) guidelines to conduct a systematic review of the applications of virtual reality in smart manufacturing. Moreover, the practical challenges and the possible future direction will also be covered.
Citation: Yu Lei, Zhi Su, Xiaotong He, Chao Cheng. Immersive virtual reality application for intelligent manufacturing: Applications and art design[J]. Mathematical Biosciences and Engineering, 2023, 20(3): 4353-4387. doi: 10.3934/mbe.2023202
[1] | Gongwei Liu . The existence, general decay and blow-up for a plate equation with nonlinear damping and a logarithmic source term. Electronic Research Archive, 2020, 28(1): 263-289. doi: 10.3934/era.2020016 |
[2] | Yi Cheng, Ying Chu . A class of fourth-order hyperbolic equations with strongly damped and nonlinear logarithmic terms. Electronic Research Archive, 2021, 29(6): 3867-3887. doi: 10.3934/era.2021066 |
[3] | Vo Van Au, Jagdev Singh, Anh Tuan Nguyen . Well-posedness results and blow-up for a semi-linear time fractional diffusion equation with variable coefficients. Electronic Research Archive, 2021, 29(6): 3581-3607. doi: 10.3934/era.2021052 |
[4] | Xu Liu, Jun Zhou . Initial-boundary value problem for a fourth-order plate equation with Hardy-Hénon potential and polynomial nonlinearity. Electronic Research Archive, 2020, 28(2): 599-625. doi: 10.3934/era.2020032 |
[5] | Shuting Chang, Yaojun Ye . Upper and lower bounds for the blow-up time of a fourth-order parabolic equation with exponential nonlinearity. Electronic Research Archive, 2024, 32(11): 6225-6234. doi: 10.3934/era.2024289 |
[6] | Abdelhadi Safsaf, Suleman Alfalqi, Ahmed Bchatnia, Abderrahmane Beniani . Blow-up dynamics in nonlinear coupled wave equations with fractional damping and external source. Electronic Research Archive, 2024, 32(10): 5738-5751. doi: 10.3934/era.2024265 |
[7] | Xiaoqiang Dai, Chao Yang, Shaobin Huang, Tao Yu, Yuanran Zhu . Finite time blow-up for a wave equation with dynamic boundary condition at critical and high energy levels in control systems. Electronic Research Archive, 2020, 28(1): 91-102. doi: 10.3934/era.2020006 |
[8] | Mohammad M. Al-Gharabli, Adel M. Al-Mahdi . Existence and stability results of a plate equation with nonlinear damping and source term. Electronic Research Archive, 2022, 30(11): 4038-4065. doi: 10.3934/era.2022205 |
[9] | Huafei Di, Yadong Shang, Jiali Yu . Existence and uniform decay estimates for the fourth order wave equation with nonlinear boundary damping and interior source. Electronic Research Archive, 2020, 28(1): 221-261. doi: 10.3934/era.2020015 |
[10] | Yuchen Zhu . Blow-up of solutions for a time fractional biharmonic equation with exponentional nonlinear memory. Electronic Research Archive, 2024, 32(11): 5988-6007. doi: 10.3934/era.2024278 |
Intelligent manufacturing (IM), sometimes referred to as smart manufacturing (SM), is the use of real-time data analysis, machine learning, and artificial intelligence (AI) in the production process to achieve the aforementioned efficiencies. Human-machine interaction technology has recently been a hot issue in smart manufacturing. The unique interactivity of virtual reality (VR) innovations makes it possible to create a virtual world and allow users to communicate with that environment, providing users with an interface to be immersed in the digital world of the smart factory. And virtual reality technology aims to stimulate the imagination and creativity of creators to the maximum extent possible for reconstructing the natural world in a virtual environment, generating new emotions, and transcending time and space in the familiar and unfamiliar virtual world. Recent years have seen a great leap in the development of intelligent manufacturing and virtual reality technologies, yet little research has been done to combine the two popular trends. To fill this gap, this paper specifically employs Preferred Reporting Items for Systematic Reviews and Meta-analysis (PRISMA) guidelines to conduct a systematic review of the applications of virtual reality in smart manufacturing. Moreover, the practical challenges and the possible future direction will also be covered.
In this paper, we deal with the following plate equation with nonlinear damping and a logarithmic source term
{utt+Δ2u+|ut|m−2ut=|u|p−2ulog|u|k,(x,t)∈Ω×R+,u=∂u∂ν=0,(x,t)∈∂Ω×R+,u(x,0)=u0(x),ut(x,0)=u1(x),x∈Ω, | (1) |
where
2<p<2(n−2)n−4ifn≥5;2<p<+∞ifn≤4. | (2) |
The logarithmic nonlinearity is of much interest in many branches of physics such as nuclear physics, optics and geophysics (see [5,6,15] and references therein). It has also been applied in quantum field theory, where this kind of nonlinearity appears naturally in cosmological inflation and in super symmetric field theories [4,13].
Let us review somework with logarithmic term which is closely related to the problem (1). Birula and Mycielski[6,7] studied the following problem
{utt−uxx+u−εulog|u|2=0,(x,t)∈[a,b]×(0,T),u(a,t)=u(b,t)=0,t∈(0,T),u(x,0)=u0(x),ut(x,0)=u1(x),x∈[a,b], | (3) |
which is a relativistic version of logarithmic quantum mechanics and can also be obtained by taking the limit
utt−Δu=ulog|u|k, | (4) |
in
utt−Δu+u+ut+|u|2u=ulog|u| | (5) |
to study the dynamics of Q-ball in theoretical physics. A numerical research was given in that work, while, there was no theoretical analysis for this problem. For the initial boundary value problem of(5), Han [17] obtained the global existence of weak solution in
utt−Δu+u+(g∗Δu)(t)+h(ut)ut+|u|2u=ulog|u|k. |
Recently, Al-Gharabli and Messaoudi [1] considered the following plate equation with logarithmic source term
{utt+Δ2u+u+h(ut)=ulog|u|k,(x,t)∈Ω×R+,u=∂u∂ν=0,(x,t)∈∂Ω×R+,u(x,0)=u0(x),ut(x,0)=u1(x),x∈Ω, | (6) |
where
utt+Δ2u+u−∫t0g(t−s)Δ2u(s)ds=ulog|u|k,(x,t)∈Ω×R+, | (7) |
they established the existence of solutions and proved an explicit and general decay rate result. However, there is no information on the finite or infinite blow up results in these researches [1,2,3].
At the same time, there are many results concerning the existence and nonexistence on evolution equation with polynomial source term. For example, for plate equation with polynomial source term
{utt+Δ2u+|ut|m−2ut=|u|p−2u,(x,t)∈Ω×R+,u=∂u∂ν=0,(x,t)∈∂Ω×R+,u(x,0)=u0(x),ut(x,0)=u1(x),x∈Ω, |
established an existence result and showed that the solution continues to exist globally if
To the best of our knowledge, there are few results on the evolution equation with the nonlinear logarithmic source term
utt−Δu+μ1ut(x,t)+μ2ut(x,t−τ)=|u|p−2ulog|u|k, | (8) |
obtained the local existence by using the semigroup theory and proved a finite time blow-up result when the initial energy is negative. Of course, these results also hold for the equation (8) without delay term (i.e.
Motivated by the above mentioned papers, our purpose in this research is to investigative the existence, energy decay and finite time blow-up of the solution to the initial boundary value problem (1). We note here that (i) the term
The rest of this article is organized as follows: Section 2 is concerned with some notation and some properties of the potential well. In Sect. 3, we present the existence and uniqueness of local solutions to (1) by using the contraction mapping principle. In Sect. 4, we prove the global existence and energy decay results. The proof of global existence result is based on the potential well theory and the continuous principle; while for energy decay result, the proof is based on the Nakao's inequality and some techniques on logarithmic nonlinearity. In Sect. 5, we prove the the finite time blow-up when the initial energy is negative. In Sect. 6, we establish the finite time blow-up result for problem (1) with
We give some material needed in the proof of our results. We use the standard Lebesgue space
Firstly, we introduce the Sobolev's embedding inequality : assume that
‖u‖p≤Cp‖Δu‖2,for u∈H20(Ω) | (9) |
where
Suppose (2) holds, we define
α∗:={2nn−4−p if n≥5,+∞ if n≤4 |
for any
Definition 2.1. A function
u(0)=u0,ut(0)=u1 |
and
⟨utt,v⟩+(Δu,Δv)+∫Ω|ut|m−2utvdx=∫Ω(|u|p−2ulog|u|k)vdx | (10) |
for any
Now, we introduce the energy functional
J(u)=J(u(t))=J(t)=12‖Δu‖2−1p∫Ω|u|plog|u|kdx+kp2‖u‖pp, | (11) |
and
I(u)=I(u(t))=I(t)=‖Δu‖2−∫Ω|u|plog|u|kdx. | (12) |
From the definitions (11) and (12), we have
J(u)=1pI(u)+(12−1p)‖Δu‖2+kp2‖u‖pp. | (13) |
The following lemmas play an important role in the studying the properties of the potential well.
Lemma 2.2. Let
(ⅰ):
(ⅱ): there exists a unique
(ⅲ):
Proof. We know
g(λ)=J(λu)==12λ2‖Δu‖2−kpλp∫Ω|u|plog|u|dx−kpλplogλ‖u‖pp+kp2λp‖u‖pp. |
It is obvious that
g′(λ)=λ(‖Δu‖2−kλp−2∫Ω|u|plog|u|dx−kλp−2logλ‖u‖pp) | (14) |
and
g′′(λ)=‖Δu‖2−k(p−1)λp−2∫Ω|u|plog|u|dx−k(p−1)λp−2logλ‖u‖pp−kλp−2‖u‖pp. |
From (14) and
g′(λ)|λ=λ∗=0, |
then we obtain
‖Δu‖2=kλ∗p−2∫Ωu2log|u|dx+kλ∗p−2logλ∗‖u‖pp. |
Substituting the above equation into
g′′(λ∗)=−k(p−2)λ∗p−2∫Ω|u|plog|u|dx−k(p−2)λ∗p−2logλ∗‖u‖pp−kλ∗p−2‖u‖pp=−(p−2)‖Δu‖2−kλ∗p−2‖u‖pp<0. |
From these and
From (12) and (14), we have
I(λu)=λddλJ(λu)=λg′(λ){>0,0<λ<λ,=0,λ=λ∗,<0,λ∗<λ<+∞. |
Then, we could define the potential well depth of the functional
d=inf{supλ≥0J(λu)|u∈H20(Ω)∖{0}}. | (15) |
We also define the well-known Nehari manifold
N={u|u∈H20(Ω)∖{0},I(u)=0}. |
As in [29,34], that the mountain pass level
d=infu∈NJ(u). |
It is easy to see that
Lemma 2.3. Assume that
r(α):=(αkCp+α∗)1p+α−2. |
Then, for any
(ⅰ) : if
(ⅱ) : if
Proof. Since
I(u)=‖∇u‖22−k∫Ω|u|plog|u|dx>‖∇u‖22−kα‖u‖p+αp+α≥‖∇u‖22−kCp+α∗α‖∇u‖p+α2=kCp+α∗α‖∇u‖22(rp+α−2(α)−‖∇u‖p+α−22). | (16) |
Obviously, the results can be obtained from the above inequality (16).
Lemma 2.4. Assume the notations in Lemma 2.2 hold, we have
0<r∗:=supα∈(0,α∗)=(αkCp+α∗)1p+α−2≤r∗:=supα∈(0,α∗)(αkBp+α)1p+α−2|Ω|αp(p+α−2)<+∞, |
where
Proof. It is obvious that
γ(α)=(αkBp+α)1p+α−2|Ω|αp(p+α−2),α∈(0,+∞). |
For any
‖u‖p≤|Ω|αp(p+α)‖u‖p+α. |
Then, noticing
C∗=supu∈H20∖{0}‖u‖p+α‖Δu‖2≥|Ω|−αp(p+α)supu∈H20∖{0}‖u‖p‖Δu‖2≥|Ω|−αP(p+α)B, |
which implies
(αkCp+α∗)1p+α−2≤(αkBp+α)1p+α−2|Ω|αp(p+α−2), |
that is
Now, we will prove
Case a. If
r∗=supα∈(0,2nn−4−p)γ(α)≤maxα∈[0,2nn−4−p]γ(α)<+∞ |
Case b. If
h(α):=log[γ(α)]=1p+α−2[logα−logk−(p+α)logB]+αp(p+α−2)log|Ω|. |
Hence
h′(α)=p2+pα−2p+pαlogk−pαlogα+2pαlogB+pαlog|Ω|−2αlog|Ω|pα(p+α−2)2. |
For simplicity, we set
g(α):=p2+pα−2p+pαlogk−pαlogα+2pαlogB+pαlog|Ω|−2αlog|Ω|, |
then
g′(α)=p+plogk−plogα−p+2plogB+plog|Ω|−2log|Ω|=plogkB2|Ω|1−2pα, |
which yields that the function
On the one hand, due to
limα→0+g(α)=p2−2p>0 |
which implies that
While on the other hand, we can deduce that
limα→+∞g(α)=limα→+∞(p2−2p+pα[1+log(kB2|Ω|1−2p)]−logα)=−∞, |
which together with
Noting the relation between
r∗=supα∈(0,+∞)σ(α)=eh(α∗)<+∞. |
Making using of the Lemmas 2.2 and 2.3, we obtain the following corollary.
