Citation: Ling Zhu. Completely monotonic integer degrees for a class of special functions[J]. AIMS Mathematics, 2020, 5(4): 3456-3471. doi: 10.3934/math.2020224
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Let
Oε={x=(x∗,xn+1)|x∗=(x1,…,xn)∈Qand0<xn+1<εg(x∗)}, |
where
γ1≤g(x∗)≤γ2,∀x∗∈¯Q. | (1) |
Denote
{dˆuε−Δˆuεdt=(H(t,x,ˆuε(t))+G(t,x))dt+m∑j=1cjˆuε∘dwj,x∈Oε,t>τ,∂ˆuε∂νε=0,x∈∂Oε, | (2) |
with the initial condition
ˆuε(τ,x)=ˆϕε(x),x∈Oε, | (3) |
where
As
{du0−1gn∑i=1(gu0yi)yidt=(H(t,(y∗,0),u0(t))+G(t,(y∗,0)))dt+m∑j=1cju0∘dwj,y∗=(y1,…,yn)∈Q,t>τ,∂u0∂ν0=0,y∗∈∂Q, | (4) |
with the initial condition
u0(τ,y∗)=ϕ0(y∗),y∗∈Q, | (5) |
where
Random attractors have been investigated in [2,5,10,19,9] in the autonomous stochastic case, and in [3,21,22,23] in the non-autonomous stochastic case. Recently, the limiting dynamical behavior of stochastic partial differential equations on thin domain was studied in [16,20,13,14,11,12,17,4]. However, in [17,13], we only investigated the limiting behavior of random attractors in
Let
We organize the paper as follows. In the next section, we establish the existence of a continuous cocycle in
Here we show that there is a continuous cocycle generated by the reaction-diffusion equation defined on
{dˆuε−Δˆuεdt=(H(t,x,ˆuε(t))+G(t,x))dt+m∑j=1cjˆuε∘dwj,x=(x∗,xn+1)∈Oε,t>τ,∂ˆuε∂νε=0,x∈∂Oε, | (6) |
with the initial condition
ˆuετ(x)=ˆϕε(x),x∈Oε, | (7) |
where
H(t,x,s)s≤−λ1|s|p+φ1(t,x), | (8) |
|H(t,x,s)|≤λ2|s|p−1+φ2(t,x), | (9) |
∂H(t,x,s)∂s≤λ3, | (10) |
|∂H(t,x,s)∂x|≤ψ3(t,x), | (11) |
where
Throughout this paper, we fix a positive number
h(t,x,s)=H(t,x,s)+λs | (12) |
for all
h(t,x,s)s≤−α1|s|p+ψ1(t,x), | (13) |
|h(t,x,s)|≤α2|s|p−1+ψ2(t,x), | (14) |
∂h(t,x,s)∂s≤β, | (15) |
|∂h(t,x,s)∂x|≤ψ3(t,x), | (16) |
where
Substituting (12) into (6) we get for
{dˆuε−(Δˆuε−λˆuε)dt=(h(t,x,ˆuε(t))+G(t,x))dt+m∑j=1cjˆuε∘dwj,x=(x∗,xn+1)∈Oε,∂ˆuε∂νε=0,x∈∂Oε, | (17) |
with the initial condition
ˆuετ(x)=ˆϕε(x),x∈Oε. | (18) |
We now transfer problem (17)-(18) into an initial boundary value problem on the fixed domain
x∗=y∗,xn+1=εg(y∗)yn+1. |
It follows from [18] that the Laplace operator in the original variable
Δxˆu(x)=|J|divy(|J|−1JJ∗∇yu(y))=1gdivy(Pεu(y)), |
where we denote by
Pεu(y)=(guy1−gy1yn+1uyn+1⋮guyn−gynyn+1uyn+1−n∑i=1yn+1gyiuyi+1ε2g(1+n∑i=1(εyn+1gyi)2)uyn+1). |
In the sequel, we abuse the notation a little bit by writing
Fε(t,y∗,yn+1,s)=F(t,y∗,εg(y∗)yn+1,s),F0(t,y∗,s)=F(t,y∗,0,s), |
where
{duε−(1gdivy(Pεuε)−λuε)dt=(hε(t,y,uε(t))+Gε(t,y))dt+m∑j=1cjuε∘dwj,y=(y∗,yn+1)∈O,Pεuε⋅ν=0,y∈∂O, | (19) |
with the initial condition
uετ(y)=ϕε(y)=ˆϕε∘T−1ε(y),y∈O, | (20) |
where
Given
θ1,t(τ)=τ+t,for allτ∈R. | (21) |
Then
Ω={ω∈C(R,R):ω(0)=0}. |
Let
θtω(⋅)=ω(⋅+t)−ω(t),ω∈Ω,t∈R. | (22) |
Then
dz+αzdt=dw(t), | (23) |
for
Lemma 2.1. There exists a
limt→±∞|ω(t)|t=0for allω∈Ω′, |
and, for such
z∗(ω)=−α∫0−∞eαsω(s)ds |
is well defined. Moreover, for
(t,ω)→z∗(θtω)=−α∫0−∞eαsθtω(s)ds=−α∫0−∞eαsω(t+s)ds+ω(t) |
is a stationary solution of (23) with continuous trajectories. In addition, for
limt→±∞|z∗(θtω)|t=0,limt→±∞1t∫t0z∗(θsω)ds=0, | (24) |
limt→±∞1t∫t0|z∗(θsω)|ds=E|z∗|<∞. | (25) |
Denote by
˜Ω=Ω′1×⋯×Ω′mand F=m⊗j=1Fj, |
Then
Denote by
SCj(t)u=ecjtu,foru∈L2(O), |
and
T(ω):=SC1(z∗1(ω))∘⋯∘SCm(z∗m(ω))=em∑j=1cjz∗j(ω)IdL2(O),ω∈Ω′. |
Then for every
T−1(ω):=SCm(−z∗m(ω))∘⋯∘SC1(−z∗1(ω))=e−m∑j=1cjz∗j(ω)IdL2(O). |
It follows that
On the other hand, since
limt→±∞1t∫t0‖T(θτω)‖2dτ=E‖T‖2=m∏j=1E(e2cjz∗j)<∞, |
and
limt→±∞1t∫t0‖T−1(θτω)‖2dτ=E‖T−1‖2=m∏j=1E(e−2cjz∗j)<∞. |
Remark 1. We now consider
Next, we define a continuous cocycle for system (19)-(20) in
{dvεdt−1gdivy(Pεvε)=(−λ+δ(θtω))vε+T−1(θtω)hε(t,y,T(θtω)vε(t))+T−1(θtω)Gε(t,y),y∈O,t>τ,Pεvε⋅ν=0,y∈∂O, | (26) |
with the initial conditions
vετ(y)=ψε(y),y∈O, | (27) |
where
Since (26) is a deterministic equation, by the Galerkin method, one can show that if
Φε(t,τ,ω,ϕε)=uε(t+τ,τ,θ−τω,ϕε)=T(θt+τω)vε(t+τ,τ,θ−τω,ψε),for all(t,τ,ω,ϕε)∈R+×R×Ω×N. | (28) |
By the properties of
Let
(Rεˆϕε)(y)=ˆϕε(T−1εy),∀ˆϕε∈L2(Oε). |
Given
ˆΦε(t,τ,ω,ˆϕε)=R−1εΦε(t,τ,ω,Rεˆϕε). |
The same change of unknown variable
{dv0dt−n∑i=11g(gv0yi)yi=(−λ+δ(θtω))v0+T−1(θtω)h0(t,y∗,T(θtω)v0(t))+T−1(θtω)G0(t,y∗),y∗∈Q,t>τ,∂v0∂ν0=0,y∗∈∂Q, | (29) |
with the initial conditions
v0τ(y∗)=ψ0(y∗),y∗∈Q, | (30) |
