Volume 2012, Article ID 875913,30pages doi:10.1155/2012/875913
Research Article
Pullback Attractors for Nonclassical Diffusion
Equations in Noncylindrical Domains
Cung The Anh
1and Nguyen Duong Toan
21Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy, Cau Giay, Hanoi, Vietnam
2Department of Mathematics, Haiphong University, 171 Phan Dang Luu, Kien An, Haiphong, Vietnam Correspondence should be addressed to Cung The Anh,[email protected]
Received 26 March 2012; Accepted 20 May 2012
Academic Editor: Ram U. Verma
Copyrightq2012 C. The Anh and N. D. Toan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The existence and uniqueness of a variational solution are proved for the following nonau-tonomous nonclassical diffusion equation ut −εΔut −Δufu gx, t, ε ∈ 0,1, in a
noncylindrical domain with homogeneous Dirichlet boundary conditions, under the assumption that the spatial domains are bounded and increase with time. Moreover, the nonautonomous dynamical system generated by this class of solutions is shown to have a pullback attractorAε,
which is upper semicontinuous atε0.
1. Introduction
In recent years, the evolution equations on noncylindrical domains, that is, spatial domains which vary in time so their Cartesian products with the time variable are noncylindrical
sets, have been investigated extensively see, e.g., 1–3. Much of the progress has been
made for nested spatial domains which expand in time. However, the results focus mainly on formulation of the problems and existence and uniqueness theory, while the existence of attractors of such systems has been less considered, except some recent works for the
reaction-diffusion equationor the heat equation 4,5. This is not really surprising since
such systems are intrinsically nonautonomous even if the equations themselves contain no time-dependent terms and require the concept of a nonautonomous attractor, which has only been introduced in recent years.
In this paper, we consider a class of nonautonomous nonclassical diffusion equations
with appropriate function spaces, and then we establish the existence and uniqueness over a finite time interval of variational solutions. Next, we show that the process of two parameter generated by such solutions has a nonautonomous pullback attractor. Finally, we study the upper semicontinuity of the obtained pullback attractor.
Let{Ωt}t∈Rbe a family of nonempty bounded open subsets ofRNsuch that
s < t⇒Ωs⊂Ωt. 1.1
From now on, we will frequently use the following notations:
Qτ,T :
t∈τ,T
Ωt× {t},
Qτ :
t∈τ,∞
Ωt× {t}, ∀τ ∈R,
τ,T
:
t∈τ,T
∂Ωt× {t},
τ
:
t∈τ,∞
∂Ωt× {t}.
1.2
In this paper we consider the following nonautonomous equation:
P
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
ut−εΔut−Δufu gx, t inQτ,
u0 on
τ
,
u|tτ uτx, x∈Ωτ,
1.3
where ε ∈ 0,1, the nonlinear term f and the external force g satisfy some conditions
specified later on. This equation is called the nonclassical diffusion equation whenε > 0,
and whenε0, it turns to be the classical reaction-diffusion equation.
Nonclassical diffusion equation arises as a model to describe physical phenomena,
such as non-Newtonian flows, soil mechanics, and heat conductionsee, e.g., 6–9. In the
last few years, the existence and long-time behavior of solutions to nonclassical diffusion
equations has attracted the attention of many mathematicians. However, to the best of our knowledge, all existing results are devoted to the study of the equation in cylindrical
domains. For example, under a Sobolev growth rate of the nonlinearity f, problem 1.3
in cylindrical domains has been studied 10–13for the autonomous case, that is the case
g not depending on timet and in 14,15 for the nonautonomous case. In this paper, we
will study the existence and long-time behavior of solutions to problem1.3in the case of
noncylindrical domains, the nonlinearityfof polynomial type satisfying some dissipativity
condition, and the external forceg depending on timet. It is noticed that this question for
problem1.3in the case ε 0, that is, for the reaction-diffusion equation, has only been
In order to study problem1.3, we make the following assumptions.
H1The functionf∈C1R,Rsatisfies that
−βα1|s|p≤fss≤βα2|s|p, 1.4
fs≥ −, 1.5
for somep ≥ 2, whereα1, α2, β, are nonnegative constants. By1.4, there exist
nonnegative constantsα1, α2, βsuch that
−βα1|s|p≤Fs≤βα2|s|p, 1.6
whereFu 0ufsdsis the primitive off.
H2The external forceg ∈L2
locRN1.
H3The initial datumuτ ∈H01Ωτ∩LpΩτis given.
Since the open set Ωt changes with time t, problem 1.3 is nonautonomous even
when the external force g is independent of time. Thus, in order to study the long-time
behavior of solutions to1.3, we use the theory of pullback attractors. This theory has been
developed for both the nonautonomous and random dynamical systems and has shown
to be very useful in the understanding of the dynamics of these dynamical systems see
16and references therein. The existence of a pullback attractor for problem1.3in the
caseε 0, that is, for the classical reaction-diffusion equation, has been derived recently
in 4. In the case ε > 0, since 1.3 contains the term −εΔut, this is essentially different
from the classical reaction-diffusion equation. For example, the reaction-diffusion equation
has some kind of “regularity”; for example, although the initial datum only belongs to a weaker topology space, the solution will belong to a stronger topology space with higher regularity, and hence we can use the compact Sobolev embeddings to obtain the existence of
attractors easily. However, for problem1.3whenε >0, because of−Δut, if the initial datum
uτbelongs toH01Ωτ∩LpΩτ, the solutionutwith the initial conditionuτ uτis always
inH01Ωt∩LpΩtand has no higher regularity, which is similar to hyperbolic equations.
This brings some difficulty in establishing the existence of attractors for the nonclassical
diffusion equations. Other difficulty arises since the considered domain is not cylindrical,
so the standard techniques used for studying evolution equations in cylindrical domains cannot be used directly. Therefore, up to now, although there are many results on attractors
for evolution equations in cylindrical domainssee, e.g., 17,18, little seems to be known
for the equations in noncylindrical domains.
