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.
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PULLBACK ATTRACTOR FOR A DYNAMIC BOUNDARY NON-AUTONOMOUS PROBLEM WITH INFINITE DELAY R. A. Samprogna1and T. Caraballo2 1Departamento de Matem´atica Centro de Ciˆencias Exatas e de Tecnologia Universidade Federal de S˜ao Carlos Caixa Postal 676, 13.565-905 S˜ao Carlos SP, Brazil 2Departamento de Ecuaciones Diferenciales y An´alisis Num´erico Universidad de Sevilla Apdo. de Correos 1160, 41080-Sevilla, Spain 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. 1. Introduction Let ⌧2Rand ⌦⇢RNbe a bounded domain with smooth boundary =@⌦andN3, consider the following dynamical boundary conditions problem with infinite delay 8 < : utpu+|u|p2u=f1(t, ut)+g1(t, x),(t, x)2(⌧,+1)⇥⌦, ut+|ru|p2@!nu=f2(t, ut)+g2(t, x),(t, x)2(⌧,+1)⇥, u(⌧+s, x)= (s, x),s2(1,0],x2⌦ (P) where !nis the outer normal to ,p2[2,+1)and pdenotes the p-Laplacian operator, defined by pu=div(|ru|p2ru). The external forces gi,i=1,2, satisfy assumptions that will be stated later, is a given function defined in the interval (1,0] and the external force field ficontaining some hereditary characteristic denoted by ut,whichisafunction defined on (1,0) by the relation ut(s)=u(t+s), s2(1,0). The interest for problems with dynamic boundary conditions has been growing over the last forty years, see [8, 14, 16]. Motivated by mathematicians’ interests and physical applications, the authors of [10] and [11] studied an autonomous version of Problem (P). After that, some works emerged of this problem, with a non-autonomous term just in perturbations gican be found in [15, 24] and [25], where the authors have established the existence of a uniform attractor and pullback attractor for the problems, respectively. In [22] the authors considered a non-autonomous term in perturbations fiand ensured the existence of solution as well as This research was partially supported by Programa Ciˆencia sem Fronteiras/CNPq, Minist´erio da Ciˆencia e Tecnologia, Brazil and by the projects MTM2015-63723-P (MINECO, Spain/ FEDER, EU) and P12-FQM- 1492 (Junta de Andaluc´ıa). Key words and phrases. pullback attractors, unbounded delays, multivalued process, dynamical boundary, p-Laplacian, asymptotic behavior of solutions. 2000 Mathematics Subject Classification. 35B41, 35K55, 37B55. 1
2 PULLBACK ATTRACTOR FOR A DYNAMIC BOUNDARY PROBLEM WITH INFINITE DELAY the existence of D-pullback attractor for the generalized process associated with a similar problem to (P) without uniqueness of solution. The delay terms appear naturally in many applications as velocity field in wind tunnel and population growth, e.g., [17]. The study of the asymptotic behaviour of problems with finite delay with uniqueness or in multivalued contexts can be found in [5], a version with infinite delays can be found in [4], both works consider autonomous and non-autonomous problems. In the work [26] the authors developed a theory of pullback attractors for multivalued process associated with infinite delay problems and they established conditions to guarantee the existence of an invariant pullback attractor for this multivalued process. Our work in this paper will be based on these results. Another thing that motivates us is that there are only a few delay problems related to operator pwhich is a very good example of a nonlinear maximal monotone