where
The same argument as above allows us to prove that problem (4) and (5) generates a continuous cocycle
Now we want to write equation (26)-(27) as an abstract evolutionary equation. We introduce the inner product
(u,v)Hg(O)=∫Oguvdy,for allu,v∈N |
and denote by
For
aε(u,v)=(J∗∇yu,J∗∇yv)Hg(O), | (31) |
where
J∗∇yu=(uy1−gy1gyn+1uyn+1,…,uyn−gyngyn+1uyn+1,1εguyn+1). |
By introducing on
‖u‖H1ε(O)=(∫O(|∇y∗u|2+|u|2+1ε2u2yn+1)dy)12, | (32) |
we see that there exist positive constants
η1∫O(|∇y∗u|2+1ε2u2yn+1)dy≤aε(u,u)≤η2∫O(∇y∗u|2+1ε2u2yn+1)dy | (33) |
and
η1‖u‖2H1ε(O)≤aε(u,u)+‖u‖2L2(O)≤η2‖u‖2H1ε(O). | (34) |
Denote by
D(Aε)={v∈H2(O),Pεv⋅ν=0on∂O} |
as defined by
Aεv=−1gdivPεv,v∈D(Aε). |
Then we have
aε(u,v)=(Aεu,v)Hg(O),∀u∈D(Aε),∀v∈H1(O). | (35) |
Using
{dvεdt+Aεvε=(−λ+δ(θtω))vε+T−1(θtω)hε(t,y,T(θtω)vε(t))+T−1(θtω)Gε(t,y),y∈O,t>τ,vετ=ψε. | (36) |
To reformulate system (29)-(30), we introduce the inner product
(u,v)Hg(Q)=∫Qguvdy∗,for allu,v∈M, |
and denote by
a0(u,v)=∫Qg▽y∗u⋅▽y∗vdy∗. |
Denote by
D(A0)={v∈H2(Q),∂v∂ν0=0on∂Q} |
as defined by
A0v=−1gn∑i=1(gvyi)yiv∈D(A0). |
Then we have
a0(u,v)=(A0u,v)Hg(Q),∀u∈D(A0),∀v∈H1(Q). |
Using
{dv0dt+A0v0=(−λ+δ(θtω))v0+T−1(θtω)h0(t,y∗,T(θtω)v0(t))+T−1(θtω)G0(t,y∗),y∗∈Q,t>τ,v0τ(s)=ψ0(s),s∈[−ρ,0]. | (37) |
Hereafter, we set
limt→−∞ect‖Bi(τ+t,θtω)‖Xi=0, |
where
Di={Bi={Bi(τ,ω):τ∈R,ω∈Ω}:Bi is tempered in Xi}. |
Our main purpose of the paper is to prove that the cocycle
limε→0supuε∈ˆAεinfu0∈A0ε−1‖uε−u0‖2H1(Oε)=0. | (38) |
To prove (38), we only need to show that the cocycle
limε→0distH(Aε(τ,ω),A0(τ,ω))=0, |
which will be established in the last section of the paper.
Furthermore, we suppose that there exists
¯γΔ=λ0−2E(|δ(ω)|)>0. | (39) |
Let us consider the mapping
γ(ω)=λ0−2|δ(ω)|. | (40) |
By the ergodic theory and (39) we have
limt→±∞1t∫t0γ(θlω)dl=Eγ=¯γ>0. | (41) |
The following condition will be needed when deriving uniform estimates of solutions:
∫τ−∞e12¯γs(‖G(s,⋅)‖2L∞(˜O)+‖φ1(s,⋅)‖2L∞(˜O)+‖ψ3(s,⋅)‖2L∞(˜O))ds<∞,∀τ∈R. | (42) |
When constructing tempered pullback attractors, we will assume
limr→−∞eσr∫0−∞e12¯γs(‖G(s+r,⋅)‖2L∞(˜O)+‖φ1(s+r,⋅)‖2L∞(˜O)+‖ψ3(s+r,⋅)‖2L∞(˜O))ds=0,∀σ>0. | (43) |
Since
∫τ−∞e12¯γs(‖G(s,⋅)‖2L∞(˜O)+‖ψ1(s,⋅)‖L∞(˜O)+‖ψ3(s,⋅)‖2L∞(˜O))ds<∞,∀τ∈R | (44) |
and
limr→−∞eσr∫0−∞e12¯γs(‖G(s+r,⋅)‖2L∞(˜O)+‖ψ1(s+r,⋅)‖2L∞(˜O)+‖ψ3(s+r,⋅)‖2L∞(˜O))ds=0, | (45) |
for any
In this section, we recall and generalize some results in [17] and derive some new uniform estimates of solutions of problem (36) or (19)-(20) which are needed for proving the existence of
Lemma 3.1. Assume that (8)-(11), (39) and (42) hold. Then for every
sup−1≤s≤0‖vε(τ+s,τ−t,θ−τω,ψε)‖2H1ε(O)≤R2(τ,ω), | (46) |
where
R2(τ,ω)=r1(ω)R1(τ,ω)+c∫0−∞e¯γr‖T−1(θrω)‖2(‖G(r+τ,⋅)‖2L∞(˜O)+‖ψ3(r+τ,⋅)‖2L∞(˜O))dr, | (47) |
where
R1(τ,ω)=c∫0−∞e∫r0γ(θlω)dl‖T−1(θrω)‖2‖G(r+τ,⋅)‖2L∞(˜O)dr+c∫0−∞e∫r0γ(θlω)dl‖T−1(θrω)‖2‖ψ1(r+τ,⋅)‖2L∞(˜O)dr, | (48) |
and
Proof. The proof is similar as that of Lemma 3.4 in [17], so we only sketch the proof here. Taking the inner product of (36) with
12ddt‖vε‖2Hg(O)≤−aε(vε,vε)+(−λ0+δ(θtω))‖vε‖2Hg(O)+(T−1(θtω)hε(t,y,T(θtω)vε(t)),vε)Hg(O)+(T−1(θtω)Gε(t,y),vε)Hg(O). | (49) |
By (13), we have
ddt‖vε‖2Hg(O)+2aε(vε,vε)+λ02‖vε‖2Hg(O)+2α1γ1‖T−1(θtω)‖2‖uε‖pLp(O)≤(−λ0+2δ(θtω))‖vε‖2Hg(O)+2λ0γ2|˜O|‖T−1(θtω)‖2‖G(t,⋅)‖2L∞(˜O)+2γ2|˜O|‖T−1(θtω)‖2‖ψ1(t,⋅)‖L∞(˜O). | (50) |
Then, we have for any
e∫στγ(θlω)dl‖vε(σ)‖2Hg(O)+2∫στe∫rτγ(θlω)dlaε(vε(r),vε(r))dr+λ02∫στe∫rτγ(θlω)dl‖vε(r)‖2Hg(O)dr+2α1γ1∫στ‖T−1(θrω)‖2e∫rτγ(θlω)dl‖uε(r)‖pLp(O)dr≤‖vε(τ)‖2Hg(O)+2λ0γ2|˜O|∫στe∫rτγ(θlω)dl‖T−1(θrω)‖2‖G(r,⋅)‖2L∞(˜O)dr+2γ2|˜O|∫στe∫rτγ(θlω)dl‖T−1(θrω)‖2‖ψ1(r,⋅)‖2L∞(˜O)dr, | (51) |
where
Thus by the similar arguments as Lemma 3.1 in [17] we get for every
‖vετ(⋅,τ−t,θ−τω,ψ)‖2L2(O)≤c∫0−∞e∫r0γ(θlω)dl‖ψ1(r+τ,⋅)‖2L∞(˜O)dr+c∫0−∞e∫r0γ(θlω)dl‖T−1(θrω)‖2‖G(r+τ,⋅)‖2L∞(˜O)dr+c∫0−∞e∫r0γ(θlω)dl‖T−1(θrω)‖2‖ψ1(r+τ,⋅)‖2L∞(˜O)dr. | (52) |
Moreover, taking the inner product of (36) with
12ddtaε(vε,vε)+‖Aεvε‖2Hg(O)≤(−λ0+δ(θtω))aε(vε,vε)+(T−1(θtω)hε(t,y,T(θtω)vε(t)),Aεvε)Hg(O)+(T−1(θtω)Gε(t,y),Aεvε)Hg(O). | (53) |
By (15)-(16) we have
ddtaε(vε,vε)+‖Aεvε‖2Hg(O)≤(c+2δ(θtω))aε(vε,vε)+c‖T−1(θtω)‖2(‖G(t,⋅)‖2L∞(˜O)+‖ψ3(t,⋅)‖2L∞(˜O)), | (54) |
The left proof is similar of that Lemma 3.4 in [17], so we omit it here.