In this paper, we first exploit the penalty method to prove the existence and uniqueness
of a variational solution satisfying the energy equality to problem1.3. Next, we prove the
existence of a pullback attractorAε for the process associated to problem1.3. Finally, we
study the continuous dependence onε of the solutions to problem 1.3, in particular we
show that the solutions of the nonclassical diffusion equations converge to the solution of
the classical reaction-diffusion equation as ε → 0. Hence using an abstract result derived
recently by Carvalho et al. 19and techniques similar to the ones used in 14, we prove the
upper semicontinuity of pullback attractorsAεinL2Ωtatε0. The last result means that
attractor A0 of the classical reaction-diffusion equations as ε → 0, in the sense of the
Hausdorffsemidistance.
The paper is organized as follows. In Section 2, for the convenience of readers, we
recall some results on the penalty method and the theory of pullback attractors. After some
preliminary results in Section 2, we proceed by a penalty method to solve approximated
problem, and then we also prove the existence and uniqueness of the solution to problem
1.3inSection 3. InSection 4, a uniform estimate for the solutions is then obtained under an
additional assumption of the external forceg, and this will lead to the proof of existence of
a pullback attractorAεin an appropriate framework. The upper semicontinuity of pullback
attractorsAεatε0 is investigated inSection 5. In the last section, we give some discussions
and related open problems.
Notations. In what follows, we will introduce some notations which are frequently used
in the paper. DenoteHr :L2ΩrandVr :H01Ωrfor eachr∈R, and denote by·,·rand
| · |r the usual inner product and associated norm inHr and by·,·and · r the usual
gradient inner product and associated norm inVr. For eachs < t, consider Vs as a closed
subspace ofVtwith the functions belonging toVsbeing trivially extended by zero. It follows
from1.1that{Vt}t∈τ,T can be considered as a family of closed subspaces ofVT for each
T > τwith
s < t⇒Vs⊂Vt. 1.7
In addition,Hrwill be identified with its topological dualHr∗by means of the Riesz theorem
andVrwill be considered as a subspace ofHr∗withv∈Vridentified with the elementfv∈Hr∗
defined by
fvh v, hr, h∈Hr. 1.8
The duality product betweenVr∗andVrwill be denoted by·,·.
2. Preliminaries
2.1. Penalty Method
To study problem1.3, for eachT > τ, we consider the following auxiliary problem:
ut−εΔut−Δufu gx, t inQτ,T,
u0 on
τ,T
,
u|tτ uτx, x∈Ωτ,
2.1
whereτ∈R,uτ :Ωτ → Randg:Qτ → Rare given functions.
The method of penalization due to Lionssee 20will be used to prove the existence
as a consequence, the existence and uniqueness of a solution to problem1.3satisfying the
energy equality a.e. inτ,∞. To begin, fixT > τand for eacht∈ τ, Tdenote by
Vt⊥:{v∈VT:v, ωT 0,∀ω∈Vt} 2.2
the orthogonal subspace ofVtwith respect the inner product inVT and byPt∈ LVTthe
orthogonal projection operator fromVTontoVT⊥, which is defined as
Ptv∈Vt⊥, v−Ptv∈Vt, 2.3
for eachv∈VT. Finally, definePt PTfor allt > T and observe thatPTis the zero of
LVT.
We will now approximatePtby operators which are more regular in time. Consider
the familypt;·,·of symmetric bilinear forms onVTdefined by
pt;v, ω: Ptv, ωT, ∀v, ω∈VT, ∀t≥τ. 2.4
It can be proved that the mapping τ,∞t→pt;v, ω∈Ris measurable for allv, ω∈VT.
Moreover,|pt;v, ω| ≤ vTωT. For each integerk≥1 and eacht≥τ, we define
pkt;v, ω:k
1/k
0
ptr;v, ωdr, ∀v, ω∈VT, ∀t≥τ, 2.5
and denote byPkt∈ LVTthe associated operator defined by
Pktv, ωT :pkt;v, ω, ∀v, ω∈VT, ∀t≥τ. 2.6
Lemma 2.1see 2,4. For any integer 1≤h≤k, anyt≥τand everyv, ω∈VT,
pkt;v, ω pkt;ω, v,
0≤pht;v, v≤pkt;v, v≤pt;v, v Ptv2T ≤ v2T,
pkt;v, v: d
dtpkt;v, v k
p
t1
k;v, v
−pt;v, v
≤0,
Pktv, zT0, ∀z∈Vt.
2.7
Moreover, for every sequence{vk} ⊂L2τ, T;VTweak convergent tovinL2τ, T;VT,
lim inf
k→∞
T
τ
pkt;vkt, vktdt≥
T
τ
pt;vt, vtdt. 2.8
LetJ:VT → VT∗be the Riesz isomorphism defined by
and for each integerk≥1 and eacht∈ τ, T, we denote
Akt:−Δ kJPkt. 2.10
Obviously,Akt ∈ LVT, VT∗, t∈ τ, T, is a family of symmetric linear operators such that
the mappingt∈ τ, T→Akt∈ LVT, VT∗is measurable, bounded, and satisfies
Aktv, vT ≥ v2T, ∀v∈VT,∀t∈ τ, T. 2.11
Letuτ∈VTbe given and for eachk≥1 consider the following problem:
ukt, vTAktukt, vT
εAktukt, v
T
fukt, v
T
gt, vT, ∀v∈VT,
ukτ, vT uτ, vT.
2.12
The idea of the penalty method is as follows: for eachk≥1 we first prove the existence
of a solutionukto problem2.12 a problem in a cylindrical domainusing standard methods
such as the Galerkin method, and then show thatukconverges to a solution to problem2.1
a problem in a noncylindrical domainin some suitable sense, and as a consequence, the
existence of a solution to problem1.3 seeSection 3for details.
2.2. Pullback Attractors
Since the open setΩtchanges with timet, problem1.3is nonautonomous even when the
external force g is independent of time. Thus, in order to study the long-time behavior of
solutions to1.3, we use the theory of pullbackD-attractors which is a modification of the
theory in 16.
