Cutaneous melanoma (SKCM) is the most invasive malignancy of skin cancer. Metastasis to distant lymph nodes or other system is an indicator of poor prognosis in melanoma patients. The aim of this study was to identify reliable prognostic biomarkers for SKCMs.
Four RNA-sequencing datasets associated with SKCMs were downloaded from the Gene Expression Omnibus (GEO) and The Cancer Genome Atlas (TCGA) database as well as corresponding clinical information. Differentially expressed genes (DEGs) were screened between primary and metastatic samples by using MetaDE tool. Weighted gene co-expression network analysis (WGCNA) was conducted to screen functional modules. A prognostic score (PS)-based predictive model and nomogram model were constructed to identify signature genes and independent clinicopathologic factors.
Based on MetaDE analysis and WGCNA, a total of 456 overlapped genes were identified as hub genes related to SKCMs progression. Functional enrichment analysis revealed these genes were mainly involved in the hippo signaling pathway, signaling pathways regulating pluripotency of stem cells, pathways in cancer. In addition, eight optimal DEGs (RFPL1S, CTSV, EGLN3, etc.) were identified as signature genes by using PS model. Cox regression analysis revealed that pathologic stage T, N and recurrence were independent prognostic factors. Three clinical factors and PS status were incorporated to construct a nomogram predictive model for estimating the three years and five-year survival probability of individuals.
The prognosis prediction model of this study may provide a promising method for decision making in clinic and prognosis predicting of SKCM patients.
Citation: Jiaping Wang. Prognostic score model-based signature genes for predicting the prognosis of metastatic skin cutaneous melanoma[J]. Mathematical Biosciences and Engineering, 2021, 18(5): 5125-5145. doi: 10.3934/mbe.2021261
[1] | Wenjun Liu, Zhijing Chen, Zhiyu Tu . New general decay result for a fourth-order Moore-Gibson-Thompson equation with memory. Electronic Research Archive, 2020, 28(1): 433-457. doi: 10.3934/era.2020025 |
[2] | Siyi Luo, Yinghui Zhang . Space-time decay rate for the 3D compressible quantum magnetohydrodynamic model. Electronic Research Archive, 2025, 33(7): 4184-4204. doi: 10.3934/era.2025189 |
[3] | 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 |
[4] | 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 |
[5] | Chahrazed Messikh, Soraya Labidi, Ahmed Bchatnia, Foued Mtiri . Energy decay for a porous system with a fractional operator in the memory. Electronic Research Archive, 2025, 33(4): 2195-2215. doi: 10.3934/era.2025096 |
[6] | Jincheng Shi, Shuman Li, Cuntao Xiao, Yan Liu . Spatial behavior for the quasi-static heat conduction within the second gradient of type Ⅲ. Electronic Research Archive, 2024, 32(11): 6235-6257. doi: 10.3934/era.2024290 |
[7] | Jie Qi, Weike Wang . Global solutions to the Cauchy problem of BNSP equations in some classes of large data. Electronic Research Archive, 2024, 32(9): 5496-5541. doi: 10.3934/era.2024255 |
[8] | 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 |
[9] | Qin Ye . Space-time decay rate of high-order spatial derivative of solution for 3D compressible Euler equations with damping. Electronic Research Archive, 2023, 31(7): 3879-3894. doi: 10.3934/era.2023197 |
[10] | 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 |
Cutaneous melanoma (SKCM) is the most invasive malignancy of skin cancer. Metastasis to distant lymph nodes or other system is an indicator of poor prognosis in melanoma patients. The aim of this study was to identify reliable prognostic biomarkers for SKCMs.
Four RNA-sequencing datasets associated with SKCMs were downloaded from the Gene Expression Omnibus (GEO) and The Cancer Genome Atlas (TCGA) database as well as corresponding clinical information. Differentially expressed genes (DEGs) were screened between primary and metastatic samples by using MetaDE tool. Weighted gene co-expression network analysis (WGCNA) was conducted to screen functional modules. A prognostic score (PS)-based predictive model and nomogram model were constructed to identify signature genes and independent clinicopathologic factors.
Based on MetaDE analysis and WGCNA, a total of 456 overlapped genes were identified as hub genes related to SKCMs progression. Functional enrichment analysis revealed these genes were mainly involved in the hippo signaling pathway, signaling pathways regulating pluripotency of stem cells, pathways in cancer. In addition, eight optimal DEGs (RFPL1S, CTSV, EGLN3, etc.) were identified as signature genes by using PS model. Cox regression analysis revealed that pathologic stage T, N and recurrence were independent prognostic factors. Three clinical factors and PS status were incorporated to construct a nomogram predictive model for estimating the three years and five-year survival probability of individuals.
The prognosis prediction model of this study may provide a promising method for decision making in clinic and prognosis predicting of SKCM patients.
The Moore-Gibson-Thompson (MGT) equation is one of the equations of nonlinear acoustics describing acoustic wave propagation in gases and liquids [13,15,30] and arising from modeling high frequency ultrasound waves [9,18] accounting for viscosity and heat conductivity as well as effect of the radiation of heat on the propagation of sound. This research field is highly active due to a wide range of applications such as the medical and industrial use of high intensity ultrasound in lithotripsy, thermotherapy, ultraound cleaning, etc. The classical nonlinear acoustics models include the Kuznetson's equation, the Westervelt equation and the Kokhlov-Zabolotskaya-Kuznetsov equation.
In order to gain a better understanding of the nonlinear MGT equation, we shall begin with the linearized model. In [15], Kaltenbacher, Lasiecka and Marchand investigated the following linearized MGT equation
τuttt+αutt+c2Au+bAut=0. | (1.1) |
For equation (1.1), they disclosed a critical parameter
τuttt+αutt+c2Au+bAut=f(u,ut,utt). | (1.2) |
They proved that the underlying PDE generates a well-posed dynamical system which admits a global and finite dimensional attractor. They also overcomed the difficulty of lacking the Lyapunov function and the lack of compactness of the trajectory.
Now, we concentrate on the stabilization of MGT equation with memory which has received a considerable attention recently. For instance, Lasiecka and Wang [17] studied the following equation:
τuttt+αutt+bAut+c2Au−∫t0g(t−s)Aw(s)ds=0, | (1.3) |
where
g′(t)≤−c0g(t), | (1.4) |
they discussed the effect of memory described by three types on decay rates of the energy when
τuttt+αutt+bAut+c2Au−∫∞0g(s)Aw(t−s)ds=0. | (1.5) |
Alves et al. [1] investigated the uniform stability of equation (1.5) encompassing three different types of memory in a history space set by the linear semigroup theory. Moreover, we refer the reader to [3,6,7,12,24,25,26,28] for other works of the equation(s) with memory.
More recently, Filippo and Vittorino [11] considered the fourth-order MGT equation
utttt+αuttt+βutt+γAutt+δAut+ϱAu=0. | (1.6) |
They investigated the stability properties of the related solution semigroup. And, according to the values of certain stability numbers depending on the strictly positive parameters
Motivated by the above results, we intend to study the following abstract version of the fourth-order Moore-Gibson-Thompson (MGT) equation with a memory term
utttt+αuttt+βutt+γAutt+δAut+ϱAu−∫t0g(t−s)Au(s)ds=0, | (1.7) |
where
u(0)=u0,ut(0)=u1,utt(0)=u2,uttt(0)=u3. | (1.8) |
A natural question that arised in dealing with the general decay of fourth-order MGT equation with memory:
● Can we get a general decay result for a class of relaxation functions satisfying
Mustafa answered this question for viscoelastic wave equations in [31,32]. Messaoudi and Hassan [29] considered the similar question for memory-type Timoshenko system in the cases of equal and non-equal speeds of wave propagation. Moreover, they extended the range of polynomial decay rate optimality from
The aim of this paper is to establish the well-posedness and answer the above mention question for fourth-order MGT equation with memory (1.7). We first use the Faedo-Galerkin method to prove the well-posedness result. We then use the idea developed by Mustafa in [31,32], taking into consideration the nature of fourth-order MGT equation, to prove new general decay results for the case
The rest of our paper is organized as follows. In Section
In this section, we consider the following assumptions and state our main results. We use
First, we consider the following assumptions as in [11] for
0<g(0)<2αϱδ(αγ−δ), ϱ−∫+∞0g(s)ds=l>0. |
g′(t)≤−ξ(t)M(g(t)),∀ t≥0. | (2.1) |
Remark 1. ([31,Remark 2.8])
(1) From assumption
g(t)→0ast→+∞andg(t)≤ϱ−lt,∀ t>0. |
Furthermore, from the assumption
g(t0)=randg(t)≤r,∀ t≥t0. |
The non-increasing property of
0<g(t0)≤g(t)≤g(0)and0<ξ(t0)≤ξ(t)≤ξ(0),∀ t∈[0,t0]. |
A combination of these with the continuity of
a≤ξ(t)M(g(t))≤d,∀ t∈[0,t0]. |
Consequently, for any
g′(t)≤−ξ(t)M(g(t))≤−a=−ag(0)g(0)≤−ag(0)g(t) |
and, hence,
g(t)≤−g(0)ag′(t),∀ t∈[0,t0]. | (2.2) |
(2) If
¯M=C2t2+(B−Cr)t+(A+C2r2−Br). |
Then, inspired by the notations in [11], we define the Hilbert spaces
Hr:=D(Ar2),r∈R. |
In order to simplify the notation, we denote the usual space
H=D(A12)×D(A12)×D(A12)×H. |
Moreover, we will denote the inner product of
After that, we introduce the following energy functional
E(t)=12[‖uttt+αutt+αϱδut‖2+δα(ϱ−G(t)ϱ)‖A12utt+αA12ut+αϱδA12u‖2+δαϱG(t)‖A12utt+αA12ut‖2+(γ−δα)‖A12utt‖2+(γ−δα)αϱδ‖A12ut‖2+2∫t0g(t−s)(A12u(t)−A12u(s),A12utt+αA12ut)ds+αϱδ(g∘A12u)(t)−α(g′∘A12u)(t)+αg(t)‖A12u‖2+(β−αϱδ)‖utt‖2+αϱδ(β−αϱδ)‖ut‖2], |
where
(g∘v)(t):=∫Ω∫t0g(t−s)(v(t)−v(s))2dsdx. |
As in [31], we set, for any
Cν=∫∞0g2(s)νg(s)−g′(s)dsandh(t)=νg(t)−g′(t). |
The following lemmas play an important role in the proof of our main results.
Lemma 2.1. ([31] )
Assume that condition
∫Ω(∫t0g(t−s)(A12u(s)−A12u(t))ds)2dx≤Cν(h∘A12u)(t),∀ t≥0. |
Lemma 2.2. (Jensen's inequality) Let
P(1k∫Ωf(x)h(x)dx)≤1k∫ΩP(f(x))h(x)dx. |
Lemma 2.3. ([2])(The generalized Young inequality) If
AB≤f∗(A)+f(B), | (2.3) |
where
f∗(s)=s(f′)−1(s)−f[(f′)−1(s)]. | (2.4) |
We are now in a position to state the well-posedness and the general decay result for problem (1.7)-(1.8).
Theorem 2.4. (Well-posedness)
Assume that
u∈C([0,T];D(A12))∩C1([0,T];H). |
Theorem 2.5. (General decay)
Let
E(t)≤k2M−11(k1∫tg−1(r)ξ(s)ds),∀ t≥g−1(t), | (2.5) |
where
Remark 2. Assume that
E(t)≤{Cexp(−˜k∫t0ξ(s)ds),ifp=1,¯k(1+∫t0ξ(s)ds)−1p−1,if1<p<2. | (2.6) |
In this section, we will prove the global existence and uniqueness of the solution of problem (1.7)-(1.8). Firstly, we give the following lemmas.
Lemma 3.1.
If
Proof. Since
2αϱδ(α−σ)(γ−δα)→2αϱδ(αγ−δ)asσ→0, |
which is trivially true.
Lemma 3.2.
Assume that
12[‖uttt+αutt+αϱδut‖2+δα(ϱ−G(t)ϱ)‖A12utt+αA12ut+αϱδA12u‖2+(γ−δα)‖A12utt‖2+(γ−δα)αϱδ‖A12ut‖2−α(g′∘A12u)(t)+αg(t)‖A12u‖2+(β−αϱδ)‖utt‖2+αϱδ(β−αϱδ)‖ut‖2]≤E(t)≤12[‖uttt+αutt+αϱδut‖2+δα(ϱ−G(t)ϱ)‖A12utt+αA12ut+αϱδA12u‖2+(γ−δα)‖A12utt‖2+(γ−δα)αϱδ‖A12ut‖2−α(g′∘A12u)(t)+2αϱδ(g∘A12u)(t)+αg(t)‖A12u‖2+(β−αϱδ)‖utt‖2+αϱδ(β−αϱδ)‖ut‖2+2δαϱG(t)‖A12utt+αA12ut‖2]. |
Proof. From the definition of
E(t)=12[‖uttt+αutt+αϱδut‖2+δα(ϱ−G(t)ϱ)‖A12utt+αA12ut+αϱδA12u‖2+δαϱG(t)‖A12utt+αA12ut‖2+(γ−δα)‖A12utt‖2+(γ−δα)αϱδ‖A12ut‖2+2∫t0g(t−s)(A12u(t)−A12u(s),A12utt+αA12ut)ds+αϱδ(g∘A12u)(t)−α(g′∘A12u)(t)+αg(t)‖A12u‖2+(β−αϱδ)‖utt‖2+αϱδ(β−αϱδ)‖ut‖2]. |
Then, we estimate the sixth term of the above equality
2|∫t0g(t−s)(A12u(t)−A12u(s),A12utt+αA12ut)ds|≤∫t0g(t−s)[αϱδ‖A12u(t)−A12u(s)‖2+δαϱ‖A12utt+αA12ut‖2]ds=αϱδ(g∘A12u)(t)+δαϱG(t)‖A12utt+αA12ut‖2. |
A combination of the above results, we complete the proof of lemma.
