Pullback attractor for a dynamic boundary non-autonomous problem with Infinite delay
Abstract
In this work we prove the existence of solution for a p-Laplacian non-autonomous problem with dynamic boundary and infinite delay. We ensure the existence of pullback attractor for the multivalued process associated to the non-autonomous problem we are concerned. Finally, we also prove the existence of a more general attractor for the problem known as D-pullback attractor.
Full text
PULLBACK ATTRACTOR FOR A DYNAMIC BOUNDARY
NON-AUTONOMOUS PROBLEM WITH INFINITE DELAY
R. A. Samp ogna1and T. Ca aballo2
1Depa amen o de Ma em´a ica
Cen o de Ciˆencias Exa as e de Tecnologia
Uni e sidade Fede al de S˜ao Ca los
Caixa Pos al 676, 13.565-905 S˜ao Ca los SP, B azil
2Depa amen o de Ecuaciones Di e enciales y An´alisis Num´e ico
Uni e sidad de Se illa
Apdo. de Co eos 1160, 41080-Se illa, Spain
Abs ac . In his wo k we p o e he exis ence o solu ion o a p-Laplacian non-au onomous
p oblem wi h dynamic bounda y and in ini e delay. We ensu e he exis ence o pullback
a ac o o he mul i alued p ocess associa ed o he non-au onomous p oblem we a e
conce ned. Finally, we also p o e he exis ence o a mo e gene al a ac o o he p oblem
known as D-pullback a ac o .
1. In oduc ion
Le ⌧2Rand ⌦⇢RNbe a bounded domain wi h smoo h bounda y =@⌦andN3,
conside he ollowing dynamical bounda y condi ions p oblem wi h in ini e delay
8
<
:
u pu+|u|p2u= 1( , u )+g1( , x),( , x)2(⌧,+1)⇥⌦,
u +| u|p2@!nu= 2( , u )+g2( , x),( , x)2(⌧,+1)⇥,
u(⌧+s, x)= (s, x),s2(1,0],x2⌦
(P)
whe e !nis he ou e no mal o ,p2[2,+1)and
pdeno es he p-Laplacian ope a o ,
de ined by pu=di (| u|p2 u). The ex e nal o ces gi,i=1,2, sa is y assump ions ha
will be s a ed la e , is a gi en unc ion de ined in he in e al (1,0] and he ex e nal
o ce ield icon aining some he edi a y cha ac e is ic deno ed by u ,whichisa unc ion
de ined on (1,0) by he ela ion u (s)=u( +s), s2(1,0).
The in e es o p oblems wi h dynamic bounda y condi ions has been g owing o e he las
o y yea s, see [8, 14, 16]. Mo i a ed by ma hema icians’ in e es s and physical applica ions,
he au ho s o [10] and [11] s udied an au onomous e sion o P oblem (P). A e ha , some
wo ks eme ged o his p oblem, wi h a non-au onomous e m jus in pe u ba ions gican
be ound in [15, 24] and [25], whe e he au ho s ha e es ablished he exis ence o a uni o m
a ac o and pullback a ac o o he p oblems, espec i ely. In [22] he au ho s conside ed
a non-au onomous e m in pe u ba ions iand ensu ed he exis ence o solu ion as well as
This esea ch was pa ially suppo ed by P og ama Ciˆencia sem F on ei as/CNPq, Minis ´e io da Ciˆencia e
Tecnologia, B azil and by he p ojec s MTM2015-63723-P (MINECO, Spain/ FEDER, EU) and P12-FQM-
1492 (Jun a de Andaluc´ıa).
Key wo ds and ph ases. pullback a ac o s, unbounded delays, mul i alued p ocess, dynamical bounda y,
p-Laplacian, asymp o ic beha io o solu ions.