We are now in a position to establish the uniform estimates for the solution
Lemma 3.2. Assume that (8)-(11), (39) and (42) hold. Then for every
sup−1≤s≤0‖uε(τ+s,τ−t,θ−τω,ϕε)‖2H1ε(O)≤r2(ω)R2(τ,ω), | (55) |
where
Lemma 3.3. Assume that (8)-(11), (39) and (42) hold. Then for every
sup−1≤s≤0‖vε(τ+s,τ−t,θ−τω,ψε)‖pLp(O)+∫ττ−ρ‖vε(r,τ−t,θ−τω,ψε)‖2p−2L2p−2(O)dr≤R3(τ,ω), | (56) |
where
Proof. The proof is similar as that of Lemma 3.6 in [14], so we omit it here.
Lemma 3.4. Assume that (8)-(11), (39) and (42) hold. Then for every
∫0−1eγMp−2s∫{y∈O: vε(s+τ,τ−t,θ−τω,ψε)≥2M}|vε(s+τ,τ−t,θ−τω,ψε)|2p−2dyds≤η, | (57) |
∫0−1eγMp−2s∫{y∈O: vε(s+τ,τ−t,θ−τω,ψε)≤−2M}|vε(s+τ,τ−t,θ−τω,ψε)|2p−2dyds≤η. | (58) |
Proof. Let
1pddt‖(vε−M)+‖pLp(O)+(p−1)∫vε≥M(vε−M)p−2aε(vε,vε)dx≤(δ(θtω)vε,(vε−M)p−1+)+(T−1(θtω)hε(t,y,T(θtω)vε),(vε−M)p−1+)+(T−1(θtω)Gε(t,y),(vε−M)p−1+). | (59) |
For the first term on the right side of (59) we have
|(δ(θtω)vε,(vε−M)p−1+)|≤1p|δ(θrω)|p∫O|vε|pdx+p−1p∫O(vε−M)p+dx. | (60) |
For the second term on the right-hand side of (59), by (8), we obtain, for
hε(t,y,T(θtω)vε) (vε−M)p−1+≤−α1‖T(θtω)‖p−1(vε)p−1(v−M)p−1+ |
+‖T(θtω)‖−1ψ1(t,y∗,εg(y∗)yn+1)(vε)−1(vε−M)p−1+ |
≤−12α1Mp−2‖T(θtω)‖p−1(vε−M)p+−12α1‖T(θtω)‖p−1(vε−M)2p−2+ |
+‖T−1(θtω)‖−1|ψ1(t,y∗,εg(y∗)yn+1)|(vε−M)p−2+ |
which implies
(T−1(θtω)hε(t,y,T(θtω)vε), (vε−M)p−1+) |
≤−12α1Mp−2‖T(θtω)‖p−2∫O(vε−M)p+dx−12α1‖T−1(θtω)‖p−2∫O(vε−M)2p−2+dx |
+‖T(θtω)‖−2∫O|ψ1(t,y∗,εg(y∗)yn+1)|(vε−M)p−2+dx |
≤−12α1Mp−2‖T(θtω)‖p−2∫O(vε−M)p+dx−12α1‖T(θtω)‖p−2∫O(vε−M)2p−2+dx |
+p−2p∫O(vε−M)p+dx+2p‖T(θtω)‖−p∫O|ψ1(t,y∗,εg(y∗)yn+1)|p2dy. | (61) |
The last term in (59) is bounded by
\begin{align} \left( {{\mathcal T}^{ - 1} \left( {\theta _{t} \omega } \right)G_\varepsilon\left( {t,y} \right),\left ( {v^\varepsilon - M} \right)_ + ^{p - 1} } \right) &\le {\frac 18} \alpha_1 \|{\mathcal T} \left( {\theta _{t}\omega } \right)\|^{p-2} \int_{\mathcal O} {\left( {v^\varepsilon - M} \right)_ + ^{2p - 2} } dx \\ &\quad + \frac{2}{\alpha_1}\|{\mathcal T} \left( {\theta _{t}\omega } \right)\|^{-p} \int_{v^\varepsilon \ge M} {\left| {G_\varepsilon\left( {t,y} \right)} \right|^2 } dy. \end{align} | (62) |
All above estimates yield
\begin{align} & \frac{d}{{dt}}\left\| {\left( {v^\varepsilon - M} \right)_ + } \right\|_{L^p(\mathcal O)}^p -(2p-3-{\frac 12}p \alpha_1 M^{p-2} \|{\mathcal T} \left( {\theta _{t}\omega } \right)\|^{p-2}) \int_{\mathcal O}(v^\varepsilon-M)_+^{p} dx\\ &\quad +{\frac 14}p \alpha_1 \|{\mathcal T} \left( {\theta _{t}\omega } \right)\|^{p-2} \int_{\mathcal O} (v^\varepsilon-M)_+^{2p-2} dx \\ &\le | {\delta\left( {\theta _r \omega } \right)} |^{p} \int_{\mathcal O} {|v^\varepsilon| ^{p} } dx+2 \| {\mathcal T} \left( {\theta _{t}\omega } \right)\|^{ - p} \int_{\mathcal O} | \psi_1(t,y^{*},\varepsilon g(y^{*})y_{n+1}) |^ {\frac p2} dy\\ &\quad + \frac{2p}{\alpha_1}\|{\mathcal T} \left( {\theta _{t}\omega } \right)\|^{-p} \int_{\mathcal O} {\left| {G_\varepsilon\left( {t,y} \right)} \right|^2 } dy. \end{align} | (63) |
Multiplying (63) by
\| \left ( v^\varepsilon(\tau, \tau-t, \omega, \psi^\varepsilon) -M \right )_+\|^p_{L^p(\mathcal O)} |
+{\frac 14} p \alpha_1 \int_{\tau -1}^{\tau} \|{\mathcal T} \left( {\theta _{\zeta}\omega } \right)\|^{p-2} e^{-\int_{\tau}^\zeta ( 2p-3 - {\frac 12} p \alpha_1 M^{p-2} \|{\mathcal T} \left( {\theta _{r}\omega } \right)\|^{p-2}) dr } |
\times \int_{\mathcal O} ( v^\varepsilon(\zeta, \tau-t, \omega, \psi^\varepsilon) -M) _+^{2p-2} dx d\zeta |
\le e^{-\int_{\tau}^{\tau -1} ( 2p-3 - {\frac 12} p \alpha_1 M^{p-2} \|{\mathcal T} \left( {\theta _{r}\omega } \right)\|^{p-2})dr } \| \left ( v^\varepsilon(\tau-1, \tau-t, \omega, \psi^\varepsilon) -M \right )_+\|^p_{L^p(\mathcal O)} |
+ \int_{\tau -1}^{\tau} | \delta (\theta_\zeta \omega)|^p e^{-\int_{\tau}^\zeta ( 2p-3 - {\frac 12} p \alpha_1 M^{p-2} \|{\mathcal T} \left( {\theta _{r}\omega } \right)\|^{p-2}) dr } \|v^\varepsilon(\zeta, \tau-t, \omega, \psi^\varepsilon)\|^p_{L^p(\mathcal O)} d\zeta |
+2|\mathcal O| \int_{\tau -1}^{\tau} \|{\mathcal T} \left( {\theta _{\zeta}\omega } \right)\|^{-p} e^{-\int_{\tau+\xi}^\zeta ( 2p-3 - {\frac 12} p \alpha_1 M^{p-2} \|{\mathcal T} \left( {\theta _{r}\omega } \right)\|^{p-2}) dr } \| \psi_1(\zeta,\cdot) \|_{L^\infty(\widetilde {\mathcal O})}^ {\frac p2} d\zeta. |
\begin{equation} +\frac{2p|\mathcal O|}{\alpha_1} \int_{\tau -1}^{\tau} \|{\mathcal T} \left( {\theta _{\zeta}\omega } \right)\|^{-p} e^{-\int_{\tau+\xi}^\zeta ( 2p-3 - {\frac 12} p \alpha_1 M^{p-2} \|{\mathcal T} \left( {\theta _{r}\omega } \right)\|^{p-2}) dr } \| G(\zeta,\cdot) \|^2_{L^\infty(\widetilde {\mathcal O})} d\zeta, \end{equation} | (64) |
where
{\frac 14} p \alpha_1 \int_{-1}^0 \|{\mathcal T} \left( {\theta _{\zeta}\omega } \right)\|^{p-2} e^{-\int_0^{\zeta} ( 2p-3 - {\frac 12} p \alpha_1 M^{p-2} \|{\mathcal T} \left( {\theta _{r}\omega } \right)\|^{p-2}) dr } |
\times\int_{\mathcal O} ( v^\varepsilon(\zeta+\tau, \tau-t,\theta_{-\tau} \omega, \psi^\varepsilon) -M) _+^{2p-2} dx d\zeta |
\le e^{-\int_0^{ -1} ( 2p-3 - {\frac 12} p \alpha_1 M^{p-2} \|{\mathcal T} \left( {\theta _{r}\omega } \right)\|^{p-2})dr } \| \left ( v^\varepsilon(\tau-1, \tau-t,\theta_{-\tau} \omega, \psi^\varepsilon) -M \right )_+\|^p_{L^p(\mathcal O)} |
+ \int_{ -1}^0 | \delta (\theta_{\zeta+\xi} \omega)|^p e^{-\int_0^{\zeta} ( 2p-3 - {\frac 12} p \alpha_1 M^{p-2} \|{\mathcal T} \left( {\theta _{r}\omega } \right)\|^{p-2}) dr } \|v^\varepsilon(\zeta+\tau, \tau-t, \theta_{-\tau}\omega, \psi^\varepsilon)\|^p_{L^p(\mathcal O)} d\zeta |
