Existence of pullback attractor for a reaction-diffusion equation in some unbounded domains with non-autonomous forcing term in H-1

EXISTENCE OF PULLBACK ATTRACTOR FOR A REACTION-DIFFUSION EQUATION IN SOME UNBOUNDED DOMAINS WITH NON-AUTONOMOUS FORCING TERM IN H−1 MAR´ IA ANGUIANO, TOM´ AS CARABALLO, & JOS´ E REAL Dp o. 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) E-mails: [email p o ec ed], ca abal[email p o ec ed], j e[email p o ec ed] To he Memo y o P o esso Vale y S. Melnik The exis ence o a pullback a ac o in L2(Ω) o he ollowing non-au onomous eac ion- di usion equa ion      ∂u ∂ − 4u= (u) + h( ),in Ω ×(τ, +∞), u= 0,on ∂Ω×(τ, +∞), u(x, τ) = uτ(x), x ∈Ω, (1) is p o ed in his pape , when he domain Ω is no necessa ily bounded bu sa is ying he Poinca ´e inequali y, and h∈L2 loc(R;H−1(Ω)). The main concep used in he p oo is he asymp o ic compac ness o he p ocess gene a ed by he p oblem. Keywo ds: pullback a ac o , asymp o ic compac ness, e olu ion p ocess, non-au onomous eac ion-di usion equa ion. Ma hema ics Subjec Classi ica ions (2000): 35B41, 35Q35, 35Q30, 35K90, 37L30 1. In oduc ion and se ing o he p oblem Le Ω ⊂RNbe an open se , no necessa ily bounded and suppose ha Ω sa is ies he Poinca ´e inequali y, i.e., he e exis s a cons an λ1>0 such ha ZΩ |u(x)|2dx ≤λ−1 1ZΩ |∇u(x)|2dx, ∀u∈H1 0(Ω) . (2) Le us conside he ollowing p oblem o a non- au onomous eac ion-di usion equa ion wi h ze o Di ichle bounda y condi ion in Ω,      ∂u ∂ − 4u= (u) + h( ),in Ω ×(τ, +∞), u= 0,on ∂Ω×(τ, +∞), u(x, τ) = uτ(x), x ∈Ω, (3) whe e τ∈R,uτ∈L2(Ω), h∈L2 loc(R;H−1(Ω)) and ∈C(R) sa is ies ha he e exis cons an s α1>0, α2>0, l≥0, and p > 2 such ha −α1|s|p≤ (s)s≤ −α2|s|p, (4) ( (s)− ( ))(s− )≤l(s− )2∀ , s ∈R. (5) 1 2M. Anguiano, T. Ca aballo & J. Real Using (4), i ollows ha | (s)| ≤ α1|s|p−1∀s∈R. (6) The aim o his pape is o show he exis ence o a pullback a ac o in he phase space L2(Ω) o he p oblem (3) in he case o open do- mains no necessa ily bounded bu sa is ying he Poinca ´e inequali y. This, and he ac ha he non-au onomous hbelongs o he space L2 loc(R;H−1(Ω)),a e he main no el ies o ou p oblem. The lack o compac ness o he injec ion H1 0(Ω) ⊂L2(Ω) (in he case o unbounded domains) implies ha he s anda d ech- niques p e iously used, pa icula ly he one in ol ing he so-called la enning p ope y (see [Kloeden & Langa, 2007], [Li & Zhong, 2007], [Song & Wu, 2007], [Wang & Zhong, 2008], amongs o he s), which ha e been success ully used when Ω is bounded and h∈L2 loc(R;L2(Ω)), do no wo k in ou case. Ins ead, we will use he asymp o ic compac - ness al eady