2000 Ma hema ics Subjec Classi ica ion. 35B41, 35K55, 37B55.
1
2 PULLBACK ATTRACTOR FOR A DYNAMIC BOUNDARY PROBLEM WITH INFINITE DELAY
he exis ence o D-pullback a ac o o he gene alized p ocess associa ed wi h a simila
p oblem o (P) wi hou uniqueness o solu ion.
The delay e ms appea na u ally in many applica ions as eloci y ield in wind unnel and
popula ion g ow h, e.g., [17]. The s udy o he asymp o ic beha iou o p oblems wi h ini e
delay wi h uniqueness o in mul i alued con ex s can be ound in [5], a e sion wi h in ini e
delays can be ound in [4], bo h wo ks conside au onomous and non-au onomous p oblems.
In he wo k [26] he au ho s de eloped a heo y o pullback a ac o s o mul i alued p ocess
associa ed wi h in ini e delay p oblems and hey es ablished condi ions o gua an ee he
exis ence o an in a ian pullback a ac o o his mul i alued p ocess. Ou wo k in his
pape will be based on hese esul s. Ano he hing ha mo i a es us is ha he e a e only
a ew delay p oblems ela ed o ope a o
pwhich is a e y good example o a nonlinea
maximal mono one ope a o .
We o ganize his wo k as ollows. In he nex sec ion, we ecall some no a ions, de ini ions
and p ope ies o sui able spaces o he s udy o P oblem (P). In Sec ion 3 we p esen some
de ini ions and a esul ha ensu es he exis ence o he pullback a ac o in mul i alued
con ex de eloped in [26]. In Sec ion 4 we p o e he exis ence o weak solu ion o P oblem
(P). Finally, Sec ion 5 is de o ed o ensu e he exis ence o pullback a ac o o ou p oblem
and hen, in Sec ion 6, we jus choose a mo e gene al uni e se o se s o be a ac ed and
show he exis ence o a mo e gene al a ac o known as D-pullback a ac o .
2. P elimina ies
In his sec ion, ollowing [9] we de ine he app op ia e spaces o s udy P oblem (P).
Conside he Lebesgue space
L () = { :k kL ()<1},
whe e k kLp()=R| |pdS1/p , o p2[1,1), dS is he su ace measu e on induced by
dx and k kL1()=in {C;| (x)|Ca.e. in }.
The phase space o be conside ed is gi en by
Xp:= Lp(⌦,dx)⇥Lp(,dS)={F=( ,g); 2Lp(⌦) and g2Lp()},
wi h he no m
kFkXp=✓Z⌦
| |pdx +Z
|g|pdS◆1/p
,
o 1 p<1,and
kFkX1:= max k kL1(⌦),kgkL1() ,
o p=+1. This space can be iden i ied wi h Lp(⌦,dµ)whe edµ =dx dS,i.e.,i A⇢⌦
is µmeasu able, hen µ(A)=|A ⌦|+S(A ).
No e ha he space X2, wi h he ollowing inne p oduc
h·,·iX2:= h·,·iL2(⌦)+h·,·iL2(),
is a sepa able Hilbe space.
Fo p2(1,1) we de ine he ac ional o de Sobole space
W11
p,p := ⇢u2Lp() : ZZ
|u(x)u(y)|p
|xy|p+N2dSxdSy<1.
PULLBACK ATTRACTOR FOR A DYNAMIC BOUNDARY PROBLEM WITH INFINITE DELAY 3
Conside he ec o subspace o W1,p(⌦)⇥W11
p,p(), gi en by
Vp={U=(u, ); u2W1,p(⌦) and =(u)},
whe e :W1,p(⌦)!W11
p,p() is he con inuous ace ope a o . In Vp,wecanconside he
usual no m kUkVp=kukW1,p(⌦)+k(u)kW11
p,p().The space Vpis densely and compac ly
con ained in he Hilbe space X2 o 2 p<+1,ascanbeseenin[10].
