This paper is concerned with the initial-boundary value problem for a class of viscoelastic plate equations on an arbitrary dimensional bounded domain. Under certain assumptions on the memory kernel and the source term, the global well-posedness of solutions and the existence of global attractors are obtained.
Citation: Yang Liu. Long-time behavior of a class of viscoelastic plate equations[J]. Electronic Research Archive, 2020, 28(1): 311-326. doi: 10.3934/era.2020018
[1] | Yang Liu . Long-time behavior of a class of viscoelastic plate equations. Electronic Research Archive, 2020, 28(1): 311-326. doi: 10.3934/era.2020018 |
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This paper is concerned with the initial-boundary value problem for a class of viscoelastic plate equations on an arbitrary dimensional bounded domain. Under certain assumptions on the memory kernel and the source term, the global well-posedness of solutions and the existence of global attractors are obtained.
In this paper, we study the following initial-boundary value problem for nonlinear viscoelastic plate equations
utt+Δ2u−∫t−∞g(t−τ)Δ2u(τ)dτ−Δut=f(u)+h(x), x∈Ω, t>0, | (1) |
u(x,t)=u0(x,t), ut(x,0)=u1(x), x∈Ω, t≤0, | (2) |
u(x,t)=Δu(x,t)=0, x∈∂Ω, t∈R, | (3) |
where
Problem 1-3 can be used to describe the vibrations of viscoelastic materials possessing a capacity of storage and dissipation of mechanical energy, see [17] for the details. And
There have been many works on the long-time behavior of viscoelastic plate equations, we refer the readers to [1,2,4,6,18,20,21,25] and the references therein. As for viscoelastic plate equations with past history, Pata [23] studied
utt+αAu−∫∞0g(τ)Au(t−τ)dτ+μut=0, t>0, |
u(t)=u0(t), ut(0)=u1, t≤0, |
where
utt+Au−∫∞0g(τ)Au(t−τ)dτ=0, t>0, |
u(−t)=u0(t), ut(0)=u1, t≥0. |
Under some assumptions on
utt+αΔ2u−∫t−∞g(t−τ)Δ2u(τ)dτ −div(|∇u|p−2∇u)−Δut+f(u)=h(x), x∈Ω, t>0, |
u(x,t)=u0(x,t), ut(x,t)=∂tu0(x,t), x∈Ω, t≤0, |
u(x,t)=Δu(x,t)=0, x∈∂Ω, t∈R, |
where
utt+αΔ2u−∫t−∞g(t−τ)Δ2u(τ)dτ+f1(ut)+f2(u)=h(x), x∈Ω, t>0, |
{u(x,0)=u0(x), ut(x,0)=u1(x), x∈Ω,u(x,−t)=ϕ0(x,t), x∈Ω, t>0, |
u(x,t)=∂u(x,t)∂n=0, x∈∂Ω, t>0, |
where
In the works mentioned above, authors introduced a variable which reflects the relative displacement history so that the corresponding problem could be turned into an autonomous system. This scheme is so-called the past history approach [12] which suggests to consider some past history variables as additional components of the phase space corresponding to the equation under study.
In the present paper, in order to study the long-time behaviour of solutions of problem 1-3, we employ the past history approach and the operator technique so that Eq. 1 can be transformed into an abstract system in the history phase space. And thus the operator technique combined with the energy estimates becomes a crucial tool for the proof of the existence of global attractors.