used in he case o non-au onomous 2D-Na ie -S okes (see [Ca aballo e al., 2006] and [Ca aballo e al., 2006b]), and which was p e i- ously used in [Rosa, 1998] o he au onomous case. We would like o emphasize ha his echnique seems o be he only one which allows o p o e he main esul o his pape (namely Theo em 4.4) con- ce ning he exis ence o pullback a ac o o ou p oblem. I is also wo h men ioning ha ou p oblem has ecei ed much a en ion o e he las yea s in he case o a bounded domain o o a less gene al e m h, as we will ecall now. In [Ca aballo e al., 2003] i is p o ed he exis ence o pullback a ac o in he space L2(Ω) (and ha i possesses ini e Hausdo dimension) when he domain is bounded and his unbounded bu wi h polynomial g ow h, i.e kh( )kL2(Ω) ≤k1| |α+k2 whe e k1, k2and αa e nonnega i e cons an s. When Ω is bounded and h∈L2 loc(R;L2(Ω)) and is ansla ion bounded, i.e. sup m∈RZm+1 m kh(s)kL2(Ω) ds < ∞,(7) he exis ence o a pullback a ac o in he space H1 0(Ω) is p o ed in [Song & Wu, 2007], while in [Li & Zhong, 2007] he ansla ion bounded condi- ion (7) is weakened o kh(s)k2 L2(Ω) ≤Meα|s|, whe e 0 ≤α≤λ1, and λ1deno es he i s eigen- alue o he Laplacian. In [Wang & Zhong, 2008], he exis ence o pullback a ac o in H1 0(Ω) is shown o a bounded domain and o a h∈L2 loc(R;L2(Ω)) such ha Z −∞ eσs kh(s)k2 L2(Ω) +kh0(s)k2 L2(Ω)ds < +∞ o all ∈Rand ce ain σ≥0. Fo a bounded domain Ω, and a ansla ion bounded unc ion h∈L2 loc(R;L2(Ω)), he exis ence o a uni o m a ac o in Lp(Ω) is demons a ed in [Song & Zhong, 2008]. Finally, he eade can ind simila esul s o se e al a ian s o ou model in he e e ences [Wang e al., 2007], [P izzi, 2003], [Mo illas & Vale o, 2005], [Sun & Zhong, 2005], among o he s. We will p o ide in his pape a su icien condi- ion ensu ing he exis ence o pullback a ac o in L2(Ω) when he domain is no necessa ily bounded and h∈L2 loc(R;H−1(Ω)). A case ha has no been conside ed in he li e a u e ye , as a as we know. 2. Exis ence and uniqueness o solu ion We s a e in his sec ion a esul on he exis ence and uniqueness o solu ion o p oblem (3). Ins ead o wo king di ec ly wi h ou equa ion, we will es- ablish a gene al esul which, in pa icula , can be applied o handle ou p oblem. 