No e ha we can iden i y u2W1,p as a couple U=(u, (u)) 2Vp.Thecon inui yo
ensu es he equi alence be ween he no ms o W1,p(⌦) and Vp. We can show ha Vpis a
e lexi e and sepa able space o 1 <p<1. Fu he mo e,
Vp⇢⇢ X2⇢(Vp)⇤ o 2 p<+1.(2.1)
3. Abs ac Resul s
In his sec ion we p esen a summa y o de ini ions and esul s om [26], whe e he au ho s
de eloped a heo y o in a ian pullback a ac o s in a mul i alued con ex .
Le (X, ⇢)beacomple eme icspace.Fo x2X,A, B ⇢Xand ">0wede ine
⇢(x, A):=in
a2A{⇢(x, a)};dis (A, B) := sup
a2A
in
b2B{⇢(a, b)};
O"(A):={z2X;⇢(z,A)<"}.
Deno e by P(X) he nonemp y subse s o X.
De ini ion 3.1. A amily o mappings U( , ⌧):X!P(X), ⌧,⌧2R, is said o be a
mul i alued p ocess i
(1) U(⌧,⌧)x={x},8⌧2R,x2X;
(2) U( , s)U(s, ⌧)x=U( , ⌧)x, 8 s⌧,⌧2R,x2X.
De ini ion 3.2. Le {U( , ⌧)}be a mul i alued p ocess on X. We say ha {U( , ⌧)}is
(1) pullback dissipa i e, i he e exis s a amily o bounded se s D={D( )} 2Rin X
such ha o any bounded se B⇢Xand each 2R, he e exis s a ⌧0=⌧0(B, )2R
such ha
U( , ⌧)B⇢D( ),8⌧⌧0.
The amily o se s Dis known as pullback abso bing amily;
(2) pullback asymp o ically uppe semicompac in Xi o each ixed 2Rand
B⇢Xbounded, any sequence {⌧n}wi h ⌧n!1,{xn}⇢B, and {yn}wi h
yn2U( , ⌧n)xn, his las sequence {yn}is p ecompac in X.
De ini ion 3.3. A amily o nonemp y compac subse s A={A( )} 2Ro Xis said o be a
pullback a ac o o he mul i alued p ocess {U( , ⌧)}i
(1) A={A( )} 2Ris in a ian , i.e.,
U( , ⌧)A(⌧)=A( ),8 ⌧,⌧2R;
(2) Ais pullback a ac ing, i.e., o e e y bounded se Bo Xand any ixed 2R,
lim
⌧!1 dis (U( , ⌧)B,A( )) = 0.
4 PULLBACK ATTRACTOR FOR A DYNAMIC BOUNDARY PROBLEM WITH INFINITE DELAY
De ini ion 3.4. Le {U( , ⌧)}be a mul i alued p ocess on X. We say ha U( , ⌧)is uppe
semicon inuous (o U.S.C.) in x o ixed ⌧,⌧2R, i xn!x, hen o any
yn2U( , ⌧)xn, he e exis a subsequence ynk2U( , ⌧)xnkand y2U( , ⌧)xsuch ha ynk!y
in X.
Theo em 3.5. ([26, Theo em 7, p. 88]) Le Xbe a Banach space and le {U( , ⌧)}be a
pullback dissipa i e, pullback asymp o ically uppe semicompac and uppe semicon inuous
mul i alued p ocess on Xwi h [⌧ D(⌧)bounded o all 2R, whe e D={D( )} 2Ris a
abso bing amily. Then {U( , ⌧}possesses a minimal pullback a ac o A={A( )} 2R.
4. Exis ence o Solu ion
Le >0be ixedandHa Hilbe space. One possibili y o deal wi h in ini e delays is
o conside he space:
C(H)=⇢'2C((1,0]; H):9lim
s!1 es'(s)2H,
which is a Banach space wi h he no m
k'k:= sup
s2(1,0]
es||'(s)||H.