This paper is organized as follows. In Section 2 some notations and assumptions on
Throughout the paper, in order to simplify the notations, we denote
‖⋅‖p:=‖⋅‖Lp(Ω), ‖⋅‖:=‖⋅‖L2(Ω). |
‖u‖≤C1‖∇u‖, ‖u‖≤C2‖Δu‖, ‖∇u‖≤C3‖Δu‖. |
As in [3,10,27], we give the following assumptions on
|f(u)−f(v)|≤b(|u|p−2+|v|p−2)|u−v|, ∀u,v∈R, |
where
2≤p<∞ if N≤4, 2≤p≤2N−4N−4 if N>4. |
Moreover,
lim sup|u|→∞F(u)|u|2≤0, | (4) |
and
lim sup|u|→∞uf(u)−ϱF(u)|u|2≤0, | (5) |
where
F(u)=∫u0f(s)ds. |
In addition, as in [7,19,22], we assume that
κ:=1−∫∞0g(t)dt>0. | (6) |
As in [2,3,8,10,27], we define the operator
Au=Δ2u, ∀u∈D(A), |
where the dense domain
D(A)={u∈H4(Ω)∩H10(Ω)|Δu∈H2(Ω)∩H10(Ω)}. |
It is easy to verify that
(u,w)Vγ=(Aγ4u,Aγ4w), ‖u‖Vγ=‖Aγ4u‖, |
(v,w)g,γ=∫∞0g(τ)(v(τ),w(τ))Vγdτ, ‖v‖2g,γ=∫∞0g(τ)‖v(τ)‖2Vγdτ, |
where
Lg,γ:=L2g(R+;Vγ)={v:R+→Vγ|∫∞0g(τ)‖v(τ)‖2Vγdτ<∞}. |
Thus
V3:=H3(Ω)={u∈H3(Ω)∩H10(Ω)|Δu∈H10(Ω)}, |
V2:=H2(Ω)∩H10(Ω), V1:=H10(Ω), V0:=L2(Ω). |
In this way, problem 1-3 can be seen as
utt+Au−∫t−∞g(t−τ)Au(τ)dτ+A12ut=f(u)+h, t>0, | (7) |
u(t)=u0(t), ut(0)=u1, t≤0. |
Now we are in a position to define the auxiliary function
vt(τ)=u(t)−u(t−τ), τ>0, t≥0. |
Thus the viscoelastic dissipation in 7 can be rewritten as
−∫t−∞g(t−τ)Au(τ)dτ=−∫∞0g(τ)Au(t−τ)dτ=−(1−κ)Au+∫∞0g(τ)Avt(τ)dτ. |
Therefore, problem 1-3 is transformed into the following system
{utt+κAu+∫∞0g(τ)Avt(τ)dτ+A12ut=f(u)+h, t>0,vtt(τ)=ut(t)−vtτ(τ), τ>0, t>0, | (8) |
with
{u(0)=u0, ut(0)=u1,v0(τ)=v0(τ), | (9) |
where
u0=u0(0),v0(τ)=u0(0)−u0(−τ), τ>0. |
Definition 2.1.
{(ut,w1)+κ∫t0(A12u,A12w1)dτ+∫t0(vs,w1)g,2ds +(A14u,A14w1)=∫t0(f(u)+h,w1)dτ+(u1,w1)+(A14u0,A14w1),(vt,w2)g,2=(u,w2)g,2−(u0,w2)g,2−∫t0(vsτ,w2)g,2ds+(v0,w2)g,2. |
for any
Thus, in order to deal with problem 1-3, we study the modified problem 8, 9. In fact, for a solution
(utt,w1)+κ(A12u,A12w1)+(vt,w1)g,2+(A14ut,A14w1)=(f(u)+h,w1). |
In view of [17,Chapter 2,Section 4], we see that
κA12u=(1+∫∞0G′(τ)dτ)A12u=A12u+(limτ→∞G(τ)−G(0))A12u, |
it follows that
(utt,w1)+(A12u,A12w1)+(limτ→∞G(τ)−G(0))(A12u,A12w1) +(vt,w1)g,2+(A14ut,A14w1)=(f(u)+h,w1). | (10) |
Since
(limτ→∞G(τ)−G(0))(A12u,A12w1)+(vt,w1)g,2=(limτ→∞G(τ)−G(0))(A12u,A12w1)+∫t0g(τ)(A12vt(τ),A12w1)dτ+∫∞tg(τ)(A12vt(τ),A12w1)dτ, |
and
vt(τ)={u(t)−u0(t−τ),τ≥t,u(t)−u(t−τ),τ<t, |
we deduce that
(limτ→∞G(τ)−G(0))(A12u,A12w1)+(vt,w1)g,2=(limτ→∞G(τ)−G(0))(A12u,A12w1)−∫t0G′(τ)(A12u(t),A12w1)dτ+∫t0G′(τ)(A12u(t−τ),A12w1)dτ−∫∞tG′(τ)(A12u(t),A12w1)dτ+∫∞tG′(τ)(A12u0(t−τ),A12w1)dτ=∫t0G′(τ)(A12u(t−τ),A12w1)dτ+∫∞tG′(τ)(A12u0(t−τ),A12w1)dτ. |
Substituting this into 10, we obtain
(utt,w1)+(A12u,A12w1)+∫t0G′(τ)(A12u(t−τ),A12w1)dτ +∫∞tG′(τ)(A12u0(t−τ),A12w1)dτ+(A14ut,A14w1)=(f(u)+h,w1). |
Due to
∫t0G′(τ)(A12u(t−τ),A12w1)dτ+∫∞tG′(τ)(A12u0(t−τ),A12w1)dτ=−∫t−∞g(t−τ)(A12u(τ),A12w1)dτ, |
we conclude that
(utt,w1)+(A12u,A12w1)−∫t−∞g(t−τ)(A12u(τ),A12w1)dτ +(A14ut,A14w1)=(f(u)+h,w1), |
which shows that
The main results of this paper are stated as follows.
