Cutaneous squamous cell carcinoma (cSCC) is one of the most frequent types of cutaneous cancer. The composition and heterogeneity of the tumor microenvironment significantly impact patient prognosis and the ability to practice precision therapy. However, no research has been conducted to examine the design of the tumor microenvironment and its interactions with cSCC.
We retrieved the datasets GSE42677 and GSE45164 from the GEO public database, integrated them, and analyzed them using the SVA method. We then screened the core genes using the WGCNA network and LASSO regression and checked the model's stability using the ROC curve. Finally, we performed enrichment and correlation analyses on the core genes.
We identified four genes as core cSCC genes: DTYMK, CDCA8, PTTG1 and MAD2L1, and discovered that RORA, RORB and RORC were the primary regulators in the gene set. The GO semantic similarity analysis results indicated that CDCA8 and PTTG1 were the two most essential genes among the four core genes. The results of correlation analysis demonstrated that PTTG1 and HLA-DMA, CDCA8 and HLA-DQB2 were significantly correlated.
Examining the expression levels of four primary genes in cSCC aids in our understanding of the disease's pathophysiology. Additionally, the core genes were found to be highly related with immune regulatory genes, suggesting novel avenues for cSCC prevention and treatment.
Citation: Jiahua Xing, Muzi Chen, Yan Han. Multiple datasets to explore the tumor microenvironment of cutaneous squamous cell carcinoma[J]. Mathematical Biosciences and Engineering, 2022, 19(6): 5905-5924. doi: 10.3934/mbe.2022276
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Cutaneous squamous cell carcinoma (cSCC) is one of the most frequent types of cutaneous cancer. The composition and heterogeneity of the tumor microenvironment significantly impact patient prognosis and the ability to practice precision therapy. However, no research has been conducted to examine the design of the tumor microenvironment and its interactions with cSCC.
We retrieved the datasets GSE42677 and GSE45164 from the GEO public database, integrated them, and analyzed them using the SVA method. We then screened the core genes using the WGCNA network and LASSO regression and checked the model's stability using the ROC curve. Finally, we performed enrichment and correlation analyses on the core genes.
We identified four genes as core cSCC genes: DTYMK, CDCA8, PTTG1 and MAD2L1, and discovered that RORA, RORB and RORC were the primary regulators in the gene set. The GO semantic similarity analysis results indicated that CDCA8 and PTTG1 were the two most essential genes among the four core genes. The results of correlation analysis demonstrated that PTTG1 and HLA-DMA, CDCA8 and HLA-DQB2 were significantly correlated.
Examining the expression levels of four primary genes in cSCC aids in our understanding of the disease's pathophysiology. Additionally, the core genes were found to be highly related with immune regulatory genes, suggesting novel avenues for cSCC prevention and treatment.
The delay effect originates from the boundary controllers in engineering. The dynamics of a system with boundary delay could be described mathematically by a differential equation with delay term subject to boundary value condition such as [20]. There are many results available in literatures on the well-posedness and pullback dynamics of fluid flow models with delays especially the 2D Navier-Stokes equations, which can be seen in [1], [2], [8] and references therein. Inspired by these works, in this paper, we study the stability of pullback attractors for 3D Brinkman-Forchheimer (BF) equation with delay, which is also a continuation of our previous work in [6]. The existence and structure of attractors are significant to understand the large time behavior of solutions for non-autonomous evolutionary equations. Furthermore, the asymptotic stability of trajectories inside invariant sets determines many important properties of trajectories. The 3D Brinkman-Forchheimer equation with delay is given below:
{∂u∂t−νΔu+αu+β|u|u+γ|u|2u+∇p=f(t,ut)+g(x,t),∇⋅u=0, u(t,x)|∂Ω=0,u|t=τ=uτ(x), x∈Ω,uτ(θ,x)=u(τ+θ,x)=ϕ(θ), θ∈(−h,0), h>0. | (1) |
Here,
(1). a general delay
or
(2). the special application of
f(t,ut)=F(u(t−ρ(t))) | (2) |
for a smooth function
The BF equation describes the conservation law of fluid flow in a porous medium that obeys the Darcy's law. The physical background of 3D BF model can be seen in [14], [9], [18], [19]. For the dynamic systems of problem
(a) For problem (1) with delay
(b) For problem (1) with special application of
(c) The asymptotic stability of trajectories inside pullback attractors is further research of the results established in [6]. However, the stability of pullback attractors for (1) with infinite delay is still unknown.
In this section, we give some notations and the equivalent abstract form of (1) in this section.
Denoting
By the Helmholz-Leray projection defined above, (1) can be transformed to the abstract equivalent form
{∂u∂t+νAu+P(αu+β|u|u+γ|u|2u)=Pf(t,ut)+Pg(t,x),u|∂Ω=0,u|t=τ=uτ(x),uτ(θ,x)=ϕ(θ,x) for θ∈(−h,0), | (3) |
then we show our results for (3) with
We also define some Banach spaces on delayed interval as
‖ϕ‖CH=supθ∈[−h,0]‖ϕ(θ)‖H, ‖ϕ‖CV=supθ∈[−h,0]‖ϕ(θ)‖V, |
respectively. The Lebesgue integrable spaces on delayed interval can be denoted as
Some assumptions on the external forces and parameters which will be imposed in our main results are the following:
‖f(t,ξ)−f(t,η)‖H≤Lf‖ξ−η‖CH, for ξ,η∈CH. |
∫tτ‖f(r,ur)−f(r,vr)‖2Hdr≤C2f∫tτ−h‖u(r)−v(r)‖2Hdr, for τ≤t. | (4) |
∫t−∞eηs‖g(s,⋅)‖2V′ds<∞. | (5) |
holds for any
Lemma 3.1. (The Gronwall inequality with differential form) Let
ddtm(t)≤v(t)m(t)+h(t), m(t=τ)=mτ, t≥τ. | (6) |
Then
m(t)≤mτe∫tτv(s)ds+∫tτh(s)e∫tsv(σ)dσds, t≥τ. | (7) |
In this part, we shall present some retarded integral inequalities from Li, Liu and Ju [5]. Consider the following retarded integral inequalities:
‖y(t)‖X≤E(t,τ)‖yτ‖X+∫tτK1(t,s)‖ys‖Xds+∫∞tK2(t,s)‖ys‖Xds+ρ, ∀ t≥τ, | (8) |
where
Let
κ(K1,K2)=supt≥τ(∫tτK1(t,s)ds+∫∞tK2(t,s)ds). |
We assume that
limt→+∞E(t+s,s)=0 | (9) |
uniformly with respect to
Lemma 3.2. (The retarded Gronwall inequality) Denoting
(1) If
‖yt‖X<μρ+ε, | (10) |
for
(2) If
‖yt‖X≤M‖y0‖Xe−λt+γρ, t≥τ | (11) |
for all bounded functions
(3) If
Proof. See Li, Liu and Ju [5].
Remark 1. (The special case:
The minimal family of pullback attractors will be stated here in preparation for our main result.
Lemma 3.3. (1) (See [7], [11]) Assume that
(|a|β−2a−|b|β−2b)⋅(a−b)≥γ0|a−b|β, |
where
(2) The following
|xq−yq|≤Cq(|x|q−1+|y|q−1)|x−y| |
for the integer
Theorem 3.4. Assume that the external forces
Proof. Step 1. Existence of local approximate solution.
By the property of the Stokes operator
Awi=λiwi, i=1,2,⋯. | (12) |
Let
{(∂tum,wj)+ν(∇um,∇wj)+(αum+β|um|um+γ|um|2um,wj)=(f(t,umt),wj)+⟨g,wj⟩,um(τ)=Pmuτ=uτm,umτ(θ,x)=Pmϕ(θ)=ϕm(θ) for θ∈[−h,0], | (13) |
Then it is easy to check that (13) is equivalent to an ordinary differential equations with unknown variable function
Step 2. Uniform estimates of approximate solutions.
Multiplying (13) by
12ddt‖um‖2H+ν‖um‖2V+α‖um‖2H+β‖um‖3L3(Ω)+γ‖um‖4L4(Ω)≤|(g(t)+f(s,umt(s)),um)|≤α‖um‖2H+ν2‖um‖2V+12ν‖g(t)‖2V′+14α‖f(t,umt)‖2H. | (14) |
Integrating in time, using the hypotheses on
‖um‖2H+ν∫tτ‖um‖2Vds+2β∫tτ‖um‖3L3(Ω)ds+2γ∫tτ‖um‖4L4(Ω)ds≤‖uτ‖2H+C2f4α∫0−h‖ϕ(s)‖2Hds+12ν∫tτ‖g(s)‖2V′ds+C2f4α∫tτ‖um‖2Hds. | (15) |
Using the Gronwall Lemma of integrable form, we conclude that
{um} is bounded in the spaceL∞(τ,T;H)∩L2(τ−h,T;V)∩L3(τ,T;L3(Ω))∩L4(τ,T;L4(Ω)). |
Step 3. Compact argument and passing to limit for deriving the global weak solutions.