Corollary 1. Assume that
(ⅰ): if
(ⅱ): if
for any
Lemma 2.5. Assume that
Proof. (ⅰ) For the case
(ⅱ) For the case
J(u)=(12−1p)‖Δu‖22+kp2‖u‖pp≥(p−22p)r2∗>0. |
We define energy for the problem (1), which obeys the following energy equality of the weak solution
E(t)+∫t0‖uτ‖mmdτ=E(0), for all t∈[0,T) | (17) |
where
E(t)=12‖ut‖2+12‖Δu‖2−1p∫Ω|u|plog|u|kdx+kp2‖u‖pp, |
E(0)=12‖u1‖2+12‖Δu0‖2−1p∫Ω|u0|plog|u0|kdx+kp2‖u0‖pp. |
It is obvious that
E(t)=12‖ut‖2+J(u). |
Taking
ddtE(t)=−‖ut‖mm. | (18) |
Now, we define the subsets of
W={u∈H10(Ω)|J(u)<d,I(u)>0},V={u∈H10(Ω)|J(u)<d,I(u)<0}, | (19) |
where
In order to establish the global existence and blow-up results of solution, we have to prove the following invariance sets of
Lemma 2.6. If
(ⅰ):
(ⅱ):
Proof. It follows from the definition of weak solution and (17) that
12‖ut‖2+J(u)≤12‖u1‖2+J(u0)<d, for any t∈[0,T). | (20) |
(ⅰ) Arguing by contradiction, we assume that there exists a number
It follows from (20) that (a) is impossible. If (b) holds, then by the definition of
(ⅱ) The proof is similar to the proof of (ⅰ). We omit it.
In this section, we are concerned with the local existence and uniqueness for the solution of the problem (1). The idea comes from [14,28,38], where the source term is polynomial. First, we give a technical lemma given in [22] which plays an important role in the uniqueness of the solution.
Lemma 3.1. ([22]) For every
j(s)=|s|p−2log|s|,p>2 |
satisfies
|j(s)|≤A+|s|p−2+ε. |
Theorem 3.2. Suppose that
Proof. For every
H:=C([0,T];H20(Ω))∩C1([0,T];L2(Ω)) |
endowed with the norm
‖u(t)‖H=(maxt∈[0,T](‖Δu(t)‖22+‖ut(t)‖22))12. |
For every given
{vtt+Δ2v+|vt|m−2vt=|u|p−2ulog|u|k,(x,t)∈Ω×R+,v=∂v∂ν=0,(x,t)∈∂Ω×R+,v(x,0)=u0(x),vt(x,0)=u1(x),x∈Ω. | (21) |
We shall prove that the problem (21) admits a unique solution
Let
u0h=h∑i=1(∫ΩΔu0Δwi)wi and u1h=h∑i=1(∫Ωu1wi)wi |
such that
vh(t)=h∑i=1γih(t)ωi, | (22) |
solves the following problem
{∫Ω(v′′h+Δ2vh+|v′h|m−2v′h−|u|p−2ulog|u|k)ηdx=0,vk(0)=u0h,v′h(0)=u1h. | (23) |
For
{γ′′ih(t)+λiγih(t)+ci|γ′ih(t)|m−2γ′ih(t)=ψi(t),γih(0)=∫Ωu0ωi,γ′ih(0)=∫Ωu1ωi, |
where
ci=‖ωi‖mm,ψi(t)=∫Ω|u(t)p−2u(t)log|u|kωidx∈C[0,T]. |
Then the above problem admits a unique local solution
Taking
‖v′h(t)‖2+‖Δvh(t)‖2+2∫t0‖v′h(τ)‖mmdτ=‖v1h‖2+‖Δv0h‖2+2∫t0∫Ω|u|p−2ulog|u|kv′h | (24) |
for every
2∫t0∫Ω|u|p−2ulog|u|kv′h |
≤2∫t0∫Ω||u|p−1log|u|k||v′h|≤∫t0∫Ω(C||u|p−1log|u||mm−1+∫t0‖v′h‖mm). | (25) |
In order to estimate (25), we focus on the logarithmic term. Here we denote
∫Ω‖u|p−1log|u||mm−1dx=∫Ω1‖u|p−1log|u||mm−1dx+∫Ω2‖u|p−1log|u||mm−1dx. |
By a simple calculation, we obtain
infs∈(0,1)sp−1logs=−1e(p−1), |
which implies
∫Ω1‖u|p−1log|u||mm−1dx≤[e(p−1)]−mm−1|Ω|. |
Let
ρ=2nn−4⋅m−1m−p+1>0forn≥5; any positive ρ,n≤4. |
By the Sobolev embedding from
∫Ω2‖u|p−1log|u||mm−1dx≤ρ−m−1m∫Ω2(|u|p−1+ρ)m−1mdx≤ρ−m−1m∫Ω2|u|2nn−4dx≤ρ−m−1m∫Ω|u|2nn−4dx=ρ−m−1m‖u‖2nn−42nn−4≤C‖u‖2nn−4H20≤C. |
The proof of the case
2∫t0∫Ω|u|p−2ulog|u|kv′h≤CT+∫t0‖v′h‖mm. | (26) |
Substituting this inequality into (24), we obtain
‖v′h(t)‖2+‖Δvh(t)‖2+∫t0‖v′h(τ)‖mmdτ≤C, | (27) |
where
vh(t) is bounded in L∞([0,T],H20(Ω)),v′h(t) is bounded in Lm[(0,T],Lm(Ω))∩L∞([0,T],L2(Ω)),[2mm]vh′′(t) is bounded in L2([0,T],H−2(Ω)). | (28) |
Hence, up to a subsequence, we could pass to the limit in (23) and obtain a weak solution
To prove the uniqueness, arguing by contradiction: if
‖wt−vt‖2+‖Δw−Δv‖2+2∫t0∫Ω(|wτ|m−2wτ−|vτ|m−2vτ)(wτ−vτ)=0. | (29) |
It follows from the following element inequality
(|φ|m−2φ−|ψ|m−2ψ)(φ−ψ)≥C|φ−ψ|m for m≥2, |
that (29) can make to be
‖wt−vt‖2+‖v−w‖2H20+C∫T0‖wτ−vτ‖mm≤0. |
Therefore, we have
Now, we are in the position to prove Theorem 3.1. For
R2:=2(‖u1‖2+‖Δu0‖2), |
and
BRT:={u∈H|u(0,x)=u0(x),ut(0,x)=u1(x),‖u‖H≤R} |
for every
Claim.
In fact, assume that
‖vt(t)‖2+‖Δv(t)‖2≤‖u1‖2+‖Δu0‖2+CR2nn−4T≤R22+CR2nn−4T |
for
Next we show that
⟨vtt,η⟩+(Δv,Δη)+∫Ω(|v1t|m−2v1t−|v2t|m−2v2t)ηdx=∫Ω(|w1|p−2w1log|w1|k−|w2|p−2w2log|w2|k)ηdx, | (30) |
for any
Taking
∫Ω(|v1t|m−2v1t−|v2t|m−2v2t)(v1t−v2t)dx≥0, |
and integrating both sides of (30) over
‖vt‖2+‖Δv‖2≤2k‖|w1|p−2w1log|w1|−|w2|p−2w2log|w2|‖‖vt‖, | (31) |
We need estimating the logarithmic term in (31) by using Lemma 3.1. By the similar argument as [22], we give the sketch of the proof.
Making use of mean value theorem, we have, for
||w1|p−2w1log|w1|−|w2|p−2w2log|w2||=k|1+(p−1)log|θw1+(1−θ)w2|||θw1+(1−θ)w2|p−2|w1−w2|. |
Then, it follows from Lemma 3.1 that
||w1|p−2w1log|w1|−|w2|p−2w2log|w2||≤k|θw1+(1−θ)w2|p−2|w1−w2|+k(p−1)A|w1−w2|+k(p−1)|θw1+(1−θ)w2|p−2+ε|w1−w2|≤k(|w1|+|w2|)p−2|w1−w2|+k(p−1)A|w1−w2|+k(p−1)(|w1|+|w2|)p−2+ε|w1−w2|. |
Since
∫Ω[(|w1|+|w2|)p−2|w1−w2|]2dx≤C(∫Ω(|w1|+|w2|)2(p−1)dx)(p−2)/(p−1)×(∫Ω|w1−w2|2(p−1)dx)1/(p−1)≤C[‖w1‖2(p−1)L2(p−1)+‖w2‖2(p−1)L2(p−1)](p−2)/(p−1)‖w1−w2‖2L2(p−1)≤C[‖w1‖2(p−1)H20(Ω)+‖w2‖2(p−1)H20(Ω)](p−2)/(p−1)‖w1−w2‖2H20(Ω)≤CR2(p−2)‖w1−w2‖2H20(Ω). |
By the similar argument, we have
∫Ω[(|w1|+|w2|)p−2+ε|w1−w2|]2dx≤C(∫Ω(|w1|+|w2|)2(p−2+ε)(p−1)/(p−2)dx)(p−2)/(p−1) |
×(∫Ω|w1−w2|2(p−1)dx)1/(p−1)≤(∫Ω(|w1|+|w2|)2(p−1)+2ε(p−1)/(p−2)dx)(p−2)/(p−1)‖w1−w2‖2L2(p−1). |
Using (2), we can choose sufficiently small
ˉp=2(p−1)+2ε(p−1)p−2≤2nn−4, |
which yields that
∫Ω[(|w1|+|w2|)p−2+ε|w1−w2|]2dx≤C[‖w1‖ˉpLˉp(Ω)+‖w2‖ˉpLˉp(Ω)](p−2)/(p−1)‖w1−w2‖2L2(p−1)≤CRˉp(p−2)/(p−1)‖w1−w2‖2H20(Ω). |
Noticing
‖|w1|p−2w1log|w1|−|w2|p−2w2log|w2|‖≤C(Rp−2+1+Rˉp(p−2)/2(p−1))‖w1−w2‖H20(Ω). |
Thus, it follows from (31) that
‖Φ(w1)−Φ(w2)‖H=‖v1−v2‖H≤C(Rp−2+1+Rˉp(p−2)/2(p−1))T‖w1−w2‖H. | (32) |
We choose
In this section, we consider the global existence and energy decay of the solution for problem (1). First, we introduce the following lemmas which play an important role in studying the decay estimate of global solution for the problem (1).
Lemma 4.1. [33] Let
ϕ(t)1+r≤ω0(ϕ(t)−ϕ(t+1))on[0,T], |
where
(ⅰ): if
ϕ(t)≤(ϕ(0)−r+ω−10r[t−1]+)−1ron[0,T]; |
(ⅱ): if
ϕ(t)≤ϕ(0)e−ω1[t−1]+on[0,T], |
where
Now, we establish the global existence and energy decay results.
Theorem 4.2. Let
E(t)≤Ke−κt,ifm=2; |
and
E(t)≤(E(0)−m−22+(m−2)τ2[t−1]+)−2m−2,ifm>2, |
where
Proof. Step 1.. Global existence. It suffices to show that
d>E(0)≥E(t)=12‖ut‖2+J(u)=12‖ut‖2+1pI(u)+(12−1p)‖Δu‖2+kp2‖u‖pp>12‖ut‖2+p−22p‖Δu‖2, | (33) |
which yields that
‖ut‖2+‖Δu‖2≤2pp−2d<+∞. |
The above inequality and the continuation principle imply the global existence, i.e.