Consider a processU·,·on a family of metric spaces{Xt, dt;t∈R}, that is, a family
{Ut, τ;−∞ < τ ≤ t < ∞}of mappingsUt, τ : Xτ → Xt such thatUτ, τx xfor all
x∈Xτ and
Ut, τ Ut, rUr, τ ∀τ≤r ≤t. 2.13
In addition, suppose D is a nonempty class of parameterized sets of the form D
{Dt;Dt⊂Xt, Dt/∅, t∈R}.
Definition 2.2see 4. The processU·,·is said to be pullbackD-asymptotically compact
if the sequence{Ut, τnxn}is relatively compact inXtfor any t ∈ R, anyD ∈ D , and any
sequences{τn}and{xn}withτn → −∞andxn∈Dτn.
Definition 2.3see 4. A familyB ∈ D is said to be pullbackD-absorbing for the process
U·,·if for anyt∈Rand anyD ∈ D , there existsτ0t,D≤tsuch that
Ut, τDτ⊂ Bt, 2.14
Remark 2.4. Note that if B ∈ D is pullbackD-absorbing for the processU·,·and Bt is
a compact subset ofXtfor anyt ∈ R, then the process U·,·is pullbackD-asymptotically
compact.
For each t ∈ R, let disttD1, D2be the Hausdorffsemi-distance between nonempty
subsetsD1andD2ofXt, which is defined as
disttD1, D2 sup
x∈D1 inf
y∈D2
dXt
x, y forD1, D2⊂Xt. 2.15
Definition 2.5 see 4. The family A {At;At ⊂ Xt, At/∅, t ∈ R} is said to be a
pullbackD-attractor forU·,·if
1atis a compact set ofXtfor allt∈R,
2Ais pullbackD-attracting, that is,
lim
τ→ −∞disttUt, τDτ, At 0 ∀D ∈ D ,∀t∈R, 2.16
3Ais invariant, that is,
Ut, τAτ At for− ∞< τ≤t <∞. 2.17
Theorem 2.6see 4. Suppose that the processU·,·is pullbackD-asymptotically compact and
thatB ∈ Dis a family of pullbackD-absorbing sets forU·,·. Then, the familyA {At;t ∈R}
defined byAt: ΛB, t , t∈R, where for eachD ∈ Dandt∈R,
ΛD, t :
s≤t
τ≤s
Ut, τDτXtclosure inXt 2.18
is a pullbackD-attractor forU·,·, which in addition satisfies
At
D∈D
ΛD, t
Xt
. 2.19
Furthermore,Ais minimal in the sense that ifC {Ct;t ∈R}is a family of nonempty sets such
thatCtis a closed subset ofXtand
lim
τ→ −∞disttUt, τBτ, Ct 0 for any t∈R, thenAt⊂Ct∀t∈R. 2.20
2.3. The Upper Semicontinuity of Pullback Attractors
We now state some results on upper semicontinuity of pullback attractors, which are slight
modifications of those in 19. Because the proof is very similar to the one in 19, so we omit
Definition 2.7. Let{Uε·,·:ε∈ 0,1}be a family of evolution processes in a family of Banach
spaces{Xt} with corresponding pullbackD-attractors {Aεt : ε ∈ 0,1, t ∈ R}. For any
bounded intervalI ∈R, we say that{Aε·}is upper semicontinuous atε0 fort∈Iif
lim sup
ε→0t∈I
disttAεt, A0t 0. 2.21
Theorem 2.8. Let{Uε·,· : ε ∈ 0,1}be a family of processes with corresponding pullback D
-attractors{Aε· :ε ∈ 0,1}. Then, for any bounded intervalI ⊂ R,{Aε· :ε∈ 0,1}is upper
semicontinuous at 0 fort∈Iif for eacht∈R, for eachT >0, and for each compact subsetKofXt−τ,
the following conditions hold:
isupτ∈ 0,Tsupx∈KdisttUεt, t−τx, U0t, t−τx → 0 asε → 0,
iiε∈ 0,1 t≤t0 Aεtis bounded for givent0,
iii0<ε≤1 Aεtis compact for eacht∈R.
3. Existence and Uniqueness of Variational Solutions
For eachT > τ, denote
Qτ,T : ΩT×τ, T,
Uτ,T :
Φ∈L∞τ, T;VT∩Lp
Qτ,T
,Φ∈L2τ, T;VT,
Φτ ΦT 0,Φt∈Vta.e. inτ, T
.
3.1
Definition 3.1. A variational solution of2.1is a functionusuch that
C1u∈L∞τ, T, VT∩LpQτ,T,u∈L2τ, T;VT,
C2for allΦ∈Uτ,T,
T
τ
−ut,ΦtT ut,ΦtTεut,ΦtTfu,ΦtTdt
T
τ
gt,ΦtTdt,
3.2
C3ut∈Vta.e. inτ, T,
C4limt↓τt−τ−1
t
τ|ur−uτ|
2
Tdr 0.
Remark 3.2. If T2 > T1 > τ and u is a variational solution of 2.1 with T T2, then the
restriction ofutoQτ,T1is a variational solution of2.1withTT1.
DenoteQτ :UT>τQτ,T.
Definition 3.3. A variational solution of1.3is a functionu : Qτ → Rsuch that for each
To prove the uniqueness of variational solutions to problem 2.1, we need the following lemmas.
Lemma 3.4see 4. Assume thatv∈L2τ, T, V
T∩LpQτ,Tand there existξ∈L2τ, T, VT∗and
η∈Lp/p−1Q
τ,Tsuch that
T
τ
vt,ΦtTdt−
T
τ
ξt,ΦtTdt−
T
τ
ηt,ΦtTdt, 3.3
for every functionΦ∈Uτ,T.
For each 0< h < T−τ, definevhby
vh:
h−1vth−vt a.e.inτ, T−h,
0 a.e.inT−h, T. 3.4
Then
lim
h↓0
T
τ
vht, ωtTdt
T
τ
ξt, ωtTdt
T
τ
ηt, ωtTdt, 3.5
for every functionω∈L2τ, T;V
T∩LpQτ,Tsuch thatωt∈Vta.e. inτ, T.