Now, we prove the well-posedness result of problem (1.7)-(1.8).
Proof of Theorem 2.1. The proof is given by Faedo-Galerkin method and combines arguments from [16,39,38]. We present only the main steps.
Step
We construct approximations of the solution
um(t)=m∑j=1amj(t)wj(x), | (3.1) |
where
∫Ωumtttt(t)wjdx+α∫Ωumttt(t)wjdx+β∫Ωumtt(t)wjdx+γ∫ΩA12umtt(t)A12wjdx+δ∫ΩA12umt(t)A12wjdx+ϱ∫ΩA12um(t)A12wjdx−∫t0g(t−s)∫ΩA12um(t)A12wjdxds=0 | (3.2) |
with initial conditions
(um(0),umt(0),umtt(0),umttt(0))=(um0,um1,um2,um3). | (3.3) |
According to the standard theory of ordinary differential equation, the finite dimensional problem (3.2)-(3.3) has a local solution
Step
Multiplying equation (3.2) by
ddtEm(t)+α(β−αϱδ)‖umtt‖2−αg′(t)2‖A12um‖2+δ2αϱg(t)‖A12umtt+αϱδA12um‖2+α2(g″∘A12um)(t)=−[α(γ−δα)−δg(t)2αϱ]‖A12umtt‖2+∫t0g′(t−s)(A12um(t)−A12um(s),A12umtt)ds+αϱ2δ(g′∘A12um)(t), |
where
Em(t)=12[‖umttt+αumtt+αϱδumt‖2+δα(ϱ−G(t)ϱ)‖A12umtt+αA12umt+αϱδA12um‖2+δαϱG(t)‖A12umtt+αA12umt‖2+(γ−δα)‖A12umtt‖2+(γ−δα)αϱδ‖A12umt‖2+2∫t0g(t−s)(A12um(t)−A12um(s),A12umtt+αA12umt)ds+αϱδ(g∘A12um)(t)−α(g′∘A12um)(t)+αg(t)‖A12um‖2+(β−αϱδ)‖umtt‖2+αϱδ(β−αϱδ)‖umt‖2]. | (3.4) |
From assumptions
∫t0g′(t−s)(A12um(t)−A12um(s),A12umtt)ds≤−(α−ε)ϱ2δ(g′∘A12um)(t)−δ2(α−ε)ϱ‖A12umtt‖2∫t0g′(t−s)ds=−(α−ε)ϱ2δ(g′∘A12um)(t)+δ(g(0)−g(t))2(α−ε)ϱ‖A12umtt‖2 |
and so
−[α(γ−δα)−δg(t)2αϱ]‖A12umtt‖2+∫t0g′(t−s)(A12um(t)−A12um(s),A12umtt)ds+αϱ2δ(g′∘A12um)(t)≤−[α(γ−δα)−δg(t)2αϱ]‖A12umtt‖2−(α−ε)ϱ2δ(g′∘A12um)(t)+δ(g(0)−g(t))2(α−ε)ϱ‖A12umtt‖2+αϱ2δ(g′∘A12um)(t)=−[α(γ−δα)−δg(0)2(α−ε)ϱ]‖A12umtt‖2+εϱ2δ(g′∘A12um)(t)−[δg(t)2(α−ε)ϱ−δg(t)2αϱ]‖A12umtt‖2≤0. | (3.5) |
Therefore, we have
ddtEm(t)+α(β−αϱδ)‖umtt‖2−αg′(t)2‖A12um‖2+δ2αϱg(t)‖A12umtt+αϱδA12um‖2+α2(g″∘A12um)(t)≤0. | (3.6) |
Integrating (3.6) from
Em(t)+∫t0[α(β−αϱδ)‖umtt‖2−αg′(τ)2‖A12um‖2+δ2αϱg(τ)‖A12umtt+αϱδA12um‖2+α2(g″∘A12um)(τ)]dτ≤Em(0). | (3.7) |
Now, since the sequences
Em(t)≤C. | (3.8) |
Therefore, using the fact
(um)m∈NisboundedinL∞(0,T;D(A12))(umt)m∈NisboundedinL∞(0,T;D(A12))(umtt)m∈NisboundedinL∞(0,T;D(A12))(umttt)m∈NisboundedinL∞(0,T;H). | (3.9) |
Consequently, we may conclude that
um⇀uweak∗inL∞(0,T;D(A12))umt⇀utweak∗inL∞(0,T;D(A12))umtt⇀uttweak∗inL∞(0,T;D(A12))umttt⇀utttweak∗inL∞(0,T;H). |
From (3.9), we get that
Since the embedding
un→ustronglyinL2(0,T;H(Ω)). |
Therefore,
un→ustronglyanda.e.on(0,T)×Ω. |
The proof now can be completed arguing as in [21].
Step
It is sufficient to show that the only weak solution of (1.7)-(1.8) with
u≡0. | (3.10) |
According to the energy estimate (3.8) and noting that
E(u(t))=0,∀t∈[0,T]. |
So, we have
‖uttt+αutt+αϱδut‖2=‖A12utt+αA12ut+αϱδA12u‖2=‖A12utt‖2=‖A12ut‖2=‖A12u‖2=‖utt‖2=0,∀t∈[0,T]. |
And this implies (3.10). Thus, we conclude that problem (1.7)-(1.8) has at most one solution.
In this section, we state and prove some lemmas needed to establish our general decay result.
Lemma 4.1.
Let
ddtE(t)≤−α(β−αϱδ)‖utt‖2−[α(γ−δα)−δg(0)2(α−ε)ϱ]‖A12utt‖2+αg′(t)2‖A12u‖2−[δg(t)2(α−ε)ϱ−δg(t)2αϱ]‖A12utt‖2−δ2αϱg(t)‖A12utt+αϱδA12u‖2−α2(g″∘A12u)(t)+εϱ2δ(g′∘A12u)≤0. |
Proof. Multiplying (1.7) by
ddtE(t)=−α(β−αϱδ)‖utt‖2−[α(γ−δα)−δg(t)2αϱ]‖A12utt‖2+αg′(t)2‖A12u‖2−δ2αϱg(t)‖A12utt+αϱδA12u‖2+∫t0g′(t−s)(A12u(t)−A12u(s),A12utt)ds−α2(g″∘A12u)(t)+αϱ2δ(g′∘A12u)(t). | (4.1) |
We proceed to show that, for a constant
|∫t0g′(t−s)(A12u(t)−A12u(s),A12utt)ds|≤−(α−ε)ϱ2δ(g′∘A12u)(t)+δ(g(0)−g(t))2(α−ε)ϱ‖A12utt‖2. | (4.2) |
Then, combining (4.1) and (4.2), we can obtain
ddtE(t)≤−α(β−αϱδ)‖utt‖2−[α(γ−δα)−δg(0)2(α−ε)ϱ]‖A12utt‖2+αg′(t)2‖A12u‖2−[δg(t)2(α−ε)ϱ−δg(t)2αϱ]‖A12utt‖2−δ2αϱg(t)‖A12utt+αϱδA12u‖2−α2(g″∘A12u)(t)+εϱ2δ(g′∘A12u). |
According to
Lemma 4.2.
Assume that
F1(t)=∫Ω(utt+αut+αϱδu)(uttt+αutt+αϱδut)dx |
satisfies the estimate
F′1(t)≤−δ2α‖A12utt+αA12ut+αϱδA12u‖2+2αλ0δ(β−αϱδ)2‖utt‖2+2αδ(γ−δα)2‖A12utt‖2+‖uttt+αutt+αϱδut‖2+2α(ϱ−l)2δ‖A12u‖2+2αδCν(h∘A12u)(t). | (4.3) |
Proof. Taking the derivative of
F′1(t)=∫Ω[−(β−αϱδ)utt](utt+αut+αϱδu)dx−δα‖A12utt+αA12ut+αϱδA12u‖2+‖uttt+αutt+αϱδut‖2−∫Ω(γ−δα)A12utt(A12utt+αA12ut+αϱδA12u)dx+∫Ω(∫t0g(t−s)A12u(s)ds)(A12utt+αA12ut+αϱδA12u)dx. |
Using Young's inequality, Lemma 2.1,
∫Ω[−(β−αϱδ)utt](utt+αut+αϱδu)dx≤2αλ0δ(β−αϱδ)2‖utt‖2+δ8αλ0‖utt+αut+αϱδu‖2≤2αλ0δ(β−αϱδ)2‖utt‖2+δ8α‖A12utt+αA12ut+αϱδA12u‖2 |
and
∫Ω(∫t0g(t−s)A12u(s)ds)(A12utt+αA12ut+αϱδA12u)dx=∫Ω(∫t0g(t−s)(A12u(s)−A12u(t))ds)(A12utt+αA12ut+αϱδA12u)dx+∫Ω(∫t0g(t−s)A12u(t)ds)(A12utt+αA12ut+αϱδA12u)dx≤2αδCν(h∘A12u)(t)+δ4α‖A12utt+αA12ut+αϱδA12u‖2+2α(ϱ−l)2δ‖A12u‖2. |
Also, we have
−∫Ω(γ−δα)A12utt(A12utt+αA12ut+αϱδA12u)dx≤2αδ(γ−δα)2‖A12utt‖2+δ8α‖A12utt+αA12ut+αϱδA12u‖2. |
Then, combining the above inequalities, we complete the proof of (4.3).
Lemma 4.3.
Assume that
F2(t)=−∫Ω(uttt+αutt+αϱδut)∫t0g(t−s)[(utt+αut+αϱδu)(t)−αϱδu(s)]dsdx |
satisfies the estimate
F′2(t)≤−G(t)4‖uttt+αutt+αϱδut‖2+[(ϱ−l)2α22ε1+4λ0g2(0)α2G(t)]‖A12ut‖2+[λ0(ϱ−l)22+(δ2α2+ϱ2)+3(ϱ−l)2]ε1‖A12utt+αA12ut+αϱδA12u‖2+[α2ϱ2(ϱ−l)2λ0ε12δ2+(ϱ−l)22ε1+3(ϱ−l)2(αϱδ)2ε1+(ϱ−l)22(αϱδ)2]‖A12u‖2+[(ϱ−l)22ε1+12ε1(γ−δα)2+12(αϱδ)2(γ−δα)2]‖A12utt‖2+[(ε1α2ϱ2λ04δ2+34ε1+1+αϱδ+2α4ϱ2λ0G(t)δ2)Cν+2α2ϱ2λ0G(t)δ2](h∘A12u)(t)+[2(β−αϱδ)2ε1+4g2(0)G(t)]‖utt‖2, | (4.4) |
where
Proof. By differentiating
\begin{align*} &F'_{2}(t)\nonumber\\ = &\int_{\Omega}\left[\beta u_{tt}+\gamma\mathcal{A}u_{tt}+\delta\mathcal{A}u_{t}+\varrho\mathcal{A}u-\int_{0}^{t}g(t-s)\mathcal{A}u(s)ds -\frac{\alpha\varrho}{\delta}u_{tt}\right] \nonumber\\ &\times\int_{0}^{t}g(t-s)\left[\left(u_{tt}+\alpha u_{t} +\frac{\alpha\varrho}{\delta}u\right)(t)-\frac{\alpha\varrho}{\delta}u(s)\right]dsdx\nonumber\\ &-g(0)\int_{\Omega}\left(u_{ttt}+\alpha u_{tt}+\frac{\alpha\varrho}{\delta}u_{t}\right)\left(u_{tt}+\alpha u_{t}\right)dx \nonumber\\ &-\int_{\Omega}\left(u_{ttt}+\alpha u_{tt}+\frac{\alpha\varrho}{\delta}u_{t}\right)\int_{0}^{t}g'(t-s)\left[\left(u_{tt}+\alpha u_{t}+\frac{\alpha\varrho}{\delta}u\right)(t)\right. \nonumber\\ &\left.-\frac{\alpha\varrho}{\delta}u(s)\right]dsdx-\int_{0}^{t}g(s)ds\left\|u_{ttt}+\alpha u_{tt}+\frac{\alpha\varrho}{\delta}u_{t}\right\|^{2} \nonumber\\ = &\int_{\Omega}\left[\left(\beta-\frac{\alpha\varrho}{\delta}\right)u_{tt}+\frac{\delta}{\alpha}\left(\mathcal{A}u_{tt} +\alpha\mathcal{A}u_{t}+\frac{\alpha\varrho}{\delta}\mathcal{A}u\right) +\left(\gamma-\frac{\delta}{\alpha}\right)\mathcal{A}u_{tt}\right. \nonumber\\ &\left.-\int_{0}^{t}g(s)ds\mathcal{A}u(t)\right] \int_{0}^{t}g(t-s)\left[\left(u_{tt}+\alpha u_{t}+\frac{\alpha\varrho}{\delta}u\right)(t)- \frac{\alpha\varrho}{\delta}u(s)\right]dsdx\nonumber\\ &-\int_{0}^{t}g(s)ds\left\|u_{ttt}+\alpha u_{tt}+\frac{\alpha\varrho}{\delta}u_{t}\right\|^{2} \nonumber\\ &-\int_{\Omega}\left(u_{ttt}+\alpha u_{tt}+\frac{\alpha\varrho}{\delta}u_{t}\right)\int_{0}^{t}g'(t-s)\left[\left(u_{tt}+\alpha u_{t}+\frac{\alpha\varrho}{\delta}u\right)(t)-\frac{\alpha\varrho}{\delta}u(s)\right]dsdx \nonumber\\ &+\left(\int_{0}^{t}g(s)ds\right)\int_{\Omega}\int_{0}^{t}g(t-s)\left(\mathcal{A}u(t)-\mathcal{A}u(s)\right)ds(u_{tt}+\alpha u_{t})dx \nonumber\\ &+\frac{\alpha\varrho}{\delta}\int_{\Omega}\left(\int_{0}^{t}g(t-s)\left(\mathcal{A}^{\frac{1}{2}}u(t)-\mathcal{A}^{\frac{1}{2}}u(s)\right)ds\right)^{2}dx \nonumber\\ &-g(0)\int_{\Omega}\left(u_{ttt}+\alpha u_{tt}+\frac{\alpha\varrho}{\delta}u_{t}\right)\left(u_{tt}+\alpha u_{t}\right)dx. \end{align*} |
Now, we estimate the terms in the right-hand side of the above identity.