Theorem 2.2. Let
Define the mapping
S(t)(u0,u1,v0)=(u(t),ut(t),vt). |
Then it is easy to see from Theorem 2.2 that
Theorem 2.3. Let
Theorem 3.1. Let
Proof. Let
We construct the approximate solutions of problem 8, 9
un(t)=n∑j=1ξjn(t)ωj, vtn(τ)=n∑j=1ζjn(t)ej(τ), n=1,2,⋯, |
which satisfy
{(untt,ωj)+κ(A12un,A12ωj)+(vtn,ωj)g,2 +(A14unt,A14ωj)=(f(un),ωj)+(h,ωj),(vtnt,ej)g,2=(unt,ej)g,2−(vtnτ,ej)g,2, j=1,2,⋯,n, | (11) |
with
{un(0)=n∑j=1ξjn(0)ωj→u0 in V3,unt(0)=n∑j=1ξ′jn(0)ωj→u1 in V1,v0n=n∑j=1ζjn(0)ej→v0 in Lg,3. | (12) |
The approximate problem 11, 12 can be reduced to an ordinary differential system in the variables
Replacing
E′n(t)+‖A14unt‖2=−(vtnτ,vtn)g,2, | (13) |
where
En(t)=12‖unt‖2+κ2‖A12un‖2+12‖vtn‖2g,2−∫ΩF(un)dx−(h,un). | (14) |
Since
(vtnτ,vtn)g,2=12∫∞0∂∂τ(g(τ)‖A12vtn(τ)‖2)dτ−12∫∞0g′(τ)‖A12vtn(τ)‖2dτ≥0. |
Hence, by integrating 13 with respect to
En(t)+∫t0‖A14unτ‖2dτ≤En(0). | (15) |
It follows from 4 in
∫ΩF(un)dx≤η‖un‖2+Cη|Ω|. |
By virtue of Cauchy's inequality with
(h,un)≤‖h‖‖un‖≤ϵC22‖A12un‖2+14ϵ‖h‖2. |
Consequently, taking sufficiently small
C1:=κ2−ηC22−ϵC22>0, |
we deduce from 14 that
En(t)≥12‖unt‖2+C1‖A12un‖2+12‖vtn‖2g,2−C2(‖h‖2+|Ω|). | (16) |
Hence, from 15, 16 and 12, it follows that
‖unt‖2+‖A12un‖2+‖vtn‖2g,2+∫t0‖A14unτ‖2dτ≤C3. | (17) |
Replacing
12ddt(‖A14unt‖2+κ‖A34un‖2+‖vtn‖2g,3)+‖A12unt‖2=(f(un),A12unt)+(h,A12unt)−(vtnτ,vtn)g,3. |
Noting that
−(vtnτ,vtn)g,3≤0, |
(f(un),A12unt)≤C4‖A12un‖2p−2+14‖A12unt‖2, |
and
(h,A12unt)≤‖h‖2+14‖A12unt‖2, |
we conclude from 17 that
‖A14unt‖2+‖A34un‖2+‖vtn‖2g,3+∫t0‖A12unτ‖2dτ≤C5. | (18) |
Therefore, there exist
un⇀u weakly star in L∞(0,T;V3), | (19) |
unt⇀ut weakly star in L∞(0,T;V1) and weakly in L2(0,T;V2), | (20) |
vtn⇀vt weakly star in L∞(0,T;Lg,3), |
for any
un→u in L2(0,T;V2). |
Moreover, from 18-20, it follows that
un→u in C([0,T];V2). | (21) |
We now claim that for any
∫t0(f(un),ωj)dτ→∫t0(f(u),ωj)dτ, | (22) |
as
Indeed, for any
(f(un)−f(u),w)≤b((|un|p−2+|u|p−2)|un−u|,w). |
If
(f(un)−f(u),w)≤b(‖un‖p−24(p−2)+‖u‖p−24(p−2))‖un−u‖4‖w‖, |
when
(f(un)−f(u),w)≤b(‖un‖p−2N(p−2)3+‖u‖p−2N(p−2)3)‖un−u‖2NN−2‖w‖2NN−4. |
Hence
(f(un)−f(u),w)≤C6(‖un‖p−2V2+‖u‖p−2V2)‖un−u‖V1‖w‖V2≤C7‖un−u‖V1‖w‖V2. | (23) |
If
|∫t0(f(un)−f(u),wj)dτ|≤C8∫t0‖un−u‖V1dτ. |
Thus assertion 22 follows from 21.