2.1. An abs ac esul Le Hbe a sepa able Hilbe space wi h scala p oduc (·,·) and no m |·|. Le Vi,i= 1, ..., m, be m≥1 e lexi e and sepa able Banach spaces such ha m [ i=1 Vi⊂H, m i=1 Viis dense in H, and Vi, i= 1, ..., m is included in Hwi h con inuous injec- ion. Pullback A ac o s o eac ion-di usion equa ions 3 By k·kiwe deno e he no m in Vi, by k·k∗i he no m in V0 i,i= 1, ..., m and, by V he space V= m i=1 Vi, wi h he no m k k= m X i=1 k ki,∀ ∈V. I is easy o see ha Vis a sepa able Banach space. We will use h·,·i o deno e he duali y p oduc be ween V0 iand Vi, o each i= 1, ..., m. We iden i y Hwi h i s dual H0using he Riesz Theo em, bu i Viis a Hilbe space, we do no iden i y Viwi h V0 i. Le u∈H, we iden i y uwi h Tu∈ ∩m i=1V0 isuch ha hTu, i= (u, ),∀ ∈Vi,∀i= 1, ..., m. Le τ∈Rbe an ini ial ime, and le Ai: (τ, ∞)× Vi→V0 i,i= 1, ..., m, be mope a o s, in gene al nonlinea , such ha A1) Fo each ∈Vi, he unc ion ∈ (τ, ∞)7−→ Ai( , )∈V0 iis Lebesgue measu able. A2) Each ope a o Aiis hemicon inuous, i.e, o all ∈(τ, ∞) and u, , w ∈Vi, he unc ion θ∈R7−→ hAi( , u +θ ), wi ∈ Ris con inuous. Suppose ha he e exis 2 ≤pi<+∞,i= 1, ..., m, and he e exis cons an s c > 0, α > 0 and λ≥0, and a nonnega i e unc ion C( )∈L1(τ, T ), o all T > τ, such ha o each i= 1, ..., m, A3) (Boundedness) kAi( , )k∗i≤c(1 + k kpi−1 i), o all ∈(τ, ∞), ∈Vi. A4) (Mono onici y) hAi( , )−Ai( , w), −wi+λ| −w|2≥0, o all ∈(τ, ∞) and o all , w ∈Vi. A5) (Coe ci i y) Fo each i he e exis s a semino m [·]iin Vi, such ha he e exis s λi≥0 o which [ ]i+λi| | is ano he no m in Vi. Mo eo e , [ ]i+λi| |and k·kia e equi alen , and hAi( , ), i+λ| |2+C( )≥α[ ]pi i, o all ∈(τ, ∞) and o all ∈Vi. Conside m unc ions hi( )∈Lp0 i(τ, T ;V0 i),∀T > τ, i = 1, ..., m, (8) and he ini ial condi ion uτ∈H. (9) I we se A( , ) = m X i=1 Ai( , ), h( ) = m X i=1 hi( ), we can conside he ollowing p oblem          u∈ m i=1 Lpi(τ, T ;Vi)∀T > τ, u0( ) + A( , u( )) = h( ),in D0(τ, ∞;V0), u(τ) = uτ. (10) The p oo o he ollowing esul is simila o ha o Theo em 1.4, Chap e 2 in [Lions, 1969]. Theo em 2.1. Assume A1)-A5), (8) and (9). Then, he e exis s a unique solu ion uo (10), such ha u∈C([τ, ∞); H),u0∈ m X i=1 Lp0 i(τ, T ;V0 i),(11) o all T > τ. 2.2. Exis ence and uniqueness o solu ion o p oblem (3) We use Theo em 2.1 o show he exis ence and uniqueness o solu ion o (3). Conside m= 2, H =L2(Ω), V1=H1 0(Ω) and V2=Lp(Ω) ∩L2(Ω) wi h p > 2, and deno e V0 1= H−1(Ω), V0 2=Lp0(Ω) + L2(Ω). Recall ha |·| deno es he no m in H, by k·k1= |∇·| we will deno e he no m in V1, and by k·k2= k·kLp(Ω) +|·| he no m in V2. I we se A1( , u) = −∆u, A2( , u) = − (u), and h1( ) = h( ), h2( ) = 0, hen, i is no di icul o apply Theo em 2.1 wi h p1= 2 and p2=p, and we ob ain 4M. Anguiano, T. Ca aballo & J. Real Theo em 2.2. Assume ha ∈C(R)sa is ies (4) and (5), and h∈L2 loc(R;H−1(Ω)). Then, o all τ∈R,uτ∈L2(Ω), he e exis s a unique solu ion u( ) = u( ;τ, uτ)o (3) such ha u∈L2(τ, T ;H1 0(Ω)) ∩Lp(τ, T ;Lp(Ω)) ∀T > τ, d d (u( ), )− h∆u( ), i=h (u( )), i +hh( ), i,in D0(τ, ∞),∀ ∈H1 0(Ω) ∩Lp(Ω) , u(τ) = uτ. Mo eo e , u∈C([τ, ∞); L2(Ω)), and usa is ies he ene gy equa ion, 1 2 d d |u( )|2+|∇u( )|2=h (u( )), u( )i +hh( ), u( )iin D0(τ, ∞).