Using Young's inequality, we obtain, for
\begin{align*} &\int_{\Omega}\left[\left(\beta-\frac{\alpha\varrho}{\delta}\right)u_{tt}+\frac{\delta}{\alpha}\left(\mathcal{A}u_{tt} +\alpha\mathcal{A}u_{t}+\frac{\alpha\varrho}{\delta}\mathcal{A}u\right)+\left(\gamma-\frac{\delta}{\alpha}\right)\mathcal{A}u_{tt} \right. \nonumber\\ &\left.-\int_{0}^{t}g(s)ds\mathcal{A}u(t)\right] \int_{0}^{t}g(t-s)\left[\left(u_{tt}+\alpha u_{t}+\frac{\alpha\varrho}{\delta}u\right)-\frac{\alpha\varrho}{\delta}u(s)\right]dsdx \nonumber\\ \leq&\left[\frac{\lambda_{0}(\varrho-l)^{2}}{2}+\left(\frac{\delta^{2}}{\alpha^{2}}+\varrho^{2}\right) +2(\varrho-l)^{2}\right]\varepsilon_{1}\left\|\mathcal{A}^{\frac{1}{2}}u_{tt}+\alpha\mathcal{A}^{\frac{1}{2}}u_{t} +\frac{\alpha\varrho}{\delta}\mathcal{A}^{\frac{1}{2}}u\right\|^{2}\nonumber\\ &+\frac{2\left(\beta-\frac{\alpha\varrho}{\delta}\right)^{2}} {\varepsilon_{1}}\|u_{tt}\|^{2}+\left[\frac{\alpha^{2}\varrho^{2}(\varrho-l)^{2}\lambda_{0}\varepsilon_{1}}{2\delta^{2}} +\frac{(\varrho-l)^{2}}{2\varepsilon_{1}}\right. \nonumber\\ &\left. +2(\varrho-l)^{2}\left(\frac{\alpha\varrho}{\delta}\right)^{2}\varepsilon_{1}+\frac{(\varrho-l)^{2}}{2}\left(\frac{\alpha\varrho}{\delta}\right)^{2}\right]\left\|\mathcal{A}^{\frac{1}{2}}u\right\|^{2} \nonumber\\ &+\left[\frac{(\varrho-l)^{2}}{2\varepsilon_{1}}+\frac{1}{2\varepsilon_{1}}\left(\gamma -\frac{\delta}{\alpha}\right)^{2}+\frac{1}{2}\left(\frac{\alpha\varrho}{\delta}\right)^{2}\left(\gamma -\frac{\delta}{\alpha}\right)^{2}\right]\left\|\mathcal{A}^{\frac{1}{2}}u_{tt}\right\|^{2} \nonumber\\ &+\frac{(\varrho-l)^{2} \alpha^{2}}{2\varepsilon_{1}}\left\|\mathcal{A}^{\frac{1}{2}}u_{t}\right\|^{2}+\left(\frac{\varepsilon_{1}\alpha^{2}\varrho^{2}\lambda_{0}}{4\delta^{2}}+\frac{1}{4\varepsilon_{1}}+1\right)C_{\nu}\left(h\circ\mathcal{A}^{\frac{1}{2}}u\right)(t) \end{align*} |
and
\begin{align*} &\left(\int_{0}^{t}g(s)ds\right)\int_{\Omega}\int_{0}^{t}g(t-s)\left(\mathcal{A}u(t)-\mathcal{A}u(s)\right)ds(u_{tt}+\alpha u_{t})dx \nonumber\\ = &\left(\int_{0}^{t}g(s)ds\right)\int_{\Omega}\int_{0}^{t}g(t-s)\left(\mathcal{A}^{\frac{1}{2}}u(t)-\mathcal{A}^{\frac{1}{2}}u(s)\right)ds(\mathcal{A}^{\frac{1}{2}}u_{tt}+\alpha \mathcal{A}^{\frac{1}{2}}u_{t})dx \nonumber\\ \leq&\frac{1}{2\varepsilon_{1}}C_{\nu}\left(h\circ\mathcal{A}^{\frac{1}{2}}u\right)(t)+\frac{(\varrho-l)^{2}}{2}\varepsilon_{1}\left\|\mathcal{A}^{\frac{1}{2}}u_{tt}+\alpha\mathcal{A}^{\frac{1}{2}}u_{t}\right\|^{2} \nonumber\\ \leq&\frac{1}{2\varepsilon_{1}}C_{\nu}\left(h\circ\mathcal{A}^{\frac{1}{2}}u\right)(t)+(\varrho-l)^{2}\varepsilon_{1} \left\|\mathcal{A}^{\frac{1}{2}}u_{tt}+\alpha\mathcal{A}^{\frac{1}{2}}u_{t}+\frac{\alpha\varrho}{\delta}\mathcal{A}^{\frac{1}{2}}u\right\|^{2} \nonumber\\ &+(\varrho-l)^{2}\left(\frac{\alpha\varrho}{\delta}\right)^{2}\varepsilon_{1}\left\|\mathcal{A}^{\frac{1}{2}}u\right\|^{2}. \end{align*} |
Also, we have
\begin{align*} \frac{\alpha\varrho}{\delta}\int_{\Omega}\left(\int_{0}^{t}g(t-s)\left(\mathcal{A}^{\frac{1}{2}}u(t)-\mathcal{A}^{\frac{1}{2}}u(s)\right)ds\right)^{2}dx\leq&\frac{\alpha\varrho} {\delta}C_{\nu}\left(h\circ\mathcal{A}^{\frac{1}{2}}u\right)(t) \end{align*} |
and
\begin{align*} &-g(0)\int_{\Omega}\left(u_{ttt}+\alpha u_{tt}+\frac{\alpha\varrho}{\delta}u_{t}\right)\left(u_{tt}+\alpha u_{t}\right)dx \nonumber\\ \leq&\frac{G(t)}{4}\left\|u_{ttt}+\alpha u_{tt}+\frac{\alpha\varrho}{\delta}u_{t}\right\|^{2}+\frac{g^{2}(0)}{G(t)}\left\|u_{tt}+\alpha u_{t}\right\|^{2} \nonumber\\ \leq&\frac{G(t)}{4}\left\|u_{ttt}+\alpha u_{tt}+\frac{\alpha\varrho}{\delta}u_{t}\right\|^{2}+\frac{2g^{2}(0)}{G(t)}\left\|u_{tt}\right\|^{2}+\frac{2\lambda_{0}g^{2}(0)\alpha^{2}}{G(t)}\left\|\mathcal{A}^{\frac{1}{2}}u_{t}\right\|^{2}. \end{align*} |
Exploiting Young's inequality and
\begin{align*} &-\int_{\Omega}\left(u_{ttt}+\alpha u_{tt}+\frac{\alpha\varrho}{\delta}u_{t}\right)\int_{0}^{t}g'(t-s)\left[\left(u_{tt}+\alpha u_{t}+\frac{\alpha\varrho}{\delta}u\right)(t)-\frac{\alpha\varrho}{\delta}u(s)\right]dsdx \nonumber\\ = &-\int_{\Omega}\left(u_{ttt}+\alpha u_{tt}+\frac{\alpha\varrho}{\delta}u_{t}\right)\int_{0}^{t}g'(t-s)\left(u_{tt}+\alpha u_{t}\right)(t)dsdx \nonumber\\ &-\frac{\alpha\varrho}{\delta}\int_{\Omega}\left(u_{ttt}+\alpha u_{tt}+\frac{\alpha\varrho}{\delta}u_{t}\right)\int_{0}^{t}g'(t-s)(u(t)-u(s))dsdx \nonumber\\ \leq&\frac{G(t)}{2}\left\|u_{ttt}+\alpha u_{tt}+\frac{\alpha\varrho}{\delta}u_{t}\right\|^{2}+\frac{2g^{2}(0)}{G(t)}\|u_{tt}\|^{2}+\frac{2\lambda_{0}\alpha^{2}g^{2}(0)}{G(t)}\left\|\mathcal{A}^{\frac{1}{2}}u_{t}\right\|^{2} \nonumber\\ &+\frac{2\alpha^{2}\varrho^{2}\lambda_{0}}{G(t)\delta^{2}}\left(\alpha^{2} C_{\nu}+1\right)\left(h\circ\mathcal{A}^{\frac{1}{2}}u\right)(t). \end{align*} |
A combination of all the above estimates gives the desired result.
As in [11], we introduce the following auxiliary functional
\begin{align*} F_{3}(t) = \int_{\Omega}\left(u_{ttt}+\alpha u_{tt}\right)u_{t}dx+\frac{\varrho}{2}\left\|\mathcal{A}^{\frac{1}{2}}u\right\|^{2}. \end{align*} |
Lemma 4.4.