Step 2.. We claim that there exists constant
I(u)≥θ‖Δu‖2. | (34) |
In fact, it follows from
\begin{equation*} \begin{split}d\leq J(\lambda_0u(t))& = \frac{1}{p} I(\lambda_0u)+\left(\frac{1}{2}-\frac{1}{p}\right)\|\Delta (\lambda_0u)\|^{2}+\frac{k}{p^{2}} \|\lambda_0u\|_p^{p}\\ & = \frac{p-2}{2p}\lambda_0^2\|\Delta u\|^2+\frac{k}{p^{2}}\lambda_0^p\|u\|^p_p\\ & = \lambda_0^p\left(\frac{p-2}{2p}\lambda_0^{2-p}\|\Delta u\|^2+\frac{k}{p^{2}} \|u\|_p^{p}\right)\\ &\leq \lambda_0^p\left(\frac{p-2}{2p}\|\Delta u\|^2+\frac{k}{p^{2}} \|u\|_p^{p}\right)\\ & < \lambda_0^pE(0), \end{split} \end{equation*} |
which implies that
\begin{equation} \lambda_0 > \left(\frac{d}{E(0)}\right)^{\frac{1}{p}} > 1. \end{equation} | (35) |
It follows from (12) that
\begin{equation*} \begin{split} 0 = I(\lambda_0u)& = \|\Delta (\lambda_0u)\|^{2}-\int_{\Omega} |\lambda_0u|^{p} \log |\lambda_0u|^{k} \mathrm{d} x\\ & = \lambda_0^2\|\Delta u\|^2-\lambda_0^pk\int_{\Omega} |u|^{p} \log |u|\mathrm{d} x -(\lambda_0^pk\log\lambda_0)\|u\|^p_p\\ & = \lambda_0^pI(u)-\lambda_0^p\|\Delta u\|^2+\lambda_0^2\|\Delta u\|^2-\left(\lambda_0^pk\log\lambda_0\right)\|u\|^p_p\\ & = \lambda_0^pI(u)-(\lambda_0^p-\lambda_0^2)\|\Delta u\|^2-\left(\lambda_0^pk\log\lambda_0\right)\|u\|^p_p. \end{split} \end{equation*} |
Combining this equality with (35), we have
\begin{equation*} \begin{split} \lambda_0^pI(u)& = (\lambda_0^p-\lambda_0^2)\|\Delta u\|^2+(\lambda_0^pk\log\lambda_0)\|u\|^p_p\\ &\geq (\lambda_0^p-\lambda_0^2)\|\Delta u\|^2, \end{split} \end{equation*} |
which implies that
\begin{equation*} I(u)\geq (1-\lambda_0^{2-p})\|\Delta u\|^2. \end{equation*} |
Hence, the inequality (34) holds with
Step 3.. Energy decay. By integrating (18) over
\begin{equation} E(t)-E(t+1) \equiv D(t)^{m}, \end{equation} | (36) |
where
\begin{equation} D(t)^{m} = \int_{t}^{t+1}\left\|u_{\tau}\right\|_{m}^{m} \mathrm{d}\tau. \end{equation} | (37) |
In view of (37) and the embedding
\begin{equation} \int_{t}^{t+1} \int_{\Omega}\left|u_{t}\right|^{2} \mathrm{d} x \mathrm{d} t \leq c(\Omega) D(t)^{2}. \end{equation} | (38) |
Thus, from (38), there exist
\begin{equation} \left\|u_{t}\left(t_{i}\right)\right\|_{2}^{2} \leq 4 c(\Omega) D(t)^{2}, \quad i = 1,2. \end{equation} | (39) |
On the other hand, multiplying
\begin{equation} \begin{split}\int_{t_{1}}^{t_{2}} I(u)\mathrm{d}t = &\int_{t_{1}}^{t_{2}}\left\|u_{t}\right\|^{2}\mathrm{d}t+\left(u_{t}\left(t_{1}\right), u\left(t_{1}\right)\right)-\left(u_{t}\left(t_{2}\right), u\left(t_{2}\right)\right)\\ &-\int_{t_{1}}^{t_{2}} \int_{\Omega}\left|u_{t}\right|^{m-2} u_{t} u\mathrm{d}x\mathrm{d}t. \end{split} \end{equation} | (40) |
It follows from (33) that
\begin{equation} \begin{split} \left| \int_{t_{1}}^{t_{2}} \int_{\Omega}|u_{t}^{m-2} u_{t} u \mathrm{d} x \mathrm{d}t \right|&\leq \int_{t_{1}}^{t_{2}}\|u\|_{m}\left\|u_{t}\right\|_{m}^{m-1} \mathrm{d}t\\ &\leq C\int_{t_{1}}^{t_{2}}\|\Delta u\|\left\|u_{t}\right\|_{m}^{m-1} \mathrm{d} t\\ & \leq C\left(\frac{2 p}{p-2}\right)^{\frac{1}{2}} \sup _{t_{1} \leq s \leq t_{2}} E(s)^{\frac{1}{2}} \int_{t_{1}}^{t_{2}}\left\|u_{t}\right\|_{m}^{m-1} \mathrm{d} t\\ &\leq C\left(\frac{2 p}{p-2}\right)^{\frac{1}{2}} \sup _{t_{1} \leq s \leq t_{2}} E(s)^{\frac{1}{2}} D(t)^{m-1}. \end{split} \end{equation} | (41) |
By using (33) and (39), we also have
\begin{equation} \left\|u_{t}\left(t_{i}\right)\right\|_{2}\left\|u\left(t_{i}\right)\right\|_{2} \leq C_{1} D(t) \sup\limits _{t_{1} \leq s \leq t_{2}} E(s)^{\frac{1}{2}}, \quad i = 1,2. \end{equation} | (42) |
Combining (38), (41) with (42), we have from (40) that
\begin{equation} \begin{split}\int_{t_{1}}^{t_{2}} I(u) \mathrm{d}t\leq & c(\Omega) D(t)^{2}+2 C_1 D(t) \sup _{t_{1} \leq s \leq t_{2}} E(s)^{\frac{1}{2}}\\ &+C\left(\frac{2 p}{p-2}\right)^{\frac{1}{2}} D(t)^{m-1} \sup _{t_{1} \leq s \leq t_{2}} E(s)^{\frac{1}{2}}. \end{split} \end{equation} | (43) |
Moreover, using (33) and (34), it is easy to see that
\begin{equation*} \|u\|_p^p\leq C_p^p\|\Delta u\|^p\leq C_p^p\left(\frac{2p}{p-2}E(0)\right)^{\frac{p-2}{2}}\frac{1}{\theta}I(u). \end{equation*} |
Thus, we deduce that
\begin{equation} E(t)\leq \frac{1}{2}\|u_t\|^2+C_2I(u), \end{equation} | (44) |
where
\begin{equation} \int_{t_{1}}^{t_{2}} E(t) \mathrm{d} t \leq \frac{1}{2} \int_{t_{1}}^{t_{2}}\left\|u_{t}\right\|_{2}^{2}\mathrm{d} t+C_{2} \int_{t_{1}}^{t_{2}} I(u) \mathrm{d} t. \end{equation} | (45) |
By integrating (18) over
\begin{equation*} E(t) = E\left(t_{2}\right)+\int_{t}^{t_{2}}\left\|u_{t}\right\|_{m}^{m} \mathrm{d} s \end{equation*} |
Since
\begin{equation*} E\left(t_{2}\right) \leq 2 \int_{t_{1}}^{t_{2}} E(t) \mathrm{d} t. \end{equation*} |
Then, in view of (36), we have
\begin{equation*} E(t) = E(t+1)+D(t)^{m} \leq E\left(t_{2}\right)+D(t)^{m} \leq 2 \int_{t_{1}}^{t_{2}} E(t)\mathrm{d}t+D(t)^{m}. \end{equation*} |
Thus, combining (38) with (45), we get that
\begin{equation*} \begin{split} E(t)\leq& \left(c(\Omega)+2 c(\Omega) C_{2}\right) D(t)^{2}+D(t)^{m}2\\ &+ C_{2}\left[2C_1 D(t)+C\left(\frac{2 p}{(p-2)}\right)^{\frac{1}{2}} D(t)^{m-1}\right]\sup\limits_{t_1\leq s\leq t_2} E(s)^{\frac{1}{2}}\\ \leq &\left(c(\Omega)+2 c(\Omega) C_{2}\right) D(t)^{2}+D(t)^{m}\\ &+2 C_{2}\left[2C_1 D(t)+C\left(\frac{2 p}{(p-2)}\right)^{\frac{1}{2}} D(t)^{m-1}\right] E(t)^{\frac{1}{2}}. \end{split} \end{equation*} |
Hence, it follows from Young's inequality that
\begin{equation} E(t) \leq C_{3}\left[D(t)^{2}+D(t)^{m}+D(t)^{2(m-1)}\right] \end{equation} | (46) |
holds with some positive constant
\begin{equation*} \begin{split}E(t)& \leq C_{3}\left[1+D(t)^{m-2}+D(t)^{2 m-4}\right] D(t)^{2} \\ &\leq C_{3}\left[1+E(0)^{\frac{m-2}{m}}+E(0)^{\frac{2 m-4}{m}}\right] D(t)^{2}, \end{split} \end{equation*} |
which implies that
\begin{equation*} E(t)^{\frac{m}{2}} \leq\left(C_{4}(E(0))\right)^{\frac{m}{2}} D(t)^{m} = \left(C_{4}(E(0))\right)^{\frac{m}{2}}\big(E(t)-E(t+1)\big), \end{equation*} |
where
\lim \limits_{E(0) \rightarrow 0} C_{4}(E(0)) = C_{3} |
Hence, the energy decay estimates hold with
\begin{equation} K = E(0)e^{\kappa},\quad \kappa = \log\frac{3C_3}{3C_3-1} \,\,\,\text{and}\,\,\, \tau = \left(C_{4}(E(0))\right)^{-\frac{m}{2}}. \end{equation} | (47) |
In this section, we will establish that the solution of problem (1) blows up in finite time provided
Lemma 5.1. Assume that (2) holds. Then there exists a positive constant
\begin{equation*} \left(\int_{\Omega}|u|^{p} \log |u|^{k} \mathrm{d} x\right)^{s / p} \leq C\left[\int_{\Omega}|u|^{p} \log |u|^{k} \mathrm{d} x+\| \Delta u\|_{2} ^{2}\right], \end{equation*} |
for any
Lemma 5.2. Assume that (2) holds. Then there exists a positive constant
\begin{equation*} \|u\|_{p}^{p} \leq C\left[\int_{\Omega}|u|^{p} \log |u|^{k} \mathrm{d} x+\|\Delta u\|^{2}\right], \end{equation*} |
for any
Lemma 5.3. Assume that (2) holds. Then there exists a positive constant
\begin{equation*} \|u\|_{p}^{s} \leq C\left[\|u\|_{p}^{p}+\|\nabla u\|_{2}^{2}\right], \end{equation*} |
for any
The proof of lemma 5.1-5.3 is similar to the proof in [22], we omit the details.
Lemma 5.4. Assume that (2) and
\begin{equation*} \|u\|_m^m\leq C\left[\left(\int_{\Omega}|u|^{p} \log |u|^{k} \mathrm{d} x\right)^{\frac{m}{p}}+\|\Delta u\|^{\frac{2m}{p}}\right], \end{equation*} |
for any
Proof. Noting
Now we are in the position to state and prove the blow up result for
Theorem 5.5. Suppose that the conditions in Lemma 5.4 hold. Then the solution to the problem (1) blows up in finite time provided that
Proof. We denote
\begin{equation*} E(t)\leq E(0) < 0,\quad \quad H'(t) = -E'(t) = \|u_t\|_m^m. \end{equation*} |
and
\begin{equation} 0 < H(0)\leq H(t)\leq \frac{1}{p}\int_{\Omega} |u|^{p} \log |u|^{k} \mathrm{d} x. \end{equation} | (48) |
We define
\begin{equation*} L(t) = H^{1-\beta}(t)+\varepsilon \int_{\Omega}uu_t\mathrm{d}x,\quad t\geq0, \end{equation*} |
where
\begin{equation} \frac{2(p-m)}{(m-1)p^2} < \beta < \frac{p-m}{2(m-1)p} < 1. \end{equation} | (49) |
By taking a derivation of
\begin{equation*} \begin{split} L'(t) = &(1-\beta)H^{-\beta}(t)H'(t)+\varepsilon\|u_t\|^2-\varepsilon\|\Delta u\|^2\\ &-\varepsilon\int_{\Omega}|u_t|^{m-2}u_tu\mathrm{d}x+\varepsilon\int_{\Omega}|u|^p\log|u|^k\mathrm{d}x. \end{split} \end{equation*} |
Adding and subtracting
\begin{equation} \begin{split} L'(t) = &(1-\beta)H^{-\beta}(t)H'(t)+\varepsilon\frac{p(1-a)+2}{2}\|u_t\|^2+\varepsilon\frac{p(1-a)-2}{2}\|\Delta u\|^2\\ &+\varepsilon p(1-a)H(t)-\varepsilon\int_{\Omega}|u_t|^{m-2}u_tu\mathrm{d}x+\varepsilon a\int_{\Omega}|u|^p\log|u|^k\mathrm{d}x\\ &+\varepsilon\frac{(1-a)k}{p}\|u\|_p^p. \end{split} \end{equation} | (50) |
In view of Young's inequality, we have
\begin{equation*} \int_{\Omega}|u_t|^{m-2}u_tu\mathrm{d}x \leq \frac{\delta^{m}}{m}\|u\|_{m}^{m}+\frac{m-1}{m} \delta^{-m /(m-1)}\left\|u_{t}\right\|_{m}^{m} \end{equation*} |
for any
\begin{equation} \begin{split} L'(t)\geq&\left[(1-\beta)H^{-\beta}(t)-\frac{m-1}{m}\varepsilon \delta^{-m /(m-1)}\right]\left\|u_{t}\right\|_{m}^{m}-\varepsilon\frac{\delta^{m}}{m}\|u\|_{m}^{m}\\ &+\varepsilon\frac{p(1-a)+2}{2}\|u_t\|^2+\varepsilon\frac{p(1-a)-2}{2}\|\Delta u\|^2+\varepsilon p(1-a)H(t)\\ &+\varepsilon a\int_{\Omega}|u|^p\log|u|^k\mathrm{d}x+\varepsilon\frac{(1-a)k}{p}\|u\|_p^p . \end{split} \end{equation} | (51) |
Since the integral is taken over the
\begin{equation} \begin{split} L'(t)\geq&\left[(1-\beta)-\frac{m-1}{m}\varepsilon M\right]H^{-\beta}(t)\left\|u_{t}\right\|_{m}^{m}-\varepsilon\frac{M^{1-m}}{m}H^{\beta(m-1)}\|u\|_{m}^{m}\\&+\varepsilon\frac{p(1-a)+2}{2}\|u_t\|^2 +\varepsilon\frac{p(1-a)-2}{2}\|\Delta u\|^2+\varepsilon p(1-a)H(t)\\ &+\varepsilon a\int_{\Omega}|u|^p\log|u|^k\mathrm{d}x+\varepsilon\frac{(1-a)k}{p}\|u\|_p^p . \end{split} \end{equation} | (52) |