For fixed
(vtnτ,ej)g,2=−∫∞0g′(τ)(A12vtn(τ),A12ej(τ))dτ−∫∞0g(τ)(A12vtn(τ),A12ejτ(τ))dτ. |
Hence
limn→∞(vtnτ,ej)g,2=(vtτ,ej)g,2. |
Consequently, for fixed
{(ut,ωj)+κ∫t0(A12u,A12ωj)dτ+∫t0(vs,ωj)g,2ds +(A14u,A14ωj)=∫t0(f(u)+h,ωj)dτ+(u1,ωj)+(A14u0,A14ωj),(vt,ej)g,2=(u,ej)g,2−(u0,ej)g,2−∫t0(vsτ,ej)g,2ds+(v0,ej)g,2. |
Moreover, it is easy to see from 12 that
Next we prove continuous dependence of
{˜utt+κA˜u+∫∞0g(τ)A˜vt(τ)dτ+A12˜ut=f(ˉu)−f(u),˜vtt=˜ut−˜vtτ, | (24) |
{˜u(0)=˜u0=ˉu0−u0, ˜ut(0)=˜u1=ˉu1−u1,˜v0(τ)=˜v0=ˉv0−v0. |
By the arguments similar to [7,Lemma 4.9], we obtain
12ddt(‖˜ut‖2+κ‖A12˜u‖2+‖˜vt‖2g,2)+‖A14˜ut‖2=(f(ˉu)−f(u),˜ut)−(˜vtτ,˜vt)g,2. | (25) |
By the arguments similar to the proof of 23, we have
(f(ˉu)−f(u),˜ut)≤C9‖A12˜u‖‖A14˜ut‖≤C94ϵ‖A12˜u‖2+C9ϵ‖A14˜ut‖2. |
Note that
12ddt(‖˜ut‖2+κ‖A12˜u‖2+‖˜vt‖2g,2)≤C294‖A12˜u‖2. |
As a consequence, by Gronwall's inequality, we obtain
‖˜ut‖2+‖A12˜u‖2+‖˜vt‖2g,2≤C10(‖˜u1‖2+‖A12˜u0‖2+‖˜v0‖2g,2). | (26) |
In particular, by taking
Proof of Theorem 2.2. For
u0m→u0 in V2, u1m→u1 in V0, v0m→v0 in Lg,2. |
According to Theorem 3.1, for any
{umtt+κAum+∫∞0g(τ)Avtm(τ)dτ+A12umt =f(um)+h, t∈(0,∞),vtmt(τ)=umt(t)−vtmτ(τ), τ∈(0,∞), t∈(0,∞), |
{um(0)=u0m, umt(0)=u1m,v0m(τ)=v0m(τ). |
Hence
Set
‖ymt‖2+‖A12ym‖2+‖zm‖2g,2≤C11(‖ymt(0)‖2+‖A12ym(0)‖2+‖zm(0)‖2g,2). | (27) |
By 27 and the arguments similar to the proof of 17, we have
um→u in C([0,T];V2), | (28) |
umt→ut in C([0,T];V0), | (29) |
vtm→vt in C([0,T];Lg,2). | (30) |
Thus
Suppose that
(u0m,u1m,v0m)∈V3×V1×Lg,3, (ˉu0m,ˉu1m,ˉv0m)∈V3×V1×Lg,3, |
such that
(u0m,u1m,v0m)→(u0,u1,v0) in V2×V0×Lg,2, | (31) |
(ˉu0m,ˉu1m,ˉv0m)→(ˉu0,ˉu1,ˉv0) in V2×V0×Lg,2. | (32) |
Set
‖˜umt‖2+‖A12˜um‖2+‖˜vtm‖2g,2≤C12(‖˜u1m‖2+‖A12˜u0m‖2+‖˜v0m‖2g,2). |
Therefore, in terms of 28-32, the conclusions of Theorem 2.2 are derived immediately.