+2|\mathcal O| \int_{ -1}^0 \|{\mathcal T} \left( {\theta _{\zeta}\omega } \right)\|^{-p} e^{-\int_0^{\zeta} ( 2p-3 - {\frac 12} p \alpha_1 M^{p-2} \|{\mathcal T} \left( {\theta _{r}\omega } \right)\|^{p-2}) dr } \| \psi_1(\zeta+\tau,\cdot) \|_{L^\infty(\widetilde {\mathcal O})}^ {\frac p2} d\zeta. |
\begin{equation} +\frac{2p|\mathcal O|}{\alpha_1} \int_{ -1}^0 \|{\mathcal T} \left( {\theta _{\zeta}\omega } \right)\|^{-p} e^{-\int_0^{\zeta} ( 2p-3 - {\frac 12} p \alpha_1 M^{p-2} \|{\mathcal T} \left( {\theta _{r}\omega } \right)\|^{p-2}) dr } \| G(\zeta+\tau,\cdot) \|^2_{L^\infty(\widetilde {\mathcal O})} d\zeta. \end{equation} | (65) |
Since
\begin{equation} c_1 \le {\frac 12} p \alpha_1 \|{\mathcal T} \left( {\theta _{r}\omega } \right)\|^{p-2} \le c_2 \quad \text{ for all } \ r\in [-\rho-1,0]. \end{equation} | (66) |
By (66) we obtain
\begin{equation} e^{c_2 M^{p-2} \zeta } \le e^{\int_\xi^{\zeta+\xi} {\frac 12} p \alpha_1 M^{p-2} \|{\mathcal T} \left( {\theta _{r}\omega } \right)\|^{p-2} dr } \le e^{c_1M^{p-2} \zeta } \quad \text{ for all } \ \zeta \in [-1,0]\,\,\text{and}\,\,\xi\in [-\rho,0]. \end{equation} | (67) |
For the left-hand side of (65), by (67) we find that there exists
{\frac 14} p \alpha_1 \int_{-1}^0 \|{\mathcal T} \left( {\theta _{\zeta}\omega } \right)\|^{p-2} e^{-\int_0^{\zeta} ( 2p-3 - {\frac 12} p \alpha_1 M^{p-2} \|{\mathcal T} \left( {\theta _{r}\omega } \right)\|^{p-2}) dr } |
\int_{\mathcal O} ( v^\varepsilon(\zeta+\tau, \tau-t,\theta_{-\tau} \omega, \psi^\varepsilon) -M) _+^{2p-2} dx d\zeta |
\begin{equation} \ge c_3 \int_{ -1}^0 e^{c_2 M^{p-2} \zeta } \int_{\mathcal O} ( v^\varepsilon(\zeta+\tau, \tau-t, \theta_{-\tau}\omega, \psi^\varepsilon) -M) _+^{2p-2} dx d\zeta. \end{equation} | (68) |
For the first term on the right-hand side of (65), by (67) we obtain
e^{-\int_0^{ -1} ( 2p-3 - {\frac 12} p \alpha_1 M^{p-2} \|{\mathcal T} \left( {\theta _{r}\omega } \right)\|^{p-2})dr } \| \left ( v^\varepsilon(\tau-1, \tau-t,\theta_{-\tau} \omega, \psi^\varepsilon) -M \right )_+\|^p_{L^p(\mathcal O)} |
\le e^{2p-3} e^{-c_1M^{p-2} } \| \left ( v^\varepsilon(\tau-1, \tau-t,\theta_{-\tau} \omega, \psi^\varepsilon) -M \right )_+\|^p_{L^p(\mathcal O)} |
\begin{equation} \le e^{2p-3} e^{-c_1M^{p-2} } \| v^\varepsilon(\tau-1, \tau-t,\theta_{-\tau} \omega, \psi^\varepsilon)\|^p_{L^p(\mathcal O)}. \end{equation} | (69) |
Similarly, for the second terms on the right-hand side of (65), we have from (67) there exists
\int_{ -1}^0 | \delta (\theta_{\zeta} \omega)|^p e^{-\int_0^{\zeta} ( 2p-3 - {\frac 12} p \alpha_1 M^{p-2} \|{\mathcal T} \left( {\theta _{r}\omega } \right)\|^{p-2}) dr } \|v^\varepsilon(\zeta+\tau, \tau-t, \theta_{-\tau}\omega, \psi^\varepsilon)\|^p_{L^p(\mathcal O)} d\zeta |
\begin{equation} \leq c_4 \int_{-1}^0 e^{ c_1M^{p-2} \zeta } \|v^\varepsilon(\zeta+\tau, \tau-t, \theta_{-\tau}\omega, \psi^\varepsilon)\|^p_{L^p(\mathcal O)} d\zeta \end{equation} | (70) |
Since
2|\mathcal O| \int_{ -1}^0 \|{\mathcal T} \left( {\theta _{\zeta}\omega } \right)\|^{-p} e^{-\int_0^{\zeta} ( 2p-3 - {\frac 12} p \alpha_1 M^{p-2} \|{\mathcal T} \left( {\theta _{r}\omega } \right)\|^{p-2}) dr } \| \psi_1(\zeta+\tau,\cdot) \|_{L^\infty(\widetilde {\mathcal O})}^ {\frac p2} d\zeta. |
+\frac{2p|\mathcal O|}{\alpha_1} \int_{ -1}^0 \|{\mathcal T} \left( {\theta _{\zeta}\omega } \right)\|^{-p} e^{-\int_0^{\zeta} ( 2p-3 - {\frac 12} p \alpha_1 M^{p-2} \|{\mathcal T} \left( {\theta _{r}\omega } \right)\|^{p-2}) dr } \| G(\zeta+\tau,\cdot) \|^2_{L^\infty(\widetilde {\mathcal O})} d\zeta |
\begin{equation} \le c_5 \int_{-1}^0 e^{ c_1M^{p-2} \zeta } d\zeta \le c_1^{-1}c_5 M^{2-p}. \end{equation} | (71) |
By (68)-(71) we get from (65) that
c_3 \int_{ -1}^0 e^{c_2 M^{p-2} \zeta } \int_{\mathcal O} ( v^\varepsilon(\zeta+\tau, \tau-t, \theta_{-\tau}\omega, \psi^\varepsilon) -M) _+^{2p-2} dy d\zeta |
\le e^{2p-3} e^{-c_1M^{p-2} } \|v^\varepsilon({\tau -1}, \tau-t, \theta_{-\tau}\omega, \psi^\varepsilon) \|^p_{L^p(\mathcal O)} |
+ c_4 \int_{-1}^0 e^{ c_1M^{p-2} \zeta } \|v^\varepsilon(\zeta+\tau, \tau-t, \theta_{-\tau}\omega, \psi^\varepsilon)\|^p_{L^p(\mathcal O)} d\zeta + c_1^{-1}c_5 M^{2-p}, |
which together with Lemma 3.2 and Lemma 3.3 implies that there exist
c_3 \int_{ -1}^0 e^{c_2 M^{p-2} \zeta } \int_{\mathcal O} ( v^\varepsilon(\zeta+\tau, \tau-t, \theta_{-\tau}\omega, \psi^\varepsilon) -M) _+^{2p-2} dx d\zeta |
\begin{equation} \le c_6 e^{-c_1M^{p-2} } + c_6 \int_{-1}^0 e^{ c_1M^{p-2} \zeta } d\zeta + c_1^{-1}c_5M^{2-p} \le c_6 e^{-c_1M^{p-2} } + c_1^{-1}(c_5+c_6) M^{2-p}. \end{equation} | (72) |
Since
\begin{equation} \int_{ -1}^0 e^{c_2 M^{p-2} \zeta } \int_{\mathcal O} ( v^\varepsilon(\zeta+\tau, \tau-t, \theta_{-\tau}\omega, \psi^\varepsilon) -M) _+^{2p-2} dy d\zeta \le \eta. \end{equation} | (73) |
Note that
\int_{ -1}^0 e^{c_2 M^{p-2} \zeta } \int_{ \{ y\in \mathcal O: \ v^\varepsilon(\zeta+\tau, \tau-t, \theta_{-\tau}\omega, \psi^\varepsilon ) \ge 2M\} } | v^\varepsilon(\zeta+\tau, \tau-t, \theta_{-\tau}\omega, \psi^\varepsilon )|^{2p-2} dy d\zeta\\\le 2^{2p-2} \int_{ -1}^0 e^{c_2 M^{p-2} \zeta } \int_{\mathcal O} ( v^\varepsilon(\zeta+\tau, \tau-t, \theta_{-\tau}\omega, \psi^\varepsilon) -M) _+^{2p-2} dx d\zeta \le 2^{2p-2} \eta. | (74) |
Similarly, one can verify that there exist