operator. We organize this work as follows. In the next section, we recall some notations, definitions and properties of suitable spaces for the study of Problem (P). In Section 3 we present some definitions and a result that ensures the existence of the pullback attractor in multivalued context developed in [26]. In Section 4 we prove the existence of weak solution for Problem (P). Finally, Section 5 is devoted to ensure the existence of pullback attractor for our problem and then, in Section 6, we just choose a more general universe of sets to be attracted and show the existence of a more general attractor known as D-pullback attractor. 2. Preliminaries In this section, following [9] we define the appropriate spaces to study Problem (P). Consider the Lebesgue space Lr() = {v:kvkLr()<1}, where kvkLp()=R|v|pdS1/p ,for p2[1,1), dS is the surface measure on induced by dx and kvkL1()=inf{C;|v(x)|Ca.e. in }. The phase space to be considered is given by Xp:= Lp(⌦,dx)⇥Lp(,dS)={F=(f,g); f2Lp(⌦) and g2Lp()}, with the norm kFkXp=✓Z⌦ |f|pdx +Z |g|pdS◆1/p , for 1 p<1,and kFkX1:= max kfkL1(⌦),kgkL1() , for p=+1. This space can be identified with Lp(⌦,dµ)wheredµ =dx dS,i.e.,ifA⇢⌦ is µmeasurable, then µ(A)=|A\⌦|+S(A\). Note that the space X2, with the following inner product h·,·iX2:= h·,·iL2(⌦)+h·,·iL2(), is a separable Hilbert space. For p2(1,1) we define the fractional order Sobolev 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 Consider the vector subspace of W1,p(⌦)⇥W11 p,p(), given by Vp={U=(u, v); u2W1,p(⌦) and v=(u)}, where :W1,p(⌦)!W11 p,p() is the continuous trace operator. In Vp,wecanconsiderthe usual norm kUkVp=kukW1,p(⌦)+k(u)kW11 p,p().The space Vpis densely and compactly contained in the Hilbert space X2for 2 p<+1,ascanbeseenin[10]. Note that we can identify u2W1,p as a couple U=(u, (u)) 2Vp.Thecontinuityof ensures the equivalence between the norms of W1,p(⌦) and Vp. We can show that Vpis a reflexive and separable space for 1 <p<1. Furthermore, Vp⇢⇢ X2⇢(Vp)⇤for 2 p<+1.(2.1) 3. Abstract Results In this section we present a summary of definitions and results from [26], where the authors developed a theory of invariant pullback attractors in a multivalued context. Let (X, ⇢)beacompletemetricspace.Forx2X,A, B ⇢Xand ">0wedefine ⇢(x, A):=inf a2A{⇢(x, a)};dist(A, B) := sup a2A inf b2B{⇢(a, b)}; O"(A):={z2X;⇢(z,A)<"}. Denote by P(X) the nonempty subsets of X. Definition 3.1. A family of mappings U(t, ⌧):X!P(X),t⌧,⌧2R, is said to be a multivalued process if (1) U(⌧,⌧)x={x},8⌧2R,x2X; (2) U(t, s)U(s, ⌧)x=U(t, ⌧)x, 8ts⌧,⌧2R,x2X. Definition 3.2. Let {U(t, ⌧)}be a multivalued process on X. We say that {U(t, ⌧)}is (1) pullback dissipative, if there exists a family of bounded sets D={D(t)}t2Rin X such that for any bounded set B⇢Xand each t2R, there exists a ⌧0=⌧0(B,t)2R such that U(t, ⌧)B⇢D(t),8⌧⌧0. The family of sets Dis known as pullback absorbing family; (2) pullback asymptotically upper semicompact in Xif for each fixed t2Rand B⇢Xbounded, any sequence {⌧n}with ⌧n!1,{xn}⇢B, and {yn}with yn2U(t, ⌧n)xn, this last sequence {yn}is precompact in X. Definition 3.3. A family of nonempty compact subsets A={A(t)}t2Rof Xis said to be a pullback attractor for the multivalued process {U(t, ⌧)}if (1) A={A(t)}t2Ris invariant, i.e., U(t, ⌧)A(⌧)=A(t),8t⌧,⌧2R; (2) Ais pullback attracting, i.e., for every bounded set Bof Xand any fixed t2R, lim ⌧!1 dist(U(t, ⌧)B,A(t)) = 0.