In this step, we shall prove
dumdt=−νAum−αum−β|um|um−γ|um|2um+P(g(t)+f(t,umt) | (16) |
and assumptions
By virtue of the Aubin-Lions Lemma, we obtain that
{um(t)⇀u(t) weakly * in L∞(τ,T;H),um(t)→u(t) stongly in L2(τ,T;H),um(t)⇀u(t) weakly in L2(τ,T;V),dum/dt⇀du/dt weakly in L2(τ,T;V′),f(⋅,um⋅)⇀f(⋅,u⋅) weakly in L2(τ,T;H),um⇀u(t) weakly in L3(τ,T;L3(Ω)),um⇀u(t) weakly in L4(τ,T;L4(Ω)) | (17) |
which coincides with the initial data
For the purpose of passing to limit in (13), denoting
∫Tτ(β|um|um−β|u|u,wj)ds≤Cλ1β‖um‖4L4(τ,T;L4(Ω))‖um−u‖4L4(τ,T;L4(Ω))+Cβ‖um−u‖L∞(τ,T;H)‖u‖2L2(τ−h,T;H) |
and
∫Tτ(γ|um|2um−γ|u|2u,wj)ds≤Cγ‖um‖2L2(τ,T;V)‖um−u‖4L4(τ,T;L4(Ω))+Cγ‖um−u‖4L4(τ,T;L4(Ω))(‖u‖2L2(τ−h,T;V)+‖um‖4L4(τ,T;L4(Ω))) | (18) |
and the convergence of delayed external force
Thus, passing to the limit of (13), we conclude that
Proposition 1. Assume that the external forces
Proof. Taking inner product of (3) with
12ddt‖A1/2u‖2H+ν‖Au‖2H+α‖A1/2u‖2H+β∫Ω|u|u⋅Audx+γ∫Ω|u|2u⋅Audx=(f(t,ut),Au)+(g(t),Au). | (19) |
According to Lemma 3.3, the nonlinear terms have the following estimates
|β(|u|u,Au)|≤ν2‖Au‖2H+β4ν‖u‖4L4 | (20) |
and
γ∫Ω|u|2u⋅Audx=γ2∫Ω|∇(|u|2)|2dx+γ∫Ω|u|2|∇u|2dx | (21) |
and
(f(t,ut),Au)+(g(t),Au)≤12ν‖f(t,ut)‖2H+12ν‖g(t)‖2H+ν2‖Au‖2H, | (22) |
hence, we conclude that
ddt‖A1/2u‖2H+2α‖A1/2u‖2H+γ∫Ω|∇(|u|2)|2dx+2γ∫Ω|u|2|∇u|2dx≤β2ν‖u‖4L4+1ν‖f(t,ut)‖2H+1ν‖g(t)‖2H. | (23) |
Letting
‖A1/2u(t)‖2H+2α∫ts‖A1/2u(r)‖2Hdr≤‖A1/2u(s)‖2H+β2ν∫ts‖u(r)‖4L4dr+2ν∫ts‖f(r,ur)‖2Hdr+2ν∫ts‖g(r)‖2Hdr | (24) |
and
∫ts‖f(r,ur)‖2Hdr≤L2f‖ϕ(θ)‖2L2H+L2f∫ts‖u(r)‖2Hdr. | (25) |
Then integrating with
‖A1/2u(t)‖2H≤∫tt−1‖A1/2u(s)‖2Hds+β2ν∫tt−1‖u(r)‖4L4dr+2L2fν‖ϕ(θ)‖2L2H+2L2fν∫tτ‖u(r)‖2Hdr+2ν∫tt−1‖g(r)‖2Hdr≤C[‖ϕ‖2L2H+‖uτ‖2H]+C∫tτ‖g‖2Hds+2L2fνλ1∫tτ‖u(r)‖2Vdr, | (26) |
which means the uniform boundedness of the global weak solution
Proposition 2. Assume the hypotheses in Theorem 3.4 hold. Then the global weak solution
Proof. Using the same energy estimates as above, we can deduce the uniqueness easily, here we skip the details.
To description of pullback attractors, the functional space
∫tτeηs‖f(s,us)‖2Hds<C2f∫tτ−heηs‖u(s)‖2Hds. | (27) |
for any
Proposition 3. For given
Lemma 3.5. Assume that
‖u(t)‖2H≤e−8ηCfα(t−τ)(‖uτ‖2H+Cf‖ϕ(r)‖2L2H)+e−8ηCfαtν−ηλ−1∫tτeηr‖g(r)‖2V′dr | (28) |
and
ν∫ts‖u(r)‖2Vdr≤‖u(s)‖2H+8Cfα‖us‖2L2H+1ν∫ts‖g(r)‖2V′dr+8Cfα∫ts‖u(r)‖2Hdr, | (29) |
β∫ts‖u(r)‖3L3(Ω)dr≤‖u(s)‖2H+8Cfα‖us‖2L2H+1ν∫ts‖g(r)‖2V′dr+8Cfα∫ts‖u(r)‖2Hdr, | (30) |
γ∫ts‖u(r)‖4L4(Ω)dr≤‖u(s)‖2H+8Cfα‖us‖2L2H+1ν∫ts‖g(r)‖2V′dr+8Cfα∫ts‖u(r)‖2Hdr. | (31) |
Proof. By the energy estimate of (1) and using Young's inequality, we arrive at
ddt‖u‖2H+2ν‖u‖2V+2α‖u‖2H+2β‖u‖3L3(Ω)+2γ‖u‖2L4(Ω)≤1ν−ηλ−1‖g‖2V′+(ν−ηλ−1)‖u‖2V+2α‖u‖2H+8α‖f(t,ut)‖2H, | (32) |
where
Multiplying the above inequality by
ddt(eηt‖u‖2H)+eηtνλ1‖u‖2H+2βeηt‖u‖3L3(Ω)+2γeηt‖u‖2L4(Ω)≤1ν−ηλ−1eηt‖g‖2V′+8Cfαeηt‖f(t,ut)‖2H. |
Thus integrating with respect to time variable, it yields
eηt‖u‖2H+νλ1∫tτeηr‖u(r)‖2Hdr≤eητ(‖uτ‖2H+Cf∫0−h‖ϕ(r)‖2Hdr)+1ν−ηλ−1∫tτeηr‖g(r)‖2V′dr+8Cfα∫tτeηr‖u(r)‖2Hdr | (33) |
and by the Gronwall Lemma, we can derive the estimate in our theorem.
Using the energy estimate of (1) again, we can check that
ddt‖u‖2H+2ν‖u‖2V+2α‖u‖2H+2β‖u‖3L3(Ω)+2γ‖u‖2L4(Ω)≤1ν‖g‖2V′+ν‖u‖2V+2α‖u‖2H+8α‖f(t,ut)‖2H, | (34) |
Integrating from
Based on Lemma 3.5, we can present the pullback dissipation based on the following universes for the tempered dynamics.
Definition 3.6. (Universe). (1) We will denote by
limτ→−∞(eητsup(ξ,ζ)∈D(τ)‖(ξ,ζ)‖2MH)=0. | (35) |
(2)
Remark 2. The universes
Proposition 4. (The
D0(t)=¯BH(0,ρH(t))×(¯BL2V(0,ρL2H(t))∩¯BCH(0,ρCH(t))) |
is the pullback
ρ2H(t)=1+e−8ηCfα(t−h)ν−ηλ−1∫t−∞eηr‖g(r)‖2V′dr,ρ2L2V(t)=1ν[1+‖uτ‖2H+8Cfα‖ϕ‖2L2H+‖g(r)‖2L2(t−h,t;V′)ν+8Cfhαρ2H(t)]. |
Moreover, the pullback
Proof. Using the estimates in Lemma 3.5, choosing any
‖u(t,τ;uτ,ϕ)‖2H≤ρ2H(t)=1+e−8ηCfα(t−h)ν−ηλ−1∫t−∞eηr‖g(r)‖2V′dr | (36) |
holds for any
Theorem 3.7. Assume that
Proof. Step 1. Weak convergence of the sequence
For arbitrary fixed
By using the similar energy estimate in Theorem 3.4 and technique in Proposition 4, there exists a pullback time
‖(un)′‖L2(t−h−1,t;V′)≤ν‖un‖L2(t−h−1,t;V)+αλ−11‖un‖L2(t−h−1,t;V)+β‖un‖L4(t−h−1,t;L4(Ω))+Cλ1,|Ω|γ‖un‖L2(t−h−1,t;V)+Cα‖f(t,unt)‖L2(t−h−1,t;H)+Cν‖g‖L2(t−h−1,t;V′). | (37) |
From the hypotheses
{un⇀u weakly * in L∞(t−3h−1,t;H),un⇀u weakly in L2(t−2h−1,t;V),(un)′⇀u′ weakly in L2(t−h−1,t;V′),um⇀u(t) weakly in L3(t−2h−1,t;L3(Ω)),um⇀u(t) weakly in L4(t−2h−1,t;L4(Ω)),un→u stongly in L2(t−h−1,t;H),un(s)→u(s) stongly in H, a.e. s∈(t−h−1,t). | (38) |
By Theorem 3.4, from the hypothesis on
f(⋅,un⋅)⇀f(⋅,u⋅) weakly in L2(t−h−1,t;H). | (39) |
Thus, from (38) and (39), we can conclude that
From the uniform bounded estimate of
\begin{eqnarray} u^n\rightarrow u\ \mbox{strongly in}\ C([t-h-1,t];H). \end{eqnarray} | (40) |
Therefore, we can conclude that
\begin{eqnarray} u^n(s_n) \rightharpoonup u(s)\ \ \mbox{weakly in } H \end{eqnarray} | (41) |
for any
\begin{eqnarray} \liminf\limits_{n\rightarrow\infty}\|u^{n}(s_n)\|_{H}\geq \|u(s)\|_{H}. \end{eqnarray} | (42) |
Step 2. The strong convergence of corresponding sequences via energy equation method:
The asymptotic compactness of sequence
\begin{eqnarray} \|u^{n}(s_n)-u(s)\|_{H}\rightarrow 0\ \mbox{as}\ n\rightarrow+\infty, \end{eqnarray} | (43) |
which is equivalent to prove (42) combining with
\begin{eqnarray} \limsup\limits_{n\rightarrow\infty}\|u^{n}(s_n)\|_H\leq \|u(s)\|_H \end{eqnarray} | (44) |
for a sequence
Using the energy estimate to all
\begin{eqnarray} &&\|u^n(s_2)\|^2_H+\nu \int^{s_2}_{s_1}\|u^n(r)\|^2_Vdr+2\beta \int^{s_2}_{s_1}\|u^n(r)\|^3_{\mathbb{L}^4(\Omega)}dr+2\gamma \int^{s_2}_{s_1}\|u^n(r)\|^4_{\mathbb{L}^4(\Omega)}\\ &\leq& \frac{2C_f^2}{\alpha}\int^{s_2}_{s_2}\|u^n_r\|^2_Hdr+\frac{8}{\nu}\int^{s_2}_{s_1}\|g(r)\|^2_{V'}dr \end{eqnarray} | (45) |
and
\begin{eqnarray} &&\|u(s_2)\|^2_H+\nu \int^{s_2}_{s_1}\|u(r)\|^2_Vdr+2\beta \int^{s_2}_{s_1}\|u(r)\|^3_{\mathbb{L}^4(\Omega)}dr+2\gamma \int^{s_2}_{s_1}\|u(r)\|^4_{\mathbb{L}^4(\Omega)}\\ &\leq& \frac{2C_f^2}{\alpha}\int^{s_2}_{s_2}\|u_r\|^2_Hdr+\frac{8}{\nu}\int^{s_2}_{s_1}\|g(r)\|^2_{V'}dr. \end{eqnarray} | (46) |