In this section, for the sake of convenience, we denote
‖S(t)(u0,u1,v0)‖2Z:=‖u‖2V2+‖ut‖2V0+‖vt‖2g,2. |
Lemma 4.1. Under the conditions of Theorem 2.3,
Proof. Let
Ut+A12U−εU−εA12u+ε2u+κAu+∫∞0g(τ)Avt(τ)dτ=f(u)+h. | (33) |
Note that
(vtτ,vt)g,2≥ρ2‖vt‖2g,2. |
Multiplying 33 by
E′1(t)+E2(t)≤0, | (34) |
where
E1(t)=12(‖U‖2+κ‖A12u‖2+‖vt‖2g,2+ε2‖u‖2−ε‖A14u‖2−2∫ΩF(u)dx−2(h,u)), |
and
E2(t)=‖A14U‖2−ε‖U‖2−ε2‖A14u‖2+ε3‖u‖2+εκ‖A12u‖2+ε∫∞0g(τ)(A12vt(τ),A12u(t))dτ−ε(f(u),u)−ε(h,u)+ρ2‖vt‖2g,2. |
Hence
E2(t)−εϱE1(t)=εκ(2−ϱ)2‖A12u‖2−ε2(1−ϱ2)‖A14u‖2+ε3(1−ϱ2)‖u‖2+4∑i=1Λi, |
where
Λ1=‖A14U‖2−ε(1+ϱ2)‖U‖2, |
Λ2=ρ2‖vt‖2g,2+ε∫∞0g(τ)(A12vt(τ),A12u(t))dτ−ϱε2‖vt‖2g,2, |
Λ3=ε(ϱ∫ΩF(u)dx−(f(u),u)), |
Λ4=−ε(1−ϱ)(h,u). |
Applying Cauchy's inequality with
Λ2≥ρ2‖vt‖2g,2−ϵ1ε(1−κ)‖A12u‖2−ε4ϵ1‖vt‖2g,2−ϱε2‖vt‖2g,2. |
It follows from 5 that, for any
Λ3≥−ε(η‖u‖2+Cη|Ω|)≥−εηC22‖A12u‖2−εCη|Ω|. |
Moreover,
Λ1≥1C21‖U‖2−ε(1+ϱ2)‖U‖2, |
and
Λ4≥−ε(1−ϱ)(ϵ2C22‖A12u‖2+14ϵ2‖h‖2). |
Consequently, by taking sufficiently small
C13:=κ(2−ϱ)2−ϵ1(1−κ)−ηC22−ϵ2(1−ϱ)C22>0, |
we deduce that
E2(t)−εϱE1(t)≥εC13‖A12u‖2−ε2(1−ϱ2)‖A14u‖2+[1C21−ε(1+ϱ2)]‖U‖2+(ρ2−ε4ϵ1−ϱε2)‖vt‖2g,2−εCη|Ω|−ε(1−ϱ)4ϵ2‖h‖2. |
Choosing
ε≤min{2C13(2−ϱ)C23, 2(2+ϱ)C21, 2ϵ1ρ1+2ϵ1ϱ}, |
we obtain
E2(t)−εϱE1(t)≥−C14(‖h‖2+|Ω|) | (35) |
and
κ‖A12u‖2−ε‖A14u‖2≥C15‖A12u‖2. | (36) |
Since
‖U‖2≥‖ut‖2−ε2‖u‖2, |
we conclude from 36 and the arguments similar to the proof of 16 that
E1(t)≥C16(‖ut‖2+‖A12u‖2+‖vt‖2g,2)−C17(‖h‖2+|Ω|). | (37) |
It follows from 34 and 35 that
E′1(t)+εϱE1(t)≤C14(‖h‖2+|Ω|), |
which yields
E1(t)≤E1(0)e−εϱt+C14εϱ(‖h‖2+|Ω|). |
This, together with 37, gives
‖S(t)(u0,u1,v0)‖2Z≤E1(0)C16e−εϱt+C14+εϱC17εϱC16(‖h‖2+|Ω|). |
Hence
Proof. We decompose
{ˆutt+κAˆu+A12ˆut+∫∞0g(τ)Aˆvt(τ)dτ=0,ˆvtt=ˆut−ˆvtτ,ˆu(0)=u0, ˆut(0)=u1, ˇv0(τ)=v0(τ), | (38) |