Assume that
\begin{align} F'_{3}(t)\leq&-\left(\frac{3\delta}{8}-\frac{\varepsilon_{2}\delta}{4}\right)\left\|\mathcal{A}^{\frac{1}{2}}u_{t}\right\|^{2}+\frac{\varepsilon_{2}\delta^{3}}{8\alpha^{2}\varrho^{2}\lambda_{0}}\left\|u_{ttt}+\alpha u_{tt}+\frac{\alpha\varrho}{\delta}u_{t}\right\|^{2} \\ &+\frac{2\gamma^{2}}{\delta}\left\|\mathcal{A}^{\frac{1}{2}}u_{tt}\right\|^{2}+\left(\frac{\varepsilon_{2}\delta^{3}}{4\varrho^{2}\lambda_{0}} +\frac{4\alpha^{2}\varrho^{2}\lambda_{0}}{\varepsilon_{2}\delta^{3}} +\frac{2\beta^{2}\lambda_{0}}{\delta}\right)\|u_{tt}\|^{2}\\ &+\frac{1}{\delta}C_{\nu}\left(h\circ\mathcal{A}^{\frac{1}{2}}u\right)(t)+\frac{2(\varrho-l)^{2}}{\delta}\left\|\mathcal{A}^{\frac{1}{2}}u\right\|^{2}, \end{align} | (4.5) |
where
Proof. Using the equation (1.7), a direct computation leads to the following identity
\begin{align} F'_{3}(t) = &\int_{\Omega}(u_{ttt}+\alpha u_{tt})u_{tt}dx+\int_{\Omega}(u_{tttt}+\alpha u_{ttt})u_{t}dx+\varrho\left(\mathcal{A}^{\frac{1}{2}}u,\mathcal{A}^{\frac{1}{2}}u_{t}\right) \\ = &(u_{ttt},u_{tt})+\alpha\|u_{tt}\|^{2}-\beta(u_{tt},u_{t})-\gamma\left(\mathcal{A}^{\frac{1}{2}}u_{tt},\mathcal{A}^{\frac{1}{2}}u_{t}\right)-\delta\left\|\mathcal{A}^{\frac{1}{2}}u_{t}\right\|^{2} \\ &+\left(\int_{0}^{t}g(t-s)\mathcal{A}^{\frac{1}{2}}u(s)ds,\mathcal{A}^{\frac{1}{2}}u_{t}\right). \end{align} | (4.6) |
Now, the first and third terms in the right-hand side of (4.6) can be estimated as follows:
\begin{align*} & (u_{ttt},u_{tt}) \nonumber\\ \leq& \frac{\varepsilon_{2}\delta^{3}}{16\alpha^{2}\varrho^{2}\lambda_{0}}\|u_{ttt}\|^{2} +\frac{4\alpha^{2}\varrho^{2}\lambda_{0}}{\varepsilon_{2}\delta^{3}}\|u_{tt}\|^{2} \nonumber\\ \leq& \frac{\varepsilon_{2}\delta^{3}}{8\alpha^{2}\varrho^{2}\lambda_{0}}\left\|u_{ttt}+\alpha u_{tt}+\frac{\alpha\varrho}{\delta}u_{t}\right\|^{2} +\frac{\varepsilon_{2}\delta^{3}}{8\alpha^{2}\varrho^{2}\lambda_{0}}\left\|\alpha u_{tt}+\frac{\alpha\varrho}{\delta}u_{t}\right\|^{2} +\frac{4\alpha^{2}\varrho^{2}\lambda_{0}}{\varepsilon_{2}\delta^{3}}\|u_{tt}\|^{2} \nonumber\\ \leq& \frac{\varepsilon_{2}\delta^{3}}{8\alpha^{2}\varrho^{2}\lambda_{0}}\left\|u_{ttt}+\alpha u_{tt}+\frac{\alpha\varrho}{\delta}u_{t}\right\|^{2} +\alpha^{2}\left(\frac{\varepsilon_{2}\delta^{3}}{4\alpha^{2}\varrho^{2}\lambda_{0}}+\frac{4\varrho^{2}\lambda_{0}}{\varepsilon_{2}\delta^{3}}\right)\|u_{tt}\|^{2} +\frac{\varepsilon_{2}\delta}{4}\left\|\mathcal{A}^{\frac{1}{2}}u_{t}\right\|^{2} \end{align*} |
and
\begin{align*} -\beta(u_{tt},u_{t})\leq\frac{2\beta^{2}\lambda_{0}}{\delta}\|u_{tt}\|^{2}+\frac{\delta}{8\lambda_{0}}\|u_{t}\|^{2}\leq\frac{2\beta^{2}\lambda_{0}}{\delta}\|u_{tt}\|^{2}+\frac{\delta}{8}\left\|\mathcal{A}^{\frac{1}{2}}u_{t}\right\|^{2}, \end{align*} |
where
Using Young's inequality and Lemma 2.1, we get
\begin{align*} -\gamma\left(\mathcal{A}^{\frac{1}{2}}u_{tt},\mathcal{A}^{\frac{1}{2}}u_{t}\right)\leq\frac{2\gamma^{2}}{\delta}\left\|\mathcal{A}^{\frac{1}{2}}u_{tt}\right\|^{2}+\frac{\delta}{8}\left\|\mathcal{A}^{\frac{1}{2}}u_{t}\right\|^{2} \end{align*} |
and
\begin{align*} & \left(\int_{0}^{t}g(t-s)\mathcal{A}^{\frac{1}{2}}u(s)ds,\mathcal{A}^{\frac{1}{2}}u_{t}\right) \nonumber\\ = & \left(\int_{0}^{t}g(t-s)\left(\mathcal{A}^{\frac{1}{2}}u(s)-\mathcal{A}^{\frac{1}{2}}u(t)+\mathcal{A}^{\frac{1}{2}}u(t)\right)ds,\mathcal{A}^{\frac{1}{2}}u_{t}\right) \nonumber\\ \leq& \frac{1}{\delta}\int_{\Omega}\left(\int_{0}^{t}g(t-s)\left(\mathcal{A}^{\frac{1}{2}}u(t)-\mathcal{A}^{\frac{1}{2}}u(s)\right)ds\right)^{2}dx \nonumber\\ &+\frac{\delta}{4}\left\|\mathcal{A}^{\frac{1}{2}}u_{t}\right\|^{2} +\frac{2(\varrho-l)^{2}}{\delta}\left\|\mathcal{A}^{\frac{1}{2}}u\right\|^{2} +\frac{\delta}{8}\left\|\mathcal{A}^{\frac{1}{2}}u_{t}\right\|^{2} \nonumber\\ \leq& \frac{1}{\delta}C_{\nu}\left(h\circ\mathcal{A}^{\frac{1}{2}}u\right)(t) +\frac{3\delta}{8}\left\|\mathcal{A}^{\frac{1}{2}}u_{t}\right\|^{2} +\frac{2(\varrho-l)^{2}}{\delta}\left\|\mathcal{A}^{\frac{1}{2}}u\right\|^{2}. \end{align*} |
Then, combining the above inequalities, we obtain the desired result.
Lemma 4.5.
Assume that
\begin{align*} F_{4}(t) = \int_{\Omega}\int_{0}^{t}f(t-s)\left|\mathcal{A}^{\frac{1}{2}}u(s)\right|^{2}dsdx \end{align*} |
satisfies the estimate
\begin{align} F'_{4}(t)\leq-\frac{1}{2}\left(g\circ\mathcal{A}^{\frac{1}{2}}u\right)(t)+3(\varrho-l)\left\|\mathcal{A}^{\frac{1}{2}}u\right\|^{2}, \end{align} | (4.7) |
where
Proof. Noting that
\begin{align*} &F'_{4}(t)\nonumber\\ = &f(0)\left\|\mathcal{A}^{\frac{1}{2}}u\right\|^{2}-\int_{\Omega}\int_{0}^{t}g(t-s)\left|\mathcal{A}^{\frac{1}{2}}u(s)\right|^{2}dsdx \nonumber\\ = &f(0)\left\|\mathcal{A}^{\frac{1}{2}}u\right\|^{2}-\int_{\Omega}\int_{0}^{t}g(t-s)\left|\mathcal{A}^{\frac{1}{2}}u(s)-\mathcal{A}^{\frac{1}{2}}u(t)\right|^{2}dsdx \nonumber\\ &-2\int_{\Omega}\mathcal{A}^{\frac{1}{2}}u\int_{0}^{t}g(t-s)\left(\mathcal{A}^{\frac{1}{2}}u(s)-\mathcal{A}^{\frac{1}{2}}u(t)\right)dsdx-\left(\int_{0}^{t}g(s)ds\right)\left\|\mathcal{A}^{\frac{1}{2}}u\right\|^{2} \nonumber\\ = &-\left(g\circ\mathcal{A}^{\frac{1}{2}}u\right)(t)-2\int_{\Omega}\mathcal{A}^{\frac{1}{2}}u\int_{0}^{t}g(t-s)\left(\mathcal{A}^{\frac{1}{2}}u(s)-\mathcal{A}^{\frac{1}{2}}u(t)\right)dsdx+f(t)\left\|\mathcal{A}^{\frac{1}{2}}u\right\|^{2}. \end{align*} |
Exploiting Young's inequality and the fact
\begin{align*} &-2\int_{\Omega}\mathcal{A}^{\frac{1}{2}}u\int_{0}^{t}g(t-s)\left(\mathcal{A}^{\frac{1}{2}}u(s)-\mathcal{A}^{\frac{1}{2}}u(t)\right)dsdx \nonumber\\ \leq&2(\varrho-l)\left\|\mathcal{A}^{\frac{1}{2}}u\right\|^{2}\nonumber\\ &+\frac{1}{2(\varrho-l)}\left(\int_{0}^{t}g(t-s)ds\right)\int_{\Omega}\int_{0}^{t}g(t-s)\left(\mathcal{A}^{\frac{1}{2}}u(s)-\mathcal{A}^{\frac{1}{2}}u(t)\right)^{2}dsdx \nonumber\\ \leq&2(\varrho-l)\left\|\mathcal{A}^{\frac{1}{2}}u\right\|^{2}+\frac{1}{2}\left(g\circ\mathcal{A}^{\frac{1}{2}}u\right)(t). \end{align*} |
Moreover, taking account of
\begin{align*} f(t)\left\|\mathcal{A}^{\frac{1}{2}}u\right\|^{2}\leq(\varrho-l)\left\|\mathcal{A}^{\frac{1}{2}}u\right\|^{2}. \end{align*} |
Combining the above estimates, we arrive at the desired result.
Lemma 4.6.
Assume that
\begin{align*} \mathcal{L}(t) = NE(t)+F_{1}(t)+N_{2}F_{2}(t)+N_{3}F_{3}(t) \end{align*} |
satisfies, for a suitable choice of
\begin{align*} \mathcal{L}(t)\sim E(t) \end{align*} |
and the estimate, for all
\begin{align} \mathcal{L'}(t)\leq&-c\left[\|u_{tt}\|^{2}+\left\|\mathcal{A}^{\frac{1}{2}}u_{tt}\right\|^{2}+\left\|u_{ttt}+\alpha u_{tt}+\frac{\alpha\varrho}{\delta}u_{t}\right\|^{2}\right. \\ &\left. +\left\|\mathcal{A}^{\frac{1}{2}}u_{tt}+\alpha\mathcal{A}^{\frac{1}{2}}u_{t}+\frac{\alpha\varrho}{\delta}u\right\|^{2}\right]-4(\varrho-l)\left\|\mathcal{A}^{\frac{1}{2}}u\right\|^{2}+\frac{1}{8}\left(g\circ\mathcal{A}^{\frac{1}{2}}u\right)(t), \end{align} | (4.8) |
where
Proof. Combining Lemmas 4.1-4.4 and recalling that
\begin{align*} &\mathcal{L'}(t)\nonumber\\ \leq&-\left[\alpha\left(\beta-\frac{\alpha\varrho}{\delta}\right)N-\frac{2\alpha\lambda_{0}}{\delta}\left(\beta-\frac{\alpha\varrho}{\delta}\right)^{2}-\left(\frac{2\left(\beta-\frac{\alpha\varrho}{\delta}\right)^{2}}{\varepsilon_{1}}+\frac{4g^{2}(0)}{G(t)}\right)N_{2}\right. \nonumber\\ &\left.-\left(\frac{\varepsilon_{2}\delta^{3}}{4\varrho^{2}\lambda_{0}}+\frac{4\alpha^{2}\varrho^{2}\lambda_{0}}{\varepsilon_{2}\delta^{3}}+\frac{2\beta^{2}\lambda_{0}}{\delta}\right)N_{3}\right]\|u_{tt}\|^{2}-\left[\left(\alpha\left(\gamma-\frac{\delta}{\alpha}\right)-\frac{\delta g(0)}{2(\alpha-\varepsilon)\varrho}\right)N\right. \nonumber\\ &\left.+\left(\frac{\delta g(t)}{2(\alpha-\varepsilon)\varrho}-\frac{\delta g(t)}{2\alpha\varrho}\right)N-\frac{2\alpha}{\delta}\left(\gamma-\frac{\delta}{\alpha}\right)^{2}-\left(\frac{(\varrho-l)^{2}}{2\varepsilon_{1}}+\frac{1}{2\varepsilon_{1}}\left(\gamma-\frac{\delta}{\alpha}\right)^{2}\right.\right. \nonumber\\ &\left.\left.+\frac{1}{2}\left(\frac{\alpha\varrho}{\delta}\right)^{2}\left(\gamma-\frac{\delta}{\alpha}\right)^{2}\right)N_{2}-\frac{2\gamma^{2}}{\delta}N_{3}\right]\left\|\mathcal{A}^{\frac{1}{2}}u_{tt}\right\|^{2}-\frac{\alpha}{2}N\left(g''\circ\mathcal{A}^{\frac{1}{2}}u\right)(t) \nonumber\\ &-\left[\left(\frac{3\delta}{8}-\frac{\varepsilon_{2}\delta}{4}\right)N_{3}-\left(\frac{(\varrho-l)^{2}\alpha^{2}}{2\varepsilon_{1}}+\frac{4\lambda_{0}g^{2}(0)\alpha^{2}}{G(t)}\right)N_{2}\right]\left\|\mathcal{A}^{\frac{1}{2}}u_{t}\right\|^{2} \nonumber\\ &-\left(\frac{G(t)}{4}N_{2}-1-\frac{\varepsilon_{2}\delta^{3}N_{3}}{8\alpha^{2}\varrho^{2}\lambda_{0}}\right)\left\|u_{ttt}+\alpha u_{tt}+\frac{\alpha\varrho}{\delta}u_{t}\right\|^{2}+\frac{\varepsilon\varrho\nu}{2\delta}N\left(g\circ\mathcal{A}^{\frac{1}{2}}u\right)(t) \nonumber\\ &-\left[\frac{\delta}{2\alpha}-\left(\frac{\lambda_{0}(\varrho-l)^{2}}{2} +\left(\frac{\delta^{2}}{\alpha^{2}}+\varrho^{2}\right) +3(\varrho-l)^{2}\right)\varepsilon_{1}N_{2}\right]\nonumber\\ &\times\left\|\mathcal{A}^{\frac{1}{2}}u_{tt} +\alpha\mathcal{A}^{\frac{1}{2}}u_{t}+\frac{\alpha\varrho}{\delta}\mathcal{A}^{\frac{1}{2}}u\right\|^{2} \nonumber\\ &-\left[-\frac{\alpha g'(t)}{2}N-\frac{2\alpha(\varrho-l)^{2}}{\delta} -\left(\frac{\alpha^{2}\varrho^{2}(\varrho-l)^{2}\lambda_{0}\varepsilon_{1}}{2\delta^{2}} +\frac{(\varrho-l)^{2}}{2\varepsilon_{1}}\right.\right. \nonumber\\ &\left.\left.+3(\varrho-l)^{2}\left(\frac{\alpha\varrho}{\delta}\right)^{2}\varepsilon_{1}+\frac{(\varrho-l)^{2}}{2}\left(\frac{\alpha\varrho}{\delta}\right)^{2}\right)N_{2} -\frac{2(\varrho-l)^{2}}{\delta}N_{3}\right]\left\|\mathcal{A}^{\frac{1}{2}}u\right\|^{2} \nonumber\\ &-\left[\frac{\varepsilon\varrho}{2\delta}N-\frac{2\alpha}{\delta}C_{\nu} -\left(\left(\frac{\varepsilon_{1}\alpha^{2}\varrho^{2}\lambda_{0}}{4\delta^{2}} \right.\right.\right.\left.+\frac{3}{4\varepsilon_{1}}+1+\frac{\alpha\varrho}{\delta} +\frac{2\alpha^{4}\varrho^{2}\lambda_{0}}{G(t)\delta^{2}}\right)C_{\nu} \nonumber\\ &\left.\left. +\frac{2\alpha^{2}\varrho^{2}\lambda_{0}}{G(t)\delta^{2}}\right)N_{2} -\frac{N_{3}}{\delta}C_{\nu}\right]\left(h\circ\mathcal{A}^{\frac{1}{2}}u\right)(t). \end{align*} |