Making using of (48), Lemma 5.4 and Young's inequality, we find
\begin{equation*} \begin{split} &H^{\beta(m-1)}\|u\|_{m}^{m}\\ \leq&\left(\int_{\Omega} |u|^{p} \log |u|^{k} \mathrm{d} x\right)^{\beta(m-1)}\|u\|_{m}^{m}\\ \leq &C\left[\left(\int_{\Omega} |u|^{p} \log |u|^{k} \mathrm{d} x\right)^{\beta(m-1)+\frac{m}{p}}+\left(\int_{\Omega} |u|^{p} \log |u|^{k} \mathrm{d} x\right)^{\beta(m-1)}\|\Delta u\|^{\frac{2m}{p}}\right]\\ \leq &C\left[\left(\int_{\Omega} |u|^{p} \log |u|^{k} \mathrm{d} x\right)^{\beta(m-1)+\frac{m}{p}}+\left(\int_{\Omega} |u|^{p} \log |u|^{k} \mathrm{d} x\right)^{\beta(m-1)\cdot\frac{p}{p-m}}+\|\Delta u\|^2\right]. \end{split} \end{equation*} |
Hence, it follows from Lemma 5.1 that
\begin{equation*} 2 < \beta(m-1)p+m\leq p \quad \text{and} \quad 2 < \frac{\beta(m-1)p^2}{p-m}\leq p. \end{equation*} |
Thus, Lemma 5.1 implies
\begin{equation} H^{\beta(m-1)}\|u\|_{m}^{m}\leq C\left(\int_{\Omega} |u|^{p} \log |u|^{k} \mathrm{d} x+\|\Delta u\|^2\right). \end{equation} | (53) |
Combining (52) and (53), we have
\begin{equation*} \begin{split} &\,\,L'(t)\\ &\geq\left[(1-\beta)-\frac{m-1}{m}\varepsilon M\right]H^{-\beta}(t)\left\|u_{t}\right\|_{m}^{m}+\varepsilon\frac{p(1-a)+2}{2}\|u_t\|^2 \end{split} \end{equation*} |
\begin{equation} \begin{split}&\,\,\,\,+\varepsilon\left[\frac{p(1-a)-2}{2}-\frac{M^{1-m}}{m}C\right]\|\Delta u\|^2+\varepsilon \left[a-\frac{M^{1-m}}{m}C\right]\int_{\Omega}|u|^p\log|u|^k\mathrm{d}x \\ &\,\,\,\,+\varepsilon p(1-a)H(t)+\varepsilon\frac{(1-a)k}{p}\|u\|_p^p. \end{split} \end{equation} | (54) |
Now, we choose
\begin{equation*} \frac{p(1-a)-2}{2} > 0 \end{equation*} |
and
\begin{equation*} \frac{p(1-a)-2}{2}-\frac{M^{1-m}}{m}C > 0\quad \text{and} \quad a-\frac{M^{1-m}}{m}C > 0. \end{equation*} |
Once
\begin{equation*} (1-\beta)-\frac{m-1}{m}\varepsilon M > 0 \quad \text{and} \quad L(0) = H^{1-\beta}(0)+\varepsilon\int_{\Omega}u_0u_1\mathrm{d}x > 0. \end{equation*} |
Thus, for some constant
\begin{equation} L'(t)\geq \gamma\left[H(t)+\|u_t\|^2+\|\Delta u\|^2+\|u\|_p^p+\int_{\Omega}|u|^p\log|u|^k\mathrm{d}x\right]. \end{equation} | (55) |
Consequently we have
\begin{equation*} L(t)\geq L(0), \,\, \text{for all} \,\,t > 0. \end{equation*} |
On the other hand, using Lemma 5.3, by the same method as in [32], we can deduce
\begin{equation} L^{\frac{1}{1-\beta}}(t)\leq C\left[H(t)+\|u_t\|^2+\|\Delta u\|^2+\|u\|_p^p\right], \quad t\geq0. \end{equation} | (56) |
Combining (55) and (56), we obtain
\begin{equation} L'(t)\geq \lambda L^{\frac{1}{1-\beta}}(t), \quad t\geq0. \end{equation} | (57) |
where
\begin{equation*} L^{\beta /(1-\beta)}(t) \geq \frac{1}{L^{-\beta /(1-\beta)}(0)-\lambda t \beta /(1-\beta)}. \end{equation*} |
which implies that
\begin{equation*} T\leq T^* = \frac{1-\beta}{\lambda \beta L^{\beta/(1-\beta)}(0)}. \end{equation*} |
This completes the proof of Theorem 5.1.
In this section, we consider the problem (1) with the linear damping term, i.e.
Lemma 6.1. [14] Let
Lemma 6.2. Suppose that
\begin{equation*} \int_{\Omega}u_0u_1\mathrm{d}x\geq0. \end{equation*} |
Let
Proof. Let
\begin{equation*} \langle u_{tt}, u\rangle = \frac{d}{dt}(u_t, u)-\|u_t\|^2 \,\,for \, a.e.t\geq0, \end{equation*} |
Moreover, by testing the equation with
\begin{equation*} \langle u_{tt}, u\rangle+\|\Delta u\|^2+(u_t, u) = \int_{\Omega}|u|^{p}\log |u|^k\mathrm{d}x, \end{equation*} |
which implies
\begin{equation*} \frac{ d}{dt}\left((u_t, u)+\frac{1}{2}\|u\|^2\right) = \|u_t\|^2-I(u). \end{equation*} |
Hence, if
\begin{equation*} G'(t)+G(t) = 2\|u_t\|^2-2I(u(t)) > 0 \,\,for\,a.e. t\in [0,T). \end{equation*} |
Therefore, it follows from Lemma 6.1 with
Lemma 6.3. Let
\begin{equation} \|u_1\|^2-2(u_1, u_0)+\Lambda E(0) < 0, \end{equation} | (58) |
where
Proof. If this was not the case, by the continuity of
\begin{equation} (u_1,u_0)\leq \|u_1\|\|u_0\|\leq \frac{1}{2}\big(\|u_1\|^2+\|u_0\|^2\big). \end{equation} | (59) |
By Lemma 6.2, (58) and (59), we deduce that
\begin{equation} F(t) = \|u(t)\|^2 > \|u_0\|^2\geq 2 (u_1,u_0)-\|u_1\|^2 > \Lambda E(0)\,\,\text{for} \,\,t\in(0, t_0), \end{equation} | (60) |
which implies
\begin{equation} F(t_0) = \|u(t_0)\|^2 > \Lambda E(0) \end{equation} | (61) |
by the continuity of
\begin{equation*} \begin{split} E(0)\geq E(t_0)& = \frac{1}{p} I(u(t_0))+\left(\frac{1}{2}-\frac{1}{p}\right)\|\Delta u(t_0)\|^{2}+\frac{k}{p^{2}} \|u(t_0)\|_p^{p}\\ &\geq \frac{p-2}{2p}\|\Delta u(t_0)\|^2 \end{split} \end{equation*} |
that is
\begin{equation*} \|\Delta u(t_0)\|^2\leq \frac{2p}{p-2}E(0). \end{equation*} |
Hence, we have
\begin{equation*} F(t_0) = \|u(t_0)\|^2\leq B_0\|\Delta u(t_0)\|^2\leq\frac{2B_0p}{p-2}E(0) = \Lambda E(0), \end{equation*} |
which is a contradiction with (61). The proof is complete.
We now present the main blow-up result for the weak solution of problem (1) with
Theorem 6.4. Assume the conditions of Lemma 6.3 hold. Then the weak solution
Proof. It follows from Lemma 6.3 that
\begin{equation*} \eta (t) = \|u\|^2+\int_0^t\|u(\tau)\|^2\mathrm{d}\tau+(T_0-t)\|u_0\|^2. \end{equation*} |
Notice
\begin{equation} \eta(t)\geq \varrho\,\,\,\text{for}\,\text{all}\,\, t\in[0,T_0]. \end{equation} | (62) |
Moreover,
\begin{equation} \eta'(t) = 2(u_t, u)+\|u\|^2-\|u_0\|^2 = 2(u_t, u)+2\int_0^t(u_{\tau}, u)\mathrm{d}\tau, \end{equation} | (63) |
hence, we have
\begin{equation} \begin{split} \eta^{\prime \prime}(t)& = 2\|u_t\|^2+2\langle u_{tt}, u\rangle+2(u_t, u)\\ & = 2\left(\|u_t\|^2-\|\Delta u\|^2+\int_{\Omega}|u|^p\log |u|^k\mathrm{d}x\right)\\ & = 2\|u_t\|^2-2I(u(t)). \end{split} \end{equation} | (64) |
It follows from (63) that
\begin{equation*} (\eta'(t))^2 = 4\left((u_t, u)^2+2(u_t, u)\int_0^t(u_{\tau}, u)\mathrm{d}\tau+\big(\int_0^t(u_{\tau}, u)\mathrm{d}\tau\big)^2\right). \end{equation*} |
By the Cauchy-Schwarz inequality, we obtain
\begin{equation*} \|u_t\|^{2}\left\|u\right\|^{2} \geq(u_t, u)^{2} \end{equation*} |
\begin{equation*} \int_{0}^{t}\|u\|^{2} \mathrm{d} \tau \int_{0}^{t}\left\|u_{\tau}\right\|^{2} \mathrm{d}\tau \geq\left(\int_{0}^{t}\left(u_{\tau}, u\right)\mathrm{d} \tau\right)^{2} \end{equation*} |
and
\begin{equation*} \begin{split} &2(u_t, u)\int_0^t(u_{\tau}, u)\mathrm{d}\tau\\ \leq&2\|u_t\|\|u\|\big(\int_0^t\|u_{\tau}\|^2\mathrm{d} \tau\big)^\frac{1}{2}\big(\int_0^t\|u\|^2\mathrm{d} \tau\big)^\frac{1}{2}\\ \leq&\|u_t\|^2\int_0^t\|u\|^2\mathrm{d} \tau+\|u\|^2\int_0^t\|u_{\tau}\|^2\mathrm{d} \tau. \end{split} \end{equation*} |
Combining the above inequalities, we have
\begin{equation} \begin{split} &\eta^{\prime}(t)^2\\ \leq &4\left(\|u_t\|^{2}\|u\|^{2}+\|u_t\|^2\int_0^t\|u\|^2\mathrm{d} \tau+\|u\|^2\int_0^t\|u_{\tau}\|^2\mathrm{d} \tau+\int_0^t\|u_{\tau}\|^2\mathrm{d} \tau\int_0^t\|u\|^2\mathrm{d} \tau\right)\\ = &4\left(\|u\|^2+\int_0^t\|u\|^2\mathrm{d} \tau\right)\left(\|u_t\|^2+\int_0^t\|u_{\tau}\|^2\mathrm{d} \tau\right)\\ \leq&4\eta(t)\left(\|u_t\|^2+\int_0^t\|u_{\tau}\|^2\mathrm{d} \tau\right). \end{split} \end{equation} | (65) |
Hence, it follows from (64) and (65) that
\begin{equation} \begin{split} & \eta^{\prime \prime}(t)\eta(t)-\frac{p+2}{4}\eta^{\prime}(t)^2\\ \geq& \eta(t)\left(\eta^{\prime \prime}(t)-(p+2)\big(\|u_t\|^2+\int_0^t\|u_{\tau}\|^2\mathrm{d} \tau\big)\right)\\ = & \eta(t)\left(-p\|u_t\|^2-2\|\Delta u\|^2+2\int_{\Omega}|u|^p\log |u|^k\mathrm{d}x-(p+2)\int_0^t\|u_{\tau}\|^2\mathrm{d} \tau\right). \end{split} \end{equation} | (66) |
Now, we define
\begin{equation*} \xi(t) = -p\|u_t\|^2-2\|\Delta u\|^2+2\int_{\Omega}|u|^p\log |u|^k\mathrm{d}x-(p+2)\int_0^t\|u_{\tau}\|^2\mathrm{d} \tau. \end{equation*} |
Noticing
\begin{equation*} \begin{split} \xi(t)& = (p-2)\|\Delta u\|^2-2pE(t)-(p+2)\int_0^t\|u_{\tau}\|^2\mathrm{d} \tau+\frac{2k}{p}\|u\|^p_p\\ & = (p-2)\|\Delta u\|^2-2pE(0)+(p-2)\int_0^t\|u_{\tau}\|^2\mathrm{d} \tau+\frac{2k}{p}\|u\|^p_p\\ &\geq (p-2)\|\Delta u\|^2-2pE(0). \end{split} \end{equation*} |
From (60) and Lemma 6.2, we deduce that
\begin{equation*} 2pE(0) < \frac{p-2}{B_0}\|u_0\|^2 < \frac{p-2}{B_0}\|u\|^2 < (p-2)\|\Delta u\|^2. \end{equation*} |
which yields that
\begin{equation*} \eta^{\prime \prime}(t)-\frac{p+2}{4}\eta^{\prime}(t)^2\geq \varrho \varsigma,\,\,\,t\in[0, T_0], \end{equation*} |
which implies that
\begin{equation*} \big(\eta^{-\frac{p-2}{4}}(t)\big)^{\prime\prime}\leq -\frac{p-2}{4}\varrho \varsigma (\eta(t))^{-\frac{p+6}{4}} < 0. \end{equation*} |
Hence, it follows that there exists a
\begin{equation*} \lim\limits_{t\rightarrow T^{*}}\eta^{-\frac{p-2}{4}}(t) = 0, \end{equation*} |
that is
\begin{equation*} \lim\limits_{t\rightarrow T^*}\eta(t) = +\infty. \end{equation*} |
In turn, this implies that
\begin{equation} \lim\limits_{t\rightarrow T^*}\|\Delta u(t)\|^2 = +\infty. \end{equation} | (67) |
In fact, if
\begin{equation*} \lim\limits_{t\rightarrow T^*} \int_0^t\|u(\tau)\|^2\mathrm{d}\tau = +\infty \end{equation*} |
so that (67) is also satisfied. Hence (67) is a contraction with
The author would like to thank Professor Hua Chen for the careful reading of this paper and for the valuable suggestions to improve the presentation of the paper. This project is supported by NSFC (No. 11801145), Key Scientific Research Foundation of the Higher Education Institutions of Henan Province, China (Grant No.19A110004 and the Fund of Young Backbone Teacher in Henan Province (NO. 2018GGJS068, 21420048).