4 PULLBACK ATTRACTOR FOR A DYNAMIC BOUNDARY PROBLEM WITH INFINITE DELAY Definition 3.4. Let {U(t, ⌧)}be a multivalued process on X. We say that U(t, ⌧)is upper semicontinuous (or U.S.C.) in xfor fixed t⌧,⌧2R, if xn!x, then for any yn2U(t, ⌧)xn, there exist a subsequence ynk2U(t, ⌧)xnkand y2U(t, ⌧)xsuch that ynk!y in X. Theorem 3.5. ([26, Theorem 7, p. 88]) Let Xbe a Banach space and let {U(t, ⌧)}be a pullback dissipative, pullback asymptotically upper semicompact and upper semicontinuous multivalued process on Xwith [⌧tD(⌧)bounded for all t2R, where D={D(t)}t2Ris a absorbing family. Then {U(t, ⌧}possesses a minimal pullback attractor A={A(t)}t2R. 4. Existence of Solution Let >0befixedandHa Hilbert space. One possibility to deal with infinite delays is to consider the space: C(H)=⇢'2C((1,0]; H):9lim s!1 es'(s)2H, which is a Banach space with the norm k'k:= sup s2(1,0] es||'(s)||H. This space was considered in [18, 26], the properties of this space that will allow us to deal with infinite delays can be found in [12]. Later we will set a more appropriate to our particular problem. Let fi:R⇥C(Li)!Li,fori=1,2, where L1=L2(⌦) and L2=L2(), and satisfies the following assumptions: (F1) for all ⇠2C(Li), the mapping R3t!fi(t, ⇠)2Liis mensurable; (F2) for each t2R,fi(t, 0) = 0; (F3) there exists Ki>0 such that 8t2R,8⇠,⌘2C(Li), kfi(t, ⇠)fi(t, ⌘)kLiKik⇠⌘kC(Li). See [20] for examples of functions with these properties. And for gi,s we have the following assumption: (G1) let g12Lp0 loc(R;Lp0(⌦)), g22Lp0 loc(R;Lp0()) where p0denotes the conjugate exponent of p,i.e.,1 p+1 p0=1. Remark 4.1. Let 2C(X2), then notice that there exists (s)2L2(⌦)and (s)2L2() for each s2(1,0] such that =( ,). Moreover, 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). Remark 4.2. For ⇠2C(W1,p(⌦)) and each s2[1,0] we have (⇠)(s)=(⇠(s)). Then, from the continuity of trace, (⇠)2C⇣W11 p,p()⌘.
PULLBACK ATTRACTOR FOR A DYNAMIC BOUNDARY PROBLEM WITH INFINITE DELAY 5 Definition 4.3 (Weak Solution to Problem (P)).Given =( ,)2C(X2),⌧2R, the couple U(t)=(u(t),w(t)) is said to be a weak solution to Problem (P) if w(t)=(u(t)) a.e. in (⌧,T)for each T>⌧, and Usatisfies (i) U2C([⌧,+1); X2)\L1(⌧,+1;X2); (ii) @tU2Lp0 loc(⌧,+1;(Vp)⇤); (iii) for all V=(v,(v)) 2Vp, h@tU, V iX2+⌦|ru|p2ru, rv↵L2(⌦)+⌦|u|p2u, v↵L2(⌦)=⌦f1(t, ut),v↵L2(⌦) +⌦f2(t, (ut)),(v)↵L2()+hg1(t),viL2(⌦)+hg2(t),(v)iL2() (4.1) a.e. in (⌧,T), for each T>⌧; (iv) U⌧= in C(X2), which means, u⌧= in C(L2(⌦)) and w⌧=in C(L2()). Before showing the existence of a weak solution to Problem (P), we obtain a priori estimates for a weak solution in the space X2. Lemma 4.4. Assume hypotheses (F1)-(F3) and (G1) are satisfied and let U(t)=(u(t),(u)(t)) be a weak solution to Problem (P)with initial delay condition 2C(X2)in ⌧2R. Then, there is a finite constant K(t, ⌧, )such that kU(t)kX2+⇥Zt ⌧ kUkp Vpds k k2 C(X2) +C1Zt ⌧⇣kg1(t)kp0 p0+kg2(t)kp0 p0,⌘ds +K(t, ⌧, ), (4.2) and kUtk2 C(X2)eC(t⌧)⇣k kC(X2)+˜ C(t⌧)⌘ +C"Zt ⌧ e2C(ts)⇣kg1(t)kp0 p0+kg2(t)kp0 p0,⌘ds, (4.3) for all t⌧, with C, C",C 1,˜ Cand ⇥positive constants independent of ⌧and t. Proof: Let Ube a weak solution of Problem (P). Take V=Uin (4.1), and from H¨older’s and Young’s inequalities we have 1 2 d dtkUk2 X2+krukp Lp(⌦)+kukp Lp(⌦)=⌦f1(t, ut),u ↵2+⌦f2(t, (ut)),(u)↵2, +hg1(t),ui2+hg2(t),(u)i2, C"kf1(t, ut)kp0 p0+C"kf2(t, (ut))kp0 p0,+C"kg1(t)kp0 p0+C"kg2(t)kp0 p0, +2"kukp p+2"k(u)kp p, C"⇣kf1(t, ut)kp0 p0+kf2(t, (ut))kp0 p0,+kg1(t)kp0 p0+kg2(t)kp0 p0,⌘+2"kUkp Vp.