This space was conside ed in [18, 26], he p ope ies o his space ha will allow us o deal
wi h in ini e delays can be ound in [12]. La e we will se a mo e app op ia e o ou
pa icula p oblem.
Le i:R⇥C(Li)!Li, o i=1,2, whe e L1=L2(⌦) and L2=L2(), and sa is ies
he ollowing assump ions:
(F1) o all ⇠2C(Li), he mapping R3 ! i( , ⇠)2Liis mensu able;
(F2) o each 2R, i( , 0) = 0;
(F3) he e exis s Ki>0 such ha 8 2R,8⇠,⌘2C(Li),
k i( , ⇠) i( , ⌘)kLiKik⇠⌘kC(Li).
See [20] o examples o unc ions wi h hese p ope ies. And o gi,s we ha e he ollowing
assump ion:
(G1) le g12Lp0
loc(R;Lp0(⌦)), g22Lp0
loc(R;Lp0()) whe e p0deno es he conjuga e exponen
o p,i.e.,1
p+1
p0=1.
Rema k 4.1. Le 2C(X2), hen no ice ha he e exis s (s)2L2(⌦)and (s)2L2()
o each s2(1,0] such ha =( ,). Mo eo e ,
k k2
C(L1)+kk2
C(L2)=sup
s2(1,0] ⇣e2sk (s)k2
L2(⌦)⌘+sup
s2(1,0] ⇣e2sk(s)k2
L2()⌘
=sup
s2(1,0] ⇣e2s⇣k (s)k2
L2(⌦)+k(s)k2
L2()⌘⌘
=sup
s2(1,0]
e2sk (s)k2
X2=k k2
C(X2).
Rema k 4.2. Fo ⇠2C(W1,p(⌦)) and each s2[1,0] we ha e (⇠)(s)=(⇠(s)). Then,
om he con inui y o ace,
(⇠)2C⇣W11
p,p()⌘.
PULLBACK ATTRACTOR FOR A DYNAMIC BOUNDARY PROBLEM WITH INFINITE DELAY 5
De ini ion 4.3 (Weak Solu ion o P oblem (P)).Gi en =( ,)2C(X2),⌧2R, he
couple U( )=(u( ),w( )) is said o be a weak solu ion o P oblem (P) i w( )=(u( )) a.e.
in (⌧,T) o each T>⌧, and Usa is ies
(i)
U2C([⌧,+1); X2) L1(⌧,+1;X2);
(ii)
@ U2Lp0
loc(⌧,+1;(Vp)⇤);
(iii) o all V=( ,( )) 2Vp,
h@ U, V iX2+⌦| u|p2 u, ↵L2(⌦)+⌦|u|p2u, ↵L2(⌦)=⌦ 1( , u ), ↵L2(⌦)
+⌦ 2( , (u )),( )↵L2()+hg1( ), iL2(⌦)+hg2( ),( )iL2()
(4.1)
a.e. in (⌧,T), o each T>⌧;
(i ) U⌧= in C(X2), which means, u⌧= in C(L2(⌦)) and w⌧=in C(L2()).
Be o e showing he exis ence o a weak solu ion o P oblem (P), we ob ain a p io i es i-
ma es o a weak solu ion in he space X2.
Lemma 4.4. Assume hypo heses (F1)-(F3) and (G1) a e sa is ied and le U( )=(u( ),(u)( ))
be a weak solu ion o P oblem (P)wi h ini ial delay condi ion 2C(X2)in ⌧2R. Then,
he e is a ini e cons an K( , ⌧, )such ha
kU( )kX2+⇥Z
⌧
kUkp
Vpds k k2
C(X2)
+C1Z
⌧⇣kg1( )kp0
p0+kg2( )kp0
p0,⌘ds +K( , ⌧, ),
(4.2)
and
kU k2
C(X2)eC( ⌧)⇣k kC(X2)+˜
C( ⌧)⌘
+C"Z
⌧
e2C( s)⇣kg1( )kp0
p0+kg2( )kp0
p0,⌘ds,
(4.3)
o all ⌧, wi h C, C",C
1,˜
Cand ⇥posi i e cons an s independen o ⌧and .