At this point, we need to choose our constants very carefully. First, we choose
\begin{align*} \varepsilon_{1} = \frac{\alpha\delta}{2N_{2}\left[\lambda_{0}\alpha^{2}(\varrho-l)^{2}+2(\delta^{2}+\alpha^{2}\varrho^{2})+6\alpha^{2}(\varrho-l)^{2}\right]}\quad\quad and \quad\quad \varepsilon_{2} = \frac{1}{N_{3}}. \end{align*} |
The above choice yields
\begin{align*} &\mathcal{L'}(t)\nonumber\\ &\leq-\left[\alpha\left(\beta-\frac{\alpha\varrho}{\delta}\right)N-\frac{2\alpha\lambda_{0}}{\delta}\left(\beta-\frac{\alpha\varrho}{\delta}\right)^{2}-\left(\frac{2\left(\beta-\frac{\alpha\varrho}{\delta}\right)^{2}}{\varepsilon_{1}}+\frac{4g^{2}(0)}{G(t)}\right)N_{2}\right. \nonumber\\ &\left.-\frac{\delta^{3}}{4\varrho^{2}\lambda_{0}}-\frac{4\alpha^{2}\varrho^{2}\lambda_{0}}{\delta^{3}}N_{3}^{2}-\frac{2\beta^{2}\lambda_{0}}{\delta}N_{3}\right]\|u_{tt}\|^{2}-\left[\left(\alpha\left(\gamma-\frac{\delta}{\alpha}\right)-\frac{\delta g(0)}{2(\alpha-\varepsilon)\varrho}\right)N\right. \nonumber\\ &\left.+\left(\frac{\delta g(t)}{2(\alpha-\varepsilon)\varrho}-\frac{\delta g(t)}{2\alpha\varrho}\right)N-\frac{2\alpha}{\delta}\left(\gamma-\frac{\delta}{\alpha}\right)^{2}-\left(\frac{(\varrho-l)^{2}}{2\varepsilon_{1}}+\frac{1}{2\varepsilon_{1}}\left(\gamma-\frac{\delta}{\alpha}\right)^{2}\right.\right. \nonumber\\ &\left.\left.+\frac{1}{2}\left(\frac{\alpha\varrho}{\delta}\right)^{2}\left(\gamma-\frac{\delta}{\alpha}\right)^{2}\right)N_{2}-\frac{2\gamma^{2}}{\delta}N_{3}\right]\left\|\mathcal{A}^{\frac{1}{2}}u_{tt}\right\|^{2}-\frac{\alpha}{2}N\left(g''\circ\mathcal{A}^{\frac{1}{2}}u\right)(t) \nonumber\\ &-\left[\frac{3\delta}{8}N_{3}-\frac{\delta}{4}-\left(\frac{(\varrho-l)^{2}\alpha^{2}}{2\varepsilon_{1}}+\frac{4\lambda_{0}g^{2}(0) \alpha^{2}}{G(t)}\right)N_{2}\right]\left\|\mathcal{A}^{\frac{1}{2}}u_{t}\right\|^{2} \nonumber\\ &+\frac{\varepsilon\varrho\nu}{2\delta}N\left(g\circ\mathcal{A}^{\frac{1}{2}}u\right)(t)-\left(\frac{G(t)}{4}N_{2}-1-\frac{\delta^{3}}{8\alpha^{2}\varrho^{2}\lambda_{0}}\right)\left\|u_{ttt}+\alpha u_{tt}+\frac{\alpha\varrho}{\delta}u_{t}\right\|^{2}\nonumber\\ & -\frac{\delta}{4\alpha}\left\|\mathcal{A}^{\frac{1}{2}}u_{tt}+\alpha\mathcal{A}^{\frac{1}{2}}u_{t}+\frac{\alpha\varrho}{\delta}\mathcal{A}^{\frac{1}{2}}u\right\|^{2} \nonumber\\ &-\left[-\frac{\alpha g'(t)}{2}N-\frac{2\alpha(\varrho-l)^{2}}{\delta}-\left(\frac{\alpha^{2}\varrho^{2}(\varrho-l)^{2} \lambda_{0}\varepsilon_{1}}{2\delta^{2}}+\frac{(\varrho-l)^{2}}{2\varepsilon_{1}}+3(\varrho-l)^{2} \left(\frac{\alpha\varrho}{\delta}\right)^{2}\varepsilon_{1}\right.\right. \nonumber\\ &\left.\left.+\frac{(\varrho-l)^{2}}{2}\left(\frac{\alpha\varrho}{\delta}\right)^{2}\right)N_{2} -\frac{2(\varrho-l)^{2}}{\delta}N_{3}\right]\left\|\mathcal{A}^{\frac{1}{2}}u\right\|^{2} -\left[\frac{\varepsilon\varrho}{2\delta}N-\frac{2\alpha}{\delta}C_{\nu} \right. \nonumber\\ &-\left(\left(\frac{\varepsilon_{1}\alpha^{2}\varrho^{2}\lambda_{0}}{4\delta^{2}}\right.\right.\left. \left.\left.+\frac{3}{4\varepsilon_{1}}+1+\frac{\alpha\varrho}{\delta}+\frac{2\alpha^{4}\varrho^{2}\lambda_{0}}{G(t)\delta^{2}}\right) C_{\nu}+\frac{2\alpha^{2}\varrho^{2}\lambda_{0}}{G(t)\delta^{2}}\right)N_{2}\right. \nonumber\\ &\left.-\frac{N_{3}}{\delta}C_{\nu}\right]\left(h\circ\mathcal{A}^{\frac{1}{2}}u\right)(t). \end{align*} |
Then, we choose
\begin{align*} \frac{G(t)}{4}N_{2}-1-\frac{\delta^{3}}{8\alpha^{2}\varrho^{2}\lambda_{0}} > 0. \end{align*} |
Next, we choose
\begin{align*} \frac{3\delta}{8}N_{3}-\frac{\delta}{4}-\left(\frac{(\varrho-l)^{2}\alpha^{2}}{2\varepsilon_{1}}+\frac{4\lambda_{0}g^{2}(0)\alpha^{2}}{G(t)}\right)N_{2} > 0. \end{align*} |
Now, as
\begin{align*} \nu C_{\nu} = \int_{0}^{\infty}\frac{\nu^{2}g(s)}{\nu g(s)-g'(s)}ds\rightarrow 0, \quad\quad as \quad\quad \nu\rightarrow 0. \end{align*} |
Hence, there is
\begin{align*} \nu C_{\nu} < \frac{1}{16\left(\frac{2\alpha}{\delta}+\left(\frac{\varepsilon_{1}\alpha^{2}\varrho^{2}\lambda_{0}}{4\delta^{2}}+\frac{3}{4\varepsilon_{1}}+1+\frac{\alpha\varrho}{\delta}+\frac{2\alpha^{4}\varrho^{2}\lambda_{0}}{G(t)\delta^{2}}\right)N_{2}+\frac{N_{3}}{\delta}\right)}. \end{align*} |
Now, let us choose
\begin{align*} \frac{\varepsilon\varrho}{4\delta}N-\frac{2\alpha^{2}\varrho^{2}\lambda_{0}}{G(t)\delta^{2}}N_{2} > 0 \quad\quad and \quad\quad \nu = \frac{\delta}{4\varepsilon\varrho N} < \nu_{0}, \end{align*} |
which means
\begin{align*} &\frac{\varepsilon\varrho}{2\delta}N-\frac{2\alpha^{2}\varrho^{2}\lambda_{0}}{G(t)\delta^{2}}N_{2}\nonumber\\ &-C_{\nu}\left(\frac{2\alpha}{\delta}+\left(\frac{\varepsilon_{1}\alpha^{2}\varrho^{2}\lambda_{0}}{4\delta^{2}}+\frac{3}{4\varepsilon_{1}} +1+\frac{\alpha\varrho}{\delta}+\frac{2\alpha^{4}\varrho^{2}\lambda_{0}}{G(t)\delta^{2}}\right)N_{2}+\frac{N_{3}}{\delta}\right) \nonumber\\ > &\frac{\varepsilon\varrho}{2\delta}N-\frac{2\alpha^{2}\varrho^{2}\lambda_{0}}{G(t)\delta^{2}}N_{2}-\frac{1}{16\nu} = \frac{\varepsilon\varrho}{4\delta}N-\frac{2\alpha^{2}\varrho^{2}\lambda_{0}}{G(t)\delta^{2}}N_{2} > 0 \end{align*} |
and
\begin{align*} &\alpha\left(\beta-\frac{\alpha\varrho}{\delta}\right)N-\frac{2\alpha\lambda_{0}}{\delta}\left(\beta-\frac{\alpha\varrho}{\delta}\right)^{2}-\left(\frac{2\left(\beta-\frac{\alpha\varrho}{\delta}\right)^{2}}{\varepsilon_{1}}+\frac{4g^{2}(0)}{G(t)}\right)N_{2} \nonumber\\ &-\frac{\delta^{3}}{4\varrho^{2}\lambda_{0}}-\frac{4\alpha^{2}\varrho^{2}\lambda_{0}}{\delta^{3}}N_{3}^{2}-\frac{2\beta^{2}\lambda_{0}}{\delta}N_{3} > 0, \nonumber\\ &\left(\alpha\left(\gamma-\frac{\delta}{\alpha}\right)-\frac{\delta g(0)}{2(\alpha-\varepsilon)\varrho}\right)N +\left(\frac{\delta g(t)}{2(\alpha-\varepsilon)\varrho}-\frac{\delta g(t)}{2\alpha\varrho}\right)N-\frac{2\alpha}{\delta}\left(\gamma-\frac{\delta}{\alpha}\right)^{2} \nonumber\\ &-\left(\frac{(\varrho-l)^{2}}{2\varepsilon_{1}}+\frac{1}{2\varepsilon_{1}}\left(\gamma-\frac{\delta}{\alpha}\right)^{2} +\frac{1}{2}\left(\frac{\alpha\varrho}{\delta}\right)^{2}\left(\gamma-\frac{\delta}{\alpha}\right)^{2}\right)N_{2}-\frac{2\gamma^{2}}{\delta}N_{3} > 0, \nonumber\\ &-\frac{\alpha g'(t)}{2}N-\frac{2\alpha(\varrho-l)^{2}}{\delta}-\left(\frac{\alpha^{2}\varrho^{2}(\varrho-l)^{2}\lambda_{0}\varepsilon_{1}}{2\delta^{2}}+\frac{(\varrho-l)^{2}}{2\varepsilon_{1}}+3(\varrho-l)^{2}\left(\frac{\alpha\varrho}{\delta}\right)^{2}\varepsilon_{1}\right. \nonumber\\ &\left.+\frac{(\varrho-l)^{2}}{2}\left(\frac{\alpha\varrho}{\delta}\right)^{2}\right)N_{2}-\frac{2(\varrho-l)^{2}}{\delta}N_{3} > 4(\varrho-l). \end{align*} |
So we arrive at, for positive constant
\begin{align*} \mathcal{L'}(t)\leq&-c\left[\|u_{tt}\|^{2}+\left\|\mathcal{A}^{\frac{1}{2}}u_{tt}\right\|^{2}+\left\|u_{ttt}+\alpha u_{tt}+\frac{\alpha\varrho}{\delta}u_{t}\right\|^{2} \right. \nonumber\\ &\left.+\left\|\mathcal{A}^{\frac{1}{2}}u_{tt}+\alpha\mathcal{A}^{\frac{1}{2}}u_{t}+\frac{\alpha\varrho}{\delta}u\right\|^{2}\right]-4(\varrho-l)\left\|\mathcal{A}^{\frac{1}{2}}u\right\|^{2}+\frac{1}{8}\left(g\circ\mathcal{A}^{\frac{1}{2}}u\right)(t). \end{align*} |
On the other hand, from Lemma 3.2, we find that
\begin{align*} &\left|\mathcal{L}(t)-NE(t)\right| \nonumber\\ \leq&\int_{\Omega}\left|u_{tt}+\alpha u_{t}+\frac{\alpha\varrho}{\delta}u\right|\left|u_{ttt}+\alpha u_{tt}+\frac{\alpha\varrho}{\delta}u_{t}\right|dx \nonumber\\ &+N_{2}\int_{\Omega}\left|u_{ttt}+\alpha u_{tt}+\frac{\alpha\varrho}{\delta}u_{t}\right|\int_{0}^{t}g(t-s) \nonumber\\ & \times\left|\left(u_{tt}+\alpha u_{t}+\frac{\alpha\varrho}{\delta}u\right)(t)-\frac{\alpha\varrho}{\delta} u(s)\right|dsdx \nonumber\\ &+N_{3}\int_{\Omega}\left|u_{ttt}+\alpha u_{tt}\right||u_{t}|dx+N_{3}\frac{\varrho}{2}\left\|\mathcal{A}^{\frac{1}{2}}u\right\|^{2} \nonumber\\ \leq&c E(t). \end{align*} |
Therefore, we can choose
In this section, we will give an estimate to the decay rate for the problem (1.7)-(1.8).