Then, we define the functionals
\begin{eqnarray} J_n(s)& = &\frac{1}{2}\|u^n\|^2_H-\int^{s}_{t-h-1}\langle g(r),u^n(r)\rangle dr-\int^{s}_{t-h-1}(f(r,u^n_r),u^n(r))dr \end{eqnarray} | (47) |
and
\begin{eqnarray} J(t)& = &\frac{1}{2}\|u(s)\|^2_H-\int^{s}_{t-h-1}\langle g(r),u(r)\rangle dr-\int^{s}_{t-h-1}(f(r,u_r),u(r))dr. \end{eqnarray} | (48) |
Combining the convergence in (38), observing that
\begin{eqnarray} &&\int^t_{t-h-1}\langle g(r),u^n(r)\rangle dr\rightarrow 2\int^t_{t-h-1}\langle g(r),u(r)\rangle dr \end{eqnarray} | (49) |
and
\begin{eqnarray} &&\int^t_{t-h-1}(f(r,u^n_r),u^n(r))dr\rightarrow 2\int^t_{t-h-1}(f(r,u_r),u(r))dr \end{eqnarray} | (50) |
as
\begin{eqnarray} J_n(s)\rightarrow J(s)\ \ \mbox{a.e.} s\in (t-h-1,t), \end{eqnarray} | (51) |
i.e., for
\begin{eqnarray} |J_n(s_k)-J(s_k)|\leq \frac{\varepsilon}{2}. \end{eqnarray} | (52) |
Since
\begin{eqnarray} |J(s_{k})-J(s)|\leq \frac{\varepsilon}{2}, \end{eqnarray} | (53) |
Choosing
\begin{eqnarray} |J_n(s_n)-J(s)|\leq |J_n(s_n)-J(s_n)|+|J(s_n)-J(s)| < \varepsilon. \end{eqnarray} | (54) |
Therefore, for any
\begin{eqnarray} \limsup\limits_{n\rightarrow\infty}J_n(s_n)\leq J(s), \end{eqnarray} | (55) |
which implies
\begin{eqnarray} \limsup\limits_{n\rightarrow \infty}\|u^n(s_n)\|_H \leq \|u(s)\|_H. \end{eqnarray} | (56) |
we conclude the strong convergence
Step 3. The strong convergence:
Combining the energy estimates in (45) and (46), noting the energy functionals
\begin{eqnarray} \|u^n(s)\|_{L^2(t-h,t;V)}\rightarrow \|u(s)\|_{L^2(t-h,t;V)}. \end{eqnarray} | (57) |
Hence jointing with the weak convergence in (38), we can derive that
Step 4. The
By using the results from Steps 2 to 4 and noting the definition of universe, we can conclude that the processes is
Remark 3. Using the similar technique, we can derive the processes
Theorem 3.8. Assume that
\begin{eqnarray} \mathcal{A}_{\mathcal{D}^{M_H}_{F}}(t) \subset \mathcal{A}_{\mathcal{D}^{M_H}_{\eta}}(t). \end{eqnarray} | (58) |
Proof. From Proposition 3, we observe that the process
Based on the universes defined in Definition 3.6, the relation between
Definition 3.9. The pullback attractors is asymptotically stable if the trajectories inside attractor reduces to a single orbit as
Theorem 3.10. Assume that
\mathit{\mbox{G}}(t)\leq K_0, |
where
\begin{equation} K_0 = \Big\{[\nu^2\lambda_1(2\nu\lambda_1+\alpha)]\Big/\Big[4C_{|\Omega|}\beta \Big(\frac{L_f^2}{\alpha}\frac{2-\frac{L^2_f}{\alpha}}{1-\frac{2L^2_f}{\alpha}}+\frac{1}{\alpha}\Big)\Big]\Big\}^{1/2},\nonumber \end{equation} |
here
Proof. Let
\begin{eqnarray} u(\tau+\theta)|_{\theta\in [-h,0]} = \phi(\theta),\ \ \ u|_{t = \tau} = u_{\tau} \end{eqnarray} | (59) |
and
\begin{eqnarray} v(\tau+\theta)|_{\theta\in [-h,0]} = \tilde{\phi}(\theta), \ \ v|_{t = \tau} = \tilde{u}_{\tau} \end{eqnarray} | (60) |
respectively. Denoting
\begin{eqnarray} (u,u_t) = U(t,\tau)(u_{\tau},\varphi)\ \ \mbox{and}\ \ (v,v_t) = U(t,\tau)(\tilde{u}_{\tau},\tilde{\varphi}) \end{eqnarray} | (61) |
as two trajectories inside the pullback attractors, letting
\begin{equation} \begin{cases} \frac{\partial w}{\partial t}+\nu A w +P\Big(\alpha w+\beta(|u|u-|v|v)+\gamma (|u|^2u-|v|^2v)\Big)&\\ \quad = P(f(t, u_t)-f(t,v_t)),& \\ w|_{\partial\Omega} = 0,& \\ w(t = \tau) = u_{\tau}-\tilde{u}_{\tau},&\\ w(\tau+\theta) = \phi(\theta)-\tilde{\phi}(\theta),\ \theta\in [-h,0].& \end{cases} \end{equation} | (62) |
Taking inner product of (62) with
\begin{eqnarray} \gamma(|u|^2u-|v|^2v, u-v)\geq \gamma \gamma_0 \|u-v\|^4_{\bf{L}^4} \end{eqnarray} | (63) |
and
\begin{eqnarray} &&\frac{1}{2}\frac{d}{d t}\|w\|^2_H+\nu\|w\|^2_V+\alpha \|w\|_H^2+\gamma \gamma_0 \|w\|^4_{\bf{L}^4}\\ &\leq& \Big|\beta(|u|u-|v|v,w)\Big|+\Big|(f(t,u_t)-f(t,v_t),w)\Big|\\ &\leq&\beta\Big(\int_{\Omega}|u|^2|w|dx+\int_{\Omega}|w||v|^2dx\Big)+\frac{\alpha}{2}\|w\|^2_H+\frac{L_f^2}{2\alpha}\|w_t\|^2_H \end{eqnarray} |
\begin{eqnarray} &\leq& \beta(\|u\|^2_{\bf{L}^4}+\|v\|^2_{\bf{L}^4})\|w\|^2_{H}+\frac{\alpha}{2}\|w\|^2_H+\frac{L_f^2}{2\alpha}\|w_t\|^2_H\\ &\leq& C_{|\Omega|}\beta(\|u\|^2_{V}+\|v\|^2_{V})\|w\|^2_{H}+\frac{\alpha}{2}\|w\|^2_H+\frac{L_f^2}{2\alpha}\|w_t\|^2_H. \end{eqnarray} | (64) |
Using the Poincaré inequality and Lemma 3.1, noting that if
\begin{eqnarray} 2\nu\lambda_1+\alpha-2C_{|\Omega|}\beta(\|u\|^2_V+\|v\|^2_V) > 0, \end{eqnarray} | (65) |
then we can obtain
\begin{eqnarray} \|w\|^2_H&\leq& e^{\int^t_{\tau}[2C_{|\Omega|}\beta(\|u\|^2_V+\|v\|^2_V)-(2\nu\lambda_1+\alpha)]ds}\Big[\|u_{\tau}-\tilde{u}_{\tau}\|^2_H+\\ &&+\frac{L_f^2}{\alpha}\int^t_{\tau}e^{-\int^t_{s}[2\nu\lambda_1+\alpha-2C_{|\Omega|}\beta(\|u\|^2_V+\|v\|^2_V)]d\sigma}\|w_t\|^2_Hds\Big]. \end{eqnarray} | (66) |
Denoting
\begin{eqnarray} E(t,\tau) = e^{-\int^t_{\tau}[2\nu\lambda_1+\alpha-2C_{|\Omega|}\beta(\|u\|^2_V+\|v\|^2_V)]ds} \end{eqnarray} | (67) |
and
\begin{eqnarray} K_1(t,s) = \frac{L_f^2}{\alpha}e^{-\int^t_{s}[2\nu\lambda_1+\alpha-2C_{|\Omega|}\beta(\|u\|^2_V+\|v\|^2_V)]d\sigma} \end{eqnarray} | (68) |
and
\begin{eqnarray} \Theta = \sup\limits_{t\geq s\geq \tau}E(t,s),\ \ \ \ \kappa(K_1,0) = \sup\limits_{t\geq\tau}\int^t_{\tau}K_1(t,s)ds, \end{eqnarray} | (69) |
by virtue of Lemma 3.2, choosing
\begin{eqnarray} \|w_t\|^2_H&\leq& M\|u_{\tau}-\tilde{u}_{\tau}\|^2_H e^{-\lambda (t-\tau)}. \end{eqnarray} | (70) |
Substituting (70) into (64), using Lemma 3.1 again, we can conclude the following estimate
\begin{eqnarray} \|w\|^2_H &\leq& \|u_{\tau}-\tilde{u}_{\tau}\|^2_He^{-\int^t_{\tau}[2\nu\lambda_1+\alpha-2C_{|\Omega|}\beta(\|u\|^2_V+\|v\|^2_V)]ds}\\ &&+\frac{L_f^2}{\alpha}M\|u_{\tau}-\tilde{u}_{\tau}\|^2_H e^{-\lambda (t-\tau)}\int^t_{\tau}e^{-\int^t_{s}[2\nu\lambda_1+\alpha-2C_{|\Omega|}\beta(\|u\|^2_V+\|v\|^2_V)]d\sigma}ds.\\ \end{eqnarray} | (71) |
From (70) and (71), if we fixed