(12) 3. P elimina ies on he heo y o pullback a ac o s Now, we will ecall he main poin s om he heo y o pullback a ac o s which will be needed in o de o p o e ou objec i e (see [Ca aballo e al., 2006] and [Ca aballo e al., 2006b] o mo e de ails). Le us conside a p ocess (also called a wo- pa ame e semig oup) Uon a me ic space X, i.e., a amily {U( , τ); −∞ < τ ≤ < +∞} o con inuous mappings U( , τ) : X→X, such ha U(τ, τ)x=x, and U( , τ) = U( , )U( , τ) o all τ≤ ≤ . (13) Suppose Dis a nonemp y class o pa ame e ized se s b D={D( ); ∈R} ⊂ P(X),whe e P(X) deno es he amily o all nonemp y subse s o X. De ini ion 3.1. The p ocess U(·,·) is said o be pullback D-asymp o ically compac i o any ∈ R, any b D∈ D,any sequence τn→ −∞,and any sequence xn∈D(τn), he sequence {U( , τn)xn}is ela i ely compac (i.e. p e-compac ) in X. De ini ion 3.2. I is said ha b B∈ D is pullback D-abso bing o he p ocess U(·,·) i o any ∈R and any b D∈ D, he e exis s a τ0( , b D)≤ such ha U( , τ)D(τ)⊂B( ) o all τ≤τ0( , b D). De ini ion 3.3. The amily b A={A( ); ∈R} ⊂ P(X) is said o be a pullback D-a ac o o U(·,·) i 1. A( ) is compac o all ∈R, 2. b Ais pullback D-a ac ing, i.e., lim τ→−∞ dis (U( , τ)D(τ), A( )) = 0, o all b D∈ D, and all ∈R, 3. b Ais in a ian , i.e., U( , τ)A(τ) = A( ), o −∞ < τ ≤ < +∞. We ha e he ollowing esul . Theo em 3.4. Suppose ha he p ocess U(·,·)is pullback D-asymp o ically compac and ha b B∈ D is a amily o pullback D-abso bing se s o U(·,·). Then, he amily b A={A( ); ∈R}⊂P(X) de ined by A( ) = Λ( b B, ), ∈R,whe e o each b D∈ D Λ( b D, ) = s≤  [ τ≤s U( , τ)D(τ) , is a pullback D-a ac o o U(·,·)which sa is ies in addi ion ha A( ) = Sb D∈D Λ( b D, ), o ∈R. Fu hemo e, b Ais minimal in he sense ha i b C={C( ); ∈R} ⊂ P(X)is a amily o closed se s such ha limτ→−∞ dis (U( , τ)B(τ), C( )) = 0, hen A( )⊂C( ). 4. Exis ence o he pullback a ac o Now, we can p o e ou main aim in his pape . Fi s , we need a con inui y esul which is es ab- lished in he nex subsec ion. 4.1. Weak Con inui y Le ∈C(R) be a unc ion, and suppose ha sa is ies (4) and (5), and h∈L2 loc(R;H−1(Ω)). Thanks o Theo em 2.2, we can de ine a p ocess {U( , τ), τ≤ }in L2(Ω), as U( , τ)uτ=u( ;τ, uτ)∀uτ∈L2(Ω) ,∀τ≤ . (14) F om he uniqueness o solu ion o p oblem (3), i ollows ha (14) de ines a p ocess in L2(Ω). In Pullback A ac o s o eac ion-di usion equa ions 5 addi ion, i can be p o ed ha he p ocess de ined