P oo : Le Ube a weak solu ion o P oblem (P). Take V=Uin (4.1), and om H¨olde ’s
and Young’s inequali ies we ha e
1
2
d
d kUk2
X2+k ukp
Lp(⌦)+kukp
Lp(⌦)=⌦ 1( , u ),u
↵2+⌦ 2( , (u )),(u)↵2,
+hg1( ),ui2+hg2( ),(u)i2,
C"k 1( , u )kp0
p0+C"k 2( , (u ))kp0
p0,+C"kg1( )kp0
p0+C"kg2( )kp0
p0,
+2"kukp
p+2"k(u)kp
p,
C"⇣k 1( , u )kp0
p0+k 2( , (u ))kp0
p0,+kg1( )kp0
p0+kg2( )kp0
p0,⌘+2"kUkp
Vp.
6 PULLBACK ATTRACTOR FOR A DYNAMIC BOUNDARY PROBLEM WITH INFINITE DELAY
Then, as he no m o Vpis equi alen o he no m o W1,p(⌦), he e is a cons an M⌦such
ha
1
2
d
d kUk2
X2+M⌦kUkp
Vp
C"⇣k 1( , u )kp0
p0+k 2( , (u ))kp0
p0,+kg1( )kp0
p0+kg2( )kp0
p0,⌘+2"kUkp
Vp.
Take ">0 such ha
⇥:=1
2(M⌦2")>0,(4.4)
and mul iplying by 2, inco po a ing he cons an s, and in eg a ing be ween ⌧ o
kU( )k2
X2+⇥Z
⌧
kUkp
Vpds kU(⌧)k2
X2
+C"Z
⌧⇣k 1( , u )kp0
p0+k 2( , (u ))kp0
p0,+kg1( )kp0
p0+kg2( )kp0
p0,⌘ds.
Thus, om Lemma 2.1 o [22], (F2) and (F3), he e a e 1,
2>0andC1,C
2>0, such
ha
kU( )k2
X2+⇥Z
⌧
kUkp
Vpds kU(⌧)k2
X2
+C"Z
⌧⇣1k 1( , u )k2
2+2k 2( , (u ))k2
2,+kg1( )kp0
p0+kg2( )kp0
p0,⌘ds
+C"( ⌧)(C1+C2)
kU(⌧)k2
X2+C"Z
⌧1K2
1ku k2
C(L1)+2K2
2k(u )k2
C(L2)ds
+C"Z
⌧⇣kg1( )kp0
p0+kg2( )kp0
p0,⌘ds +C"( ⌧)(C1+C2).
Take K:= max{1K2
1,
2K2
2}and le
C:= C"K(4.5)
and ˜
C:= C"(C1+C2). F om Rema k 4.1 we ha e
kU( )k2
X2+⇥Z
⌧
kUkp
Vpds kU(⌧)k2
X2
+C"Z
⌧⇣kg1( )kp0
p0+kg2( )kp0
p0,⌘ds +CZ
⌧
kUsk2
C(X2)ds +˜
C( ⌧).
(4.6)
o ⌧.