Remark 3.5. Ifτ < T < T andΦ ∈ L2τ, T;V
T∩LpΩT ×τ, T, with Φ ∈ L2τ, T;HT
satisfies Φτ ΦT 0 and Φt ∈ Vta.e. in τ, T, then the trivial extension Φ of Φ
satisfiesΦ ∈Uτ,T, withΦ Φ. Using the open setsΩt: ΩtT−T, τ ≤t≤T, it is easy to see
that under the conditions of3.5, one also has
lim
h↓0
T−h
τ
vht, ωtTdt
T
τ
ξt, ωtTdt
T
τ
ηt, ωtTdt, 3.6
for everyτ ≤T≤T and every functionω∈L2τ, T;V
T∩LpQτ,Tsuch thatωt∈Vta.e. in
τ, T.
Lemma 3.6see 4. Letvi∈L2τ, T;VT ∩LpQτ,T, i1,2, be two functions such thatvit∈
Vta.e. inτ, Tfori 1,2. Assume that there existξi ∈ L2τ, T;VT∗, ηi ∈ Lp/p−1Qτ,T, i 1,2
such that
T
τ
vit,Φt
Tdt−
T
τ
ξit,ΦtTdt−
T
τ
ηit,Φt
Tdt, i1,2, 3.7
for every functionΦ ∈ Uτ,T. Then, for every pairτ ≤ s < t ≤ T of Lebesgue points of the inner
v1t, v2tT−v1s, v2sT
t
s
ξ1r, v2rTdr
t
s
ξ2r, v1rTdr
t
s
η1r, v2rTdr
t
s
η2r, v1rTdr
lim
h↓0h
−1
t−h
s
v1rh−v1r, v2rh−v2rTdr.
3.8
Ifuis a variational solution of problem2.1, thenτis the Lebesgue point of|u|2
T since
the conditionC4is satisfied. The next corollary gives an obvious consequence of3.8.
Corollary 3.7. Ifuis a variational solution of 2.1, then for every Lebesgue pointt∈τ, Tof|u|2T it holds
|ut|2Tεut2T2
t
τ
ur2Tdr2
t
τ
fur, urTdr
|uτ|2Tεuτ2T2
t
τ
gr, urTdrlim
h↓0h
−1
t−h
τ
|urh−ur|2Tdr.
3.9
Proof. Ifuis a variational solution of2.1, then we have
T
τ
−ut,ΦtTε
∂u
∂t,Φt
T
ut,ΦtTfut,ΦtT
dt T τ
gt,ΦtTdt.
3.10
ApplyingLemma 3.6withv1v2u, we get
|ut|2T− |uτ|2T −
t
τ
ε∂u
∂t ur, ur
T dr− t τ ε∂u
∂t ur, ur
T dr − t τ
fur−gr, urTdr−
t
τ
fur−gr, urTdr
lim
h→0h
−1
t−h
τ
|urh−ur|2Tdr
−2
t
τ
ur2Tdr−2
t
τ
fur, urTdr2
t
τ
gr, urTdr
−εut2Tεuτ2Thlim→0h−1
t−h
τ
|urh−ur|2Tdr.
3.11
The aim of this section is to obtain a variational solution of2.1such that
|ut|2Tεut2T2
t
τ
ur2Tdr2
t
τ
fur, urTdr
|uτ|2Tεuτ2T2
t
τ
gr, urTdr.
3.12
We will say that u satisfies the energy equality in τ, T if 3.12 is satisfied a.e. in τ, T.
Analogously, if u is a variational solution of 1.3, we will say that u satisfies the energy
equality a.e. inτ,∞if for eachT > τthe restriction ofutoQτ,Tsatisfies the energy equality
3.12a.e. inτ, T.
For any functionv∈L2τ, T;H
Tand anyt∈τ, T, we put
ηv,Tt:lim sup
h↓0
h−1
t−h
τ
|vrh−vr|2Tdr. 3.13
Then ηv,T is a nondecreasing function. By Corollary 3.7, a variational solution u of 1.3
satisfies the energy equality a.e. inτ, Tif and only ifηu,Tt 0 for a.e.t ∈ τ, T. In fact,
using the continuity of the following mapping:
t∈ τ, T→ |uτ|2Tεuτ2T2
t
τ
gr, urT− ur2T−fur, urTdr, 3.14
one can see that a variational solutionuof1.3satisfies the energy equality a.e. inτ, Tif
and only ifηu,TT 0.
The next lemma provides a sufficient condition foruto satisfy the energy equality a.e.
inτ, T.
Lemma 3.8. Letube a variational solution of 2.1and suppose that there exists a sequence{tn} ⊂
τ, Tof Lebesgue points of|u|2
Tsuch thattn → T and
lim sup
n−→∞
|utn|2Tεutn2T
≤ |uτ|2Tεuτ2T2
T
τ
gr, urT− urT2 −fur, urTdr.
3.15
Proof. It is sufficient to prove thatηu,TT 0. Sincetn → T andηu,T is nondecreasing, by Corollary 3.7, we have
ηu,TT≤lim sup
n↑∞ ηu,Ttn lim supn→∞ |utn|
2
Tεutn2T− |uτ|2T−εuτ2T
−2
tn
τ
gr, urT− urT2 −fur, urTdr
!
≤ lim sup
n→∞
|utn|2Tεutn2T
− |uτ|2T−εuτ2T
−2
T
τ
gr, urT− urT2 −fur, urTdr
≤ 0.
3.16
This completes the proof.
Proposition 3.9. Let u, u be two variational solutions of 2.1 corresponding to the initial data
uτ, uτ ∈Vτ∩LpΩτ, respectively, which satisfy the energy equality a.e. inτ, T. Then,
|ut−ut|2Tεut−ut2T2
t
τ
ur−ur2Tdr
≤e2t−τ|uτ−uτ|2Tεuτ−uτ2T
a.e. t∈τ, T.
3.17
Hence, it implies the uniqueness of variational solutions to 2.1 satisfying the energy equality in
τ, T.
Proof. Sinceu, usatisfy the energy equation,ηu,Tt ηu,Tt 0 for allt∈τ, Tand
|ut−ut|2Tεut−ut2T2
t
τ
ur−ur2Tdr
2
t
τ
fur−fur, ur−urTdr
≤ |uτ−uτ|2Tεuτ−uτ2T−2 limh↓0 h−1
t−h
τ
urh−ur, urh−urTdr.