\begin{equation} 2\nu\lambda_1+\alpha > 2C_{|\Omega|}\beta \langle \|u\|^2_V+\|v\|^2_V\rangle_{\leq t}, \end{equation} | (72) |
where
\begin{equation} \langle h \rangle_{\leq t} = \limsup\limits_{\tau\rightarrow -\infty}\frac{1}{t-\tau}\int^t_{\tau}h(r)dr. \end{equation} | (73) |
Since
\begin{eqnarray} &&\frac{1}{2}\frac{d}{dt}\|u\|^2_H+\nu\|A^{1/2}u\|^2_H+\alpha\|u\|^2_H+\beta \|u\|^3_{\bf{L}^3}+\gamma \|u\|^4_{\bf{L}^4}\\ &\leq&\alpha \|u\|^2_H+\frac{1}{2\alpha}\Big[\|f(t,u_t)\|^2_H+\|g\|^2_{H}\Big]\\ &\leq&\alpha \|u\|^2_H+\frac{L_f^2}{2\alpha}\|u_t\|^2_H+\frac{1}{2\alpha}\|g\|^2_{H}. \end{eqnarray} | (74) |
Using the Poincaré inequality and Lemma 3.1, then we can obtain
\begin{eqnarray} \|u\|^2_H&\leq& e^{-2\nu\lambda_1(t-\tau)}\|u_{\tau}\|^2_H+\\ &&+\frac{L_f^2}{\alpha}\int^t_{\tau}e^{-2\nu\lambda_1(t-s)}\|u_s\|^2_Hds+\frac{1}{\alpha}\int^t_{\tau}e^{-2\nu\lambda_1(t-s)}\|g\|^2_Hds. \end{eqnarray} | (75) |
Denoting
\begin{eqnarray} E(t,\tau) = e^{-2\nu\lambda_1(t-\tau)} \end{eqnarray} | (76) |
and
\begin{eqnarray} K_1(t,s) = \frac{L_f^2}{\alpha}e^{-2\nu\lambda_1(t-s)} \end{eqnarray} | (77) |
and
\begin{equation} \rho = \frac{1}{\alpha}\int^t_{\tau}e^{-2\nu\lambda_1(t-s)}\|g\|^2_Hds, \end{equation} | (78) |
letting
\begin{eqnarray} \Theta = \sup\limits_{t\geq s\geq \tau}E(t,s),\ \ \ \ \kappa(K_1,0) = \sup\limits_{t\geq\tau}\int^t_{\tau}K_1(t,s)ds, \end{eqnarray} | (79) |
by virtue of Lemma 3.2, choosing
\begin{eqnarray} \|u_t\|^2_H&\leq& \hat{M}\|u_{\tau}\|^2_H e^{-\lambda (t-\tau)}+\frac{2-\frac{L^2_f}{\alpha}}{1-\frac{2L^2_f}{\alpha}}\int^t_{\tau}e^{-2\nu\lambda_1(t-s)}\|g\|^2_Hds\\ &\leq& \hat{M}\|u_{\tau}\|^2_H e^{-\lambda (t-\tau)}+\frac{2-\frac{L^2_f}{\alpha}}{1-\frac{2L^2_f}{\alpha}}\int^t_{\tau}\|g\|^2_Hds. \end{eqnarray} | (80) |
Substituting (80) into (75), using Lemma 3.1 again, we can conclude the following estimate
\begin{eqnarray} \|u\|^2_H &\leq& C\|u_{\tau}\|^2_He^{-\lambda (t-\tau)}+\Big(\frac{L_f^2}{\alpha}\frac{2-\frac{L^2_f}{\alpha}}{1-\frac{2L^2_f}{\alpha}}+\frac{1}{\alpha}\Big)\int^t_{\tau}\|g\|^2_Hds. \end{eqnarray} | (81) |
Integrating (74) from
\begin{eqnarray} &&\|u\|^2_H+2\nu\int^t_{\tau}\|u\|^2_Vds+2\beta\int^t_{\tau} \|u\|^3_{\bf{L}^3}ds+2\gamma \int^t_{\tau}\|u\|^4_{\bf{L}^4}ds\\ &\leq& \Big[\frac{1}{\alpha}\|\phi\|^2_{L^2_H}+\|u_{\tau}\|^2_H\Big]+\frac{L_f^2}{\alpha}\int^{t}_{\tau}\|u_t(s)\|^2_Hds+\frac{1}{\alpha}\int^t_{\tau}\|g\|^2_{H}ds. \end{eqnarray} | (82) |
By the estimate of (80) and (81), we derive
\begin{eqnarray} \int^t_{\tau}\|u(r)\|^4_{\bf{L}^4}dr\leq C\Big[\frac{1}{\alpha}\|\phi\|^2_{L^2_H}+\|u_{\tau}\|^2_H\Big]+\Big(\frac{L_f^2}{\alpha}\frac{2-\frac{L^2_f}{\alpha}}{1-\frac{2L^2_f}{\alpha}}+\frac{1}{\alpha}\Big)\int^t_{\tau}\|g\|^2_{H}ds \end{eqnarray} | (83) |
and
\begin{eqnarray} \int^t_{\tau}\|u(r)\|^2_{V}dr\leq C\Big[\frac{1}{\alpha}\|\phi\|^2_{L^2_H}+\|u_{\tau}\|^2_H\Big]+\Big(\frac{L_f^2}{\alpha}\frac{2-\frac{L^2_f}{\alpha}}{1-\frac{2L^2_f}{\alpha}}+\frac{1}{\alpha}\Big)\int^t_{\tau}\|g\|^2_{H}ds. \end{eqnarray} | (84) |
Combining (72), (73) with (84), we conclude that
\begin{eqnarray} && \langle \|u\|^2_V+\|v\|^2_V\rangle|_{\leq t}\leq 2 \Big(\frac{L_f^2}{\alpha}\frac{2-\frac{L^2_f}{\alpha}}{1-\frac{2L^2_f}{\alpha}}+\frac{1}{\alpha}\Big)\langle \|g\|^2_H\rangle|_{\leq t} \end{eqnarray} | (85) |
and hence the asymptotic stability holds provided that
\begin{eqnarray} 4C_{|\Omega|}\beta \Big(\frac{L_f^2}{\alpha}\frac{2-\frac{L^2_f}{\alpha}}{1-\frac{2L^2_f}{\alpha}}+\frac{1}{\alpha}\Big)\langle \|g\|^2_H\rangle|_{\leq t} \leq 2\nu\lambda_1+\alpha. \end{eqnarray} | (86) |
If we define the generalized Grashof number as
\begin{eqnarray} G(t)\leq \Big\{(2\nu\lambda_1+\alpha)\Big/\Big[4C_{|\Omega|}\nu^2\beta\lambda_1 \Big(\frac{L_f^2}{\alpha}\frac{2-\frac{L^2_f}{\alpha}}{1-\frac{2L^2_f}{\alpha}}+\frac{1}{\alpha}\Big)\Big]\Big\}^{1/2} = K_0, \end{eqnarray} | (87) |
which completes the proof for our first result.
Remark 4. Theorem 3.10 is a further research for the existence of pullback attractor in [6].
We first state some hypothesis on the external forces and sub-linear operator.
\Big|\frac{d\rho}{dt}\Big|\leq\rho^{\ast} < 1, \ \ \forall t\geq 0. |
\begin{eqnarray} \|F(y)\|^2_H\leq a(t)\|y\|^2_H+b(t), \ \ \forall t\geq\tau, y\in H. \end{eqnarray} | (88) |
\begin{eqnarray} \|F(u)-F(v)\|_H\leq L(R)\kappa^\frac{1}{2}(t)\|u-v\|_H, \ u,v\in H. \end{eqnarray} | (89) |
holds for
\begin{eqnarray} \int^{t}_{-\infty}e^{ms}\|g(s,\cdot)\|^{2}_Hds < \infty, \ \ \forall t\in\mathbb{R}. \end{eqnarray} | (90) |
\begin{eqnarray} \frac{\nu}{2}-\frac{\|a\|_{L^q_{loc}(\mathbb{R})}}{1-\rho^\ast} > 0. \end{eqnarray} | (91) |
In this part, the well-posedness and pullback attractors for problem (1) with sub-linear operator will be stated for our discussion in sequel.
Assume that the initial date
\begin{equation} \begin{cases} u(t)+\int^t_\tau P(\nu Au+\alpha u+\beta|u|u+\gamma |u|^2u)ds &\\ \quad = u(\tau) +\int^t_\tau P\Big(F\big(u(s-\rho(s))\big)+g(s,x)\Big)ds,& \\ w|_{\partial\Omega} = 0,& \\ u(t = \tau) = u_{\tau},&\\ u(\tau+t) = \phi(t),\ t\in [-h,0],& \end{cases} \end{equation} | (92) |
which possesses a global mild solution as the following theorem.
Theorem 4.1. Assume that the external forces
\begin{eqnarray} &&\|u(t)\|^{2}_H+2\nu\int^t_{\tau}\|u(s)\|^{2}_Vds+2\alpha \int^t_{\tau}\|u(s)\|^{2}_Hds\\ &&+2\beta\int^t_{\tau}\|u(s)\|^{3}_{\bf{L}^3}ds+2\gamma\int^t_{\tau}\|u(s)\|^{4}_{\bf{L}^4}ds\\ & = &\|u_{\tau}\|^{2}_H+2\int^t_{\tau}\Big[\big(F(u(s-\rho(s))),u(s)\big)+2(g(s,x),u(s))\Big]ds. \end{eqnarray} | (93) |
Moreover, we can define a continuous process
Proof. Using the Galerkin method and compact argument as in Section 3.3, we can easily derive the result.
After obtaining the existence of the global well-posedness, we establish the existence of the pullback attractors to (1) with sub-linear operator.
Theorem 4.2. (The pullback attractors in
Proof. Using the similar technique as in Section3.3, we can obtain the existence of pullback attractors, here we skip the details.