Proof of Theorem 2.2. Our proof starts with the observation that, for any
\begin{align*} &\int_{0}^{t_{0}}g(s)\int_{\Omega}\left|\mathcal{A}^{\frac{1}{2}}u(t)-\mathcal{A}^{\frac{1}{2}}u(t-s)\right|^{2}dxds\nonumber\\ \leq&-\frac{g(0)}{a}\int_{0}^{t_{0}}g'(s)\int_{\Omega}\left|\mathcal{A}^{\frac{1}{2}}u(t)-\mathcal{A}^{\frac{1}{2}}u(t-s)\right|^{2}dxds \nonumber\\ \leq&-\frac{g(0)}{a}\int_{0}^{t}g'(s)\int_{\Omega}\left|\mathcal{A}^{\frac{1}{2}}u(t)-\mathcal{A}^{\frac{1}{2}}u(t-s)\right|^{2}dxds \nonumber\\ \leq&-c E'(t), \end{align*} |
which are derived from (2.2) and Lemma 4.1 and can be used in (4.8).
Taking
\begin{align*} &\mathcal{L'}(t)\nonumber\\ \leq&-c\left[\|u_{tt}\|^{2}+\left\|\mathcal{A}^{\frac{1}{2}}u_{tt}\right\|^{2}+\left\|u_{ttt}+\alpha u_{tt}+\frac{\alpha\varrho}{\delta}u_{t}\right\|^{2}+\left\|\mathcal{A}^{\frac{1}{2}}u_{tt}+\alpha\mathcal{A}^{\frac{1}{2}}u_{t}+\frac{\alpha\varrho}{\delta}u\right\|^{2}\right] \nonumber\\ &-4(\varrho-l)\left\|\mathcal{A}^{\frac{1}{2}}u\right\|^{2}+\frac{1}{8}\left(g\circ\mathcal{A}^{\frac{1}{2}}u\right)(t) \nonumber\\ \leq&-mE(t)+c\left(g\circ\mathcal{A}^{\frac{1}{2}}u\right)(t) \nonumber\\ \leq&-mE(t)-cE'(t)+c\int_{t_{0}}^{t}g(s)\int_{\Omega}\left|\mathcal{A}^{\frac{1}{2}}u(t)-\mathcal{A}^{\frac{1}{2}}u(t-s)\right|^{2}dxds, \end{align*} |
where
\begin{align} \mathcal{F'}(t)& = \mathcal{L'}(t)+cE'(t)\\ & \leq-mE(t)+c\int_{t_{0}}^{t}g(s)\int_{\Omega}\left|\mathcal{A}^{\frac{1}{2}}u(t)-\mathcal{A}^{\frac{1}{2}}u(t-s)\right|^{2}dxds. \end{align} | (5.1) |
We consider the following two cases relying on the ideas presented in [31].
(ⅰ)
We multiply (5.1) by
\begin{align*} \xi(t)\mathcal{F'}(t)\leq&-m\xi(t)E(t)+c\xi(t)\int_{t_{0}}^{t}g(s)\int_{\Omega}\left|\mathcal{A}^{\frac{1}{2}}u(t)-\mathcal{A}^{\frac{1}{2}}u(t-s)\right|^{2}dxds \nonumber\\ \leq&-m\xi(t)E(t)+c\int_{t_{0}}^{t}\xi(s)g(s)\int_{\Omega}\left|\mathcal{A}^{\frac{1}{2}}u(t)-\mathcal{A}^{\frac{1}{2}}u(t-s)\right|^{2}dxds \nonumber\\ \leq&-m\xi(t)E(t)-c\int_{t_{0}}^{t}g'(s)\int_{\Omega}\left|\mathcal{A}^{\frac{1}{2}}u(t)-\mathcal{A}^{\frac{1}{2}}u(t-s)\right|^{2}dxds \nonumber\\ \leq&-m\xi(t)E(t)-c\int_{0}^{t}g'(s)\int_{\Omega}\left|\mathcal{A}^{\frac{1}{2}}u(t)-\mathcal{A}^{\frac{1}{2}}u(t-s)\right|^{2}dxds \nonumber\\ \leq&-m\xi(t)E(t)-cE'(t). \end{align*} |
Therefore,
\begin{align*} \xi(t)\mathcal{F'}(t)+cE'(t)\leq-m\xi(t)E(t). \end{align*} |
As
\begin{align*} \xi(t)\mathcal{F}(t)+cE(t)\sim E(t) \end{align*} |
and
\begin{align*} (\xi\mathcal{F}+cE)'(t)\leq -m\xi(t)E(t), \quad\quad \forall\ t\geq t_{0}. \end{align*} |
It follows immediately that
\begin{align*} E'(t)\leq -m\xi(t)E(t), \quad\quad \forall\ t\geq t_{0}. \end{align*} |
We may now integrate over
\begin{align*} E(t)\leq k_{2}\exp\left(-k_{1}\int_{t_{0}}^{t}\xi(s)ds\right), \quad\quad \forall\ t\geq t_{0}. \end{align*} |
By the continuity of
\begin{align*} E(t)\leq k_{2}\exp\left(-k_{1}\int_{0}^{t}\xi(s)ds\right), \quad\quad \forall\ t > 0. \end{align*} |
(ⅱ)
First, we define the functional
\begin{align*} L(t) = \mathcal{L}(t)+F_{4}(t). \end{align*} |
Obviously,
\begin{align*} L'(t) = &\mathcal{L}'(t)+F_{4}'(t) \nonumber\\ \leq&-c\left[\|u_{tt}\|^{2}+\left\|\mathcal{A}^{\frac{1}{2}}u_{tt}\right\|^{2}+\left\|u_{ttt}+\alpha u_{tt}+\frac{\alpha\varrho}{\delta}u_{t}\right\|^{2} \right. \nonumber\\ &\left.+\left\|\mathcal{A}^{\frac{1}{2}}u_{tt}+\alpha\mathcal{A}^{\frac{1}{2}}u_{t}+\frac{\alpha\varrho}{\delta}u\right\|^{2}\right]-(\varrho-l)\left\|\mathcal{A}^{\frac{1}{2}}u\right\|^{2}-\frac{3}{8}\left(g\circ\mathcal{A}^{\frac{1}{2}}u\right)(t) \nonumber\\ \leq& -b E(t). \end{align*} |
Therefore, integrating the above inequality over
\begin{align*} -L(t_{0})\leq L(t)-L(t_{0})\leq-b\int_{t_{0}}^{t}E(s){\rm d}s. \end{align*} |
It is sufficient to show that
\begin{align} \int_{0}^{\infty}E(s){\rm d}s < \infty \end{align} | (5.2) |
and
\begin{align*} E(t)\leq \frac{c}{t-t_{0}}, \quad\quad\quad \forall t > t_{0}. \end{align*} |
Now, we define a functional
\begin{align*} \lambda(t): = -\int_{t_{0}}^{t}g'(s)\left\|\mathcal{A}^{\frac{1}{2}}u(t)-\mathcal{A}^{\frac{1}{2}}u(t-s)\right\|^{2}{\rm d}s. \end{align*} |
Clearly, we have
\begin{align} \lambda(t)\leq&-\int_{0}^{t}g'(s)\left\| \mathcal{A}^{\frac{1}{2}}u(t)-\mathcal{A}^{\frac{1}{2}}u(t-s)\right\|^{2}ds \\ \leq&-cE'(t), \quad\quad\quad \forall\ t\geq t_{0}. \end{align} | (5.3) |
After that, we define another functional
\begin{align*} I(t): = q\int_{t_{0}}^{t}\left\|\mathcal{A}^{\frac{1}{2}}u(t)-\mathcal{A}^{\frac{1}{2}}u(t-s)\right\|^{2}{\rm d}s. \end{align*} |
Now, the following inequality holds under Lemma 4.1 and (5.2) that
\begin{align} \int_{t_{0}}^{t}\left\|\mathcal{A}^{\frac{1}{2}}u(t)-\mathcal{A}^{\frac{1}{2}}u(t-s)\right\|^{2}{\rm d}s \leq & 2\int_{t_{0}}^{t}\left(\left\|\mathcal{A}^{\frac{1}{2}}u(t)\right\|^{2}+\left\|\mathcal{A}^{\frac{1}{2}}u(t-s)\right\|^{2}\right)ds \\ \leq& 4\int_{t_{0}}^{t}\left(E(t)+E(t-s)\right)ds \\ \leq& 8\int_{t_{0}}^{t}E(0)ds \\ < & \infty. \end{align} | (5.4) |
Then (5.4) allows for a constant
\begin{align} 0 < I(t) < 1; \end{align} | (5.5) |
otherwise we get an exponential decay from (5.1).
Moreover, recalling that
\begin{align*} M(\theta x)\leq \theta M(x), \quad\quad\quad for \quad\quad 0\leq \theta\leq 1 \quad\quad and \quad\quad x\in(0,r]. \end{align*} |
From assumptions
\begin{align*} \lambda(t) = &-\int_{t_{0}}^{t}g'(s)\left\|\mathcal{A}^{\frac{1}{2}}u(t)-\mathcal{A}^{\frac{1}{2}}u(t-s)\right\|^{2}{\rm d}s \nonumber\\ = &\frac{1}{qI(t)}\int_{t_{0}}^{t}I(t)(-g'(s))q\left\|\mathcal{A}^{\frac{1}{2}}u(t)-\mathcal{A}^{\frac{1}{2}}u(t-s)\right\|^{2}{\rm d}s \nonumber\\ \geq& \frac{1}{qI(t)}\int_{t_{0}}^{t}I(t)\xi(s)M(g(s))q\left\|\mathcal{A}^{\frac{1}{2}}u(t)-\mathcal{A}^{\frac{1}{2}}u(t-s)\right\|^{2}{\rm d}s \nonumber\\ \geq&\frac{\xi(t)}{qI(t)}\int_{t_{0}}^{t}M(I(t)g(s))q\left\|\mathcal{A}^{\frac{1}{2}}u(t)-\mathcal{A}^{\frac{1}{2}}u(t-s)\right\|^{2}{\rm d}s \nonumber\\ \geq&\frac{\xi(t)}{q}M\left(\frac{1}{I(t)}\int_{t_{0}}^{t}I(t)g(s)q\left\|\mathcal{A}^{\frac{1}{2}}u(t)-\mathcal{A}^{\frac{1}{2}}u(t-s)\right\|^{2}{\rm d}s\right) \nonumber\\ = &\frac{\xi(t)}{q}M\left(q\int_{t_{0}}^{t}g(s)\left\|\mathcal{A}^{\frac{1}{2}}u(t)-\mathcal{A}^{\frac{1}{2}}u(t-s)\right\|^{2}{\rm d}s\right). \end{align*} |
According to
\begin{align*} \lambda(t)\geq&\frac{\xi(t)}{q}\overline{M}\left(q\int_{t_{0}}^{t}g(s)\left\|\mathcal{A}^{\frac{1}{2}}u(t)-\mathcal{A}^{\frac{1}{2}}u(t-s)\right\|^{2}{\rm d}s\right). \end{align*} |
In this way,
\begin{align*} \int^{t}_{t_{0}}g(s)\left\|\mathcal{A}^{\frac{1}{2}}u(t)-\mathcal{A}^{\frac{1}{2}}u(t-s)\right\|^{2}{\rm d}s\leq\frac{1}{q}\overline{M}^{-1}\left(\frac{q\lambda(t)}{\xi(t)}\right) \end{align*} |
and (5.1) becomes
\begin{align} \mathcal{F'}(t) \leq&-mE(t)+c\int_{t_{0}}^{t}g(s)\int_{\Omega}\left|\mathcal{A}^{\frac{1}{2}}u(t)-\mathcal{A}^{\frac{1}{2}}u(t-s)\right|^{2}dxds \\ \leq&-mE(t)+c\overline{M}^{-1}\left(\frac{q\lambda(t)}{\xi(t)}\right), \quad\quad\quad \forall\ t\geq t_{0}. \end{align} | (5.6) |
Let
\begin{align*} \mathcal{F}_{1}(t): = \overline{M}'\left(\varepsilon_{0}\frac{E(t)}{E(0)}\right)\mathcal{F}(t)+E(t), \quad\quad\quad \forall\ t\geq 0. \end{align*} |
Then, recalling that
\begin{align} \mathcal{F}'_{1}(t) \leq& -mE(t)\overline{M}'\left(\varepsilon_{0}\frac{E(t)}{E(0)}\right)+c\overline{M}'\left(\varepsilon_{0}\frac{E(t)}{E(0)}\right)\overline{M}^{-1}\left(\frac{q\lambda(t)}{\xi(t)}\right)+E'(t). \end{align} | (5.7) |
Taking account of Lemma 2.3, we obtain
\begin{align} &\overline{M}'\left(\varepsilon_{0}\frac{E(t)}{E(0)}\right)\overline{M}^{-1}\left(\frac{q\lambda(t)}{\xi(t)}\right) \\ \leq&\overline{M}^{*}\left(\overline{H}'\left(\varepsilon_{0}\frac{E(t)}{E(0)}\right)\right)+\overline{M}\left(\overline{M}^{-1}\left(\frac{q\lambda(t)}{\xi(t)}\right)\right) \\ = &\overline{M}^{*}\left(\overline{M}'\left(\varepsilon_{0}\frac{E(t)}{E(0)}\right)\right)+\frac{q\lambda(t)}{\xi(t)} \end{align} | (5.8) |
where
\begin{align} &\overline{M}^{*}\left(\overline{M}'\left(\varepsilon_{0}\frac{E(t)}{E(0)}\right)\right) \\ = &\overline{M}'\left(\varepsilon_{0}\frac{E(t)}{E(0)}\right)\left(\overline{M}'\right)^{-1}\left(\overline{M}'\left(\varepsilon_{0}\frac{E(t)}{E(0)}\right)\right)-\overline{M}\left[\left(\overline{M}'\right)^{-1}\left(\overline{M}'\left(\varepsilon_{0}\frac{E(t)}{E(0)}\right)\right)\right] \\ = &\varepsilon_{0}\frac{E(t)}{E(0)}\overline{M}'\left(\varepsilon_{0}\frac{E(t)}{E(0)}\right)-\overline{M}\left(\varepsilon_{0}\frac{E(t)}{E(0)}\right) \\ \leq&\varepsilon_{0}\frac{E(t)}{E(0)}\overline{M}'\left(\varepsilon_{0}\frac{E(t)}{E(0)}\right). \end{align} | (5.9) |
So, combining (5.7), (5.8) and (5.9), we obtain
\begin{align*} \mathcal{F}'_{1}(t)\leq-(mE(0)-c\varepsilon_{0})\frac{E(t)}{E(0)}\overline{M}'\left(\varepsilon_{0}\frac{E(t)}{E(0)}\right)+c\frac{q\lambda(t)}{\xi(t)}+E'(t). \end{align*} |
From this, we multiply the above inequality by
\begin{align*} \xi(t)\mathcal{F}'_{1}(t)\leq-(mE(0)-c\varepsilon_{0})\xi(t)\frac{E(t)}{E(0)}\overline{M}'\left(\varepsilon_{0}\frac{E(t)}{E(0)}\right)+cq\lambda(t)+\xi(t)E'(t). \end{align*} |
Then, using the fact that, as
\begin{align*} \xi(t)\mathcal{F}'_{1}(t) \leq&-(mE(0)-c\varepsilon_{0})\xi(t)\frac{E(t)}{E(0)}M'\left(\varepsilon_{0}\frac{E(t)}{E(0)}\right)-cE'(t). \end{align*} |