\begin{equation} \int_{ -1}^0 e^{c_2 M^{p-2} \zeta } \int_{ \{ y\in \mathcal O: \ v^\varepsilon(\zeta+\tau, \tau-t, \theta_{-\tau}\omega, \psi^\varepsilon ) \le - 2M \} } | v^\varepsilon(\zeta+\tau, \tau-t, \theta_{-\tau}\omega, \psi^\varepsilon )|^{2p-2} dy d\zeta \le 2^{2p-2} \eta. \end{equation} | (75) |
Then Lemma 3.4 follows from (3) and (75) immediately.
Note that
\begin{equation*} 0\leq \lambda_1^\varepsilon \leq\lambda_2^\varepsilon\leq\ldots\leq\lambda_n^\varepsilon\leq\cdots \to +\infty, \end{equation*} |
and their associated eigenfunctions
It follows from Corollary 9.7 in [8] that the eigenvalues and the eigenfunctions of
Next, we introduce the spectral projections. We use
\begin{equation*} P^\varepsilon_n (u) = \sum\limits_{i = 1}^m(u,\varpi_i^\varepsilon)_{Y_\varepsilon}\varpi_i^\varepsilon \quad\text{for}\; u\in Y_\varepsilon. \end{equation*} |
We use
\begin{equation} a_\varepsilon \left( {u,u} \right) = \left( {A_\varepsilon u,u} \right)_{H_g \left(\mathcal O \right)} \le \lambda ^\varepsilon _n \left( {u,u} \right)_{H_g \left( \mathcal O \right)}, \quad \forall u \in P_n^\varepsilon D\left( {A_\varepsilon ^{1/2} } \right). \end{equation} | (76) |
and
\begin{equation} a_\varepsilon \left( {u,u} \right) = \left( {A_\varepsilon u,u} \right)_{H_g \left(\mathcal O \right)} \ge \lambda _{m + 1}^\varepsilon \left( {u,u} \right)_{H_g \left(\mathcal O \right)} ,\quad u \in Q_m^\varepsilon D\left( {A_\varepsilon ^{1/2} } \right). \end{equation} | (77) |
Let
Lemma 3.5. Assume that (8)-(11), (39) and (42) hold. Then for every
\left\| { u^\varepsilon_2\left( {\tau,\tau-t,\theta _{ - \tau} \omega , \phi^\varepsilon} \right)} \right\|_{H^1(\mathcal O)} \le \eta. |
Proof. Taking the inner product (36) with
\begin{align} & \frac{1}{2}\frac{d}{{dt}}a_\varepsilon(v_2^\varepsilon,v_2^\varepsilon) + \left\| {A_\varepsilon v_2^\varepsilon } \right\|^2 \leq \left( {\delta \left( {\theta _{t } \omega } \right)v_2^\varepsilon , A_\varepsilon v_2^\varepsilon } \right) \\ &\quad + \left( {Q_n^\varepsilon{\mathcal T}^{ - 1} \left( {\theta _{t } \omega } \right)h_\varepsilon\left(t, {y,{\mathcal T}\left( {\theta _{t } \omega } \right)v^\varepsilon} \right), A_\varepsilon v_2^\varepsilon } \right) \\ &\quad+\left(Q_n^\varepsilon {{\mathcal T}^{ - 1} \left( {\theta _{t } \omega } \right)G_\varepsilon\left( {t,y} \right), A_\varepsilon v_2^\varepsilon } \right). \end{align} | (78) |
For the first term on the right-hand side of (78), we have
\begin{equation} \left( {\delta \left( {\theta _{t } \omega } \right)v_2^\varepsilon ,A_\varepsilon v_2^\varepsilon } \right) \le \frac{1}{8}\left\| {A_\varepsilon v_2^\varepsilon } \right\|^2 + 2| {\delta \left( {\theta _{t } \omega } \right)} |^2 \left\| {v_2^\varepsilon } \right\|^2. \end{equation} | (79) |
For the superlinear term, we have from (9) that
\begin{align} &\left( {Q_n^\varepsilon{\mathcal T}^{ - 1} \left( {\theta _{t } \omega } \right) h_\varepsilon\left(t, {y,{\mathcal T}\left( {\theta _{t } \omega } \right)v^\varepsilon} \right), A_\varepsilon v_2^\varepsilon } \right) \\ & \le \frac{1}{8}\left\| {A_\varepsilon v_2^\varepsilon } \right\|^2 + 2\left\| {{\mathcal T}^{ - 1} \left( {\theta _{t } \omega } \right)} \right\|^2 \int_{\mathcal O} {\left| {h_\varepsilon\left(t, {y,{\mathcal T}\left( {\theta _{t} \omega } \right)v^\varepsilon} \right)} \right|} ^2 dy \\ &\le \frac{1}{8}\left\| {A_\varepsilon v_2^\varepsilon } \right\|^2 + 2\alpha_2\left\| {{\mathcal T}^{ - 1} \left( {\theta _{t } \omega } \right)} \right\|^2 \int_{\mathcal O} {\left( {\left| {{\mathcal T}\left( {\theta _{t } \omega } \right)v^\varepsilon} \right|^{p - 1} + \psi _2 \left( t,y^{*},\varepsilon g(y^{*})y_{n+1} \right)} \right)} ^2 dy \\ &\le \frac{1}{8}\left\| {A_\varepsilon v_2^\varepsilon } \right\|^2 + 4\alpha_2\left\| {{\mathcal T}\left( {\theta _{t } \omega } \right)} \right\|^{2p - 4} \left\| v \right\|_{2p - 2}^{2p - 2} + 4\alpha_2|\mathcal O|\left\| {{\mathcal T}^{ - 1} \left( {\theta _{t } \omega } \right)} \right\|^2\| \psi_2(t,\cdot) \|^2_{L^\infty(\widetilde {\mathcal O})} . \end{align} | (80) |
For the last term on the right-hand side of (78), we have
\begin{align} \left( {Q_n^\varepsilon T^{ - 1} \left( {\theta _t \omega } \right) G_\varepsilon \left( {t,y} \right),A_\varepsilon v_2^\varepsilon } \right) \le \frac{1}{8}\left\| {A_\varepsilon v_2^\varepsilon } \right\|^2 + 2|\mathcal O|\left\| {T^{ - 1} \left( {\theta _t \omega } \right)} \right\|^2 \left\| {G\left( {t, \cdot } \right)} \right\|_{L^\infty \left( \widetilde {\mathcal O} \right)}^2 \end{align} | (81) |
Noting that
\begin{align} & \frac{d}{{dt}}a_\varepsilon(v_2^\varepsilon,v_2^\varepsilon) + \lambda _{n + 1}^\varepsilon a_\varepsilon(v_2^\varepsilon,v_2^\varepsilon) \le 4 {\delta^2 \left( {\theta _{t } \omega } \right)} \left\| {v_2^\varepsilon } \right\|^2 \\ &\quad + 8\alpha_2\left\| {{\mathcal T}\left( {\theta _{t } \omega } \right)} \right\|^{2p - 4} \left\| v^\varepsilon \right\|_{2p - 2}^{2p - 2} \\ &\quad + c\left\| {{\mathcal T}^{ - 1} \left( {\theta _{t } \omega } \right)} \right\|^2(\| \psi_2(t,\cdot) \|^2_{L^\infty(\widetilde {\mathcal O})} +\left\| {G\left( {t,\cdot} \right)} \right\|^2_{L^\infty(\widetilde {\mathcal O})}). \end{align} | (82) |
Taking