by (14) is con inuous in L2(Ω). Mo eo e , Uis weakly con inuous, and mo e exac ly he ollowing esul holds ue. We will deno e by “*” he weak con e gence in he co - esponding indica ed space, while “→” will deno e he s ong con e gence, as usual. P oposi ion 4.1. Le {uτn} ⊂ L2(Ω) be a se- quence con e ging weakly in L2(Ω) o an elemen uτ∈L2(Ω). Then, o all T > τ, i ollows U( , τ)uτn* U ( , τ)uτin L2(Ω) ∀ ≥τ,(15) U(·, τ)uτn* U (·, τ)uτin L2(τ, T;H1 0(Ω)),(16) U(·, τ)uτn* U (·, τ)uτin Lp(τ, T;Lp(Ω)),(17) (U(·, τ)uτn)* (U(·, τ)uτ)in Lp0(τ, T;Lp0(Ω)). (18) I Ωis a bounded se , hen U(·, τ)uτn−→ U(·, τ)uτin L2(τ, T ;L2(Ω)).(19) P oo . F om he assump ions, we deduce ha he e exis s a posi i e cons an Csuch ha |uτn| ≤ C∀n≥1.(20) Fix τ∈R, and se un( ) = U( , τ)uτn, u( ) = U( , τ)uτ. (21) Using (4), i ollows d d |un( )|2+ 2 |∇un( )|2≤ −2α2kun( )kp Lp(Ω) +2 hh( ), uni. In eg a ing be ween τand , we ob ain |un( )|2+ 2 Z τ |∇un(s)|2ds + 2α2Z τ kun(s)kp Lp(Ω) ds ≤ |uτn|2+ 2 Z τ hh(s), unids ∀ ≥τ. (22) On he o he hand, Z τ hh(s), unids ≤Z τ khkH−1(Ω) |∇un|ds ≤1 2ZT τ khk2 H−1(Ω) ds +1 2Z τ |∇un|2ds, which, join ly wi h (20) and (22) imply |un( )|2+Z τ |∇un(s)|2ds + 2α2Z τ kun(s)kp Lp(Ω) ds ≤C2+Z τ khk2 H−1(Ω) ds ∀ > τ. We deduce ha {un}is bounded in L2(τ, T ;H1 0(Ω))∩Lp(τ, T ;Lp(Ω))∩C([τ, T ]; L2(Ω)), (23) o all T > τ. Fix T > τ. In pa icula , {un(T)}is bounded in L2(Ω). On he o he hand, om (6) we ha e k (un( ))kp0 Lp0(Ω) ≤αp0 1kun( )kp Lp(Ω) , whence (un) is bounded in Lp0(τ, T;Lp0(Ω)). Then, he e exis s a subsequence {uµ}⊂{un} such ha uµ∗ * weak-s a in L∞(τ, T ;L2(Ω)), uµ* in Lp(τ, T ;Lp(Ω)), (24) uµ(T)* ξ in L2(Ω) , (25) uµ* in L2(τ, T ;H1 0(Ω)), (26) and (uµ)* χ in Lp0(τ, T ;Lp0(Ω)). (27) Now (26) imply ha ∆uµ*∆ in L2(τ, T ;H−1(Ω)). F om (26), (27), and hanks o he equa ion u0 µ( ) = ∆uµ( ) + (uµ( ))+h( ), (28) i is a s anda d ma e o p o e ha we can pick an elemen in he equi alence class o sa is ying ( ) = uτ+Z τ (∆ (s) + χ(s) + h(s))ds, (29) o all ∈[τ, T ]. We a e now in posi ion o show ha ξ= (T) and χ( ) = ( ( )). Le w∈H1 0(Ω) ∩Lp(Ω). In eg a ing (28) be- ween τand T, we ob ain (uµ(T), w) = uτµ, w +ZT τ h∆uµ(s) + (uµ(s))+h(s), wids, 6M. Anguiano, T. Ca aballo & J. Real and hus (ξ, w) = (uτ, w) + ZT τ h∆ (s) + χ(s) + h(s), wids, when µ−→ +∞. By densi y and using (29), i ollows (T) = ξ. (30) To p o e ha χ( ) = ( ( )),we a gue simila ly o [Rosa, 1998]. In eg a ing he equali y d ds (uµ(s), w) = −(∇uµ(s),∇w) +h (uµ(s)), wi+hh(s), wi, be ween and +a, wi h a∈(0, T −τ), ∈ (τ, T −a),and using he H¨olde inequali y, we ob- ain (uµ( +a)−uµ( ), w) ≤Z +a |∇uµ(s)| |∇w|ds +Z +a k (uµ(s)kLp0(Ω) kwkLp(Ω) ds +Z +a kh(s)kH−1(Ω) |∇w|ds ≤ |∇w|a1/2kuµkL2(τ,T ;H1 0(Ω)) +kwkLp(Ω) a1/p k (uµ)kLp0(τ,T ;Lp0(Ω)) +|∇w|a1/2khkL2(τ,T ;H−1(Ω)) , and hanks o (23), we deduce ha he e exis s a cons an C(1) such ha (uµ( +a)−uµ( ), w) ≤C(1)(a1/2+a1/p)(|∇w|+kwkLp(Ω)). I we ake in he las inequali y w=uµ( +a)− uµ( )∈H1 0(Ω) ∩Lp(Ω) a.e. ∈(τ, T −a),we ob ain |uµ( +a)−uµ( )|2 ≤C(1)(a1/2+a1/p)|∇uµ( +a)− ∇uµ( )| +C(1)(a1/2+a1/p)kuµ( +a)−uµ( )kLp(Ω) , a.e. ∈(τ, T −a). In eg a ing be ween τand T−a, ZT−a τ |uµ( +a)−uµ( )|2d ≤C(1)(a1/2+a1/p)ZT−a τ |∇uµ( +a)|d +C(1)(a1/2+a1/p)ZT−a τ |∇uµ( )|d +C(1)(a1/2+a1/p)ZT−a τ kuµ( +a)kLp(Ω) d +C(1)(a1/2+a1/p)ZT−a τ kuµ( )kLp(Ω) d , i ollows ZT−a τ |uµ( +a)−uµ( )|2d ≤2C(1)(a1/2+a1/p)ZT τ |∇uµ(s)|ds + 2C(1)(a1/2+a1/p)ZT τ kuµ(s)kLp(Ω) ds, and using he H¨olde inequali y, we ob ain ZT−a τ |uµ( +a)−uµ( )|2d ≤2C(1)(a1/2+a1/p)(T−τ)1/2kuµkL2(τ,T ;H1 0(Ω)) + 2C(1)(a1/2+a1/p)(T−τ)1/p0kuµkLp(τ,T ;Lp(Ω)) . Thanks o (23) we deduce ha he e exis s a con- s an e CTsuch ha ZT−a τ |uµ( +a)−uµ( )|2d ≤e CT(a1/2+a1/p), o all µ, and all a∈(0, T −τ),and hus lim a→0sup µZT−a τ |uµ( +a)−uµ( )|2d = 0. (31) Now, o all m∈Z,m≥1, we deno e Ωm= Ω ∩x∈RN:|x|RN< m, whe e |·|RNdeno es he Euclidean no m in RN. Le φ∈C1([0,+∞)) be a unc ion such ha 0≤φ(s)≤1, φ(s)=1 ∀s∈[0,1], and φ(s) = 0∀s≥2. Fo each µand m≥1, we de ine µ,m(x, ) = φ |x|2 RN m2!uµ(x, ) . Pullback A ac o s o eac ion-di usion equa ions 7 F om (23), o all m≥1, we ob ain ha { µ,m}µ≥1is bounded in L2(τ, T ;H1 0(Ω2m)) ∩ Lp(τ, T ;Lp(Ω2m)) ∩L∞(τ, T ;L2(Ω2m)). In pa icula , lim a→0sup µZτ+a τ | µ,m( )|2 L2(Ω2m)d + +ZT T−a | µ,m( )|2 L2(Ω2m)d = 0. (32) On he o he hand, om (31) we deduce ha o all m≥1, lim a→0sup µZT−a τ | µ,m( +a)− µ,m( )|2 L2(Ω2m)d = 0. (33) Mo eo e , as Ω2mis a bounded se , hen H1 0(Ω2m) is included in L2(Ω2m) wi h compac injec ion. Then, by he compac ness Theo em 13.3 o [Temam, 1983] wi h X=L2(Ω2m), Y=H1 0(Ω2m), = 2 and G={ µ,m}µ≥1, we ob ain ha { µ,m}µ≥1 is ela i ely compac in L2τ, T;L2(Ω2m), and hus, aking in o accoun ha µ,m(x, ) = uµ(x, ) o all x∈Ωm,we deduce ha , in pa icula , o all m≥1 nuµ|Ωmoµ≥1is p e-compac in L2τ, T ;L2(Ωm). (34) I is no di icul o conclude om (34), (26) and (2), ia a diagonal p ocedu e, he exis ence o a subsequence {uµ µ}µ≥1⊂ {uµ}µ≥1such ha uµ µ→ a.e. in Ωm×(τ, T) as µ−→ ∞ ∀m≥1. Then, as is con inuous, (uµ µ)→ ( ) a.e. in Ωm×(τ, T ) , and as { (uµ µ)}is bounded in Lp0(Ωm×(τ, T)),by Lemma 1.3, Chap e 1 in [Lions, 1969], we ob ain (uµ µ)* ( ) weakly in Lp0τ, T;Lp0(Ωm). F om (27) (uµ)* χ|Ωm×(τ,T )weakly in Lp0τ, T ;Lp0(Ωm). By he