Consequently, defining
\begin{align} \mathcal{F}_{2}(t)\sim E(t), \end{align} | (5.10) |
and with a suitable choice of
\begin{align} \mathcal{F}'_{2}(t)\leq-k\xi(t)\left(\frac{E(t)}{E(0)}\right)M'\left(\varepsilon_{0}\frac{E(t)}{E(0)}\right). \end{align} | (5.11) |
Define
\begin{align*} R(t) = \frac{\lambda_{1}\mathcal{F}_{2}(t)}{E(0)}, \quad\quad\lambda_{1} > 0\quad\quad\quad and \quad\quad\quad M_{2}(t) = tM'(\varepsilon_{0} t). \end{align*} |
Moreover, it suffices to show that
\begin{align} \mathcal{F}'_{2}(t) \leq -k\xi(t)M_{2}\left(\frac{E(t)}{E(0)}\right). \end{align} | (5.12) |
According to (5.10) and (5.12), there exist
\begin{align} \lambda_{2}R(t)\leq E(t)\leq\lambda_{3}R(t). \end{align} | (5.13) |
Then, it follows that there exists
\begin{align} k_{1}\xi(t)\leq-\frac{R'(t)}{M_{2}(R(t))}, \quad\quad\quad \forall\ t\geq t_{0}. \end{align} | (5.14) |
Next, we define
\begin{align*} M_{1}(t): = \int_{t}^{r}\frac{1}{sM'(s)}{\rm d}s. \end{align*} |
And based on the properties of
Now, we integrate (5.14) over
\begin{align*} -\int_{t_{0}}^{t}\frac{R'(s)}{M_{2}(R(s))}{\rm d}s\geq k_{1}\int_{t_{0}}^{t}\xi(s){\rm d}s \end{align*} |
so
\begin{align*} k_{1}\int_{t_{0}}^{t}\xi(s){\rm d}s\leq M_{1}\left(\varepsilon_{0}R(t)\right)-M_{1}\left(\varepsilon_{0}R(t_{0})\right), \end{align*} |
which implies that
\begin{align*} M_{1}\left(\varepsilon_{0}R(t)\right)\geq k_{1}\int_{t_{0}}^{t}\xi(s){\rm d}s. \end{align*} |
It is easy to obtain that
\begin{align} R(t)\leq\frac{1}{\varepsilon_{0}}M^{-1}_{1}\left(k_{1}\int_{t_{0}}^{t}\xi(s){\rm d}s\right), \quad\quad\quad \forall\ t\geq t_{0}. \end{align} | (5.15) |
A combining of (5.13) and (5.15) gives the proof.
The authors are grateful to the anonymous referees and the editor for their useful remarks and comments.
[1] | D. Burns, J. George, D. Aucoin, J. Bower, N. Bower, The pathogenesis and clinical management of cutaneous melanoma: an evidence-based review, J. Med. Imaging Radiat. Sci., 50 (2019), 460-469. |
[2] | R. L. Siegel, K. D. Miller, A. Jemal, Cancer statistics, CA. Cancer J. Clin., 70 (2020), 7-30. |
[3] | T. Crosby, R. Fish, B. Coles, M. Mason, Systemic treatments for metastatic cutaneous melanoma, Cochrane Database Syst. Rev., 2 (2018), CD001215. |
[4] |
L. C. van Kempen, M. Redpath, C. Robert, A. Spatz, Molecular pathology of cutaneous melanoma, Melanoma Manag. , 1 (2014), 151-164. doi: 10.2217/mmt.14.23
![]() |
[5] |
C. Lugassy, S. Zadran, L. A. Bentolila, M. Wadehra, R. Prakash, S. T. Carmichael, et al., Angiotropism, pericytic mimicry and extravascular migratory metastasis in melanoma: an alternative to intravascular cancer dissemination, Cancer Microenviron. , 7 (2014), 139-152. doi: 10.1007/s12307-014-0156-4
![]() |
[6] |
S. L. V. Es, M. Colman, J. F. Thompson, S. W. McCarthy, R. A. Scolyer, Angiotropism is an independent predictor of local recurrence and in-transit metastasis in primary cutaneous melanoma, Am. J. Surg. Pathol. , 32 (2008), 1396-1403. doi: 10.1097/PAS.0b013e3181753a8e
![]() |
[7] | L. Mervic, Time course and pattern of metastasis of cutaneous melanoma differ between men and women, PLoS One., 7 (2012), e32955. |
[8] |
N. R. Adler, A. Haydon, C. A. McLean, J. W. Kelly, V. J. Mar, Metastatic pathways in patients with cutaneous melanoma, Pigment Cell Melanoma Res. , 30 (2017), 13-27. doi: 10.1111/pcmr.12544
![]() |
[9] | I. J. Fiddler, Melanoma metastasis, Cancer Control, 2 (1995), 398-404. |
[10] |
C. Haqq, M. Nosrati, D. Sudilovsky, J. Crothers, D. Khodabakhsh, B. L. Pulliam, et al., The gene expression signatures of melanoma progression, Proc. Natl. Acad. Sci. U. S. A. , 102 (2005), 6092-6097. doi: 10.1073/pnas.0501564102
![]() |
[11] | S. Mandruzzato, A. Callegaro, G. Turcatel, S. Francescato, M. C. Montesco, V. Chiarion-Sileni, et al., A gene expression signature associated with survival in metastatic melanoma, J. Transl. Med. , 4 (2006), 1479-5876. |
[12] |
B. Huang, W. Han, Z. F. Sheng, G. L. Shen, Identification of immune-related biomarkers associated with tumorigenesis and prognosis in cutaneous melanoma patients, Cancer Cell Int. , 20 (2020), 020-01271. doi: 10.1186/s12935-020-1101-x
![]() |
[13] |
M. Liao, F. Zeng, Y. Li, Q. Gao, M. Yin, G. Deng, et al., A novel predictive model incorporating immune-related gene signatures for overall survival in melanoma patients, Sci. Rep. , 10 (2020), 12462. doi: 10.1038/s41598-020-69330-2
![]() |
[14] |
O. Kabbarah, C. Nogueira, B. Feng, R. M. Nazarian, M. Bosenberg, M. Wu, et al., Integrative genome comparison of primary and metastatic melanomas, PLoS One, 5 (2010), 0010770. doi: 10.1371/journal.pone.0010770
![]() |
[15] | A. I. Riker, S. A. Enkemann, O. Fodstad, S. Liu, S. Ren, C. Morris, et al., The gene expression profiles of primary and metastatic melanoma yields a transition point of tumor progression and metastasis, BMC Med. Genomics, 1 (2008), 1755-8794. |
[16] |
H. Cirenajwis, H. Ekedahl, M. Lauss, K. Harbst, A. Carneiro, Molecular stratification of metastatic melanoma using gene expression profiling : Prediction of survival outcome and benefit from molecular targeted therapy, Oncotarget, 6 (2015), 12297-12309. doi: 10.18632/oncotarget.3655
![]() |
[17] |
R. Cabrita, M. Lauss, A. Sanna, M. Donia, G. Jönsson, Tertiary lymphoid structures improve immunotherapy and survival in melanoma, Nature, 577 (2020), 561-565. doi: 10.1038/s41586-019-1914-8
![]() |
[18] | V. Nicolaidou, C. Papaneophytou, C. Koufaris, Detection and characterisation of novel alternative splicing variants of the mitochondrial folate enzyme MTHFD2, Mol. Biol. Rep., 47 (2020), 1-8. |
[19] | C. Qi, L. Hong, Z. Cheng, Q. Yin, Identification of metastasis-associated genes in colorectal cancer using metaDE and survival analysis, Oncol. Lett. , 11 (2015), 568-574. |
[20] |
X. Wang, D. D. Kang, K. Shen, C. Song, S. Lu, L. C. Chang, et al., An R package suite for microarray meta-analysis in quality control, differentially expressed gene analysis and pathway enrichment detection, Bioinformatics, 28 (2012), 2534-2536. doi: 10.1093/bioinformatics/bts485
![]() |
[21] |
X. Zhai, Q. Xue, Q. Liu, Y. Guo, Z. Chen, Colon cancer recurrenceassociated genes revealed by WGCNA coexpression network analysis, Mol. Med. Rep. , 16 (2017), 6499-6505. doi: 10.3892/mmr.2017.7412
![]() |
[22] | P. Langfelder and S. Horvath, WGCNA: an R package for weighted correlation network analysis, BMC Bioinf. , 9 (2008), 1471-2105. |
[23] |
J. Cao, S. Zhang, A Bayesian extension of the hypergeometric test for functional enrichment analysis, Biometrics. , 70 (2014), 84-94. doi: 10.1111/biom.12122
![]() |
[24] |
P. Shannon, A. Markiel, O. Ozier, N. S. Baliga, J. T. Wang, D. Ramage, et al., Cytoscape: a software environment for integrated models of biomolecular interaction networks, Genome Res. , 13 (2003), 2498-2504. doi: 10.1101/gr.1239303
![]() |
[25] |
D. W. Huang, B. T. Sherman, R. A. Lempicki, Systematic and integrative analysis of large gene lists using DAVID bioinformatics resources, Nat. Protoc. , 4 (2009), 44-57. doi: 10.1038/nprot.2008.211
![]() |
[26] |
P. Wang, Y. Wang, B. Hang, X. Zou, J. H. Mao, A novel gene expression-based prognostic scoring system to predict survival in gastric cancer, Oncotarget, 7 (2016), 55343-55351. doi: 10.18632/oncotarget.10533
![]() |
[27] |
R. Tibshirani, The lasso method for variable selection in the Cox model, Stat. Med. , 16 (1997), 385-395. doi: 10.1002/(SICI)1097-0258(19970228)16:4<385::AID-SIM380>3.0.CO;2-3
![]() |
[28] | J. J. Goeman, L1 penalized estimation in the Cox proportional hazards model, Biom. J. , 52 (2010), 70-84. |
[29] |
K. H. Eng, E. Schiller, K. Morrel, On representing the prognostic value of continuous gene expression biomarkers with the restricted mean survival curve, Oncotarget, 6 (2015), 36308-36318. doi: 10.18632/oncotarget.6121
![]() |
[30] |
W. Liang, L. Zhang, G. Jiang, Q. Wang, J. He, Development and validation of a nomogram for predicting survival in patients with resected non-small-cell lung cancer, J. Clin. Oncol. , 33 (2015), 861-869. doi: 10.1200/JCO.2014.56.6661
![]() |
[31] | C. Zhang, F. Wang, F. Guo, C. Ye, B. Yang, A 13-gene risk score system and a nomogram survival model for predicting the prognosis of clear cell renal cell carcinoma, Urol. Oncol. , 38 (2020), 74. e1-74. e11. |
[32] |
A. Subramanian, P. Tamayo, V. K. Mootha, S. Mukherjee, B. L. Ebert, M. A. Gillette, et al., Gene set enrichment analysis: a knowledge-based approach for interpreting genome-wide expression profiles, Proc. Natl. Acad. Sci. U. S. A. , 102 (2005), 15545-15550. doi: 10.1073/pnas.0506580102
![]() |
[33] |
X. Zhang, L. Yang, P. Szeto, G. K. Abali, Y. Zhang, A. Kulkarni, et al., The Hippo pathway oncoprotein YAP promotes melanoma cell invasion and spontaneous metastasis, Oncogene, 39 (2020), 5267-5281. doi: 10.1038/s41388-020-1362-9
![]() |
[34] | Z. Kozovska, V. Gabrisova and L. Kucerova, Malignant melanoma: diagnosis, treatment and cancer stem cells, Neoplasma, 63 (2016), 510-517. |
[35] |
H. Moon, L. R. Donahue, E. Choi, P. O. Scumpia, W. E. Lowry, J. K. Grenier, et al., Melanocyte Stem Cell Activation and Translocation Initiate Cutaneous Melanoma in Response to UV Exposure, Cell Stem. Cell, 21 (2017), 665-678. doi: 10.1016/j.stem.2017.09.001