\begin{align} & a_\varepsilon( v_2^\varepsilon \left( \tau,\tau-t,\theta_{-\tau}\omega,\psi^\varepsilon \right),v_2^\varepsilon \left( \tau,\tau-t,\theta_{-\tau}\omega,\psi^\varepsilon \right)) \\ &\le\int_{\tau - 1 }^{\tau} {e^{\lambda _{n + 1}^\varepsilon \left( {r - \tau } \right)} } a_\varepsilon( v_2^\varepsilon \left( r,\tau-t,\theta_{-\tau}\omega,\psi^\varepsilon \right),v_2^\varepsilon \left( r,\tau-t,\theta_{-\tau}\omega,\psi^\varepsilon \right)) dr\\ & + 4\delta^2\int_{{\tau} - 1}^{\tau} {e^{\lambda _{n + 1}^\varepsilon \left( {r - \tau } \right)} \delta ^2 \left( {\theta _{r - \tau } \omega } \right)} a_\varepsilon( v_2^\varepsilon \left( r,\tau-t,\theta_{-\tau}\omega,\psi^\varepsilon \right),v_2^\varepsilon \left( r,\tau-t,\theta_{-\tau}\omega,\psi^\varepsilon \right)) dr \\ & + 8\alpha_2\int_{{\tau} - 1 }^{\tau} {e^{\lambda _{n + 1}^\varepsilon \left( {r - \tau } \right)} \left\| {{\mathcal T}\left( {\theta _{r - \tau } \omega } \right)} \right\|^{2p - 4} \left\| {v^\varepsilon\left( r,\tau-t,\theta_{-\tau}\omega,\psi^\varepsilon \right)} \right\|_{2p - 2}^{2p - 2} } dr \\ &+c\int_{{\tau} - 1 }^{\tau}e^{\lambda _{n + 1}^\varepsilon \left( {r - \tau } \right)} \left\| {{\mathcal T}^{ - 1} \left( {\theta _{r - \tau } \omega } \right)} \right\|^2 (\| \psi_2(r,\cdot) \|^2_{L^\infty(\widetilde {\mathcal O})}) dr \\ & + c\int_{{\tau} - 1 }^{\tau} {e^{\lambda _{n + 1}^\varepsilon \left( {r - \tau } \right)} \left\| {{\mathcal T}^{ - 1} \left( {\theta _{r - \tau } \omega } \right)} \right\|^2 \left\| {G\left( {r,\cdot} \right)} \right\|^2_{L^\infty(\widetilde {\mathcal O})} } dr. \end{align} | (83) |
Since
\begin{align} &a_\varepsilon( v_2^\varepsilon \left( \tau,\tau-t,\theta_{-\tau}\omega,\psi^\varepsilon \right),v_2^\varepsilon \left( \tau,\tau-t,\theta_{-\tau}\omega,\psi^\varepsilon \right))\\ & \le c\int_{ - 1 }^0 {e^{\lambda _{n + 1}^\varepsilon r} \left\| {v^\varepsilon\left( r+\tau,\tau-t,\theta_{-\tau}\omega,\psi^\varepsilon \right)} \right\|_{2p - 2}^{2p - 2} } dr \\ &\quad + c\int_{ - 1 }^{0} {e^{\lambda _{n + 1}^\varepsilon r} } a_\varepsilon( v^\varepsilon \left(r+ \tau,\tau-t,\theta_{-\tau}\omega,\psi^\varepsilon \right),v^\varepsilon \left( r+\tau,\tau-t,\theta_{-\tau}\omega,\psi^\varepsilon \right)) dr \end{align} |
\begin{align} &\quad + c\int_{ - 1 }^{0} {e^{\lambda _{n + 1}^\varepsilon r} } dr\\ & \le c\int_{ - 1 }^0 {e^{(\lambda _{n + 1}^0-1) r} \left\| {v^\varepsilon\left( r+\tau,\tau-t,\theta_{-\tau}\omega,\psi^\varepsilon \right)} \right\|_{2p - 2}^{2p - 2} } dr \\ &\quad + c\int_{ - 1 }^{0} {e^{(\lambda _{n + 1}^0-1) r} } a_\varepsilon( v^\varepsilon \left( r+\tau,\tau-t,\theta_{-\tau}\omega,\psi^\varepsilon \right),v^\varepsilon \left( r+\tau+s,\tau-t,\theta_{-\tau}\omega,\psi^\varepsilon \right)) dr\\ &\quad + c\int_{ - 1 }^0 {e^{(\lambda _{n + 1}^0-1) r} \left\| {v^\varepsilon\left( {r+\tau - \rho_0(r+\tau+s),\tau-t, \theta_{-\tau}\omega,\psi^\varepsilon } \right)} \right\|^2 } dr\\ &\quad + c\int_{ - 1 }^{0} {e^{(\lambda _{n + 1}^0-1) r} } dr. \end{align} | (84) |
Given
\begin{align} & c\int_{ - 1 }^0 {e^{(\lambda _{n + 1}^0-1) r} \left\| {v^\varepsilon\left( r+\tau,\tau-t,\theta_{-\tau}\omega, \psi^\varepsilon \right)} \right\|_{2p - 2}^{2p - 2} } dr \\ &\leq c\int_{ - 1 }^0 {e^{(\lambda _{n + 1}^0-1) r} \int_{\{y\in \mathcal O:|v^\varepsilon|\geq 2M\}}| {v^\varepsilon\left( r+\tau, \tau-t,\theta_{-\tau}\omega,\psi^\varepsilon \right)} |^{2p - 2}dy } dr \\ &\quad+ c\int_{ - 1 }^0 {e^{(\lambda _{n + 1}^0-1) r} \int_{\{y\in \mathcal O:|v^\varepsilon| < 2M\}} | {v^\varepsilon\left( r+\tau,\tau-t, \theta_{-\tau}\omega,\psi^\varepsilon \right)} |^{2p - 2}dy } dr \\ &\leq c\int_{ - 1 }^0 {e^{\gamma M^{p-2} r} \int_{\{y\in \mathcal O:|v^\varepsilon|\geq 2M\}}| {v^\varepsilon\left( r+\tau, \tau-t,\theta_{-\tau}\omega,\psi^\varepsilon \right)} |^{2p - 2}dy } dr \\ &\quad+ c\int_{ - 1 }^0 {e^{(\lambda _{n + 1}^0-1) r} \int_{\{y\in \mathcal O:|v^\varepsilon| < 2M\}} | {v^\varepsilon\left( r+\tau,\tau-t, \theta_{-\tau}\omega,\psi^\varepsilon \right)} |^{2p - 2}dy } dr \\ &\leq \eta+c2^{2p-2} M^{2p-2}|\mathcal O| \int_{ - 1 }^0 {e^{(\lambda _{n + 1}^0-1) r} } dr \leq \eta+c2^{2p-2} M^{2p-2}|\mathcal O| \frac{1}{\lambda _{n + 1}^0-1} . \end{align} | (85) |
For the last three terms on the right-hand side of (84), by Lemma 3.1, we find that there exist
\begin{align} & c\int_{ - 1 }^{0} {e^{(\lambda _{n + 1}^0-1) r} } a_\varepsilon( v^\varepsilon \left( r+\tau,\tau-t,\theta_{-\tau}\omega,\psi^\varepsilon \right),v^\varepsilon \left( r+\tau,\tau-t,\theta_{-\tau}\omega,\psi^\varepsilon \right)) dr\\ &\quad+ c\int_{ - 1 }^{0} {e^{(\lambda _{n + 1}^0-1) r} } dr \leq c_1 \int_{ - 1 }^{0} {e^{(\lambda _{n + 1}^0-1) r} } dr \leq c_1 \frac{1}{\lambda _{n + 1}^0-1} . \end{align} | (86) |
Since
a_\varepsilon( v_2^\varepsilon \left( \tau+s,\tau-t,\theta_{-\tau}\omega,\psi^\varepsilon \right),v_2^\varepsilon \left( \tau+s,\tau-t,\theta_{-\tau}\omega,\psi^\varepsilon \right)) \le 2\eta, |
which together
In this subsection, we establish the existence of
Lemma 4.1. Suppose (8)-(11), (39) and (43) hold. Then the cocycle
Proof. We first notice that, by Lemma 3.2,
\begin{equation} K \left( {\tau ,\omega } \right) = \left\{ {u \in H^1(\mathcal O) :\left\| u \right\|_{H^1(\mathcal O) }^2 \le L\left( {\tau ,\omega } \right)} \right\}, \end{equation} | (87) |
where
\Phi_\varepsilon \left( {t,\tau - t,\theta _{ - t} \omega ,D \left( {\tau - t,\theta _{ - t} \omega } \right)} \right) \subseteq K\left( {\tau ,\omega } \right). |
Thus we find that
Lemma 4.2. Assume that (8)-(11), (39) and (43) hold. Then, the cocycle
Proof. We will show that for every