uniqueness o he weak limi , we ha e χ= ( ) a.e. in Ωm×(τ, T )∀m≥1, and hus, aking in o accoun ha ∞ [ m=1 Ωm= Ω, we ob ain χ= ( ) a.e. in Ω ×(τ, T ) . (35) F om (29) and (35), and by he uniqueness o so- lu ions we ha e ( ) = u( ) o all ∈[τ, T ]. And hen, i we conside (30) in (26) and (35) in (27), we ha e uµ* u in L2(τ, T ;H1 0(Ω)), uµ* u in Lp(τ, T ;Lp(Ω)), uµ(T)* u(T) in L2(Ω) , (uµ)* (u) in Lp0(τ, T ;Lp0(Ω)). Then, by a con adic ion a gumen we deduce un* u in L2(τ, T ;H1 0(Ω)), un* u in Lp(τ, T ;Lp(Ω)), un(T)* u(T) in L2(Ω) , (un)* (u) in Lp0(τ, T ;Lp0(Ω)), and, as T > τ has been aken a bi a ily, he i s pa o he p oo is inished. Now, i Ω is bounded, we deduce om (34) ha uµ|Ωµ≥1is p e-compac in L2τ, T ;L2(Ω). (36) Finally, (19) ollows om (36). Rema k 4.2. F om he p oo , i is clea ha o any m≥1, U(·, τ)uτn→U(·, τ)uτin L2(τ, T ;L2(Ωm)). Mo eo e , i is possible o p o e ha U( , τ)uτn→U( , τ)uτin L2(Ωm), o all > τ. Rema k 4.3. No ice ha all he esul s ob ained in he p e ious analysis hold ue o a gene al non- emp y open subse Ω ⊂RN. 8M. Anguiano, T. Ca aballo & J. Real 4.2. The exis ence o he global pullback a - ac o Le Rλ1be he se o all unc ions :R→(0,+∞) such ha lim →−∞ eλ1 2( ) = 0, and deno e by Dλ1 he class o all amilies b D={D( ) : ∈R}⊂P(L2(Ω) ) such ha D( )⊂ B(0, b D( )), o some b D∈ Rλ1, whe e B(0, b D( )) deno es he closed ball in L2(Ω) cen e ed a ze o wi h adius b D( ). Now, we can p o e he ollowing esul . Theo em 4.4. Suppose ha Ωsa is ies (2), and suppose ha ∈C(R)sa is ies (4) and (5) wi h l= 0. Le h∈L2 loc(R;H−1(Ω)) such ha Z −∞ eλ1skh(s)k2 H−1(Ω) ds < +∞ ∀ ∈R. Then, he e exis s a unique global pullback Dλ1- a ac o o he p ocess U, which belongs o Dλ1, and is de ined by (14). P oo . Le τ∈R, and uτ∈L2(Ω) be ixed, and deno e u( ) = u( ;τ, uτ) = U( , τ)uτ∀ ≥τ. Taking in o accoun (4) and he ene gy equali y, d d eλ1 |u( )|2+ 2eλ1 |∇u( )|2 =λ1eλ1 |u( )|2+ 2eλ1 h (u( )), u( )i + 2eλ1 hh( ), u( )i(37) ≤λ1eλ1 |u( )|2 +eλ1 |∇u( )|2+eλ1 kh( )k2 H−1(Ω) , and hus, om (2), we ob ain d d eλ1 |u( )|2≤eλ1 kh( )k2 H−1(Ω) . In eg a ing be ween τand , i ollows eλ1 |u( )|2≤Z τ eλ1skh(s)k2 H−1(Ω) ds +eλ1τ|uτ|2 ≤Z −∞ eλ1skh(s)k2 H−1(Ω) ds +eλ1τ|uτ|2. Le b D∈ Dλ1be gi en. Then |U( , τ)uτ|2≤e−λ1 Z −∞ eλ1skh(s)k2 H−1(Ω) ds +eλ1(τ− ) 2 D(τ), (38) o all uτ∈D(τ) and o all ≥τ. Deno e by Rλ1( ) he nonnega i e numbe gi en o each ∈Rby R2 λ1( ) = e−λ1 Z −∞ eλ1skh(s)k2 H−1(Ω) ds + 1. (39) Obse e ha lim →−∞ eλ1 R2 λ1( ) = 0, and, consequen ly, Rλ1∈ Rλ1. Now, conside he amily b Bλ1o closed balls in L2(Ω) b Bλ1={Bλ1( ) : ∈R}, de ined by Bλ1( ) =  ∈L2(Ω) : | | ≤ Rλ1( ). I