{ˇutt+κAˇu+A12ˇut+∫∞0g(τ)Aˇvt(τ)dτ=Φ,ˇvtt=ˇut−ˇvtτ,ˇu(0)=0, ˇut(0)=0, ˇv0(τ)=0, | (39) |
where
Φ=f(u)+h. |
Let
{ψtt+κAψ+A12ψt+∫∞0g(τ)Aφt(τ)dτ=AδΦ,φtt=ψt−φtτ,ψ(0)=0, ψt(0)=0, φ0(τ)=0. | (40) |
Let
Ψt+A12Ψ−εΨ−εA12ψ+ε2ψ+κAψ+∫∞0g(τ)Aφt(τ)dτ=AδΦ. | (41) |
Multiplying 41 by
E′3(t)+2‖A14Ψ‖2−2ε‖Ψ‖2−2ε2‖A14ψ‖2+2ε3‖ψ‖2+2εκ‖A12ψ‖2 +2ε(φt(τ),ψ(t))g,2=2(AδΦ,Ψ)−2(φtτ,φt)g,2, |
where
E3(t)=‖Ψ‖2+κ‖A12ψ‖2−ε‖A14ψ‖2+ε2‖ψ‖2+‖φt‖2g,2. |
Hence
E′3(t)+2∑i=1Λi−2ε2‖A14ψ‖2+2ε3‖ψ‖2+2εκ‖A12ψ‖2≤2(AδΦ,Ψ), | (42) |
where
Λ1=2‖A14Ψ‖2−2ε‖Ψ‖2, |
and
Λ2=2ε(φt(τ),ψ(t))g,2+ρ‖φt‖2g,2. |
Note that
Λ1≥‖A14Ψ‖2+(1C21−2ε)‖Ψ‖2, |
and
Λ2≥(ρ−ε2ϵ)‖φt‖2g,2−2εϵ(1−κ)‖A12ψ‖2. |
Consequently, taking
ϵ≤κ(2−σ1)2(1−κ), |
with some constant
E′3(t)+‖A14Ψ‖2+(1C21−2ε)‖Ψ‖2−2ε2‖A14ψ‖2+2ε3‖ψ‖2 +σ1εκ‖A12ψ‖2+(ρ−ε2ϵ)‖φt‖2g,2≤2(AδΦ,Ψ). |
We further choose
ε<min{1(σ2+2)C21, 2ϵρ1+2ϵσ2, (σ1−σ2)κ2C23, κC23}, |
with some constant
E′3(t)+σ2εE3(t)+‖A14Ψ‖2≤2(AδΦ,Ψ) | (43) |
and
E3(t)≥‖ψt‖2+(κ−εC23)‖A12ψ‖2+‖φt‖2g,2. | (44) |
Applying Hölder's inequality and Cauchy's inequality to the right side of 43, we get
E′3(t)+σ2εE3(t)≤‖Φ‖2V4δ−1. | (45) |
For any
(f(u),w)≤b‖|u|p−1‖2NN+2(1−4δ)‖w‖2NN−2(1−4δ)≤C18‖u‖p−1V2‖w‖V1−4δ. |
Since
‖Φ‖V4δ−1≤‖f(u)‖V4δ−1+‖h‖V4δ−1, |
we deduce from 45 that
E3(t)≤E3(0)e−σ2εt+C19. |
Combining this with 40
‖ˇu‖2V2+4δ+‖ˇut‖2V4δ+‖ˇvt‖2g,2+4δ≤C20, ∀t∈[0,∞). | (46) |
Taking into account
ˇvt(τ)={ˇu(t),τ≥t,ˇu(t)−ˇu(t−τ),0<τ<t, |
we get
ˇvtτ(τ)={0,τ≥t,ˇut(t−τ),0<τ<t. |
Let
‖ˆu‖2V2+‖ˆut‖2V0+‖ˆvt‖2g,2≤C21e−σ2εt, ∀t∈[0,∞). |
This means that
sup(u0,u1,v0)∈B‖S1(t)(u0,u1,v0)‖Z→0, |
as
The author would like to thank the editors and the referees for the valuable comments and suggestions.
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