\begin{equation} \left\| { u^\varepsilon_2\left( {\tau,\tau-t_n,\theta _{ - \tau} \omega , \phi^\varepsilon} \right)} \right\|_{H^1(\mathcal O)} = \left\| { Q_{m_0}u^\varepsilon\left( {\tau,\tau-t_n,\theta _{ - \tau} \omega , \phi^\varepsilon} \right)} \right\|_{H^1(\mathcal O)} < {\frac {\eta}{4}}. \end{equation} | (88) |
On the other hand, by Lemma 3.2 we find that the sequence
Theorem 4.3. Assume that (8)-(11), (39) and (43) hold. Then, the cocycle
Proof. First, we know from Lemma 4.1 that
Analogous results also hold for the solution of (4)-(5). In particular, we have:
Theorem 4.4. Assume that (8)-(11), (39) and (43) hold. Then, the cocycle
The following estimates are needed when we derive the convergence of pullback attractors. By the similar proof of that of Theorem 5.1 in [14], we get the following lemma.
Lemma 5.1. Assume that (8)-(11) and (39) hold. Then for every
\begin{eqnarray} &&\int_{\tau}^{t} {{\left\| {v^\varepsilon \left( {r,\tau , \omega ,\psi^\varepsilon } \right)} \right\|_{H_\varepsilon ^1(\mathcal O) }^2 }} dr \le c \left\| \psi^\varepsilon \right\|_{\mathcal N}^2 \\ &&\mathit{\mbox{}} +c\int_{ \tau }^{\tau+T} \left( {\left\| {G\left( { r,\cdot} \right)} \right\|_{L^\infty( { {\widetilde {\mathcal O}} } )}^2 + \left\| {\psi_1 \left( { r,\cdot } \right)} \right\|_{L^\infty( { {\widetilde {\mathcal O}} } )}^2 } \right)dr, \end{eqnarray} |
where
Similarly, one can prove
Lemma 5.2. Assume that (8)-(11) and (39) hold. Then for every
\begin{eqnarray} &&\int_{\tau}^{t} {{\left\| {v^0 \left( {r,\tau , \omega ,\psi^0 } \right)} \right\|_{H ^1(\mathcal Q) }^2 }} dr \\ &\le& c \left\| \psi^0 \right\|_{\mathcal M}^2+c \int_{ \tau }^{\tau+T} \left( {\left\| {G\left( { r,\cdot} \right)} \right\|_{L^\infty( { {\widetilde {\mathcal O}} } )}^2 + \left\| {\psi_1 \left( { r,\cdot } \right)} \right\|_{L^\infty( { {\widetilde {\mathcal O}} } )}^2 } \right)dr, \end{eqnarray} |
where
In the sequel, we further assume the functions
\begin{equation} \left\| {G_\varepsilon(t,\cdot) - G_0(t,\cdot) } \right\|_{L^2 \left( \mathcal O \right)} \le \kappa_1(t)\varepsilon \end{equation} | (89) |
and
\begin{equation} \left\| {H_\varepsilon(t,\cdot,s) - H_0(t,\cdot,s) } \right\|_{L^2 \left( \mathcal O \right)} \le \kappa_2(t)\varepsilon , \end{equation} | (90) |
where
By (12) and (90) we have, for all
\begin{equation} \left\| {h_\varepsilon(t,\cdot,s) - h_0(t,\cdot,s) } \right\|_{L^2 \left( \mathcal O \right)} \le \kappa_2(t)\varepsilon. \end{equation} | (91) |
Since
Theorem 5.3. Suppose (8)-(11), (39), and (89)-(90) hold. Given
\mathop {\lim }\limits_{n \to \infty} \left\| {\Phi _{\varepsilon_n} \left( {t,\tau,\omega ,\phi^{\varepsilon_n} } \right) - \Phi _{0 } \left( {t,\tau,\omega ,\phi^0} \right)} \right\|_{\mathcal N} = 0. |
Proof. Since
\begin{eqnarray} && \left\| {v^{\varepsilon_n} \left( t \right)}-{v^0 \left( t \right)} \right\|_{\mathcal N}^2 \le c \left\| \phi^{\varepsilon_n}-\phi^0 \right\|_{{\mathcal N}}^2 +c\mathop {\max }\limits_{\nu \in \left[ {\tau ,t} \right]} \xi \left( {\theta _\nu \omega } \right) \int_\tau ^t { \left\| {{v^{\varepsilon_n} \left( s \right)}- {v^0 \left( s \right)}} \right\|_{{\mathcal N}}^2 } ds \end{eqnarray} |
\begin{eqnarray} &&\mbox{} + c{\varepsilon_n} \mathop {\max }\limits_{\nu \in \left[ {\tau ,t} \right]} \left\| {{\mathcal T}^{ - 1} \left( {\theta _{\nu} \omega } \right)} \right\| \int_\tau ^t {\left( \left\| {v^{\varepsilon_n}(s) } \right\|_{H_{\varepsilon_n} ^1 \left( {\mathcal O} \right)}^2 + \left\| v^0(s) \right\|_{H ^1 \left( {\mathcal Q} \right)}^2 \right)} ds \\ &&\mbox{} + c{\varepsilon_n} \mathop {\max }\limits_{\nu \in \left[ {\tau ,t} \right]} \left\| {{\mathcal T}^{ - 1} \left( {\theta _{\nu} \omega } \right)} \right\| \int_\tau ^t {\left(\kappa^2_1(s)+\kappa^2_2(s) \right)} ds \\ &&\mbox{}+ c{\varepsilon_n} \int_\tau ^t {\left( {\left\| {v^{\varepsilon_n}(s) } \right\|_{H_{\varepsilon_n} ^1 \left( \mathcal O \right)}^2 + \left\| {v^0(s) } \right\|_{H^1 \left( \mathcal Q \right)}^2 } \right)} ds, \end{eqnarray} | (92) |
where
\begin{eqnarray} \left\| {{v^{\varepsilon_n} \left( t \right)}-{v^0 \left( t \right)}} \right\|_{\mathcal N }^2 & \le& e^{c(1+ \mathop {\max }\limits_{\nu \in \left[ {\tau ,\tau+T} \right]} \xi \left( {\theta _\nu \omega } \right)) T} \left\|\phi^{\varepsilon_n}-\phi^0 \right\|_{\mathcal N }^2 \\ &&\mbox{} +\varrho{\varepsilon_n} e^{c(1+ \mathop {\max }\limits_{\nu \in \left[ {\tau ,\tau+T} \right]} \xi \left( {\theta _\nu \omega } \right)) T} [ \| {\psi ^0 } \|_{\mathcal M}^2+\| {\psi ^{\varepsilon_n} } \|_{\mathcal N}^2 \\ &&\mbox{}+\int_\tau ^{\tau+T} {\left(\kappa^2_1(s)+\kappa^2_2(s) \right)} ds \\ &&\mbox{}+ {\int_{ \tau }^{\tau+T} {( {\| {G( { s,\cdot} )} \|_{L^\infty ( {\widetilde{\mathcal O}} )}^2 + \| {\psi_1 ( { s,\cdot} )} \|_{L^\infty ( {\widetilde{\mathcal O} } )}^2 } ) }ds} ]. \end{eqnarray} | (93) |
Notice that, for all
\begin{eqnarray} &&\| {u^{\varepsilon_n} ( {t,\tau,\omega ,\phi^\varepsilon } ) - u^0( {t,\tau,\omega ,\phi^0 } )} \|_{\mathcal N}^2 \\ &\leq& \mathop {\max }\limits_{\nu \in [ {\tau ,\tau + T} ]} \| {{\mathcal T}( {\theta _\nu \omega } )} \|^2 \| {v^{\varepsilon_n} ( {t,\tau,\omega ,{ {\mathcal T}}^{ - 1} ( {\theta _\tau \omega } )\phi^\varepsilon } ) - v^0( {t,\tau,\omega ,{ {\mathcal T}}^{ - 1} ( {\theta _\tau \omega } )\phi^0 } }\|_{\mathcal N }^2, \end{eqnarray} |
which together with (93) implies the desired results.