6 PULLBACK ATTRACTOR FOR A DYNAMIC BOUNDARY PROBLEM WITH INFINITE DELAY Then, as the norm of Vpis equivalent to the norm of W1,p(⌦), there is a constant M⌦such that 1 2 d dtkUk2 X2+M⌦kUkp Vp C"⇣kf1(t, ut)kp0 p0+kf2(t, (ut))kp0 p0,+kg1(t)kp0 p0+kg2(t)kp0 p0,⌘+2"kUkp Vp. Take ">0 such that ⇥:=1 2(M⌦2")>0,(4.4) and multiplying by 2, incorporating the constants, and integrating between ⌧to t kU(t)k2 X2+⇥Zt ⌧ kUkp Vpds kU(⌧)k2 X2 +C"Zt ⌧⇣kf1(t, ut)kp0 p0+kf2(t, (ut))kp0 p0,+kg1(t)kp0 p0+kg2(t)kp0 p0,⌘ds. Thus, from Lemma 2.1 of [22], (F2) and (F3), there are 1, 2>0andC1,C 2>0, such that kU(t)k2 X2+⇥Zt ⌧ kUkp Vpds kU(⌧)k2 X2 +C"Zt ⌧⇣1kf1(t, ut)k2 2+2kf2(t, (ut))k2 2,+kg1(t)kp0 p0+kg2(t)kp0 p0,⌘ds +C"(t⌧)(C1+C2) kU(⌧)k2 X2+C"Zt ⌧1K2 1kutk2 C(L1)+2K2 2k(ut)k2 C(L2)ds +C"Zt ⌧⇣kg1(t)kp0 p0+kg2(t)kp0 p0,⌘ds +C"(t⌧)(C1+C2). Take K:= max{1K2 1, 2K2 2}and let C:= C"K(4.5) and ˜ C:= C"(C1+C2). From Remark 4.1 we have kU(t)k2 X2+⇥Zt ⌧ kUkp Vpds kU(⌧)k2 X2 +C"Zt ⌧⇣kg1(t)kp0 p0+kg2(t)kp0 p0,⌘ds +CZt ⌧ kUsk2 C(X2)ds +˜ C(t⌧). (4.6) for t⌧.
PULLBACK ATTRACTOR FOR A DYNAMIC BOUNDARY PROBLEM WITH INFINITE DELAY 7 Further kUtk2 C(X2)max (sup l2(1,⌧t] e2lk (l+t⌧)k2 X2, sup l2(⌧t,0] e2l✓kU(⌧)k2 X2+C"Zt+l ⌧⇣kg1(s)kp0 p0+kg2(s)kp0 p0,⌘ds +CZt+l ⌧ kUsk2 C(X2)ds +˜ C(t⌧)◆ max (sup l2(1,⌧t] e2lk (l+t⌧)k2 X2, kU(⌧)k2 X2+C"Zt ⌧⇣kg1(s)kp0 p0+kg2(s)kp0 p0,⌘ds +CZt ⌧ kUsk2 C(X2)ds +˜ C(t⌧), and, note that sup l2(1,⌧t] e2lk (l+t⌧)k2 X2=sup l0 e2(l(t⌧))k (l))k2 X2=e2(t⌧)k kC(X2)k kC(X2), and kU(⌧)kX2=k (0)kX2k kC(X2). From Gronwall’s Lemma kUtk2 C(X2)eC(t⌧)⇣k kC(X2)+˜ C(t⌧)⌘+C"eC(t⌧)Zt ⌧⇣kg1(s)kp0 p0+kg2(s)kp0 p0,⌘ds, ensuring estimate (4.3) for all t⌧, with this estimate and (4.6) we can deduce estimate (4.2). ⌅ Theorem 4.5. Let 2C(X2)and ⌧2R. Assume (F1)-(F3) and (G1) hold true. Then there exists at least one weak solution for Problem (P)with initial delay condition in ⌧. Proof: We will define some appropriate operators to reformulate expression (4.1) in order to have a simpler functional formulation of our problem, see [10] and [22] for examples of the same method. Then let, for U, V 2Vp, the following operator p(U, V )=⌦|ru|p2ru, rv↵2+⌦|u|p2u, v↵2. For each U2Vpwe have pU:= p(U, ·)2(Vp)⇤and the operator p:Vp!(Vp)⇤is a maximal monotone operator, see [22]. And we define F(t, Ut)=✓f1(t, ut) f2(t, (ut)) ◆,G(t)=✓g1(t) g2(t)◆and @tU=✓ut (u)t◆ in the usual way, see [22] for more details. In this way, finding a weak solution of Problem (P) is equivalent to find a function Uwith regularities of weak solution definition, and satisfying the following functional equation @tU+pU=F(t, Ut)+G(t)(4.7)