[1] | L. K. Johnson, Smart intelligence, Foreign Policy, (1992), 53–69. |
[2] | J. Wang, C. Xu, J. Zhang, R. Zhong, Big data analytics for intelligent manufacturing systems: A review, J. Manuf Syst., (2021). https://doi.org/10.1016/j.jmsy.2021.03.005 |
[3] |
W. H. Zijm, Towards intelligent manufacturing planning and control systems, OR-Spektrum, 22 (2000), 313–345. https://doi.org/10.1007/s002919900032 doi: 10.1007/s002919900032
![]() |
[4] | W. Qi H. Su, A cybertwin based multimodal network for ecg patterns monitoring using deep learning, IEEE Trans. Industr. Inform., (2022). https://doi.org/10.1109/TII.2022.3159583 |
[5] |
L. Monostori, J. Prohaszka, A step towards intelligent manufacturing: Modelling and monitoring of manufacturing processes through artificial neural networks, CIRP Ann., 42 (1993), 485–488. https://doi.org/10.1016/S0007-8506(07)62491-3 doi: 10.1016/S0007-8506(07)62491-3
![]() |
[6] | X. Yao, J. Zhou, J. Zhang, C. R. Boër, From intelligent manufacturing to smart manufacturing for industry 4.0 driven by next generation artificial intelligence and further on, in 2017 5th international conference on enterprise systems (ES). IEEE, (2017), 311–318. https://doi.org/10.1109/ES.2017.58 |
[7] |
J. Yi, C. Lu, G. Li, A literature review on latest developments of harmony search and its applications to intelligent manufacturing, Math. Biosci. Eng., 16 (2019), 2086–2117. https://doi.org/10.3934/mbe.2019102 doi: 10.3934/mbe.2019102
![]() |
[8] |
S. Shan, X. Wen, Y. Wei, Z. Wang, Y. Chen, Intelligent manufacturing in industry 4.0: A case study of sany heavy industry, Syst. Res. Behav. Sci., 37 (2020), 679–690. https://doi.org/10.1002/sres.2709 doi: 10.1002/sres.2709
![]() |
[9] |
H. Yoshikawa, Manufacturing and the 21st century intelligent manufacturing systems and the renaissance of the manufacturing industry, Technol. Forecast Soc. Change, 49 (1995), 195–213. https://doi.org/10.1016/0040-1625(95)00008-X doi: 10.1016/0040-1625(95)00008-X
![]() |
[10] | J. Zheng, K. Chan, I. Gibson, Virtual reality, IEEE Potent., 17 (1998), 20–23. |
[11] |
M. J. Schuemie, P. Van Der Straaten, M. Krijn, C. A. Van Der Mast, Research on presence in virtual reality: A survey, Cyberpsychol. & Behav., 4 (2001), 183–201. https://doi.org/10.1089/109493101300117884 doi: 10.1089/109493101300117884
![]() |
[12] | C. Anthes, R. J. García-Hernández, M. Wiedemann, D. Kranzlmüller, State of the art of virtual reality technology, in IEEE Aerosp. Conf.. (2016), 1–19. 10.1109/AERO.2016.7500674 |
[13] | F. Biocca, B. Delaney, Immersive virtual reality technology, Communication in the age of virtual reality, 15 (1995). https://doi.org/10.4324/9781410603128 |
[14] | T. Mazuryk, M. Gervautz, Virtual reality-history, applications, technology and future, 1996. |
[15] |
N.-N. Zhou, Y.-L. Deng, Virtual reality: A state-of-the-art survey, Int. J. Autom. Comput., 6 (2009), 319–325. https://doi.org/10.1007/s11633-009-0319-9 doi: 10.1007/s11633-009-0319-9
![]() |
[16] |
J. Egger, T. Masood, Augmented reality in support of intelligent manufacturing–a systematic literature review, Comput. Ind. Eng., 140 (2020), 106195. https://doi.org/10.1016/j.cie.2019.106195 doi: 10.1016/j.cie.2019.106195
![]() |
[17] |
B.-H. Li, B.-C. Hou, W.-T. Yu, X.-B. Lu, C.-W. Yang, Applications of artificial intelligence in intelligent manufacturing: a review, Front. Inform. Tech. El., 18 (2017), 86–96. https://doi.org/10.1631/FITEE.1601885 doi: 10.1631/FITEE.1601885
![]() |
[18] |
B. He, K.-J. Bai, Digital twin-based sustainable intelligent manufacturing: A review, Adv. Manuf., 9 (2021), 1–21. https://doi.org/10.1007/s40436-020-00302-5 doi: 10.1007/s40436-020-00302-5
![]() |
[19] | G.-J. Cheng, L.-T. Liu, X.-J. Qiang, Y. Liu, Industry 4.0 development and application of intelligent manufacturing, in 2016 international conference on information system and artificial intelligence (ISAI). IEEE, (2016), 407–410. https://doi.org/10.1109/ISAI.2016.0092 |
[20] |
G. Y. Tian, G. Yin, D. Taylor, Internet-based manufacturing: A review and a new infrastructure for distributed intelligent manufacturing, J. Intell. Manuf., 13 (2002), 323–338. https://doi.org/10.1023/A:1019907906158 doi: 10.1023/A:1019907906158
![]() |
[21] |
H. Su, W. Qi, J. Chen, D. Zhang, Fuzzy approximation-based task-space control of robot manipulators with remote center of motion constraint, IEEE Trans. Fuzzy Syst., 30 (2022), 1564–1573. https://doi.org/10.1109/TFUZZ.2022.3157075 doi: 10.1109/TFUZZ.2022.3157075
![]() |
[22] | M.-S. Yoh, The reality of virtual reality, in Proceedings seventh international conference on virtual systems and multimedia. IEEE, (2001), 666–674. https://doi.org/10.1109/VSMM.2001.969726 |
[23] |
V. Antoniou, F. L. Bonali, P. Nomikou, A. Tibaldi, P. Melissinos, F. P. Mariotto, et al., Integrating virtual reality and gis tools for geological mapping, data collection and analysis: An example from the metaxa mine, santorini (greece), Appl. Sci., 10 (2020), 8317. https://doi.org/10.3390/app10238317 doi: 10.3390/app10238317
![]() |
[24] |
A. Kunz, M. Zank, T. Nescher, K. Wegener, Virtual reality based time and motion study with support for real walking, Proced. CIRP, 57 (2016), 303–308. https://doi.org/10.1016/j.procir.2016.11.053 doi: 10.1016/j.procir.2016.11.053
![]() |
[25] | M. Serras, L. G.-Sardia, B. Simes, H. lvarez, J. Arambarri, Dialogue enhanced extended reality: Interactive system for the operator 4.0, Appl. Sci., 10 (2020). https://doi.org/10.3390/app10113960 |
[26] |
A. G. da Silva, M. V. M. Gomes, I. Winkler, Virtual reality and digital human modeling for ergonomic assessment in industrial product development: A patent and literature review, Appl. Sci., 12 (2022), 1084. https://doi.org/10.3390/app12031084 doi: 10.3390/app12031084
![]() |
[27] |
J. Kim, J. Jeong, Design and implementation of opc ua-based vr/ar collaboration model using cps server for vr engineering process, Appl. Sci., 12 (2022), 7534. https://doi.org/10.3390/app12157534 doi: 10.3390/app12157534
![]() |
[28] |
J.-d.-J. Cordero-Guridi, L. Cuautle-Gutiérrez, R.-I. Alvarez-Tamayo, S.-O. Caballero-Morales, Design and development of a i4. 0 engineering education laboratory with virtual and digital technologies based on iso/iec tr 23842-1 standard guidelines, Appl. Sci., 12 (2022), 5993. https://doi.org/10.3390/app12125993 doi: 10.3390/app12125993
![]() |
[29] |
H. Heinonen, A. Burova, S. Siltanen, J. Lähteenmäki, J. Hakulinen, M. Turunen, Evaluating the benefits of collaborative vr review for maintenance documentation and risk assessment, Appl. Sci., 12 (2022), 7155. https://doi.org/10.3390/app12147155 doi: 10.3390/app12147155
![]() |
[30] |
V. Settgast, K. Kostarakos, E. Eggeling, M. Hartbauer, T. Ullrich, Product tests in virtual reality: Lessons learned during collision avoidance development for drones, Designs, 6 (2022), 33. https://doi.org/10.3390/designs6020033 doi: 10.3390/designs6020033
![]() |
[31] |
D. Mourtzis, J. Angelopoulos, N. Panopoulos, Smart manufacturing and tactile internet based on 5g in industry 4.0: Challenges, applications and new trends, Electronics-Switz, 10 (2021), 3175. https://doi.org/10.3390/electronics10243175 doi: 10.3390/electronics10243175
![]() |
[32] |
Y. Saito, K. Kawashima, M. Hirakawa, Effectiveness of a head movement interface for steering a vehicle in a virtual reality driving simulation, Symmetry, 12 (2020), 1645. https://doi.org/10.3390/sym12101645 doi: 10.3390/sym12101645
![]() |
[33] |
Y.-P. Su, X.-Q. Chen, T. Zhou, C. Pretty, G. Chase, Mixed-reality-enhanced human–robot interaction with an imitation-based mapping approach for intuitive teleoperation of a robotic arm-hand system, Appl. Sci., 12 (2022), 4740. https://doi.org/10.3390/app12094740 doi: 10.3390/app12094740
![]() |
[34] |
F. Arena, M. Collotta, G. Pau, F. Termine, An overview of augmented reality, Computers, 11 (2022), 28. https://doi.org/10.3390/computers11020028 doi: 10.3390/computers11020028
![]() |
[35] | P. C. Thomas, W. David, Augmented reality: An application of heads-up display technology to manual manufacturing processes, in Hawaii international conference on system sciences, 2. ACM SIGCHI Bulletin New York, NY, USA, 1992. |
[36] |
J. Safari Bazargani, A. Sadeghi-Niaraki, S.-M. Choi, Design, implementation, and evaluation of an immersive virtual reality-based educational game for learning topology relations at schools: A case study, Sustainability-Basel, 13 (2021), 13066. https://doi.org/10.3390/su132313066 doi: 10.3390/su132313066
![]() |
[37] |
K. Židek, J. Pitel', M. Balog, A. Hošovskỳ, V. Hladkỳ, P. Lazorík, et al., CNN training using 3d virtual models for assisted assembly with mixed reality and collaborative robots, Appl. Sci., 11 (2021), 4269. https://doi.org/10.3390/app11094269 doi: 10.3390/app11094269
![]() |
[38] | S. Mandal, Brief introduction of virtual reality & its challenges, Int. J. Sci. Eng. Res., 4 (2013), 304–309. |
[39] | D. Rose, N. Foreman, Virtual reality. The Psycho., (1999). |
[40] | G. Riva, C. Malighetti, A. Chirico, D. Di Lernia, F. Mantovani, A. Dakanalis, Virtual reality, in Rehabilitation interventions in the patient with obesity. Springer, (2020), 189–204. |
[41] |
J. N. Latta, D. J. Oberg, A conceptual virtual reality model, IEEE Comput. Graph. Appl., 14 (1994), 23–29. https://doi.org/10.1109/38.250915 doi: 10.1109/38.250915
![]() |
[42] | J. Lanier, Virtual reality: The promise of the future. Interactive Learning International, 8 (1992), 275–79. |
[43] | S. Serafin, C. Erkut, J. Kojs, N. C. Nilsson, R. Nordahl, Virtual reality musical instruments: State of the art, design principles, and future directions, Comput. Music. J., 40 (2016). https://doi.org/10.1162/COMJ_a_00372 |
[44] |
W. Qi, H. Su, A. Aliverti, A smartphone-based adaptive recognition and real-time monitoring system for human activities, IEEE Trans. Hum. Mach. Syst., 50 (2020), 414 - 423. https://doi.org/10.1109/THMS.2020.2984181 doi: 10.1109/THMS.2020.2984181
![]() |
[45] |
P. Kopacek, Intelligent manufacturing: present state and future trends, J. Intell. Robot. Syst., 26 (1999), 217–229. https://doi.org/10.1023/A:1008168605803 doi: 10.1023/A:1008168605803
![]() |
[46] |
Y. Feng, Y. Zhao, H. Zheng, Z. Li, J. Tan, Data-driven product design toward intelligent manufacturing: A review, Int. J. Adv. Robot. Syst., 17 (2020), 1729881420911257. https://doi.org/10.1177/1729881420911257 doi: 10.1177/1729881420911257
![]() |
[47] |
H. Su, W. Qi, Y. Hu, H. R. Karimi, G. Ferrigno, E. De Momi, An incremental learning framework for human-like redundancy optimization of anthropomorphic manipulators, IEEE Trans. Industr. Inform., 18 (2020), 1864–1872. https://doi.org/10.1109/TII.2020.3036693 doi: 10.1109/TII.2020.3036693
![]() |