PULLBACK ATTRACTOR FOR A DYNAMIC BOUNDARY PROBLEM WITH INFINITE DELAY 7
Fu he
kU k2
C(X2)max (sup
l2(1,⌧ ]
e2lk (l+ ⌧)k2
X2,
sup
l2(⌧ ,0] e2l✓kU(⌧)k2
X2+C"Z +l
⌧⇣kg1(s)kp0
p0+kg2(s)kp0
p0,⌘ds
+CZ +l
⌧
kUsk2
C(X2)ds +˜
C( ⌧)◆
max (sup
l2(1,⌧ ]
e2lk (l+ ⌧)k2
X2,
kU(⌧)k2
X2+C"Z
⌧⇣kg1(s)kp0
p0+kg2(s)kp0
p0,⌘ds
+CZ
⌧
kUsk2
C(X2)ds +˜
C( ⌧),
and, no e ha
sup
l2(1,⌧ ]
e2lk (l+ ⌧)k2
X2=sup
l0
e2(l( ⌧))k (l))k2
X2=e2( ⌧)k kC(X2)k kC(X2),
and kU(⌧)kX2=k (0)kX2k kC(X2). F om G onwall’s Lemma
kU k2
C(X2)eC( ⌧)⇣k kC(X2)+˜
C( ⌧)⌘+C"eC( ⌧)Z
⌧⇣kg1(s)kp0
p0+kg2(s)kp0
p0,⌘ds,
ensu ing es ima e (4.3) o all ⌧, wi h his es ima e and (4.6) we can deduce es ima e
(4.2).
⌅
Theo em 4.5. Le 2C(X2)and ⌧2R. Assume (F1)-(F3) and (G1) hold ue. Then
he e exis s a leas one weak solu ion o P oblem (P)wi h ini ial delay condi ion in ⌧.
P oo : We will de ine some app op ia e ope a o s o e o mula e exp ession (4.1) in o de
o ha e a simple unc ional o mula ion o ou p oblem, see [10] and [22] o examples o
he same me hod. Then le , o U, V 2Vp, he ollowing ope a o
p(U, V )=⌦| u|p2 u, ↵2+⌦|u|p2u, ↵2.
Fo each U2Vpwe ha e pU:= p(U, ·)2(Vp)⇤and he ope a o p:Vp!(Vp)⇤is a
maximal mono one ope a o , see [22].
And we de ine
F( , U )=✓ 1( , u )
2( , (u )) ◆,G( )=✓g1( )
g2( )◆and @ U=✓u
(u) ◆
in he usual way, see [22] o mo e de ails.
In his way, inding a weak solu ion o P oblem (P) is equi alen o ind a unc ion Uwi h
egula i ies o weak solu ion de ini ion, and sa is ying he ollowing unc ional equa ion
@ U+pU=F( , U )+G( )(4.7)
8 PULLBACK ATTRACTOR FOR A DYNAMIC BOUNDARY PROBLEM WITH INFINITE DELAY
in Lp0(0,T;(Vp)⇤), see Rema k 4.6 in [22].
In o de o ind a weak solu ion o P oblem (P), we use he Faedo-Gale kin app oxima ion.
Since X2is sepa able and Vpis dense in X2, he e is a o hono mal basis o X2con ained in
Vp.Wedeno esuchbasisby{n=(n,
n)2X2;n2N}.
Le
Kn=span{1, ..., n},K
1=[1
n=1Kn,
and P n:X2!Knbe he o hogonal p ojec ion.
Gi en 2C(X2)andT>⌧we wan o ind a solu ion Un=Pn
i=1 di( )i2Kn o
an ndimensional e sion o p oblem (4.7), which is equi alen o ind a solu ion o he
ollowing sys em o o dina y di↵e en ial equa ions
⇢h@ Un,ii+hpUn,ii=hF( , U
n),ii+hG( ),ii
hUn(⌧+s),Vi=hP n (s),Vi, o s2(1,0],
o all 1 inand a.e. in [⌧,T], whe e h·,·ideno e he dual p oduc be ween (Vp)⇤and
Vp.
The abo e sys em o o dina y unc ional di↵e en ial equa ions wi h in ini e delay ul ils
he condi ions o exis ence and uniqueness o local solu ion es ablished in Theo em 1.1 o
[13]. A p io i es ima es ensu e ha solu ions do exis o all ime in [⌧,T].