On the other hand,
"" ""
"h−1
t−h
τ
urh−ur, urh−urTdr"""""
2
≤ h−1
t−h
τ
|urh−ur|2dr !
h−1
t−h
τ
|urh−ur|2dr !
,
3.19
so
lim
h↓0 h
−1
t−h
τ
urh−ur, urh−urTdr0. 3.20
Using this and1.5in3.12, one can conclude
|ut−ut|2Tεut−ut2T2
t
τ
ur−ur2Tdr
≤ |uτ−uτ|2Tεuτ−uτ2T−2
t
τ
fur−fur, ur−urTdr
≤ |uτ−uτ|2Tεuτ−uτ2T2
t
τ
|ur−ur|2Tdr.
3.21
By an application of Gronwall’s inequality, we get3.17.
The method of penalization will now be used to prove the existence and uniqueness
of a variational solution to problem 2.1 satisfying an energy equality a.e. in τ, T and,
as a consequence, the existence and uniqueness of a variational solution to problem 1.3
satisfying the energy equality a.e. inτ,∞.
Theorem 3.10. Letg ∈L2τ, T;H
T, uτ ∈Vτ∩LpΩτbe given. Then problem2.1has a unique
variational solution satisfying the energy equality a.e. inτ, T.
Proof. We divide the proof into two steps.
Step 1. Existence of a weak solution to problem2.12.
We will use the Galerkin methodsee 20. Take an orthonormal Hilbert basis{ej}
of HT formed by elements of VT ∩LpΩT such that the vector space generated by{ej} is
dense inVT andLpΩT. Then, one takes a sequence{uτm}converging touτ inVT, with{uτm}
in the vector space spanned by themfirst{ej}. For each integerm ≥ 1, one considers the
approximationukmt
m
j1γkm,jtej, defined as a solution of
uk
mt, ej
T
Aktukmt, ej
Tε
#
Aktukmt, ej
$
T
fukmt, ej
T
gt, ej
T,
ukmτ, ej
T
uτm, ej
T.
Multiplying3.22byγk
m,jtand summing fromj1 tom, we obtain
uk
mt, u
kmt
T
#
Aktukmt, u
kmt
$
ε#Aktukmt, u
kmt
$
T
fukmt, u
kmt
T
gt, uk
mt
T, 3.23 or "" "u
kmt
"" "2 T 1 2 d
dtukmt
2
Tε%%%ukmt
%%
%2
Tk
Pktukmt, ukmt
T
εkPktukmt, u
kmt
T
fukmt, u
kmt
T
gt, uk
mt
T.
3.24
Thus,
gt, uk
mt
T ≥
""
"u
kmt
"" "2
Tε
%%
%u
kmt
%% %2 T 1 2 d dt
ukmt
2
TkPktukmt, ukmtT
εkPktukmt, u
kmt
T
fukmt, u
kmt
T
"""u
kmt
"" "2
Tε
%%
%u
kmt
%%
%2
Tεk
Pktukmt, u
kmt
T 1 2 d dt &
ukmt
2
TkPktukmt, ukmtT2
ΩT
Fukmx, tdx
' .
3.25
We have
gt, ukmt
T≤
1 2
""
gt""2T"u"" kmt"""2
T
, 3.26
so
""gt""2T ≥2ε%%%uk
mt
%% %2 T d dt &
ukmt
2
TkPktukmt, ukmtT2
ΩT
Fukmx, tdx
'
2εkPktukmt, ukmt
T
""
"u
kmt
"" "2
T.
Integrating3.27on τ, t, τ≤t≤T, we obtain 2ε t τ %% %u
kmr
%%
%2
Tdr2εk
t
τ
Pkrukmr, ukmr
Tdr t τ "" "u
kmr
"" "2
Tdr
ukmt
2
TkPktukmt, ukmtT2
ΩT
Fukmx, tdx
≤
t
τ
""gr""2Tdrukmτ
2
TkPkτukmτ, ukmτT2
ΩT
Fukmx, τdx.
3.28
Since
ΩT
Fukmx, tdx≥ −β|ΩT|α1ukmt
p LpΩ
T,
ΩT
Fukmx, τdx≤β|ΩT|α2uτm
p LpΩT,
3.29 we have 2ε t τ %% %u
kmr
%%
%2
Tdr2εk
t
τ
Pkrukmr, u
kmr
Tdr t τ "" "u
kmr
"" "2
Tdr
ukmt
2
TkPktukmt, ukmtT2α1ukmt p LpΩ
T
≤
t
τ
""gr""2Tdrukmτ
2
TkPkτukmτ, ukmτT4β|ΩT|2α2uτm p LpΩT.
3.30
From3.30, we deduce that
{ukm}is bounded inL∞τ, T;VT∩L
pQ τ,T,
{ukm} ukweakly inL∞τ, T;VT∩L
pQ τ,T,
{uk
m}is bounded inL
2τ, T;V
T,
{uk
m} u
kweakly inL2τ, T;VT.
Since{ukm}is bounded inL∞τ, T;VT∩LpQτ,T, one can check that{fukm}is bounded in
Lqτ, T;LqΩ
Twithqp/p−1, hencefukm ηinL
qτ, T;LqΩ
T. We now prove that
ηfuk.
Indeed, we have
VT⊂⊂HT ⊂VT∗,
{ukm}is bounded inL∞τ, T;VT,
{uk
m}is bounded inL
2τ, T;V∗
By the Aubin-Lions lemma 20, Chapter 1, {ukm} is relatively compact in L2τ, T;HT.
Therefore, one can assume thatukm → ukstrongly inL2τ, T;HT, soukm → uka.e. inQτ,T.
Sincefis continuous,fukm → fuka.e. inQτ,T. Applying Lemma 1.3 in 20, we have
fukm fuk weakly inL
qτ, T;LqΩ
T. 3.31
This implies thatukis a weak solution of problem2.12.
Step 2. Existence of a variational solution to2.1satisfying the energy equality.