Theorem 4.3. We assume that the external forces
Then the trajectories inside pullback attractors
\begin{equation} \mathit{\mbox{G}}(t)\leq \tilde{K}_0, \end{equation} | (94) |
where
\begin{equation} \tilde{K}_0 = \Big\{(2\nu\lambda_1+\alpha)\Big/\Big[2C_{|\Omega|}\beta\nu\lambda_1 (\frac{1}{\alpha^2(1-\rho^*)}+\|\tilde{a}(t)\|_{L^1})\Big]\Big\}^{1/2} > 0,\nonumber \end{equation} |
here
Proof. Step 1. The inequality for asymptotic stability of trajectories.
Let
\begin{eqnarray} u(\theta+\tau)|_{\theta\in [-h,0]} = \phi(\theta)|_{\theta\in[-h,0]},\ \ u|_{t = \tau} = u_{\tau} \end{eqnarray} | (95) |
and
\begin{eqnarray} v(\theta+\tau)|_{\theta\in [-h,0]} = \tilde{\phi}(\theta)|_{\theta\in[-h,0]},\ \ v|_{t = \tau} = \tilde{u}_{\tau} \end{eqnarray} | (96) |
respectively, then
\begin{eqnarray} (u,u_t) = (U(t,\tau)u_{\tau},U(t,\tau)\phi),\ \ (v,v_t) = (U(t,\tau)\tilde{u}_{\tau},U(t,\tau)\tilde{\phi}). \end{eqnarray} | (97) |
If we denote
\begin{equation} \begin{cases} \frac{\partial w}{\partial t}+\nu A w +P\Big(\alpha w+\beta(|u|u-|v|v)+\gamma (|u|^2u-|v|^2v)\Big)&\\ \quad = P\Big(F\big(u(t-\rho(t))\big)-F\big(v(t-\rho(t))\big)\Big),& \\ w|_{\partial\Omega} = 0,& \\ w(t = \tau) = u_{\tau}-\tilde{u}_{\tau},&\\ w(\tau+\theta) = \phi(\theta)-\tilde{\phi}(\theta),\ \theta\in [-h,0].& \end{cases} \end{equation} | (98) |
Multiplying (98) with
\begin{eqnarray} &&\frac{1}{2}\frac{d}{d t}\|w\|^2_H+\nu\|w\|^2_V+\alpha \|w\|_H^2+\gamma \gamma_0 \|w\|^4_{\bf{L}^4}\\ &\leq& |\beta(|u|u-|v|v,w)|+\Big|\Big(F\big(u(t-\rho(t))\big)-F\big(v(t-\rho(t))\big),w\Big)\Big|\\ &\leq& C_{|\Omega|}\beta(\|u\|^2_{V}+\|u\|^2_{V})\|w\|^2_{H}+\frac{\alpha}{2}\|w\|^2_H\\ &&+\frac{1}{\alpha}\|F\big(u(t-\rho(t))\big)-F\big(v(t-\rho(t))\big)\|^2_H\\ &\leq& C_{|\Omega|}\beta(\|u\|^2_{V}+\|u\|^2_{V})\|w\|^2_{H}+\frac{\alpha}{2}\|w\|^2_H\\ &&+\frac{L^2(R)\kappa(t)}{\alpha}\|w(t-\rho(t))\|^2_H. \end{eqnarray} | (99) |
Using the Poincaré inequality and Lemma 3.1, noting that if
\begin{eqnarray} 2\nu\lambda_1+\alpha-2C_{|\Omega|}\beta(\|u\|^2_V+\|v\|^2_V) > 0, \end{eqnarray} | (100) |
then we can obtain
\begin{eqnarray} \|w\|^2_H&\leq& e^{\int^t_{\tau}[2C_{|\Omega|}\beta(\|u\|^2_V+\|v\|^2_V)-(2\nu\lambda_1+\alpha)]ds}\Big[\|u_{\tau}-\tilde{u}_{\tau}\|^2_H+\\ &&+\frac{L^2(R)\|\kappa(t)\|_{L^{\infty}}}{\alpha}\int^t_{\tau}e^{-\int^t_{s}[2\nu\lambda_1+\alpha-2C_{|\Omega|}\beta(\|u\|^2_V+\|v\|^2_V)]d\sigma}\\ && \quad \times\|w(t-\rho(t))\|^2_Hds\Big]. \end{eqnarray} | (101) |
Denoting
\begin{eqnarray} E(t,\tau) = e^{-\int^t_{\tau}[2\nu\lambda_1+\alpha-2C_{|\Omega|}\beta(\|u\|^2_V+\|v\|^2_V)]ds} \end{eqnarray} | (102) |
and
\begin{eqnarray} K_1(t,s) = \frac{L^2(R)\|\kappa(t)\|_{L^{\infty}}}{\alpha}e^{-\int^t_{s}[2\nu\lambda_1+\alpha-2C_{|\Omega|}\beta(\|u\|^2_V+\|v\|^2_V)]d\sigma} \end{eqnarray} | (103) |
and
\begin{eqnarray} \Theta = \sup\limits_{t\geq s\geq \tau}E(t,s),\ \ \ \ \kappa(K_1,0) = \sup\limits_{t\geq\tau}\int^t_{\tau}K_1(t,s)ds, \end{eqnarray} | (104) |
by virtue of Lemma 3.2, choosing
\begin{eqnarray} \|w(t-\rho(t))\|^2_H&\leq& \tilde{M}\|u_{\tau}-\tilde{u}_{\tau}\|^2_H e^{-\tilde{\lambda} (t-\tau)}. \end{eqnarray} | (105) |
Substituting (105) into (99), using Lemma 3.1 again, we can conclude the following estimate
\begin{eqnarray} \|w\|^2_H &\leq& \|u_{\tau}-\tilde{u}_{\tau}\|^2_He^{-\int^t_{\tau}[2\nu\lambda_1+\alpha-2C_{|\Omega|}\beta(\|u\|^2_V+\|v\|^2_V)]ds}\\ &&+\frac{L^2(R)\|\kappa(t)\|_{L^{\infty}}}{\alpha}\tilde{M}\|u_{\tau}-\tilde{u}_{\tau}\|^2_H e^{-\tilde{\lambda} (t-\tau)}\\ && \quad \times\int^t_{\tau}e^{-\int^t_{s}[2\nu\lambda_1+\alpha-2C_{|\Omega|}\beta(\|u\|^2_V+\|v\|^2_V)]d\sigma}ds. \end{eqnarray} | (106) |
From the result in last section, we can find that the pullback attractors is asymptotically stable as
\begin{equation} 2\nu\lambda_1+\alpha > 2C_{|\Omega|}\beta \langle \|u\|^2_V+\|v\|^2_V\rangle_{\leq t}. \end{equation} | (107) |
Step 2.Some energy estimate for (1) with sub-linear operator.
Multiplying (3) with
\begin{eqnarray} &&\frac{1}{2}\frac{d}{dt}\|u\|^2_H+\nu\|A^{1/2}u\|^2_H+\alpha\|u\|^2_H+\beta \|u\|^3_{\bf{L}^3}+\gamma \|u\|^4_{\bf{L}^4}\\ &\leq&\alpha \|u\|^2_H+\frac{1}{2\alpha}\Big[\|f\big(t,u(t-\rho(t))\big)\|^2_H+\|g\|^2_{H}\Big]. \end{eqnarray} | (108) |
Moreover, let
\begin{equation} d\theta = (1-\rho'(s))ds,\ a(t)\rightarrow \tilde{a}(\bar{t})\in L^p(\tau,T), \end{equation} | (109) |
which means
\begin{align} &\int^t_\tau\|f(s,u(s-\rho(s)))\|^2_Hds\\ \leq&\int^t_\tau a(s)\|u(s-\rho(s))\|^2_Hds+\int^T_\tau b(s)ds\\ \leq& \ \dfrac{1}{1-\rho^*}\int_{\tau-\rho(\tau)}^{t-\rho(t)} \tilde{a}(s)\|u(s)\|^2_Hds+\int^t_\tau b(s)ds\\ \leq& \ \dfrac{1}{1-\rho^*}\left(\int_{-\rho(\tau)}^{0}\tilde{a}(t+\tau)\|\phi(t)\|^2_Hdt +\int^t_\tau\tilde{a}(s)\|u(s)\|^2_Hds\right)+\int^t_\tau b(s)ds\\ \leq& \dfrac{1}{1-\rho^*}\left(\|\phi(t)\|^2_{L^{2q}_{H}}\|\tilde{a}\|_{L^q(\tau-h,\tau)} +\int^t_\tau\tilde{a}(s)\|u(s)\|^2_Hds\right)+\int^t_\tau b(s)ds, \end{align} | (110) |
Integrating (108) with time variable from
\begin{eqnarray} &&\|u\|^2_H+2\nu\int^t_{\tau}\|u\|^2_Vds+2\beta\int^t_{\tau} \|u\|^3_{\bf{L}^3}ds+2\gamma \int^t_{\tau}\|u\|^4_{\bf{L}^4}ds\\ &\leq& \dfrac{\|\tilde{a}\|_{L^q(\tau-h,\tau)}}{\alpha(1-\rho^*)}\|\phi(t)\|^2_{L^{2q}_{H}}+\|u_{\tau}\|^2_H+\frac{1}{\alpha(1-\rho^*)}\int^{t}_{\tau}\tilde{a}(s)\|u(s)\|^2_Hds\\ &&+\frac{1}{\alpha}\int^t_{\tau}\|g\|^2_{H}ds+\frac{1}{\alpha}\int^t_\tau b(s)ds, \end{eqnarray} | (111) |
then we can achieve that
\begin{eqnarray} \|u(t)\|^2_H&\leq& \Big[\dfrac{\|\tilde{a}\|_{L^q(\tau-h,\tau)}}{\alpha(1-\rho^*)}\|\phi(t)\|^2_{L^{2q}_{H}}+\|u_{\tau}\|^2_H\Big]e^{-\chi_{\sigma}(t,\tau)}\\ &&+\frac{1}{\alpha}\int^t_{\tau}\|g\|^2_{H}e^{-\chi_{\sigma}(t,s)}ds+\frac{1}{\alpha}\int^t_{\tau}b(s)e^{-\chi_{\sigma}(t,s)}ds, \end{eqnarray} | (112) |
where the new variable index
\begin{eqnarray} \chi_{\sigma}(t,s) = (2\nu\lambda_1-\sigma)(t-s)-\frac{1}{\alpha(1-\rho^*)}\int^t_{s}\tilde{a}(r)dr, \end{eqnarray} | (113) |
which satisfies the relations
\begin{eqnarray} \chi_{\sigma}(0,t)-\chi_{\sigma}(0,s) = -\chi_{\sigma}(t,s) \end{eqnarray} | (114) |
and
\begin{eqnarray} \chi_{\sigma}(0,r)\leq \chi_{\sigma}(0,t)+\Big(2\nu\lambda_1-\delta\Big)h,\ \ \mbox{if}\ 2\nu\lambda_1+\alpha-\delta > 0 \end{eqnarray} | (115) |
for
Moreover, using the variable index introduced above, we can conclude that
\begin{eqnarray} &&2\nu\int^t_{\tau}\|u(r)\|^2_{V}dr\\ &\leq& \dfrac{\|\tilde{a}\|_{L^q(\tau-h,\tau)}}{\alpha(1-\rho^*)}\|\phi(t)\|^2_{L^{2q}_{H}}+\|u_{\tau}\|^2_H\\ &&+\frac{1}{\alpha}\int^t_{\tau}\|g\|^2_{H}ds+\frac{1}{\alpha}\int^t_\tau b(s)ds\\ &&+\frac{1}{\alpha(1-\rho^*)}\Big[\dfrac{\|\tilde{a}\|_{L^q(\tau-h,\tau)}}{\alpha(1-\rho^*)}\|\phi(t)\|^2_{L^{2q}_{H}}+\|u_{\tau}\|^2_H\Big]\int^t_{\tau}\tilde{a}(s)e^{-\chi_{\sigma}(s,\tau)}ds\\ &&+\frac{1}{\alpha^2(1-\rho^*)}\int^t_{\tau}\|g(s)\|^2_{H}ds\int^t_{\tau}\tilde{a}(s)ds+\frac{\|b\|_{L^1(\tau,T)}}{\alpha^2(1-\rho^*)}\int^t_{\tau}\tilde{a}(s)ds. \end{eqnarray} | (116) |
Step 3. The sufficient condition for asymptotic stability of trajectories inside pullback attractors.