The next result is concerned with uniform compactness of attractors with respect to
Lemma 5.4. Assume that (8)-(11), (39) and (43) hold. If
\mathop {\lim }\limits_{n \to \infty } \left\| {u^{\varepsilon_n} - u} \right\|_{H^1(\mathcal O)} = 0. |
Proof. Take a sequence
\begin{equation} u^{\varepsilon_n} = \Phi _{\varepsilon _n } \left( {t_n ,\tau - t_n ,\theta _{ - t_n } \omega ,\phi^{\varepsilon_n} } \right). \end{equation} | (94) |
By Lemma 4.1, we have
\begin{equation} \left\| { {Q_{N_1}^{\varepsilon_n} } u^{\varepsilon_n}\left( {\tau,\tau-t_n,\theta _{ - \tau} \omega , \phi^{\varepsilon_n}} \right)} \right\|_{H^1(\mathcal O)} \le \eta. \end{equation} | (95) |
By Lemma 3.2, we have
\begin{equation} \|{ {P_{N_1}^{\varepsilon_n} } u^{\varepsilon_n}\left( {\tau,\tau-t_n,\theta _{ - \tau} \omega , \phi^{\varepsilon_n}} \right)}\|_{H^1(\mathcal O)} < M. \end{equation} | (96) |
It follows from (95) and (96) that
\begin{equation} \mathop {\lim }\limits_{n \to \infty } \left\| {u^{\varepsilon_n} - u } \right\|_{H^1(\mathcal O)} = 0. \end{equation} | (97) |
This completes the proof.
Now we are in a position to prove the main result of this paper.
Theorem 5.5. Assume that (8)-(11), (39), (43), and (89)-(90) hold. The attractors
\mathop {\lim }\limits_{\varepsilon \to 0} \mathit{\text{dist}}_{H^1(\mathcal O)} \left( {\mathcal{A}_\varepsilon \left(\tau, \omega \right),\mathcal{A}_0 \left(\tau, \omega \right)} \right) = 0. |
Proof. Given
\begin{equation} \| u\|^2_{H^1_\varepsilon ( {\mathcal{O}})} \le L(\tau, \omega) \quad \mbox{for all } \ 0 < \varepsilon < \varepsilon_0 \ \ \mbox{and} \ u \in \mathcal{A}_\varepsilon (\tau, \omega), \end{equation} | (98) |
where
\begin{equation} \text{dist}_{H^1(\mathcal O)} \left( {z_n ,\mathcal{A}_0 \left(\tau, \omega \right)} \right) \ge \delta\quad \text{for all}\quad n\in \mathbb{N}. \end{equation} | (99) |
By Lemma 5.4 there exists
\begin{equation} \mathop {\lim }\limits_{n \to \infty } \left\| {z_n - z_ * } \right\|_{H^1(\mathcal O)} = 0. \end{equation} | (100) |
By the invariance property of the attractor
\begin{equation} z_n = \Phi _{\varepsilon _n } \left( {t,\tau-t,\theta _{- t} \omega ,y_n^t } \right). \end{equation} | (101) |
By Lemma 5.4 again there exists
\begin{equation} \mathop {\lim }\limits_{n \to \infty } \left\| {y_n^t - y_*^t } \right\|_{H^1(\mathcal O)} = 0. \end{equation} | (102) |
It follows from Theorem 5.3 that for every
\begin{eqnarray} \mathop {\lim }\limits_{n \to \infty } \Phi _{\varepsilon _n } \left( {t,\tau-t,\theta _{- t} \omega ,y_n^t } \right) = \Phi _0 \left( {t,\tau-t,\theta _{ - t} \omega ,y_ * ^t } \right)\quad \text{in}\quad \mathcal N. \end{eqnarray} | (103) |
By (100), (101), (103) and uniqueness of limits we obtain
\begin{equation} z_ * = \Phi _0 \left( {t,\tau-t,\theta _{ - t} \omega ,y_ * ^t } \right)\quad \text{in}\quad H^1(\mathcal O). \end{equation} | (104) |
Notice that
\begin{equation} \mathop {\lim \sup }\limits_{n \to \infty } \left\| {y_n^t } \right\|_{H^1(\mathcal O)} \le \left\| {K\left( {\tau - t,\theta _{- t} \omega } \right)} \right\|_{H^1(\mathcal O)} \le L\left( {\tau - t,\theta _{ - t} \omega } \right). \end{equation} | (105) |
By (102) and (105) we get, for every
\begin{equation} \left\| {y_ * ^t } \right\|_{H^1(\mathcal Q)} \le L\left( {\tau - t,\theta _{- t} \omega } \right). \end{equation} | (106) |
By
\begin{eqnarray} && \text{dist}_{H^1(\mathcal Q)} \left( {z_ * ,\mathcal A_0 \left( {\tau ,\omega } \right)} \right) \\ & = & \text{dist}_{H^1(\mathcal Q)} \left( {\Phi _0 \left( {t,\tau - t,\theta _{ - t} \omega ,y_ * ^t } \right),\mathcal A_0 \left( {\tau ,\omega } \right)} \right) \\ &\le& \text{dist}_{H^1(\mathcal Q)} \left( {\Phi _0 \left( {t,\tau - t, \theta _{ - t} \omega ,K_0 \left( {\tau - t,\theta _{ - t} \omega } \right)} \right),\mathcal A_0 \left( {\tau ,\omega } \right)} \right)\\ && \to 0,\quad \text{as}\,\,t \to \infty . \end{eqnarray} | (107) |
This implies that
\text{dist}_{H^1(\mathcal O)} \left( {z_n ,\mathcal{A}_0 \left( \tau,\omega \right)} \right) \le \text{dist}_{H^1(\mathcal O)} \left( {z_n ,z_ * } \right) \to 0, |
a contradiction with (99). This completes the proof.
The authors would like to thank the anonymous referee for the useful suggestions and comments.
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