![]() |
[36] |
E. Seroussi, D. Kedra, H. Q. Pan, M. Peyrard, C. Schwartz, P. Scambler, et al., Duplications on human chromosome 22 reveal a novel Ret Finger Protein-like gene family with sense and endogenous antisense transcripts, Genome Res. , 9 (1999), 803-814. doi: 10.1101/gr.9.9.803
![]() |
[37] |
J. Bonnefont, T. Laforge, O. Plastre, B. Beck, S. Sorce, C. Dehay, et al., Primate-specific RFPL1 gene controls cell-cycle progression through cyclin B1/Cdc2 degradation, Cell Death Differ. , 18 (2011), 293-303. doi: 10.1038/cdd.2010.102
![]() |
[38] |
X. Zhang, S. Sun, J. K. Pu, A. C. Tsang, D. Lee, V. O. Man, et al., Long non-coding RNA expression profiles predict clinical phenotypes in glioma, Neurobiol. Dis. , 48 (2012), 1-8. doi: 10.1016/j.nbd.2012.06.004
![]() |
[39] |
M. Toss, I. Miligy, K. Gorringe, K. Mittal, R. Aneja, I. Ellis, et al., Prognostic significance of cathepsin V (CTSV/CTSL2) in breast ductal carcinoma in situ, J. Clin. Pathol. , 73 (2020), 76-82. doi: 10.1136/jclinpath-2019-205939
![]() |
[40] |
C. -L. Lin, T. -W. Hung, T. -H. Ying, C. -J. Lin, Y. -H. Hsieh, C. -M. Chen, Praeruptorin B mitigates the metastatic ability of human renal carcinoma cells through targeting CTSC and CTSV expression, Int. J. Mol. Sci. , 21 (2020), 2919. doi: 10.3390/ijms21082919
![]() |
[41] | Q. L. Liu, Q. L. Liang, Z. Y. Li, Y. Zhou, W. T. Ou, Z. G. Huang, Function and expression of prolyl hydroxylase 3 in cancers, Arch Med. Sci. , 9 (2013), 589-593. |
[42] |
N. Pescador, Y. Cuevas, S. Naranjo, M. Alcaide, D. Villar, M. O. Landázuri, et al., Identification of a functional hypoxia-responsive element that regulates the expression of the egl nine homologue 3 (egln3/phd3) gene, Biochem. J. , 390 (2005), 189-197. doi: 10.1042/BJ20042121
![]() |
[43] |
J. Rodriguez, A. Herrero, S. Li, N. Rauch, A. Quintanilla, K. Wynne, et al., PHD3 regulates p53 protein stability by hydroxylating proline 359, Cell Rep. , 24 (2018), 1316-1329. doi: 10.1016/j.celrep.2018.06.108
![]() |
[44] |
J. M. Roda, Y. Wang, L. A. Sumner, G. S. Phillips, C. B. Marsh, T. D. Eubank, Stabilization of HIF-2α induces sVEGFR-1 production from tumor-associated macrophages and decreases tumor growth in a murine melanoma model, J. Immunol. , 189 (2012), 3168-3177. doi: 10.4049/jimmunol.1103817
![]() |
[45] |
A. Reustle, M. Di Marco, C. Meyerhoff, A. Nelde, J. S. Walz, S. Winter, et al., Integrative -omics and HLA-ligandomics analysis to identify novel drug targets for ccRCC immunotherapy, Genome Med. , 12 (2020), 32-32. doi: 10.1186/s13073-020-00731-8
![]() |
[46] |
Y. Wang, X. Li, W. Liu, B. Li, D. Chen, F. Hu, et al., MicroRNA-1205, encoded on chromosome 8q24, targets EGLN3 to induce cell growth and contributes to risk of castration-resistant prostate cancer, Oncogene, 38 (2019), 4820-4834. doi: 10.1038/s41388-019-0760-3
![]() |
[47] |
S. Li, J. Rodriguez, W. Li, P. Bullova, S. M. Fell, O. Surova, et al., EglN3 hydroxylase stabilizes BIM-EL linking VHL type 2C mutations to pheochromocytoma pathogenesis and chemotherapy resistance, Proc. Natl. Acad. Sci. U. S. A. , 116 (2019), 16997-17006. doi: 10.1073/pnas.1900748116
![]() |
[48] | T. W. Bebee, J. W. Park, K. I. Sheridan, C. C. Warzecha, B. W. Cieply, A. M. Rohacek, et al., The splicing regulators Esrp1 and Esrp2 direct an epithelial splicing program essential for mammalian development, Elife, 15 (2015), 08954. |
[49] |
K. Horiguchi, K. Sakamoto, D. Koinuma, K. Semba, A. Inoue, S. Inoue, et al., TGF-β drives epithelial-mesenchymal transition through δEF1-mediated downregulation of ESRP, Oncogene, 31 (2012), 3190-3201. doi: 10.1038/onc.2011.493
![]() |
[50] |
J. Ueda, Y. Matsuda, K. Yamahatsu, E. Uchida, Z. Naito, M. Korc, et al., Epithelial splicing regulatory protein 1 is a favorable prognostic factor in pancreatic cancer that attenuates pancreatic metastases, Oncogene, 33 (2014), 4485-4495. doi: 10.1038/onc.2013.392
![]() |
[51] | B. Wang, Y. Li, C. Kou, J. Sun, X. Xu, Mining database for the clinical significance and prognostic value of ESRP1 in cutaneous malignant melanoma, Biomed. Res. Int. , 5 (2020), 4985014. |
[52] |
A. Sawant, J. A. Hensel, D. Chanda, B. A. Harris, G. P. Siegal, A. Maheshwari, et al., Depletion of plasmacytoid dendritic cells inhibits tumor growth and prevents bone metastasis of breast cancer cells, J. Immunol. , 189 (2012), 4258-4265. doi: 10.4049/jimmunol.1101855
![]() |
[53] |
A. E. Boyce, J. A. McGrath, T. Techanukul, D. F. Murrell, C. W. Chow, L. McGregor, et al., Ectodermal dysplasia-skin fragility syndrome due to a new homozygous internal deletion mutation in the PKP1 gene, Australas. J. Dermatol. , 53 (2012), 61-65. doi: 10.1111/j.1440-0960.2011.00846.x
![]() |
[54] |
I. Hofmann, Plakophilins and their roles in diseased states, Cell Tissue Res. , 379 (2020), 5-12. doi: 10.1007/s00441-019-03153-0
![]() |
[55] |
P. Lee, S. Jiang, Y. Li, J. Yue, X. Gou, S. Y. Chen, et al., Phosphorylation of Pkp1 by RIPK4 regulates epidermal differentiation and skin tumorigenesis, Embo. J. , 36 (2017), 1963-1980. doi: 10.15252/embj.201695679
![]() |
[56] |
Y. Bao, Y. Guo, Y. Yang, X. Wei, S. Zhang, Y. Zhang, et al., PRSS8 suppresses colorectal carcinogenesis and metastasis, Oncogene, 38 (2019), 497-517. doi: 10.1038/s41388-018-0453-3
![]() |
[57] |
Y. Bao, Q. Wang, Y. Guo, Z. Chen, K. Li, Y. Yang, et al., PRSS8 methylation and its significance in esophageal squamous cell carcinoma, Oncotarget, 7 (2016), 28540-28555. doi: 10.18632/oncotarget.8677
![]() |
[58] |
A. Tamir, A. Gangadharan, S. Balwani, T. Tanaka, U. Patel, A. Hassan, et al., The serine protease prostasin (PRSS8) is a potential biomarker for early detection of ovarian cancer, J. Ovarian Res. , 9 (2016), 016-0228. doi: 10.1186/s13048-016-0226-y
![]() |
[59] |
A. Maurichi, R. Miceli, H. Eriksson, J. Newton-Bishop, J. Nsengimana, M. Chan, et al., Factors affecting sentinel node metastasis in thin (T1) cutaneous melanomas: development and external validation of a predictive nomogram, J. Clin. Oncol. , 38 (2020), 1591-1601. doi: 10.1200/JCO.19.01902
![]() |
[60] | B. Hu, Q. Wei, C. Zhou, M. Ju, L. Wang, L. Chen, et al., Analysis of immune subtypes based on immunogenomic profiling identifies prognostic signature for cutaneous melanoma, Int. Immunopharmacol. , 6 (2020), 107162. |
1. | Zhiyu Tu, Wenjun Liu, Well‐posedness and exponential decay for the Moore–Gibson–Thompson equation with time‐dependent memory kernel, 2023, 0170-4214, 10.1002/mma.9133 | |
2. | Marina Murillo‐Arcila, Well‐posedness for the fourth‐order Moore–Gibson–Thompson equation in the class of Banach‐space‐valued Hölder‐continuous functions, 2023, 46, 0170-4214, 1928, 10.1002/mma.8618 | |
3. | Carlos Lizama, Marina Murillo-Arcila, Well-posedness for a fourth-order equation of Moore–Gibson–Thompson type, 2021, 14173875, 1, 10.14232/ejqtde.2021.1.81 | |
4. | Danhua Wang, Wenjun Liu, Well-posedness and decay property of regularity-loss type for the Cauchy problem of the standard linear solid model with Gurtin–Pipkin thermal law, 2021, 123, 18758576, 181, 10.3233/ASY-201631 | |
5. | Yang Liu, Wenke Li, A class of fourth-order nonlinear parabolic equations modeling the epitaxial growth of thin films, 2021, 14, 1937-1632, 4367, 10.3934/dcdss.2021112 | |
6. | Danhua Wang, Wenjun Liu, Kewang Chen, Well-posedness and decay property for the Cauchy problem of the standard linear solid model with thermoelasticity of type III, 2023, 74, 0044-2275, 10.1007/s00033-023-01964-4 | |
7. | Doaa Atta, Ahmed E. Abouelregal, Hamid M. Sedighi, Rasmiyah A. Alharb, Thermodiffusion interactions in a homogeneous spherical shell based on the modified Moore–Gibson–Thompson theory with two time delays, 2023, 1385-2000, 10.1007/s11043-023-09598-9 | |
8. | İbrahim TEKİN, Identification of the time-dependent lowest term in a fourth order in time partial differential equation, 2023, 72, 1303-5991, 500, 10.31801/cfsuasmas.1127250 | |
9. | Husam Alfadil, Ahmed E. Abouelregal, Marin Marin, Erasmo Carrera, Goufo-Caputo fractional viscoelastic photothermal model of an unbounded semiconductor material with a cylindrical cavity, 2023, 1537-6494, 1, 10.1080/15376494.2023.2278181 | |
10. | Danhua Wang, Wenjun Liu, Global Existence and Decay Property for the Cauchy Problem of the Nonlinear MGT Plate Equation, 2024, 89, 0095-4616, 10.1007/s00245-024-10126-5 | |
11. | Flank D. M. Bezerra, Lucas A. Santos, Maria J. M. Silva, Carlos R. Takaessu, A Note on the Spectral Analysis of Some Fourth-Order Differential Equations with a Semigroup Approach, 2023, 78, 1422-6383, 10.1007/s00025-023-01999-z | |
12. | Carlos Lizama, Marina Murillo-Arcila, On the existence of chaos for the fourth-order Moore–Gibson–Thompson equation, 2023, 176, 09600779, 114123, 10.1016/j.chaos.2023.114123 | |
13. | Wen-jun Liu, Zhi-yu Tu, Equivalence between the internal observability and exponential decay for the Moore-Gibson-Thompson equation, 2024, 39, 1005-1031, 89, 10.1007/s11766-024-4133-5 | |
14. | Ahmed E. Abouelregal, Marin Marin, Holm Altenbach, Thermally stressed thermoelectric microbeam supported by Winkler foundation via the modified Moore–Gibson–Thompson thermoelasticity theory, 2023, 103, 0044-2267, 10.1002/zamm.202300079 |