is s aigh o wa d o check ha b Bλ1∈ Dλ1, and mo eo e , by (38), he amily b Bλ1is pullback Dλ1-abso bing o he p ocess U. Acco ding o Theo em 3.4, o inish he p oo o he heo em we only ha e o p o e ha Uis pullback Dλ1-asymp o ically compac . Le us ix b D∈ Dλ1, a sequence τn→ −∞, a se- quence uτn∈D(τn), and ∈R. We ha e o p o e ha om he sequence {U( , τn)uτn}we can ex ac a subsequence ha con e ges in L2(Ω). As he amily b Bλ1is pullback Dλ1-abso bing, o each in ege k≥0, he e exis s a τD(k)≤ −k such ha U( −k, τ)D(τ)⊂Bλ1( −k)∀τ≤τD(k). (40) Again, by a diagonal p ocedu e, i is no di icul o conclude om (40), ha he e exis a subse- quence τn0, uτn0⊂ {(τn, uτn)}, and a sequence {wk;k≥0} ⊂ L2(Ω) such ha o all k≥0, and wk∈Bλ1( −k), U( −k, τn0)uτn0* wkin L2(Ω) . (41) Pullback A ac o s o eac ion-di usion equa ions 9 Obse e ha , by P oposi ion 4.1. w0=weak −lim n0→∞ U( , τn0)uτn0 =weak −lim n0→∞ U( , −k)U( −k, τn0)uτn0 =U( , −k)weak −lim n0→∞ U( −k, τn0)uτn0. i.e., U( , −k)wk=w0∀k≥0. (42) Then, by he lowe semi-con inui y o he no m, using (41) we ob ain |w0| ≤ lim in n0→∞ U( , τn0)uτn0. (43) I we now p o e ha also lim sup n0→∞ U( , τn0)uτn0≤ |w0|, (44) hen we will ha e lim n0→∞ U( , τn0)uτn0=|w0|. And his, oge he wi h he weak con e gence, will imply he s ong con e gence in L2(Ω) o U( , τn0)uτn0 o w0. In o de o p o e (44), conside [u] := |∇u|2−λ1 2|u|2− h (u), ui. (45) F om (37), and in eg a ing be ween τand , eλ1 |u( )|2−eλ1τ|uτ|2=−2Z τ eλ1s[u(s)] ds +2 Z τ eλ1shh(s), u(s)ids, i.e., |U( , τ)uτ|2=eλ1(τ− )|uτ|2(46) + 2 Z τ eλ1(s− )(hh(s), U(s, τ)uτi − [U(s, τ)uτ]) ds. F om (46) i is immedia e ha o all k≥0 and all τn0≤ −k, U( , τn0)uτn02 =U( , −k)U( −k, τn0)uτn02(47) =U( −k, τn0)uτn02e−λ1k + 2 Z −k eλ1(s− )h(s), U(s, −k)U( −k, τn0)uτn0ds −2Z −k eλ1(s− )U(s, −k)U( −k, τn0)uτn0ds. As, hanks o (40), U( −k,τn0)uτn0∈Bλ1( −k)∀τn0≤τD(k), k≥0, we ha e lim sup n0→∞ U( −k, τn0)uτn02e−λ1k ≤R2 λ1( −k)e−λ1k∀k≥0. (48) On he o he hand, om (41) and P oposi ion 4.1 we deduce ha U(·, −k)U( −k, τn0)uτn0* U(·, −k)wk(49) in L2( −k, ;H1 0(Ω)). Taking in o accoun ha , in pa icula , eλ1(s− )h(s)∈L2( −k, ;H−1(Ω)), we ob ain om (49), lim n0→∞ Z −k eλ1(s− )h(s), U(s, −k)U( −k, τn0)uτn0ds =Z −k eλ1(s− )hh(s), U(s, −k)wkids. (50) Now we will p o e ha Z −k eλ1(s− )[U(s, −k)wk]ds (51) ≤lim in n0→∞ Z −k eλ1(s− )U(s, −k)U( −k, τn0)uτn0ds. Deno e Jk( ) = J(1) k( ) + J(2) k( ), whe e J(1) k( ) = Z −k eλ1(s− )|∇ (s)|2−λ1 2| (s)|2ds, and J(2) k( ) = −Z −k eλ1(s− )h ( ), ids, o all ∈L2( −k, ;H1 0(Ω)) ∩Lp( −k, ;Lp(Ω)). Then, we wan o p o e Jk(U(·, −k)wk) ≤lim in n0→∞ Jk(U(·, −k)U( −k, τn0)uτn0),