From3.30, we have
k
T
τ
Pkrukmr, ukmrTdr
≤T−τ
t
τ
""gr""2Tdrukmτ
2
TkPkτukmτ, ukmτT
4β|ΩT|2α2 uτm p LpΩ
T
! .
3.32
Consider the functionΦ:L2τ, T;V
T → Rdefined by
Φv
T
τ
Pktvt, vtTdt, v∈L2τ, T;VT. 3.33
It is easy to see that Φ is a continuous and convex function. It follows that
T
τPktukmt, ukmtTdt is weakly lower semicontinuous in L2τ, T;VT. Moreover,
{ukm} ukweakly inL2τ, T;VT, hence
k
T
τ
Pktukt, uktTdt
≤klim inf
m→ ∞
T
τ
Pktukmt, ukmtTdt
≤T−τ
t
τ
""gr""2
Tdruτ
2
TkPkτuτ, uτT4β|ΩT| 2α2 uτpLpΩ T
! .
Since{uk
m} u
kweakly inL2τ, T;VT, then, reasoning as above,
2εk
T
τ
Pktukt, ukt
Tdt
≤2klim inf
m→ ∞
T
τ
Pktukmt, ukmtTdt
≤ t
τ
""gr""2Tdruτ2TkPkτuτ, uτT4β|ΩT|2α2uτpLpΩ
T
! .
3.35
From the facts thatukm ukweakly inL∞τ, T;VT,ukm ukweakly inL2τ, T;VTand the
weak lower semicontinuity of the norm, we deduce that
2ε
t
τ
%%u
kr%%
2
Tdr2εk
t
τ
Pkrukr, ukr
Tdr
t
τ
""u
kr""
2
Tdr
ukt2Tk
T
τ
Pktukt, uktT2α1uktpLpΩ T
≤5T−τ
t
τ
""gr""2Tdruτ2TkPkτuτ, uτT4β|ΩT|2α2 uτpLpΩ
T
!
C.
3.36
Sinceuτ ∈Vτ∩LpΩτ,Pkτuτ, uτT0 for allk≥1,3.36gives
{uk}is bounded inL∞τ, T;VT∩Lpτ, T;LpΩT,
{uk}is bounded inL2τ, T;V
T,
{uk} uweakly inL∞τ, T;VT∩Lpτ, T;LpΩT,
{uk} uweakly inL2τ, T;V
T.
FromLemma 2.1, we have
T
τ
Ptut2Tdt≤lim inf
k→ ∞
T
τ
Pktukt, uktTdt≤lim infk→ ∞
C
k 0,
that is, Ptut 0 a.e.inτ, Torut∈Vta.e.inτ, T,
T
τ
%%Ptut%%2
Tdt≤lim infk→ ∞
T
τ
Pktukt, ukt
Tdt≤lim infk→ ∞
C
k 0,
that is, Ptut 0 a.e.inτ, T.
3.37
Moreover,3.36and the equality
ukt−uks
t
s
give
|ukt−uks|T ≤C1/2|t−s|1/2 ∀s, t∈ τ, T, ∀k≥1. 3.39
It follows from3.36thatuktT ≤ Cfor allt ∈ τ, Tand eachk ≥ 1. Since the injection
ofVT intoHT is compact, the set{v ∈ VT : v2T ≤ C}is compact inHT. By3.39and the
Arzela-Ascoli theorem, there exists a subsequence, that will be still denoted by{uk}, such
that
uk → u inC τ, T;HTask−→∞. 3.40
So, the conditionC4is satisfied.
On the other hand,{uk}is bounded inL∞τ, T;VTand{uk}is bounded inL2τ, T;VT,
applying the Aubin-Lions lemma and Lemma 1.3 in 20, Chapter 1, one has
fuk fu weakly inLqτ, T, LqΩT. 3.41
Sinceukis the weak solution of the problem
ukt, vTAktukt, vT
εAktukt, v
T
fukt, v
T
gt, vT, ∀v∈VT,
ukτ, vT uτ, vT,
3.42
taking to the limit ask → ∞and using the fact thatPtut 0, Ptut 0 a.e. inτ, T,
we can conclude thatuis the solution of2.1.
Now, we will show thatusatisfies the energy equality inτ, T. Multiplying3.22by
γkm,j and summing fromj 1 tom, we obtain
uk
mt, ukmt
TAktukmt, ukmtT
ε#Aktukmt, ukmt
$
T
fukmt, ukmt
T
gt, ukmt
T.
3.43
Hence, we get
1 2
d
dt|ukmt|
2
Tukmt
2
TkPktukmt, ukmtT
2 d
dtukmt
2
Tk
Pktukmt, ukmt
T
fukmt, ukmt
T
gt, ukmt
T,
or
|ukmt|
2
T2
t
τ
ukmr
2
Tdr2k
T
τ
Pkrukmr, ukmrTdr
ukmt
2
T2k
t
τ
Pkrukmr, ukmr
Tdr2
t
τ
fukmr, ukmr
Tdr 2 t τ
gr, ukmr
Tdr|uτm|
2
Tεuτm
2
T.
3.45
Since
Pktukmt, ukmt
T ≥
1 2
d
dtPktukmt, ukmtT,
Pktukmt, ukmtT≥0,
|ukmt|
2
T2
t
τ
ukmr
2
Tdrukmt
2
T2
t
τ
fukmr, ukmr
Tdr ≤2 t τ
gr, ukmr
Tdr|uτm|
2
Tεuτm
2
TεkPktuτm, uτmT,
3.46
lettingm → ∞, we obtain
|ukT|2T2
T
τ
ukr2TdrukT2T2
T
τ
fukr, ukr
Tdr ≤2 T τ
gr, ukr
Tdr|uτ|
2
Tεuτ2T.
3.47
Now
T
τ
fukr, ukr
Tdr T τ
fukr−fur, ukr−ur
Tdr
T
τ
fukr, ur
Tdr T τ
fur, ukr
Tdr−
T
τ
fur, urTdr
≥ −
T
τ
|ukr−ur|2Tdr
T
τ
fukr, ur
Tdr T τ
fur, ukr
Tdr−
T
τ
fur, urTdr.