[48] | E. Hozdić, Smart factory for industry 4.0: A review, Int. J. Adv. Manuf. Technol., 7 (2015), 28–35. |
[49] | R. Burke, A. Mussomeli, S. Laaper, M. Hartigan, B. Sniderman, The smart factory: Responsive, adaptive, connected manufacturing, Deloitte Insights, 31 (2017), 1–10. |
[50] |
R. Y. Zhong, X. Xu, E. Klotz, S. T. Newman, Intelligent manufacturing in the context of industry 4.0: a review, Engineering-Prc, 3 (2017), 616–630. https://doi.org/10.1016/J.ENG.2017.05.015 doi: 10.1016/J.ENG.2017.05.015
![]() |
[51] | A. Kusiak, Intelligent manufacturing, System, Prentice-Hall, Englewood Cliffs, NJ, (1990). |
[52] |
G. Rzevski, A framework for designing intelligent manufacturing systems, Comput. Ind., 34 (1997), 211–219. https://doi.org/10.1016/S0166-3615(97)00056-0 doi: 10.1016/S0166-3615(97)00056-0
![]() |
[53] | E. Oztemel, Intelligent manufacturing systems, in Artificial intelligence techniques for networked manufacturing enterprises management. Springer, (2010), pp. 1–41. https://doi.org/10.1007/978-1-84996-119-6_1 |
[54] |
J. Zhou, P. Li, Y. Zhou, B. Wang, J. Zang, L. Meng, Toward new-generation intelligent manufacturing, Engineering-Prc, 4 (2018), 11–20. https://doi.org/10.1016/j.eng.2018.01.002 doi: 10.1016/j.eng.2018.01.002
![]() |
[55] |
R. Y. Zhong, X. Xu, E. Klotz, S. T. Newman, Intelligent manufacturing in the context of industry 4.0: a review, Engineering-Prc, 3 (2017), 616–630. https://doi.org/10.1016/J.ENG.2017.05.015 doi: 10.1016/J.ENG.2017.05.015
![]() |
[56] |
H. S. Kang, J. Y. Lee, S. Choi, H. Kim, J. H. Park, J. Y. Son, B. H. Kim, S. D. Noh, Smart manufacturing: Past research, present findings, and future directions, Int. J. Pr. Eng. Man-Gt., 3 (2016), 111–128. https://doi.org/10.1007/s40684-016-0015-5 doi: 10.1007/s40684-016-0015-5
![]() |
[57] | R. Jardim-Goncalves, D. Romero, A. Grilo, Factories of the future: challenges and leading innovations in intelligent manufacturing, Int. J. Comput. Integr. Manuf., 30 (2017), 4–14. |
[58] | A. Kusiak, Smart manufacturing, Int. J. Prod. Res., 56 (2018), 508–517. https://doi.org/10.1080/00207543.2017.1351644 |
[59] |
B. Wang, F. Tao, X. Fang, C. Liu, Y. Liu, T. Freiheit, Smart manufacturing and intelligent manufacturing: A comparative review, Engineering-Prc, 7 (2021), 738–757. https://doi.org/10.1016/j.eng.2020.07.017 doi: 10.1016/j.eng.2020.07.017
![]() |
[60] |
P. Zheng, Z. Sang, R. Y. Zhong, Y. Liu, C. Liu, K. Mubarok, et al., Smart manufacturing systems for industry 4.0: Conceptual framework, scenarios, and future perspectives, Front. Mech. Eng., 13 (2018), 137–150. https://doi.org/10.1007/s11465-018-0499-5 doi: 10.1007/s11465-018-0499-5
![]() |
[61] | P. Osterrieder, L. Budde, T. Friedli, The smart factory as a key construct of industry 4.0: A systematic literature review, Int. J. Prod. Econ., 221 107476. https://doi.org/10.1016/j.ijpe.2019.08.011 |
[62] | D. Guo, M. Li, R. Zhong, G. Q. Huang, Graduation intelligent manufacturing system (gims): an industry 4.0 paradigm for production and operations management, Ind. Manage. Data Syst., (2020). https://doi.org/10.1108/IMDS-08-2020-0489 |
[63] |
A. Barari, M. de Sales Guerra Tsuzuki, Y. Cohen, M. Macchi, Intelligent manufacturing systems towards industry 4.0 era, J. Intell. Manuf., 32 (2021), 1793–1796. https://doi.org/10.1007/s10845-021-01769-0 doi: 10.1007/s10845-021-01769-0
![]() |
[64] | C. Christo, C. Cardeira, Trends in intelligent manufacturing systems, in 2007 IEEE International Symposium on Industrial Electronics-Switz.. IEEE, (2007), 3209–3214. https://doi.org/10.1109/ISIE.2007.4375129 |
[65] |
M.-P. Pacaux-Lemoine, D. Trentesaux, G. Z. Rey, P. Millot, Designing intelligent manufacturing systems through human-machine cooperation principles: A human-centered approach, Comput. Ind. Eng., 111 (2017), 581–595. https://doi.org/10.1016/j.cie.2017.05.014 doi: 10.1016/j.cie.2017.05.014
![]() |
[66] |
W. F. Gaughran, S. Burke, P. Phelan, Intelligent manufacturing and environmental sustainability, Robot. Comput. Integr. Manuf., 23 (2007), 704–711. https://doi.org/10.1016/j.rcim.2007.02.016 doi: 10.1016/j.rcim.2007.02.016
![]() |
[67] | Y. Boas, Overview of virtual reality technologies, in Inter. Mult. Confer., 2013 (2013). |
[68] |
lvaro Segura, H. V. Diez, I. Barandiaran, A. Arbelaiz, H. lvarez, B. Simes, J. Posada, A. Garca-Alonso, R. Ugarte, Visual computing technologies to support the operator 4.0, Comput. Ind. Eng., 139 (2020), 105550. https://doi.org/10.1016/j.cie.2018.11.060 doi: 10.1016/j.cie.2018.11.060
![]() |
[69] | D. Romero, J. Stahre, T. Wuest, O. Noran, P. Bernus, Fasth, Fast-Berglund, D. Gorecky, Towards an operator 4.0 typology: A human-centric perspective on the fourth industrial revolution technologies, 10 (2016). |
[70] | H. Qiao, J. Chen, X. Huang, A survey of brain-inspired intelligent robots: Integration of vision, decision, motion control, and musculoskeletal systems, " IEEE T. Cybernetics, 52 (2022), 11267 - 11280. https://doi.org/10.1109/TCYB.2021.3071312 |
[71] | F. Firyaguna, J. John, M. O. Khyam, D. Pesch, E. Armstrong, H. Claussen, H. V. Poor et al., Towards industry 5.0: Intelligent reflecting surface (irs) in smart manufacturing, arXiv preprint arXiv: 2201.02214, (2022). https://doi.org/10.1109/MCOM.001.2200016 |
[72] | A. M. Almassri, W. Wan Hasan, S. A. Ahmad, A. J. Ishak, A. Ghazali, D. Talib, C. Wada, Pressure sensor: state of the art, design, and application for robotic hand, J. Sensors, 2015 (2015). https://doi.org/10.1155/2015/846487 |
[73] | B. Munari, Design as art. Penguin UK, (2008). |
[74] | B. De La Harpe, J. F. Peterson, N. Frankham, R. Zehner, D. Neale, E. Musgrave, R. McDermott, Assessment focus in studio: What is most prominent in architecture, art and design? IJADE., 28 (2009), 37–51. https://doi.org/10.1111/j.1476-8070.2009.01591.x |
[75] | C. Gray, J. Malins, Visualizing research: A guide to the research process in art and design. Routledge, (2016). |
[76] | M. Barnard, Art, design and visual culture: An introduction. Bloomsbury Publishing, (1998). |
[77] | C. Crouch, Modernism in art, design and architecture. Bloomsbury Publishing, (1998). |
[78] | M. Biggs, The role of the artefact in art and design research, Int. J. Des. Sci. Technol., 2002. |
[79] | H. Su, W. Qi, Y. Schmirander, S. E. Ovur, S. Cai, X. Xiong, A human activity-aware shared control solution for medical human–robot interaction, Assembly Autom., (2022) ahead-of-print. https://doi.org/10.1108/AA-12-2021-0174 |
[80] | R. D. Gandhi, D. S. Patel, Virtual reality–opportunities and challenges, Virtual Real., 5 (2018). |
[81] |
A. J. Trappey, C. V. Trappey, M.-H. Chao, C.-T. Wu, Vr-enabled engineering consultation chatbot for integrated and intelligent manufacturing services, J. Ind. Inf. Integrat., 26 (2022), 100331. https://doi.org/10.1016/j.jii.2022.100331 doi: 10.1016/j.jii.2022.100331
![]() |
[82] |
K. Valaskova, M. Nagy, S. Zabojnik, G. Lăzăroiu, Industry 4.0 wireless networks and cyber-physical smart manufacturing systems as accelerators of value-added growth in slovak exports, Mathematics-Basel, 10 (2022), 2452. https://doi.org/10.3390/math10142452 doi: 10.3390/math10142452
![]() |
[83] |
J. de Assis Dornelles, N. F. Ayala, A. G. Frank, Smart working in industry 4.0: How digital technologies enhance manufacturing workers' activities, Comput. Ind. Eng., 163 (2022), 107804. https://doi.org/10.1016/j.cie.2021.107804 doi: 10.1016/j.cie.2021.107804
![]() |
[84] |
V. Tripathi, S. Chattopadhyaya, A. K. Mukhopadhyay, S. Sharma, C. Li, S. Singh, W. U. Hussan, B. Salah, W. Saleem, A. Mohamed, A sustainable productive method for enhancing operational excellence in shop floor management for industry 4.0 using hybrid integration of lean and smart manufacturing: An ingenious case study, Sustainability-Basel, 14 (2022), 7452. https://doi.org/10.3390/su14127452 doi: 10.3390/su14127452
![]() |
[85] |
S. M. M. Sajadieh, Y. H. Son, S. D. Noh, A conceptual definition and future directions of urban smart factory for sustainable manufacturing, Sustainability-Basel, 14 (2022), 1221. https://doi.org/10.3390/su14031221 doi: 10.3390/su14031221
![]() |
[86] |
Y. H. Son, G.-Y. Kim, H. C. Kim, C. Jun, S. D. Noh, Past, present, and future research of digital twin for smart manufacturing, J. Comput. Des. Eng., 9 (2022), 1–23. https://doi.org/10.1093/jcde/qwab067 doi: 10.1093/jcde/qwab067
![]() |
[87] |
G. Moiceanu, G. Paraschiv, Digital twin and smart manufacturing in industries: A bibliometric analysis with a focus on industry 4.0, Sensors-Basel, 22 (2022), 1388. https://doi.org/10.3390/s22041388 doi: 10.3390/s22041388
![]() |
[88] |
K. Cheng, Q. Wang, D. Yang, Q. Dai, M. Wang, Digital-twins-driven semi-physical simulation for testing and evaluation of industrial software in a smart manufacturing system, Machines, 10 (2022), 388. https://doi.org/10.3390/machines10050388 doi: 10.3390/machines10050388
![]() |
[89] |
S. Arjun, L. Murthy, P. Biswas, Interactive sensor dashboard for smart manufacturing, Procedia Comput. Sci., 200 (2022), 49–61. https://doi.org/10.1016/j.procs.2022.01.204 doi: 10.1016/j.procs.2022.01.204
![]() |
[90] |
J. Yang, Y. H. Son, D. Lee, S. D. Noh, Digital twin-based integrated assessment of flexible and reconfigurable automotive part production lines, Machines, 10 (2022), 75. https://doi.org/10.3390/machines10020075 doi: 10.3390/machines10020075
![]() |
[91] |
J. Friederich, D. P. Francis, S. Lazarova-Molnar, N. Mohamed, A framework for data-driven digital twins for smart manufacturing, Comput. Ind., 136 (2022), 103586. https://doi.org/10.1016/j.compind.2021.103586 doi: 10.1016/j.compind.2021.103586
![]() |
[92] |
L. Li, B. Lei, C. Mao, Digital twin in smart manufacturing, J. Ind. Inf. Integr., 26 (2022), 100289. https://doi.org/10.1016/j.jii.2021.100289 doi: 10.1016/j.jii.2021.100289
![]() |
[93] |
D. Nåfors, B. Johansson, Virtual engineering using realistic virtual models in brownfield factory layout planning, Sustainability-Basel, 13 (2021), 11102. https://doi.org/10.3390/su131911102 doi: 10.3390/su131911102
![]() |
[94] |
A. Geiger, E. Brandenburg, R. Stark, Natural virtual reality user interface to define assembly sequences for digital human models, Appl. System Innov., 3 (2020), 15. https://doi.org/10.3390/asi3010015 doi: 10.3390/asi3010015
![]() |
[95] |
G. Gabajova, B. Furmannova, I. Medvecka, P. Grznar, M. Krajčovič, R. Furmann, Virtual training application by use of augmented and virtual reality under university technology enhanced learning in slovakia, Sustainability-Basel, 11 (2019), 6677. https://doi.org/10.3390/su11236677 doi: 10.3390/su11236677
![]() |
[96] |