Combining (107) with (116), we conclude that
\begin{eqnarray} &&2C_{|\Omega|}\beta\langle \|u\|^2_V+\|v\|^2_V\rangle|_{\leq t}\\ &\leq& \frac{2C_{|\Omega|}\beta}{\nu}\Big[\Big(\frac{1}{\alpha^2(1-\rho^*)}+\int^t_{\tau}\tilde{a}(s)ds\Big)\langle \|g(t)\|_H^2\rangle|_{\leq t} \\ &&+\frac{1}{\alpha}\langle \|b(t)\|_{L^1}\rangle|_{\leq t}+\frac{\|b\|_{L^1(\tau,T)}}{\alpha^2(1-\rho^*)}\langle \|\tilde{a}(t)\|_{L^1}\rangle|_{\leq t}\Big]. \end{eqnarray} | (117) |
and hence the asymptotic stability holds provided that
\begin{eqnarray} &&\Big(\frac{1}{\alpha^2(1-\rho^*)}+\|\tilde{a}(t)\|_{L^1}\Big)\langle \|g(t)\|_H^2\rangle|_{\leq t}+\frac{1}{\alpha}\langle \|b(t)\|_{L^1}\rangle|_{\leq t}+\frac{\|b\|_{L^1(\tau,T)}}{\alpha^2(1-\rho^*)}\langle \|\tilde{a}(t)\|_{L^1}\rangle|_{\leq t}\\ &&\leq \frac{\nu(2\nu\lambda_1+\alpha)}{2C_{|\Omega|}\beta}. \end{eqnarray} | (118) |
If we define the generalized Grashof number as
\begin{eqnarray} G(t)\leq \Big\{(2\nu\lambda_1+\alpha)\Big/\Big[2C_{|\Omega|}\beta\nu\lambda_1 (\frac{1}{\alpha^2(1-\rho^*)}+\|\tilde{a}(t)\|_{L^1})\Big]\Big\}^{1/2} = \tilde{K}_0, \end{eqnarray} | (119) |
which completes the proof for our first result.
Remark 5. If we denote
\begin{eqnarray} \limsup\limits_{\tau\rightarrow -\infty}\frac{1}{t-\tau}\int^t_{\tau}b(r)dr = b_0\in [0,+\infty) \end{eqnarray} | (120) |
and
\begin{eqnarray} \limsup\limits_{\tau\rightarrow -\infty}\frac{1}{t-\tau}\int^t_{\tau}\tilde{a}(r)dr = \tilde{a}_0\in [0,+\infty), \end{eqnarray} | (121) |
such that there exists some
\begin{eqnarray} \frac{\nu(2\nu\lambda_1+\alpha)}{2C_{|\Omega|}\beta} > \frac{b_0}{\alpha}+\frac{\|b\|_{L^1(\tau,T)}\tilde{\alpha}_0}{\alpha^2(1-\rho^*)}+\delta \end{eqnarray} | (122) |
holds. Then more precise sufficient condition for the asymptotic stability of pullback attractors is
\begin{eqnarray} G(t)\leq \Big[\frac{\frac{\nu(2\nu\lambda_1+\alpha)}{2C_{|\Omega|}\beta}-\frac{b_0}{\alpha}-\frac{\|b\|_{L^1(\tau,T)}\tilde{\alpha}_0}{\alpha^2(1-\rho^*)}}{\nu^2\lambda_1(\frac{1}{\alpha^2(1-\rho^*)}+\|\tilde{a}(t)\|_{L^1})}\Big]^{1/2} \end{eqnarray} | (123) |
which has smaller upper boundedness than (119).
The structure and stability of 3D BF equations with delay are investigated in this paper. A future research in the pullback dynamics of (1) is to study the geometric property of pullback attractors, such as the fractal dimension.
Xin-Guang Yang was partially supported by the Fund of Young Backbone Teacher in Henan Province (No. 2018GGJS039) and Henan Overseas Expertise Introduction Center for Discipline Innovation (No. CXJD2020003). Xinjie Yan was partly supported by Excellent Innovation Team Project of "Analysis Theory of Partial Differential Equations" in China University of Mining and Technology (No. 2020QN003). Ling Ding was partly supported by NSFC of China (Grant No. 1196302).
The authors want to express their most sincere thanks to refrees for the improvement of this manuscript. The authors also want to thank Professors Tomás Caraballo (Universidad de Sevilla), Desheng Li (Tianjin University) and Shubin Wang (Zhengzhou University) for fruitful discussion on this subject.
[1] |
C. Fitzmaurice, D. Abate, N. Abbasi, H. Abbastabar, F. Abd-Allah, O. Abdel-Rahman, et al., Global, regional, and national cancer incidence, mortality, years of life lost, years lived with disability, and disability-adjusted life-years for 29 cancer groups, 1990 to 2017: a Systematic analysis for the global burden of disease study, JAMA Oncol., 5 (2019), 1749–1768. https://doi.org/10.1001/jamaoncol.2019.2996 doi: 10.1001/jamaoncol.2019.2996
![]() |
[2] |
H. Sung, J. Ferlay, R. L. Siegel, M. Laversanne, I. Soerjomataram, A. Jemal, et al., Global cancer statistics 2020: GLOBOCAN estimates of incidence and mortality worldwide for 36 cancers in 185 countries, CA Cancer J. Clin., 71 (2021), 209–249. https://doi.org/10.3322/caac.21660 doi: 10.3322/caac.21660
![]() |
[3] |
P. S. Karia, J. Han, C. D. Schmults, Cutaneous squamous cell carcinoma: estimated incidence of disease, nodal metastasis, and deaths from disease in the United States, 2012, J. Am. Acad. Dermatol., 68 (2013), 957–966. https://doi.org/10.1016/j.jaad.2012.11.037 doi: 10.1016/j.jaad.2012.11.037
![]() |
[4] |
J. M. Janus, R. F. L. O'Shaughnessy, C. Harwood, T. Maffucci, Phosphoinositide 3-Kinase-Dependent signalling pathways in cutaneous squamous cell carcinomas, Cancers, 9 (2017), 86. https://doi.org/10.3390/cancers9070086 doi: 10.3390/cancers9070086
![]() |
[5] |
M. Piipponen, R. Riihilä, L. Nissinen, V. Kähäri, The role of p53 in progression of cutaneous squamous cell carcinoma, Cancers, 13 (2021), 4507. https://doi.org/10.3390/cancers13184507 doi: 10.3390/cancers13184507
![]() |
[6] |
A. Boutros, F. Cecchi, E. Tanda, E. Croce, R. Gili1, L. Arecco, et al., Immunotherapy for the treatment of cutaneous squamous cell carcinoma, Front. Oncol., 11 (2021), 733917. https://doi.org/10.3389/fonc.2021.733917 doi: 10.3389/fonc.2021.733917
![]() |
[7] |
Y. Sawada, M. Nakamura, Daily lifestyle and cutaneous malignancies, Int. J. Mol. Sci., 22 (2021), 5227. https://doi.org/10.3390/ijms22105227 doi: 10.3390/ijms22105227
![]() |
[8] | K, Suozzi, J. Turban, M. Girardi, Cutaneous photoprotection: a review of the current status and evolving strategies, Yale J. Biol. Med., 93 (2020), 55–67. |
[9] | C. Flower, D. Gaskin, S. Bhamjee, Z. Bynoe, High-risk variants of cutaneous squamous cell carcinoma in patients with discoid lupus erythematosus: a case series, Lupus, 22 (2013), 736–739. https://doi.org/10.1177%2F0961203313490243 |
[10] |
K. K. Das, A. Chakaraborty, A. Rahman, S. Khandkar, Incidences of malignancy in chronic burn scar ulcers: experience from Bangladesh, Burns, 41 (2015), 1315–1321. https://doi.org/10.1016/j.burns.2015.02.008 doi: 10.1016/j.burns.2015.02.008
![]() |
[11] |
T. J. Knackstedt, L. K. Collins, Z. Li, S. Yan, F. Samie, Squamous cell carcinoma arising in hypertrophic lichen planus: a review and analysis of 38 cases, Dermatol. Surg., 41 (2015), 1411–1418. http://doi.org/10.1097/DSS.0000000000000565 doi: 10.1097/DSS.0000000000000565
![]() |
[12] |
J. Xing, Z. Jia, Y. Xu, M. Chen, Z. Yang, Y. Chen, et al., KLF9 (Kruppel Like Factor 9) induced PFKFB3 (6-Phosphofructo-2-Kinase/Fructose-2, 6-Biphosphatase 3) downregulation inhibits the proliferation, metastasis and aerobic glycolysis of cutaneous squamous cell carcinoma cells, Bioengineered, 12 (2021), 7563–7576. https://doi.org/10.1080/21655979.2021.1980644 doi: 10.1080/21655979.2021.1980644
![]() |
[13] |
J. G. Newman, M. A. Hall, S. J. Kurley, R. Cook, A. S. Farberg, J. L. Geiger, et al., Adjuvant therapy for high-risk cutaneous squamous cell carcinoma: 10-year review, Head Neck, 43 (2021), 2822–2843. https://doi.org/10.1002/hed.26767 doi: 10.1002/hed.26767
![]() |
[14] | J. Pang, H. Pan, C. Yang, P. Meng, W. Xie, J. Li, et al., Prognostic value of immune-related multi-incRNA signatures associated with tumor microenvironment in esophageal cancer, Front. Genet., 12 (2021), 722601. https://dx.doi.org/10.3389%2Ffgene.2021.722601 |
[15] |