This inequality and3.47give
|ukT|2T2
T
τ
ukr2TdrukT2T
≤2
T
τ
gr, ukr
Tdr|uτ|
2
Tεuτ2T2l
T
τ
|ukr−ur|2Tdr
−
T
τ
fukr, ur
Tdr−
T
τ
fur, ukr
Tdr
T
τ
fur, urTdr.
3.49
Sinceuk uweakly inL2τ, T;VT, we get
|uT|2TεuTT2 ≤ |uτ|2Tεuτ2T−2
T
τ
fur, urTdr
−2
T
τ
ur2Tdr2
T
τ
gr, urTdr.
3.50
ApplyingLemma 3.8withtn T for alln, one concludes thatusatisfies the energy equality
onτ, T, and the desired uniqueness follows fromProposition 3.9.
4. Existence of Pullback
D
-Attractors
The aim of this section is to establish the existence of a pullback attractor for problem1.3.
Suppose thatg∈L2locRN1. Then, according toTheorem 3.10, for eachτ∈Rand each
uτ ∈Vτ∩LpΩτgiven, there exists a unique variational solutionu·;τ, uτof problem1.3
satisfying the energy equality a.e. inτ, Tfor allT > τ.
Define
Ut, τuτ :ut;τ, uτ, −∞< τ ≤t <∞, uτ ∈Vτ∩LpΩτ. 4.1
It is easy to check that the family of mappings{Ut, τ;−∞< τ≤t <∞}is a processU·,·.
A uniform estimate inVT∩LpΩTwill be obtained now for the variational solutions
of 1.3 satisfying the energy equality, and since the compactness of the embedding VT ∩
LpΩ
T → HT, this will immediately imply the existence of a pullback attractor for the
processU·,·. The proof requires the following lemma.
Lemma 4.1see 21. LetX⊂Ybe Banach spaces such thatXis reflexive and the injection ofXin
Y is compact. Suppose that{vn}is a bounded sequence inL∞t0, T;Xsuch thatvn vweakly in
Lpt0, T;Xfor somep∈ 1,∞andv∈C0 t0, T;Y. Then,vt∈Xfor allt∈ t0, Tand
vtX≤lim inf
Proposition 4.2. Suppose that the assumptions inTheorem 3.10hold andg∈L2locRN1satisfies
Cg,Tsup
t≤T
t
t−1 ""gr""2
Tdr <∞. 4.3
Then, for anyuτ ∈ Vτ ∩LpΩτgiven, the corresponding variational solutionuof 1.3satisfying
the energy equality inτ, Talso satisfies
utpLpΩTut2T
≤C
e−σTt−τ
uτ2τuτpLpΩ τ
1 1
1−e−σTCg,T
, ∀t∈ τ1, T,
4.4
whereσT min{λ1,T/2,1/ε1, α1/α2},λ1,T >0 is the first eigenvalue of the operator−ΔinΩT
with the homogeneous Dirichlet condition,α1,α2 are the constants inH1, and the constant Cis
independent oft, τ, ε.
Proof. Assume thatukm are the Galerkin approximations ofukdefined by3.22. From3.44
and3.24, we have
1 2
d dt
|ukmt|
2
T ε1ukmt
2
T
"""u
kmt
"" "2
Tukmt
2
T
ε%%%uk
mt
%%
%2
T ε1k
Pktukmt, ukmt
TkPktukmt, ukmtT
εkPktukmt, u
kmt
T
fukmt, u
kmt ukmt
T
gt, ukmt
T
gt, uk
mt
T.
4.5
Moreover,
Pktukmt, ukmt
T ≥
1 2
d
dtPktukmt, ukmtT,
fukmt, ukmt
T
d dt
ΩT
Fukmx, tdx,
gt, ukmt
T ≤
1
4η1""gt""
2
Tη1|ukmt|
2
T, ∀η1>0,
gt, uk
mt
T≤
1
4η2
""gt""2Tη2"""ukmt
"" "2
T, ∀η2>0,
so
1 2
d
dt |ukmt|
2
T ε1ukmt
2
T ε1kPktukmt, ukmtT
2
ΩT
Fukmx, tdx
!
"""u
kmt
"" "2
Tukmt
2
Tε%%%ukmt
%%
%2
T
εkPktukmt, u
kmt
TkPktukmt, ukmtT
fukmt, ukmt
T
≤ 1
4η1
""gt""2
Tη1|ukmt|
2
T
1
4η2
""gt""2
Tη2"""ukmt
"" "2
T, ∀η1, η2>0.
4.7
Since
fukmt, ukmt
T
ΩT
fukmtukmtdx
≥
ΩT
−βα1|ukmt|
pdx−β|Ω
T|α1ukmt p LpΩ
T,
4.8
we have
1 2
d
dt |ukmt|
2
T ε1ukmt
2
T ε1kPktukmt, ukmtT
2Ω
TFukmx, tdx
!
1−η2"""ukmt
"" "2
T
1
2ukmt
2
T
λ1,T
2 −η1
|ukmt|
2
T
ε%%%uk
mt
%%
%2
Tεk
Pktukmt, u
kmt
T
kPktukmt, ukmtTα1ukmt p LpΩ
T
≤β|ΩT|
1 4η1 1 4η2 ""
gt""2T.
4.9
Denote
ykmt:|ukmt|
2
T ε1ukmt
2
T ε1kPktukmt, ukmtT
2
ΩT
Fukmx, tdx.
Chooseη2<1 andη1small enough such thatσT<min{1/ε1, α1/α2, λ1,T −2η1}, we have
σTykmt σT |ukmt|
2
T ε1ukmt
2
T ε1kPktukmt, ukmtT
2
ΩT
Fukmx, tdx
!
≤σT |ukmt|
2
T ε1ukmt
2
T ε1kPktukmt, ukmtT
2
ΩT
βα2|ukmt|
pdx2β|Ω T|
!