W. Qi, S. E. Ovur, Z. Li, A. Marzullo, R. Song, Multi-sensor guided hand gesture recognition for a teleoperated robot using a recurrent neural network, IEEE Robot Autom Lett., 6 (2021), 6039–6045. https://doi.org/10.1109/LRA.2021.3089999 doi: 10.1109/LRA.2021.3089999
![]() |
[97] |
L. Pérez, S. Rodríguez-Jiménez, N. Rodríguez, R. Usamentiaga, D. F. García, Digital twin and virtual reality based methodology for multi-robot manufacturing cell commissioning, Appl. Sci., 10 (2020), 3633. https://doi.org/10.3390/app10103633 doi: 10.3390/app10103633
![]() |
[98] |
J. Mora-Serrano, F. Muñoz-La Rivera, I. Valero, Factors for the automation of the creation of virtual reality experiences to raise awareness of occupational hazards on construction sites, Electronics-Switz., 10 (2021), 1355. https://doi.org/10.3390/electronics10111355 doi: 10.3390/electronics10111355
![]() |
[99] |
C. McDonald, K. A. Campbell, C. Benson, M. J. Davis, C. J. Frost, Workforce development and multiagency collaborations: A presentation of two case studies in child welfare, Sustainability-Basel, 13 (2021), 10190. https://doi.org/10.3390/su131810190 doi: 10.3390/su131810190
![]() |
[100] |
Z. Xu, N. Zheng, Incorporating virtual reality technology in safety training solution for construction site of urban cities, Sustainability-Basel, 13 (2020), 243. https://doi.org/10.3390/su13010243 doi: 10.3390/su13010243
![]() |
[101] |
L. Frizziero, L. Galletti, L. Magnani, E. G. Meazza, M. Freddi, Blitz vision: Development of a new full-electric sports sedan using qfd, sde and virtual prototyping, Inventions, 7 (2022), 41. https://doi.org/10.3390/inventions7020041 doi: 10.3390/inventions7020041
![]() |
[102] | N. Lyons, Deep learning-based computer vision algorithms, immersive analytics and simulation software, and virtual reality modeling tools in digital twin-driven smart manufacturing, Econom. Manag. Financ. Markets, 17 (2022). |
[103] |
H. Qiao, S. Zhong, Z. Chen, H. Wang, Improving performance of robots using human-inspired approaches: A survey, Sci. China Inf. Sci., 65 (2022), 221201. https://doi.org/10.1007/s11432-022-3606-1 doi: 10.1007/s11432-022-3606-1
![]() |
[104] |
H. Su, A. Mariani, S. E. Ovur, A. Menciassi, G. Ferrigno, E. De Momi, Toward teaching by demonstration for robot-assisted minimally invasive surgery, IEEE Trans. Autom, 18 (2021), 484 - 494. https://doi.org/10.1109/TASE.2020.3045655 doi: 10.1109/TASE.2020.3045655
![]() |
[105] |
H. Su, W. Qi, Z. Li, Z. Chen, G. Ferrigno, E. De Momi, Deep neural network approach in EMG-based force estimation for human–robot interaction, IEEE Trans. Artif. Intell., 2 (2021), 404 - 412. https://doi.org/10.1109/TAI.2021.3066565 doi: 10.1109/TAI.2021.3066565
![]() |
[106] |
A. A. Malik, T. Masood, A. Bilberg, Virtual reality in manufacturing: immersive and collaborative artificial-reality in design of human-robot workspace, Int. J. Comput. Integr. Manuf., 33 (2020), 22–37. https://doi.org/10.1080/0951192X.2019.1690685 doi: 10.1080/0951192X.2019.1690685
![]() |
[107] |
A. Corallo, A. M. Crespino, M. Lazoi, M. Lezzi, Model-based big data analytics-as-a-service framework in smart manufacturing: A case study, Robot. Comput. Integr. Manuf., 76 (2022), 102331. https://doi.org/10.1016/j.rcim.2022.102331 doi: 10.1016/j.rcim.2022.102331
![]() |
[108] |
Y.-M. Tang, G. T. S. Ho, Y.-Y. Lau, S.-Y. Tsui, Integrated smart warehouse and manufacturing management with demand forecasting in small-scale cyclical industries, Machines, 10 (2022), 472. https://doi.org/10.3390/machines10060472 doi: 10.3390/machines10060472
![]() |
[109] | M. Samardžić, D. Stefanović, U. Marjanović, Transformation towards smart working: Research proposal, in 2022 21st International Symposium INFOTEH-JAHORINA (INFOTEH). IEEE, (2022), 1–6. https://doi.org/10.1109/INFOTEH53737.2022.9751256 |
[110] |
T. Caporaso, S. Grazioso, G. Di Gironimo, Development of an integrated virtual reality system with wearable sensors for ergonomic evaluation of human–robot cooperative workplaces, Sensors-Basel, 22 (2022), 2413. https://doi.org/10.3390/s22062413 doi: 10.3390/s22062413
![]() |
[111] |
W. Qi, N. Wang, H. Su, A. Aliverti DCNN based human activity recognition framework with depth vision guiding, Neurocomputing, 486 (2022), 261–271. https://doi.org/10.1016/j.neucom.2021.11.044 doi: 10.1016/j.neucom.2021.11.044
![]() |
[112] |
W. Zhu, X. Fan, Y. Zhang, Applications and research trends of digital human models in the manufacturing industry, VRIH, 1 (2019), 558–579. https://doi.org/10.1016/j.vrih.2019.09.005 doi: 10.1016/j.vrih.2019.09.005
![]() |
[113] |
O. Robert, P. Iztok, B. Borut, Real-time manufacturing optimization with a simulation model and virtual reality, Procedia Manuf., 38 (2019), 1103–1110. https://doi.org/10.1016/j.promfg.2020.01.198 doi: 10.1016/j.promfg.2020.01.198
![]() |
[114] |
I. Kačerová, J. Kubr, P. Hořejší, J. Kleinová, Ergonomic design of a workplace using virtual reality and a motion capture suit, Appl. Sci., 12 (2022), 2150. https://doi.org/10.3390/app12042150 doi: 10.3390/app12042150
![]() |
[115] |
M. Woschank, D. Steinwiedder, A. Kaiblinger, P. Miklautsch, C. Pacher, H. Zsifkovits, The integration of smart systems in the context of industrial logistics in manufacturing enterprises, Procedia Comput. Sci., 200 (2022), 727–737. https://doi.org/10.1016/j.procs.2022.01.271 doi: 10.1016/j.procs.2022.01.271
![]() |
[116] |
A. Umbrico, A. Orlandini, A. Cesta, M. Faroni, M. Beschi, N. Pedrocchi, A. Scala, P. Tavormina, S. Koukas, A. Zalonis et al., Design of advanced human–robot collaborative cells for personalized human–robot collaborations, Appl. Sci., 12 (2022), 6839. https://doi.org/10.3390/app12146839 doi: 10.3390/app12146839
![]() |
[117] |
W. Qi, A. Aliverti, A multimodal wearable system for continuous and real-time breathing pattern monitoring during daily activity, IEEE JBHI., 24 (2019), 2199–2207. https://doi.org/10.1109/JBHI.2019.2963048 doi: 10.1109/JBHI.2019.2963048
![]() |
[118] |
J. M. Runji, Y.-J. Lee, C.-H. Chu, User requirements analysis on augmented reality-based maintenance in manufacturing, J. Comput. Inf. Sci. Eng., 22 (2022), 050901. https://doi.org/10.1115/1.4053410 doi: 10.1115/1.4053410
![]() |
[119] | D. Wuttke, A. Upadhyay, E. Siemsen, A. Wuttke-Linnemann, Seeing the bigger picture? ramping up production with the use of augmented reality, Manuf. Serv. Oper. Manag., (2022). https://doi.org/10.1287/msom.2021.1070 |
[120] |
M. Catalano, A. Chiurco, C. Fusto, L. Gazzaneo, F. Longo, G. Mirabelli, L. Nicoletti, V. Solina, S. Talarico, A digital twin-driven and conceptual framework for enabling extended reality applications: A case study of a brake discs manufacturer, Procedia Comput. Sci., 200 (2022), 1885–1893. https://doi.org/10.1016/j.procs.2022.01.389 doi: 10.1016/j.procs.2022.01.389
![]() |
[121] | J. S. Devagiri, S. Paheding, Q. Niyaz, X. Yang, S. Smith, Augmented reality and artificial intelligence in industry: Trends, tools, and future challenges, Expert Syst. Appl., (2022), 118002. https://doi.org/10.1016/j.eswa.2022.118002 |
[122] |
P. T. Ho, J. A. Albajez, J. Santolaria, J. A. Yagüe-Fabra, Study of augmented reality based manufacturing for further integration of quality control 4.0: A systematic literature review, Appl. Sci., 12 (2022), 1961. https://doi.org/10.3390/app12041961 doi: 10.3390/app12041961
![]() |
[123] |
Z.-H. Lai, W. Tao, M. C. Leu, Z. Yin, Smart augmented reality instructional system for mechanical assembly towards worker-centered intelligent manufacturing, J. Manuf. Syst., 55 (2020), 69–81. https://doi.org/10.1016/j.jmsy.2020.02.010 doi: 10.1016/j.jmsy.2020.02.010
![]() |
[124] |
J. Xiong, E.-L. Hsiang, Z. He, T. Zhan, S.-T. Wu, Augmented reality and virtual reality displays: emerging technologies and future perspectives, Light Sci. Appl., 10 (2021), 1–30. https://doi.org/10.1038/s41377-021-00658-8 doi: 10.1038/s41377-021-00658-8
![]() |
[125] |
M.-G. Kim, J. Kim, S. Y. Chung, M. Jin, M. J. Hwang, Robot-based automation for upper and sole manufacturing in shoe production, Machines, 10 (2022), 255. https://doi.org/10.3390/machines10040255 doi: 10.3390/machines10040255
![]() |
[126] |
P. Grefen, I. Vanderfeesten, K. Traganos, Z. Domagala-Schmidt, J. van der Vleuten, Advancing smart manufacturing in europe: Experiences from two decades of research and innovation projects, Machines, 10 (2022), 45. https://doi.org/10.3390/machines10010045 doi: 10.3390/machines10010045
![]() |
[127] |
Y. Zhou, J. Zang, Z. Miao, T. Minshall, Upgrading pathways of intelligent manufacturing in china: Transitioning across technological paradigms, Engineering-Prc, 5 (2019), 691–701. https://doi.org/10.1016/j.eng.2019.07.016 doi: 10.1016/j.eng.2019.07.016
![]() |
[128] |
K. S. Kiangala, Z. Wang, An experimental safety response mechanism for an autonomous moving robot in a smart manufacturing environment using q-learning algorithm and speech recognition, Sensors-Basel, 22 (2022), 941. https://doi.org/10.3390/s22030941 doi: 10.3390/s22030941
![]() |
[129] |
S. Fernandes, Which way to cope with covid-19 challenges? contributions of the iot for smart city projects, Big Data Cogn. Comput., 5 (2021), 26. https://doi.org/10.3390/bdcc5020026 doi: 10.3390/bdcc5020026
![]() |
[130] | C. Thomay, U. Bodin, H. Isakovic, R. Lasch, N. Race, C. Schmittner, G. Schneider, Z. Szepessy, M. Tauber, Z. Wang, Towards adaptive quality assurance in industrial applications, in 2022 IEEE/IFIP NOMS.. IEEE, (2022), 1–6. https://doi.org/10.1109/NOMS54207.2022.9789928 |
[131] |
D. Stadnicka, P. Litwin, D. Antonelli, Human factor in intelligent manufacturing systems-knowledge acquisition and motivation, Proced. CIRP, 79 (2019), 718–723. https://doi.org/10.1016/j.procir.2019.02.023 doi: 10.1016/j.procir.2019.02.023
![]() |
[132] |
H.-X. Li, H. Si, Control for intelligent manufacturing: A multiscale challenge, Engineering-Prc, 3 (2017), 608–615. https://doi.org/10.1016/J.ENG.2017.05.016 doi: 10.1016/J.ENG.2017.05.016
![]() |
[133] |
T. Kalsoom, N. Ramzan, S. Ahmed, M. Ur-Rehman, Advances in sensor technologies in the era of smart factory and industry 4.0, Sensors-Basel, 20 (2020), 6783. https://doi.org/10.3390/s20236783 doi: 10.3390/s20236783
![]() |
[134] |
J. Radianti, T. A. Majchrzak, J. Fromm, I. Wohlgenannt, A systematic review of immersive virtual reality applications for higher education: Design elements, lessons learned, and research agenda, Comput. Educ., 147 (2020), COMPUT EDUC103778. https://doi.org/10.1016/j.compedu.2019.103778 doi: 10.1016/j.compedu.2019.103778
![]() |
[135] |
D. Kamińska, T. Sapiński, S. Wiak, T. Tikk, R. E. Haamer, E. Avots, A. Helmi, C. Ozcinar, G. Anbarjafari, Virtual reality and its applications in education: Survey, Information, 10 (2019), 318. https://doi.org/10.3390/info10100318 doi: 10.3390/info10100318
![]() |
[136] |