8 PULLBACK ATTRACTOR FOR A DYNAMIC BOUNDARY PROBLEM WITH INFINITE DELAY in Lp0(0,T;(Vp)⇤), see Remark 4.6 in [22]. In order to find a weak solution to Problem (P), we use the Faedo-Galerkin approximation. Since X2is separable and Vpis dense in X2, there is a orthonormal basis of X2contained in Vp.Wedenotesuchbasisby{n=(n, n)2X2;n2N}. Let Kn=span{1, ..., n},K 1=[1 n=1Kn, and Prn:X2!Knbe the orthogonal projection. Given 2C(X2)andT>⌧we want to find a solution Un=Pn i=1 di(t)i2Knfor an ndimensional version of problem (4.7), which is equivalent to find a solution of the following system of ordinary di↵erential equations ⇢h@tUn,ii+hpUn,ii=hF(t, Ut n),ii+hG(t),ii hUn(⌧+s),Vi=hPrn (s),Vi,for s2(1,0], for all 1 inand a.e. in [⌧,T], where h·,·idenote the dual product between (Vp)⇤and Vp. The above system of ordinary functional di↵erential equations with infinite delay fulffils the conditions for existence and uniqueness of local solution established in Theorem 1.1 of [13]. A priori estimates ensure that solutions do exist for all time in [⌧,T]. Estimate (4.3) of Lemma 4.4 ensures that for 2C(X2)andR>0 such that k kC(X2) R,thereexistsaconstantC=C(⌧,T,R), but independent of nand t2(⌧,T), such that kUt nk2 C(X2)C(⌧,T,R).(4.8) In particular, the previous limit and estimate (4.2) imply the existence of another constant (relabelled the same) C=C(⌧,T,R)suchthat ⇢kUn(t)kL1(⌧,T;X2)C kUn(t)kLp(⌧,T;Vp)C. (4.9) Then, this guarantees that pUnis bounded in Lp0(⌧,T;(Vp)⇤), see [22] for more details. Hypotheses (F2), (F3), (4.8) and recalling that X2⇢(Vp)⇤continuously imply that F(t, Ut n) is bounded in Lp0(⌧,T;(Vp)⇤). Note that, @tUn=pUn+F(t, Ut n)+G(t)inLp0(⌧,T;(Vp)⇤).(4.10) Therefore, the limits of pUnand F(t, Ut n) ensure that there exists a constant (relabelled the same) C(⌧,T,R)suchthat k@tUnkLp0(⌧,T;(Vp)⇤)C. (4.11) The limits in (4.9) and (4.11) ensure that there is a subsequence (which we relabel the same) {Un},andanelementU2L1(⌧,T;X2)\Lp(⌧,T;Vp)with@tU2Lp0(⌧,T;(Vp)⇤), such that 8 < : Un ⇤ *Uin L1(⌧,T;X2), Un*Uin Lp(⌧,T;Vp), @tUn ⇤ *@ tUin Lp0(⌧,T;(Vp)⇤). (4.12)
PULLBACK ATTRACTOR FOR A DYNAMIC BOUNDARY PROBLEM WITH INFINITE DELAY 9 From compactness results, see Theorems 1.4 and 1.5 page 32 of [7], the sequences in fact have the following convergences ⇢Un!Uin Lp(⌧,T;X2), Un!Uin C([⌧,T]; X2).(4.13) Note that, Prn ! inC(X2), and thanks to the strong convergence in C([⌧,T]; X2) yield that Ut n!Utin C(X2)8tT, see, for instance, [18, 19] and [20] for details about both convergences. The above convergence and hypotheses (F2) and (F3) imply that F(t, Ut n)!F(t, Ut)inLp0(⌧,T;X2), which together with convergences (4.12) and the theory of maximal monotone operators allow us to deduce that pUn ⇤ * pUin Lp0(⌧,T;(Vp)⇤), see [22] for details. Therefore, Uis solution of the limit equation of (4.10) in the weak star topology of Lp0(⌧,T;(Vp)⇤). This ensures that Uis a weak solution of Problem (P) in the interval (1,T] with initial condition U⌧= . ⌅ The existence of solution allows us define the multivalued process {U(t, ⌧)}on C(X2)by U(t, ⌧) =Ut|U(·) is a solution of Problem (P)withU⌧= 2C(X2) . Indeed, item (2) of Definition 3.1 follows from concatenation and translation of solutions, see [4] and [5] for details. Lemma 4.6. The multivalued process {U(t, ⌧)}is upper-semicontinuous in C(X2). Proof: Let ⌧2R,{ n}n2Nand such that n! inC(X2), and let {Yn}n2Nsuch that Y· n2U(·,⌧) n. Given