≤σT
|ukmt|
2
T ε1ukmt
2
T ε1kPktukmt, ukmtT
2α2ukmt p LpΩ
Tdx
≤21−η2"""uk
mt
"" "2
T2
1
2ukmt
2
T2
λ
1,T
2 −η1
|ukmt|
2
T2ε%%%ukmt
%%
%2
T
2εkPktukmt, u
kmt
T2kPktukmt, ukmtT2α1ukmt p LpΩ
T. 4.11
Hence, we have
d
dtykmt σTykmt≤C
1""gt""2T. 4.12
By Gronwall’s lemma, we get
ykmt≤e
−σTt−τy
kmτ C 1e
−σTt
t
τ
eσTs""gs""2
Tds
!
. 4.13
Now, observe that
ykmτ |uτm|
2
T ε1uτm
2
T ε1kPkτuτm, uτmT
2
ΩT
Fuτmdx2β|ΩT|
≤
1
λ1,T
ε1
uτm
2
2
ΩT
βα2 |uτm|
pdx2β|Ω T|
CTuτm
2
T ε1kPkτuτm, uτmT2α2uτm p LpΩ
T4
β|ΩT|
≤CT
1uτm
2
Tuτm p LpΩ
T
ε1kPkτuτm, uτmT.
4.14
Since
2
ΩT
Fukmx, tdx2β|ΩT| ≥2α1ukm
p LpΩ
T, 4.15
so combining with4.13and4.14, we have
2α1 ukm
p LpΩ
Tukmt
2
T
≤C e−σTt−τ
1uτm
2
Tuτm p LpΩ
T
1e−σTt
t
τ
eσTs""gs""2
Tds
!
e−σTt−τε1kP
kτuτm, uτmT,
4.16
whereCis independent oft, τ, ε, andk.
Now, it is known thatukm u∗k-weakly inL∞τ, T;VTasm → ∞. Hence, by4.16
andLemma 4.1, we can conclude that
2α1 ukpLpΩ
Tukt
2
T
≤C e−σTt−τ
1uτ2τuτpLpΩ
τ
1e−σTt
t
τ
eσTs""gs""2
Tds
!
e−σTt−τε1kP
kτuτ, uττ
C e−σTt−τ
1uτ2τuτpLpΩ
τ
1e−σTt
t
τ
eσTs""gs""2
Tds
! .
4.17
Finally, sinceuk uinL2τ, T;VTask → ∞, we get
2α1utpLpΩ
Tut
2
T
≤C e−σTt−τ
1uτ2τuτpLpΩ
τ
1e−σTt
t
τ
eσTs""gs""2
Tds
!
≤C
e−σTt−τ
uτ2τuτpLpΩ τ
1 1
1−eσTCg,T
,
where we have used the fact that
t
τ
e−σTt−s""gs""2
Tds≤
t
t−1
e−σTt−s""gs""2
Tds
t−1
t−2
e−σTt−s""gs""2
Tds· · ·
≤1e−σT e−2σT· · ·
Cg,T
1
1−e−σTCg,T.
4.19
LetRbe the set of allrtsuch that
lim
t→ −∞e
tσt
rt2trtpLpΩ t
0. 4.20
Denote byDthe class of all familiesD {Dt;Dt∈Vt∩LpΩt, Dt/∅, t∈R}such that
Dt⊂Brtfor somert∈ R.
For eacht∈Rdefine
r02t 2C
1 1
1−e−σtCg,t
, 4.21
and consider the family of closed ballsB{Bt;t∈R}, where
Bt {v∈Vt:vt≤r0t}, t∈R. 4.22
Then using4.4, it is not difficult to check thatBis pullback D-absorbing for the process
U·,·. Moreover, by the compactness of the injection ofVtintoHt, it is clear thatBtis a
compact set ofHtfor anyt∈R. Then, it follows fromTheorem 2.6that the processU·,·has
a pullbackD-attractorAε{Aεt:t∈R}in a family of spaces{Ht}.
5. The Upper Semicontinuity of Pullback
D
-Attractors at
ε
0
It is proved in 4, whenε 0, the existence of a pullbackD-attractorA0 {A0t: t∈ R}
in a family of spaces{Ht} for problemP0. The aim of this section is to prove the upper
semicontinuity of pullback attractorsAεatε0 in{Ht}, that is,
lim
ε→0supt∈I disttAεt, A0t 0, 5.1
whereIis an arbitrary bounded interval inR.
The following lemma is the key of this section.
Lemma 5.1. For eacht∈R, eachT >0, and each compact subset K ofVt−T, we have
|Uεt, τuτ−U0t, τuτ|2t ≤C
√
ε, ∀τ∈ t−T, t, ∀uτ ∈K, 5.2
Proof. DenoteUεt, τuτbyut, andU0t, τuτbyvt. Letwt ut−vt, we have
wt−εΔut−Δwfu−fv 0. 5.3
Multiplying this equation bywand integrating overΩt, we get
1 2
d
dt|w|
2
t−εΔut, wtw2t
fu−fv, wt0. 5.4
We have
fu−fv, wt
Ωt
fu−fvu−v≥ −|u−v|2t,
εΔut, wt−ε∇ut,∇wt≤εutt· wt.
5.5
Applying5.5in5.4, we have
d
dt|w|
2
t ≤2|w|2t2εutt· wt. 5.6
Hence
|wt|2t ≤2ε
t
τ
e2t−sutst· wst≤2εe2T
⎛
⎝t
τ
uts2t
⎞ ⎠
1/2⎛
⎝t
τ
ws2t ⎞ ⎠
1/2
. 5.7
Now, we estimate the term on the right-hand side of5.7. Multiplying the first equation in
1.3byutand integrating overΩt, we obtain
|ut|2t εut2t
1 2
d
dtu
2
t
d dt
Ωt
Fu≤
Ωt
gtut. 5.8
Using Cauchy’s inequality, we conclude that
d
dt ut
2
t 2
Ωt
Fut
!
2εutt2t ≤
1 2""gt""
2
t. 5.9
Integrating5.9fromτ tot,τ∈ t−T, t, we find that
u2t 2
Ωt
Fut 2ε
t
τ
utst2≤ uτ2t2
Ωt
Fuτ
1 2
t
τ