Y. Pan, H. Han, K. E. Labbe, H. Zhang, W. Wong, Recent advances in preclinical models for lung squamous cell carcinoma, Oncogene, 40 (2021), 2817–2829. https://doi.org/10.1038/s41388-021-01723-7 doi: 10.1038/s41388-021-01723-7
![]() |
[16] |
A. Elmusrati, J. Wang, C. Y. Wang, Tumor microenvironment and immune evasion in head and neck squamous cell carcinoma, Int. J. Oral. Sci., 13 (2021), 24. https://doi.org/10.1038/s41368-021-00131-7 doi: 10.1038/s41368-021-00131-7
![]() |
[17] |
T. Suwa, M. Kobayashi, J. M. Nam, H, Harada, Tumor microenvironment and radioresistance, Exp. Mol. Med., 53 (2021), 1029–1035. https://doi.org/10.1038/s12276-021-00640-9 doi: 10.1038/s12276-021-00640-9
![]() |
[18] | S. Paget, The distribution of secondary growths in cancer of the breast, Cancer Metastasis Rev., 8 (1889), 98–101. |
[19] |
H. Wang, M. M. H. Yung, H. Y. S. Ngan, K. Chan, D. W. Chan, The impact of the tumor microenvironment on macrophage polarization in cancer metastatic progression, Int. J. Mol. Sci., 22 (2021), 6560. https://doi.org/10.3390/ijms22126560 doi: 10.3390/ijms22126560
![]() |
[20] |
J. Zhuyan, M. Chen, T. Zhu, X. Bao, T. Zhen, K. Xing, et al., Critical steps to tumor metastasis: alterations of tumor microenvironment and extracellular matrix in the formation of pre-metastatic and metastatic niche, Cell Biosci., 10 (2020), 89. https://doi.org/10.1186/s13578-020-00453-9 doi: 10.1186/s13578-020-00453-9
![]() |
[21] |
Y. Xie, F. Xie, L. Zhang, X. Zhou, J. Huang, F. Wang, et al., Targeted anti-tumor immunotherapy using tumor infiltrating cells, Adv. Sci., e2101672. https://doi.org/10.1002/advs.202101672 doi: 10.1002/advs.202101672
![]() |
[22] |
M. Akhtar, A. Haider, S. Rashid, A. Ai-Nabet, Paget's " Seed and Soil" theory of cancer metastasis: an idea whose time has come, Adv. Anat. Pathol., 26 (2019), 69–74. https://doi.org/10.1097/PAP.0000000000000219 doi: 10.1097/PAP.0000000000000219
![]() |
[23] |
G. Yan, L. Li, S. Zhu, Y. Wu, Y. Zhu, L. Zhu, et al., Single-cell transcriptomic analysis reveals the critical molecular pattern of UV-induced cutaneous squamous cell carcinoma, Cell Death Dis., 13 (2022), 23. https://doi.org/10.1038/s41419-021-04477-y doi: 10.1038/s41419-021-04477-y
![]() |
[24] |
A. Ji, A. Rubin, K. Thrane, S. Jiang, D. L. Reynolds, R. M. Meyers, et al., Multimodal analysis of composition and spatial architecture in human squamous cell carcinoma, Cell, 182 (2020), 497–514. https://doi.org/10.1016/j.cell.2020.05.039 doi: 10.1016/j.cell.2020.05.039
![]() |
[25] |
C. B. Steen, C. L. Liu, A. A. Alizadeh, A. M. Newman, Profiling cell type abundance and expression in bulk tissues with CIBERSORTx, Methods Mol. Biol., 2117 (2020), 135–157. https://doi.org/10.1007/978-1-0716-0301-7_7 doi: 10.1007/978-1-0716-0301-7_7
![]() |
[26] |
J. L. Sevilla, V. Segura, A. Podhorski, E. Guruceaga, J. M. Mato, L. A. Martinez-Cruz, et al., (2005) Correlation between gene expression and GO semantic similarity, IEEE/ACM Trans. Comput. Biol. Bioinform., 2 (2005), 330–338. https://doi.org/10.1109/TCBB.2005.50 doi: 10.1109/TCBB.2005.50
![]() |
[27] |
S. Jain, G. D. Bader, An improved method for scoring protein-protein interactions using semantic similarity within the gene ontology, BMC Bioinf., 11 (2010), 562. https://doi.org/10.1186/1471-2105-11-562 doi: 10.1186/1471-2105-11-562
![]() |
[28] | X. Guo, C. D. Shriver, H. Hu, M. N. Liebman, Analysis of metabolic and regulatory pathways through Gene Ontology-derived semantic similarity measures, in AMIA Annual Symposium Proceedings, American Medical Informatics Association, (2005), 972. |
[29] |
P. M. Tedder, J. R. Bradford, C. J. Needham, G. A. McConkey, A. J. Bulpitt, D. R. Westhead, Gene function prediction using semantic similarity clustering and enrichment analysis in the malaria parasite Plasmodium falciparum, Bioinformatics, 26 (2010), 2431–2437. https://doi.org/10.1093/bioinformatics/btq450 doi: 10.1093/bioinformatics/btq450
![]() |
[30] |
G. Yu, F. Li, Y. Qin, X. Bo, Y. Wu, S. Wang, GOSemSim: an R package for measuring semantic similarity among GO terms and gene products, Bioinformatics, 26 (2010), 976–978. https://doi.org/10.1093/bioinformatics/btq064 doi: 10.1093/bioinformatics/btq064
![]() |
[31] |
J. Z. Wang, Z. Du, R. Payattakool, P. S. Yu, C. F. Chen, A new method to measure the semantic similarity of GO terms, Bioinformatics, 23 (2007), 1274–1281. https://doi.org/10.1093/bioinformatics/btm087 doi: 10.1093/bioinformatics/btm087
![]() |
[32] |
E. Rognoni, M. Widmaier, M. Jakobson, R. Ruppert, S. Ussar, D. Katsougkri, Kindlin-1 controls Wnt and TGF-β availability to regulate cutaneous stem cell proliferation, Nat. Med., 20 (2014), 350–359. https://doi.org/10.1038/nm.3490 doi: 10.1038/nm.3490
![]() |
[33] | M. Lai, R. Pampena, L. Cornacchia, G. Odorici, A. Piccerillo, G. Pellacani, et al., Cutaneous squamous cell carcinoma in patients with chronic lymphocytic leukemia: a systematic review of the literature, Int. J. Dermatol., 2021 (2021). https://doi.org/10.1111/ijd.15813 |
[34] |
H. B. Jie, P. J. Schuler, S. C. Lee, R. M. Srivastava, A. Argiris, S. Ferrone, et al., CTLA-4⁺ regulatory T cells increased in cetuximab-treated head and neck cancer patients suppress NK cell cytotoxicity and correlate with poor prognosis, Cancer Res., 75 (2015), 2200–2210. https://doi.org/10.1158/0008-5472.CAN-14-2788 doi: 10.1158/0008-5472.CAN-14-2788
![]() |
[35] |
S. Z. Lin, K. J. Chen, Z. Y. Xu, H. Chen, L. Zhou, H. Y. Xie, et al., Prediction of recurrence and survival in hepatocellular carcinoma based on two Cox models mainly determined by FoxP3+ regulatory T cells, Cancer Prev. Res., 6 (2013), 594–602. https://doi.org/10.1158/1940-6207.CAPR-12-0379 doi: 10.1158/1940-6207.CAPR-12-0379
![]() |
[36] |
B. Azzimonti, E. Zavattaro, M. Provasi, M. Vidali, A. Conca, E. Catalano, et al., Intense Foxp3+ CD25+ regulatory T-cell infiltration is associated with high-grade cutaneous squamous cell carcinoma and counterbalanced by CD8+/Foxp3+ CD25+ ratio, Br. J. Dermatol., 172 (2014), 64–73. https://doi.org/10.1111/bjd.13172 doi: 10.1111/bjd.13172
![]() |
[37] | S. M. Gorsch, V. A. Memoli, T. A. Stukel, L. I. Gold, B. A. Arrick, Immunohistochemical staining for transforming growth factor beta 1 associates with disease progression in human breast cancer, Cancer Res., 52 (1992), 6949–6952. |
[38] |
M. Ponzoni, F. Pastorino, D. Di Paolo, P. Perri, C. Brignole, Targeting macrophages as a potential therapeutic intervention: impact on inflammatory diseases and cancer, Int. J. Mol. Sci., 19 (2018), 1953. https://doi.org/10.3390/ijms19071953 doi: 10.3390/ijms19071953
![]() |
[39] |
L. Nissinen, M. Farshchian, P. Riihilä, V. Kähäre, New perspectives on role of tumor microenvironment in progression of cutaneous squamous cell carcinoma, Cell Tissue Res., 365 (2016), 691–702. https://doi.org/10.1007/s00441-016-2457-z doi: 10.1007/s00441-016-2457-z
![]() |
[40] |
J. S. Pettersen, J. Fuentes-Duculan, M. Suárez-Fariñas, K. C. Pierson, A. Pitts-Kiefer, L. Fan, et al., Tumor-associated macrophages in the cutaneous SCC microenvironment are heterogeneously activated, J. Invest. Dermatol., 131 (2011), 1322–1330. https://doi.org/10.1038/jid.2011.9 doi: 10.1038/jid.2011.9