Es ima e (4.3) o Lemma 4.4 ensu es ha o 2C(X2)andR>0 such ha k kC(X2)
R, he eexis sacons an C=C(⌧,T,R), bu independen o nand 2(⌧,T), such ha
kU
nk2
C(X2)C(⌧,T,R).(4.8)
In pa icula , he p e ious limi and es ima e (4.2) imply he exis ence o ano he cons an
( elabelled he same) C=C(⌧,T,R)such ha
⇢kUn( )kL1(⌧,T;X2)C
kUn( )kLp(⌧,T;Vp)C. (4.9)
Then, his gua an ees ha pUnis bounded in Lp0(⌧,T;(Vp)⇤), see [22] o mo e de ails.
Hypo heses (F2), (F3), (4.8) and ecalling ha X2⇢(Vp)⇤con inuously imply ha F( , U
n)
is bounded in Lp0(⌧,T;(Vp)⇤). No e ha ,
@ Un=pUn+F( , U
n)+G( )inLp0(⌧,T;(Vp)⇤).(4.10)
The e o e, he limi s o pUnand F( , U
n) ensu e ha he e exis s a cons an ( elabelled
he same) C(⌧,T,R)such ha
k@ UnkLp0(⌧,T;(Vp)⇤)C. (4.11)
The limi s in (4.9) and (4.11) ensu e ha he e is a subsequence (which we elabel he
same) {Un},andanelemen U2L1(⌧,T;X2) Lp(⌧,T;Vp)wi h@ U2Lp0(⌧,T;(Vp)⇤),
such ha
8
<
:
Un
⇤
*Uin L1(⌧,T;X2),
Un*Uin Lp(⌧,T;Vp),
@ Un
⇤
*@
Uin Lp0(⌧,T;(Vp)⇤).
(4.12)
PULLBACK ATTRACTOR FOR A DYNAMIC BOUNDARY PROBLEM WITH INFINITE DELAY 9
F om compac ness esul s, see Theo ems 1.4 and 1.5 page 32 o [7], he sequences in ac
ha e he ollowing con e gences
⇢Un!Uin Lp(⌧,T;X2),
Un!Uin C([⌧,T]; X2).(4.13)
No e ha ,
P n ! inC(X2),
and hanks o he s ong con e gence in C([⌧,T]; X2) yield ha
U
n!U in C(X2)8 T,
see, o ins ance, [18, 19] and [20] o de ails abou bo h con e gences.
The abo e con e gence and hypo heses (F2) and (F3) imply ha
F( , U
n)!F( , U )inLp0(⌧,T;X2),
which oge he wi h con e gences (4.12) and he heo y o maximal mono one ope a o s
allow us o deduce ha
pUn
⇤
*
pUin Lp0(⌧,T;(Vp)⇤),
see [22] o de ails.
The e o e, Uis solu ion o he limi equa ion o (4.10) in he weak s a opology o
Lp0(⌧,T;(Vp)⇤). This ensu es ha Uis a weak solu ion o P oblem (P) in he in e al
(1,T] wi h ini ial condi ion U⌧= .
⌅
The exis ence o solu ion allows us de ine he mul i alued p ocess {U( , ⌧)}on C(X2)by
U( , ⌧) =U |U(·) is a solu ion o P oblem (P)wi hU⌧= 2C(X2) .
Indeed, i em (2) o De ini ion 3.1 ollows om conca ena ion and ansla ion o solu ions,
see [4] and [5] o de ails.
Lemma 4.6. The mul i alued p ocess {U( , ⌧)}is uppe -semicon inuous in C(X2).
P oo : Le ⌧2R,{ n}n2Nand such ha n! inC(X2), and le {Yn}n2Nsuch
ha Y·
n2U(·,⌧) n.
Gi en T>⌧, obse e ha , as
n! inC(X2), gi en R>0, excep o a ini e numbe
o elemen s, we ha e ha { n}⇢BC(X2)( ,R). Then, om Lemma 4.4 he sequence {Yn}
is bounded in L1(⌧,T;X2)andLp(⌧,T;Vp).