T. Joda, G. Gallucci, D. Wismeijer, N. U. Zitzmann, Augmented and virtual reality in dental medicine: A systematic review, Comput. Biol. Med., 108 (2019), 93–100. https://doi.org/10.1016/j.compbiomed.2019.03.012 doi: 10.1016/j.compbiomed.2019.03.012
![]() |
[137] | C. Li, Y. Chen, Y. Shang, A review of industrial big data for decision making in intelligent manufacturing, J. Eng. Sci. Technol., (2021). https://doi.org/10.1016/j.jestch.2021.06.001 |
[138] |
L. Zhou, Z. Jiang, N. Geng, Y. Niu, F. Cui, K. Liu, N. Qi, Production and operations management for intelligent manufacturing: a systematic literature review, Int. J. Prod. Res., 60 (2022), 808–846. https://doi.org/10.1080/00207543.2021.2017055 doi: 10.1080/00207543.2021.2017055
![]() |
[139] |
L. Adriana Crdenas-Robledo, scar Hernndez-Uribe, C. Reta, J. Antonio Cantoral-Ceballos, Extended reality applications in industry 4.0. a systematic literature review, Telemat. Inform., 73 (2022), 101863. https://doi.org/10.1016/j.tele.2022.101863 doi: 10.1016/j.tele.2022.101863
![]() |
[140] |
Z. Wang, X. Bai, S. Zhang, M. Billinghurst, W. He, P. Wang, W. Lan, H. Min, Y. Chen, A comprehensive review of augmented reality-based instruction in manual assembly, training and repair, Robot. Comput. Integr. Manuf., 78 (2022), 102407. https://doi.org/10.1016/j.rcim.2022.102407 doi: 10.1016/j.rcim.2022.102407
![]() |
[141] |
N. Kumar, S. C. Lee, Human-machine interface in smart factory: A systematic literature review, Technol. Forecast. Soc. Change, 174 (2022), 121284. https://doi.org/10.1016/j.techfore.2021.121284 doi: 10.1016/j.techfore.2021.121284
![]() |
[142] | M. Javaid, A. Haleem, R. P. Singh, R. Suman, Enabling flexible manufacturing system (fms) through the applications of industry 4.0 technologies, Int. Things Cyber-Phys. Syst., (2022). https://doi.org/10.1016/j.iotcps.2022.05.005 |
[143] | A. Künz, S. Rosmann, E. Loria, J. Pirker, The potential of augmented reality for digital twins: A literature review, in 2022 IEEE Conference on Virtual Reality and 3D User Interfaces (VR). IEEE, (2022), 389–398. https://doi.org/10.1109/VR51125.2022.00058 |
[144] |
I. Shah, C. Doshi, M. Patel, S. Tanwar, W.-C. Hong, R. Sharma, A comprehensive review of the technological solutions to analyse the effects of pandemic outbreak on human lives, Medicina (Kaunas), 58 (2022), 311. https://doi.org/10.3390/medicina58020311 doi: 10.3390/medicina58020311
![]() |
[145] |
R. P. Singh, M. Javaid, R. Kataria, M. Tyagi, A. Haleem, R. Suman, Significant applications of virtual reality for covid-19 pandemic, Diabetes Metab. Syndr., 14 (2020), 661–664. https://doi.org/10.1016/j.dsx.2020.05.011 doi: 10.1016/j.dsx.2020.05.011
![]() |
[146] |
A. O. Kwok, S. G. Koh, Covid-19 and extended reality (xr), Curr. Issues Tour., 24 (2021), 1935–1940. https://doi.org/10.1080/13683500.2020.1798896 doi: 10.1080/13683500.2020.1798896
![]() |
[147] | G. Czifra, Z. Molnár et al., Covid-19, industry 4.0, Research papers faculty of materials science and technology slovak university of technology, 28 (2020), 36–45. https://doi.org/10.2478/rput-2020-0005 |
[148] | Q. Yu-ming, D. San-peng et al., Research on intelligent manufacturing flexible production line system based on digital twin, in 2020 35th Youth Academic Annual Conference of Chinese Association of Automation (YAC), IEEE, (2020), 854–862. https://doi.org/10.1109/YAC51587.2020.9337500 |
[149] |
L. O. Alpala, D. J. Quiroga-Parra, J. C. Torres, D. H. Peluffo-Ordóñez, Smart factory using virtual reality and online multi-user: Towards a metaverse for experimental frameworks, Appl. Sci., 12 (2022), 6258. https://doi.org/10.3390/app12126258 doi: 10.3390/app12126258
![]() |
[150] |
E. Chang, H. T. Kim, B. Yoo, Virtual reality sickness: A review of causes and measurements, Int. J. Hum-Comput. Int., 36 (2020), 1658–1682. https://doi.org/10.1080/10447318.2020.1778351 doi: 10.1080/10447318.2020.1778351
![]() |
[151] |
H. Su, W. Qi, C. Yang, J. Sandoval, G. Ferrigno, E. De Momi, Deep neural network approach in robot tool dynamics identification for bilateral teleoperation, IEEE Robot. Autom. Lett., 5 (2020), 2943–2949. https://doi.org/10.1109/LRA.2020.2974445 doi: 10.1109/LRA.2020.2974445
![]() |
[152] |
H. Su, Y. Hu, H. R. Karimi, A. Knoll, G. Ferrigno, E. De Momi, Improved recurrent neural network-based manipulator control with remote center of motion constraints: Experimental results, Neural Netw., 131 (2020), 291–299. https://doi.org/10.1016/j.neunet.2020.07.033 doi: 10.1016/j.neunet.2020.07.033
![]() |
[153] |
S. Phuyal, D. Bista, R. Bista, Challenges, opportunities and future directions of smart manufacturing: A state of art review, Sustain. Fut., 2 (2020), 100023. https://doi.org/10.1016/j.sftr.2020.100023 doi: 10.1016/j.sftr.2020.100023
![]() |
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2. | Vo Van Au, Jagdev Singh, Anh Tuan Nguyen, Well-posedness results and blow-up for a semi-linear time fractional diffusion equation with variable coefficients, 2021, 29, 2688-1594, 3581, 10.3934/era.2021052 | |
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4. | Jinxing Liu, Xiongrui Wang, Jun Zhou, Huan Zhang, Blow-up phenomena for the sixth-order Boussinesq equation with fourth-order dispersion term and nonlinear source, 2021, 14, 1937-1632, 4321, 10.3934/dcdss.2021108 | |
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6. | Xiatong Li, Zhong Bo Fang, Blow-up phenomena for a damped plate equation with logarithmic nonlinearity, 2023, 71, 14681218, 103823, 10.1016/j.nonrwa.2022.103823 | |
7. | Jorge Ferreira, Erhan Pışk˙ın, Nazlı Irkıl, Carlos Raposo, Blow up results for a viscoelastic Kirchhoff-type equation with logarithmic nonlinearity and strong damping, 2021, 25, 1450-5932, 125, 10.5937/MatMor2102125F | |
8. | Nouri Boumaza, Billel Gheraibia, Gongwei Liu, Global Well-posedness of Solutions for the p-Laplacian Hyperbolic Type Equation with Weak and Strong Damping Terms and Logarithmic Nonlinearity, 2022, 26, 1027-5487, 10.11650/tjm/220702 | |
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15. | Mohammad M. Al-Gharabli, Aissa Guesmia, Salim A. Messaoudi, Some Existence and Exponential Stability Results for a Plate Equation with Strong Damping and a Logarithmic Source Term, 2022, 0971-3514, 10.1007/s12591-022-00625-8 | |
16. | Wenjun Liu, Jiangyong Yu, Gang Li, Global existence, exponential decay and blow-up of solutions for a class of fractional pseudo-parabolic equations with logarithmic nonlinearity, 2021, 14, 1937-1632, 4337, 10.3934/dcdss.2021121 | |
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18. | Nazlı Irkıl, Erhan Pişkin, Praveen Agarwal, Global existence and decay of solutions for a system of viscoelastic wave equations of Kirchhoff type with logarithmic nonlinearity, 2022, 45, 0170-4214, 2921, 10.1002/mma.7964 | |
19. | Rongting Pan, Yunzhu Gao, Qiu Meng, Guotao Wang, Properties of Weak Solutions for a Pseudoparabolic Equation with Logarithmic Nonlinearity of Variable Exponents, 2023, 2023, 2314-4785, 1, 10.1155/2023/7441168 | |
20. | Nazlı Irkıl, Erhan Pişkin, 2023, Chapter 15, 978-3-031-21483-7, 163, 10.1007/978-3-031-21484-4_15 | |
21. | Nazlı Irkil, Erhan Pişkin, 2023, 9781119879671, 67, 10.1002/9781119879831.ch4 | |
22. | Gongwei Liu, Mengyun Yin, Suxia Xia, Blow-Up Phenomena for a Class of Extensible Beam Equations, 2023, 20, 1660-5446, 10.1007/s00009-023-02469-0 | |
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25. | Bhargav Kumar Kakumani, Suman Prabha Yadav, Global existence and asymptotic behaviour for a viscoelastic plate equation with nonlinear damping and logarithmic nonlinearity, 2023, 18758576, 1, 10.3233/ASY-231859 | |
26. | Qingqing Peng, Zhifei Zhang, Stabilization and blow-up for a class of weakly damped Kirchhoff plate equation with logarithmic nonlinearity, 2023, 0019-5588, 10.1007/s13226-023-00518-8 | |
27. | Amir Peyravi, Lifespan estimates and asymptotic stability for a class of fourth-order damped p-Laplacian wave equations with logarithmic nonlinearity, 2023, 29, 1405-213X, 10.1007/s40590-023-00570-8 | |
28. | Xiang-kun Shao, Nan-jing Huang, Donal O'Regan, Infinite time blow-up of solutions for a plate equation with weak damping and logarithmic nonlinearity, 2024, 535, 0022247X, 128144, 10.1016/j.jmaa.2024.128144 | |
29. | Xiatong Li, Zhong Bo Fang, Infinite time blow‐up with arbitrary initial energy for a damped plate equation, 2024, 0025-584X, 10.1002/mana.202300275 | |
30. | Qingqing Peng, Zhifei Zhang, Stabilization and Blow-up in an Infinite Memory Wave Equation with Logarithmic Nonlinearity and Acoustic Boundary Conditions, 2024, 37, 1009-6124, 1368, 10.1007/s11424-024-3132-1 | |
31. | Hongwei Zhang, Xiao Su, Shuo Liu, Global existence and blowup of solutions to a class of wave equations with Hartree type nonlinearity, 2024, 37, 0951-7715, 065011, 10.1088/1361-6544/ad3f67 | |
32. | Mohammad Kafini, Different aspects of blow-up property for a nonlinear wave equation, 2024, 11, 26668181, 100879, 10.1016/j.padiff.2024.100879 | |
33. | Gongwei Liu, Yi Peng, Xiao Su, Well‐posedness for a class of wave equations with nonlocal weak damping, 2024, 0170-4214, 10.1002/mma.10300 | |
34. | Radhouane Aounallah, Abdelbaki Choucha, Salah Boulaaras, Asymptotic behavior of a logarithmic-viscoelastic wave equation with internal fractional damping, 2024, 0031-5303, 10.1007/s10998-024-00611-3 | |
35. | Abdelbaki Choucha, Salah Boulaaras, Rashid Jan, Ahmed Himadan Ahmed, Global existence and decay of a viscoelastic wave equation with logarithmic source under acoustic, fractional, and nonlinear delay conditions, 2024, 2024, 1687-2770, 10.1186/s13661-024-01959-8 | |
36. | Khadijeh Baghaei, Blow up phenomenon for a plate equation with logarithmic source term, 2025, 0003-6811, 1, 10.1080/00036811.2024.2448658 | |
37. | Nazlı Irkıl, On the local existence of solutions to p-Laplacian equation with logarithmic nonlinearity and nonlinear damping term, 2024, 25, 1787-2405, 759, 10.18514/MMN.2024.4360 | |
38. | Mohammad Shahrouzi, Existence, decay and blow-up results for a plate viscoelastic equation with variable-exponent logarithmic terms, 2024, 38, 0354-5180, 7051, 10.2298/FIL2420051S | |
39. | Mohammad Shahrouzi, Salah Boulaaras, Rashid Jan, Well-posedness and blow-up of solutions for the p(l)-biharmonic wave equation with singular dissipation and variable-exponent logarithmic source, 2025, 16, 1662-9981, 10.1007/s11868-025-00680-z | |
40. | Khadijeh Baghaei, Blow up phenomenon for a plate equation with logarithmic source term and positive initial energy, 2025, 2193-5343, 10.1007/s40065-025-00523-1 | |
41. | Faramarz Tahamtani, Mohammad Shahrouzi, The Lifespan of Solutions for a Boussinesq‐Type Model, 2025, 0170-4214, 10.1002/mma.11067 |