![]() |
[41] |
M. Takahara, S. Chen, M. Kido, S. Takeuchi, H. Uchi, Y. Tu, et al., Stromal CD10 expression, as well as increased dermal macrophages and decreased Langerhans cells, are associated with malignant transformation of keratinocytes, J. Cutan. Pathol., 36 (2009), 668–674. https://doi.org/10.1111/j.1600-0560.2008.01139.x doi: 10.1111/j.1600-0560.2008.01139.x
![]() |
[42] |
D. Moussai, H. Mitsui, J. S. Pettersen, K. C. Pierson, K. R. Shah, M. Suárez- Fariñas, et al., The human cutaneous squamous cell carcinoma microenvironment is characterized by increased lymphatic density and enhanced expression of macrophage-derived VEGF-C, J. Invest. Dermatol., 131 (2011), 229–236. https://doi.org/10.1038/jid.2010.266 doi: 10.1038/jid.2010.266
![]() |
[43] | C. A. Janeway, J. Ron, M. E. Katz, The B cell is the initiating antigen-presenting cell in peripheral lymph nodes, J. Immunol., 138 (1987), 1051–1055. |
[44] |
D. P. Harris, L. Haynes, P. C. Sayles, D. K. Duso, S. M. Eaton, N. M. Lepak, et al., Reciprocal regulation of polarized cytokine production by effector B and T cells, Nat. Immunol., 1 (2000), 475–482. https://doi.org/10.1038/82717 doi: 10.1038/82717
![]() |
[45] |
A. Sarvaria, J. A. Madrigal, A. Saudemont, B cell regulation in cancer and anti-tumor immunity, Cell Mol. Immunol., 14 (2017), 662–674. https://doi.org/10.1038/cmi.2017.35 doi: 10.1038/cmi.2017.35
![]() |
[46] |
P. Andreu, M. Johansson, N. Affara, F. Pucci, T. Tan, S. Junankar, et al., FcRgamma activation regulates inflammation-associated squamous carcinogenesis, Cancer Cell, 17 (2010), 121–134. https://doi.org/10.1016/j.ccr.2009.12.019 doi: 10.1016/j.ccr.2009.12.019
![]() |
[47] |
K. W. de Visser, L. V. Korets, L. M. Coussens, De novo carcinogenesis promoted by chronic inflammation is B lymphocyte dependent, Cancer Cell, 7 (2005), 411–423. https://doi.org/10.1016/j.ccr.2005.04.014 doi: 10.1016/j.ccr.2005.04.014
![]() |
[48] |
T. Schioppa, R. Moore, R. G. Thompson, F. R. Balkwill, B regulatory cells and the tumor-promoting actions of TNF-α during squamous carcinogenesis, Proc. Natl. Acad. Sci., 108 (2011), 10662–10667. https://doi.org/10.1073/pnas.1100994108 doi: 10.1073/pnas.1100994108
![]() |
[49] |
G. Crawford, M. D. Hayes, R. C. Seoane, S. Ward, T. Dalessandri, C. Lai, et al., Epithelial damage and tissue γδ T cells promote a unique tumor-protective IgE response, Nat. Immunol., 19 (2018), 859–870. https://doi.org/10.1038/s41590-018-0161-8 doi: 10.1038/s41590-018-0161-8
![]() |
[50] |
T. Zhou, R. Qin, S. Shi, H. Zhang, C. Niu, G. Ju, et al., DTYMK promote hepatocellular carcinoma proliferation by regulating cell cycle, Cell Cycle, 20 (2021), 1681–1691. https://doi.org/10.1080/15384101.2021.1958502 doi: 10.1080/15384101.2021.1958502
![]() |
[51] | Y. Guo, W. Luo, S. Huang, W. Zhao, H. Chen, Y. Ma, et al., DTYMK expression predicts prognosis and chemotherapeutic response and correlates with immune infiltration in hepatocellular carcinoma, J. Hepatocell Carcinoma, 8 (2021), 871–885. https://dx.doi.org/10.2147%2FJHC.S312604 |
[52] |
T. Jeon, M. J. Ko, Y. R. Seo, S. J. Jung, D. Seo, S. Y. Park, et al., Silencing CDCA8 suppresses hepatocellular carcinoma growth and stemness via restoration of ATF3 tumor suppressor and inactivation of AKT/β-catenin signaling, Cancers, 13 (2021), 1055. https://doi.org/10.3390/cancers13051055 doi: 10.3390/cancers13051055
![]() |
[53] |
G. Vlotides, T. Eigler, S. Melmed, Pituitary tumor-transforming gene: physiology and implications for tumorigenesis, Endocr. Rev., 28 (2007), 165–186. https://doi.org/10.1210/er.2006-0042 doi: 10.1210/er.2006-0042
![]() |
[54] | H. Hong, Z. Jin, T. Qian, X. Xu, X. Zhu, Q. Fei, et al., Falcarindiol enhances cisplatin chemosensitivity of hepatocellular carcinoma via down-regulating the STAT3-modulated PTTG1 pathway, Front. Pharmacol., 12 (2021), 656697. https://dx.doi.org/10.3389%2Ffphar.2021.656697 |
[55] |
S. W. Chen, H. F. Zhou, H. J. Zhang, R. He, Z. Huang, Y. Dang, et al., The clinical significance and potential molecular mechanism of PTTG1 in esophageal squamous cell carcinoma, Front. Genet., 11 (2021), 583085. https://doi.org/10.3389/fgene.2020.583085 doi: 10.3389/fgene.2020.583085
![]() |
[56] |
Z. Chen, K. Cao, Y. Hou, F. Lu, L. Li, L. Wang, et al., PTTG1 knockdown enhances radiation-induced antitumour immunity in lung adenocarcinoma, Life Sci., 277 (2021), 119594. https://doi.org/10.1016/j.lfs.2021.119594 doi: 10.1016/j.lfs.2021.119594
![]() |
[57] |
J. E. Noll, K. Vandyke, D. R. Hewett, K. M. Mrozik, R. J. Bala, S. A. Williams, et al., PTTG1 expression is associated with hyperproliferative disease and poor prognosis in multiple myeloma, J. Hematol. Oncol., 8 (2015), 106. https://doi.org/10.1186/s13045-015-0209-2 doi: 10.1186/s13045-015-0209-2
![]() |
[58] |
R. Wei, Z. Wang, Y. Zhang, B. Wang, N. Shen, E. Li, et al., Bioinformatic analysis revealing mitotic spindle assembly regulated NDC80 and MAD2L1 as prognostic biomarkers in non-small cell lung cancer development, BMC Med. Genomics, 13 (2020), 112. https://doi.org/10.1186/s12920-020-00762-5 doi: 10.1186/s12920-020-00762-5
![]() |
[59] |
M. Vleugel, T. A. Hoek, E. Tromer, T. Sliedrecht, V. Groenewold, M. Omerzu, et al., Dissecting the roles of human BUB1 in the spindle assembly checkpoint, J. Cell Sci., 128 (2015), 2975–2982. https://doi.org/10.1242/jcs.169821 doi: 10.1242/jcs.169821
![]() |
[60] |
Y. H. Ko, J. H. Roh, Y. I. Son, M. K. Chung, J. Y. Jang, H. Byun, et al., Expression of mitotic checkpoint proteins BUB1B and MAD2L1 in salivary duct carcinomas, J. Oral Pathol. Med., 39 (2010), 349–355. https://doi.org/10.1111/j.1600-0714.2009.00835.x doi: 10.1111/j.1600-0714.2009.00835.x
![]() |
[61] |
M. Abal, A. Obrador-Hevia, K. P. Janssen, L. Casadome, M. Menendez, S. Carpentier, et al., APC inactivation associates with abnormal mitosis completion and concomitant BUB1B/MAD2L1 up-regulation, Gastroenterology, 132 (2007), 2448–2458. https://doi.org/10.1053/j.gastro.2007.03.027 doi: 10.1053/j.gastro.2007.03.027
![]() |
[62] |
Y. Wang, Z. Zhou, L. Chen, Y. Li, Z. Zhou, X. Chu, Identification of key genes and biological pathways in lung adenocarcinoma via bioinformatics analysis, Mol. Cell Biochem., 476 (2021), 931–939. https://doi.org/10.1007/s11010-020-03959-5 doi: 10.1007/s11010-020-03959-5
![]() |
[63] |
R. Marima, R. Hull, C. Penny, Z. Dlamini, Mitotic syndicates Aurora Kinase B (AURKB) and mitotic arrest deficient 2 like 2 (MAD2L2) in cohorts of DNA damage response (DDR) and tumorigenesis, Mutat. Res. Rev. Mutat. Res., 787 (2021), 108376. https://doi.org/10.1016/j.mrrev.2021.108376 doi: 10.1016/j.mrrev.2021.108376
![]() |
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15. | Keqin Su, Xin-Guang Yang, Alain Miranville, He Yang, Dynamics and robustness for the 2D Navier–Stokes equations with multi-delays in Lipschitz-like domains, 2023, 134, 18758576, 513, 10.3233/ASY-231845 | |
16. | Lingrui Zhang, Xue-zhi Li, Keqin Su, Dynamical behavior of Benjamin-Bona-Mahony system with finite distributed delay in 3D, 2023, 31, 2688-1594, 6881, 10.3934/era.20233348 |