T>⌧, observe that, as n! inC(X2), given R>0, except for a finite number of elements, we have that { n}⇢BC(X2)( ,R). Then, from Lemma 4.4 the sequence {Yn} is bounded in L1(⌧,T;X2)andLp(⌧,T;Vp). Then, similarly to the proof of Theorem 4.5, we can ensure the existence of an element Y·2U(·,⌧) such that Yt n!Ytin C(X2)foralltT. Therefore, as T>⌧is arbitrary, it follows that {U(t, ⌧)}is upper-semicontinuous. ⌅ 5. Pullback Attractor for Problem (P) In this section we develop some estimates to show that the multivalued process generated by solutions of Problem (P) possesses a pullback absorbing family and it is pullback asymptotically upper semicompact. Therefore, we can ensure the existence of pullback attractor for the problem.
16 PULLBACK ATTRACTOR FOR A DYNAMIC BOUNDARY PROBLEM WITH INFINITE DELAY [24] YANG, L.; YANG, M.; KLOEDEN, P. E. Pullback attractors for non-autonomous quasilinear parabolic equations with dynamical boundary conditions. Disc. and Cont. Dynamical Systems B, Vol. 17, No. 7, 1-11, (2012). [25] YANG, L.; YANG, M.; WU, J. On uniform attractors for non-autonomous p-Laplacian equation with a dynamic boundary condition. Disc. and Cont. Dynamical Systems B, Vol. 17, No. 7, 1-11, (2012). [26] YEJUAN, W.; KLOEDEN, P. E. Pullback attractors of a multi-valued process generated by parabolic di↵erential equations with unbounded delays. Nonlinear Analysis 90, 8695, (2013).
Response for Reviewers First of all, we want to thank the reviewers for his dedication and his great contributions to this work. One of the reviewers noted that we had not actually shown that the pullback absorbing family Ddefined in Lemma 5.4 have the property that [⌧tD(⌧)isboundendfor all t2R.TohavethispropertywehadtochangeourG2hypothesisto (G2) sup r0 erZr 1 es⇣kg1(s)kp0 p0+kg2(s)kp0 p0,⌘ds < +1, like in [1]. With that change the small final section loses its meaning and had to be removed. But the most important results of the work remain valid. Furthermore, we have made other minor changes, which are pointed below: 1. page 10, line 3: We added the phrase: ”Consider in this section p>2” at the end of the paragraph; 2. page 6 (line 10), page 8 (line -12) and page 9 (line 10): we have replaced “(H2) and (H3)” by “(F2) and (F3)”; 3. page 6 (line 6) and page 10 (line 4):we have replaced “⇥:= (M⌦2")>0” by “⇥:= 1 2(M⌦2")>0”; 4. page 4, line 10: we have replaced “Banach space” by “Hilbert space” 5. page 12, at the beginning of the demonstration of Lemma 5.5: we have replaced “U(·;⌧n, n)” by “U(t0;⌧n, n)” 6. page 4, line 15: we removed the expressionwe ”another spaces”; 7. page 5, line 8: we removed the ”{”; 8. page 11, line -10: we have replaced “comma” by “period” References [1] YEJUAN, W.; KLOEDEN, P. E. Pullback attractors of a multi-valued process generated by parabolic di↵erential equations with unbounded delays. Nonlinear Analysis 90, 86–95, (2013). 1 had the interesting and helpful suggestions which allowed us to greatly improved the presentation of this paper. bounded Proof