Then, simila ly o he p oo o Theo em 4.5, we can ensu e he exis ence o an elemen
Y·2U(·,⌧) such ha Y
n!Y in C(X2) o all T.
The e o e, as T>⌧is a bi a y, i ollows ha {U( , ⌧)}is uppe -semicon inuous.
⌅
5. Pullback A ac o o P oblem (P)
In his sec ion we de elop some es ima es o show ha he mul i alued p ocess gene a ed
by solu ions o P oblem (P) possesses a pullback abso bing amily and i is pullback asymp-
o ically uppe semicompac . The e o e, we can ensu e he exis ence o pullback a ac o
o he p oblem.
16 PULLBACK ATTRACTOR FOR A DYNAMIC BOUNDARY PROBLEM WITH INFINITE DELAY
[24] YANG, L.; YANG, M.; KLOEDEN, P. E. Pullback a ac o s o non-au onomous quasilinea pa abolic
equa ions wi h dynamical bounda y condi ions. Disc. and Con . Dynamical Sys ems B, Vol. 17, No. 7,
1-11, (2012).
[25] YANG, L.; YANG, M.; WU, J. On uni o m a ac o s o non-au onomous p-Laplacian equa ion wi h
a dynamic bounda y condi ion. Disc. and Con . Dynamical Sys ems B, Vol. 17, No. 7, 1-11, (2012).
[26] YEJUAN, W.; KLOEDEN, P. E. Pullback a ac o s o a mul i- alued p ocess gene a ed by pa abolic
di↵e en ial equa ions wi h unbounded delays. Nonlinea Analysis 90, 8695, (2013).
Response o Re iewe s
Fi s o all, we wan o hank he e iewe s o his dedica ion and his g ea con i-
bu ions o his wo k.
One o he e iewe s no ed ha we had no ac ually shown ha he pullback ab-
so bing amily Dde ined in Lemma 5.4 ha e he p ope y ha [⌧ D(⌧)isboundend o
all 2R.Toha e hisp ope ywehad ochangeou G2hypo hesis o
(G2) sup
0
e Z
1
es⇣kg1(s)kp0
p0+kg2(s)kp0
p0,⌘ds < +1,
like in [1]. Wi h ha change he small inal sec ion loses i s meaning and had o be
emo ed. Bu he mos impo an esul s o he wo k emain alid.
Fu he mo e, we ha e made o he mino changes, which a e poin ed below:
1. page 10, line 3: We added he ph ase: ”Conside in his sec ion p>2” a he end
o he pa ag aph;
2. page 6 (line 10), page 8 (line -12) and page 9 (line 10): we ha e eplaced “(H2) and
(H3)” by “(F2) and (F3)”;
3. page 6 (line 6) and page 10 (line 4):we ha e eplaced “⇥:= (M⌦2")>0” by
“⇥:= 1
2(M⌦2")>0”;
4. page 4, line 10: we ha e eplaced “Banach space” by “Hilbe space”
5. page 12, a he beginning o he demons a ion o Lemma 5.5: we ha e eplaced
“U(·;⌧n, n)” by “U( 0;⌧n, n)”
6. page 4, line 15: we emo ed he exp essionwe ”ano he spaces”;
7. page 5, line 8: we emo ed he ”{”;
8. page 11, line -10: we ha e eplaced “comma” by “pe iod”
Re e ences
[1] YEJUAN, W.; KLOEDEN, P. E. Pullback a ac o s o a mul i- alued p ocess gene -
a ed by pa abolic di↵e en ial equa ions wi h unbounded delays. Nonlinea Analysis 90,
86–95, (2013).
1
had
he in e es ing and help ul sugges ions which allowed us o
g ea ly imp o ed he p esen a ion o his pape .
bounded
P oo