Pullback attractor for a non-autonomous reaction-diffusion equation in some unbounded domains

Bol. So . Esp. Ma . Apl. n o 51(2010), 917 PULLBACK ATTRACTOR FOR A NON-AUTONOMOUS REACTION-DIFFUSION EQUATION IN SOME UNBOUNDED DOMAINS MARÍA ANGUIANO Dp o. Euaiones Di e eniales y Análisis Numé io Uni e sidad de Se illa, Apdo. de Co eos 1160, 41080-Se illa (Spain) anguianous.es Abs a The exis ene o a pullbak a a o in L2(Ω) o he ollowing non- au onomous ea ion-diusion equa ion 8 > < > : ∂u ∂ − △u= (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 neessa ily bounded bu sa is ying he Poina é inequali y, and h∈L2 loc(R;H−1(Ω)) . The main onep used in he p o o is he asymp o i ompa ness o he p o ess gene a ed by he p oblem. Key wo ds: pul lbak a a o , asymp o i ompa ness, e olu ion p oess, non-au onomous ea ion-diusion equa ion. AMS sub je lassia ions: 35B41, 35Q35, 35Q30, 35K90, 37L30. 1 In o du ion and se ing o he p oblem Le Ω⊂RN b e an op en se , no neessa ily bounded and suppose ha Ω sa ises he Poina é inequali y, i.e., he e exis s a ons an λ1>0 suh ha ZΩ|u(x)|2dx ≤λ−1 1ZΩ|∇u(x)|2dx, ∀u∈H1 0(Ω) . (2) Le us onside he ollowing p oblem o a non-au onomous ea ion- diusion equa ion wi h ze o Di ihle b ounda y ondi ion in Ω ,      ∂u ∂ −△u= (u) + h( ), in Ω×(τ, +∞) , u= 0, on ∂Ω×(τ, +∞) , u(x, τ) = uτ(x), x ∈Ω , (3) 9 10 M. Anguiano whe e τ∈R , uτ∈L2(Ω) , h∈L2 loc(R;H−1(Ω)) and ∈C(R) sa ises ha he e exis ons an s α1>0 , α2>0 , l≥0 , and p > 2 suh ha −α1|s|p≤ (s)s≤ −α2|s|p , (4) ( (s)− ( ))(s− )≤l(s− )2∀ , s ∈R . (5) The aim o his pape is o show he exis ene o a pullbak a a o in he phase spae L2(Ω) o he p oblem (3) in he ase o op en domains no neessa ily b ounded bu sa is ying he Poina é inequali y. This, and he a ha he non- au onomous h b elongs o he spae L2 loc(R;H−1(Ω)), a e he main no el ies o ou p oblem. The lak o ompa ness o he inje ion H1 0(Ω) ⊂L2(Ω) (in he ase o unb ounded domains) implies ha he s anda d ehniques p e iously used, pa iula ly he one in ol ing he so-alled a enning p op e y (see [6℄, [7℄, [12℄, [14℄, amongs o he s), whih ha e b een suess ully used when Ω is b ounded and h∈L2 loc(R;L2(Ω)) , do no wo k in ou ase. Ins ead, we will use he asymp o i ompa ness al eady used in he ase o non-au onomous 2D-Na ie -S okes (see [1℄ and [2℄, see also [5℄ o a lose esul ), and whih was p e iously used in [11℄ o he au onomous ase. We would like o emphasize ha his ehnique seems o be he only one whih allows o p o e he main esul o his pap e (namely Theo em 4) one ning he exis ene o pullbak a a o o ou p oblem. I is also wo h men ioning ha ou p oblem has eei ed muh a en ion o e he las yea s in he ase o a b ounded domain o o a less gene al e m h (see [3℄, [7℄, [12℄, [14℄). Finally, he eade an nd simila esul s o se e al a ian s o ou model in he e e enes [9℄, [10℄, among o he s. 2 Exis ene and uniqueness o solu ion We s a e in his se ion a esul on he exis ene and uniqueness o solu ion o p oblem (3). By |·| we deno e he no m in L2(Ω) , by |∇·| he no m in H1 0(Ω) and by k·k∗ he no m in H−1(Ω) . We will use (·,·) o deno e he sala p o du in L2(Ω) and we will use h·,·i o deno e he duali y p o du b e ween H−1(Ω) and H1 0(Ω) . Theo em 1 Suppose ha Ω sa ises (2). Assume ha ∈C(R) sa ises (4) and (5), and h∈L2 loc(R;H−1(Ω)) . Then, o al l τ∈R , uτ∈L2(Ω) , he e exis s a unique solu ion u( ) = u( ;τ, uτ) o (3) suh 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 D′(τ, ∞),∀ ∈H1 0(Ω) ∩Lp(Ω) , u(τ) = uτ. A ea ion-diusion equa ion in some unb ounded domains 11 Mo eo e , u∈C([τ, ∞); L2(Ω)), and u sa ises he ene gy equa ion, 1 2 d d |u( )|2+|∇u( )|2=h (u( )), u( )i +hh( ), u( )i in D′(τ, ∞). (6) P oo . The p o o o his Theo em an b e done by he me ho d o mono ony (see [8℄).  3 P elimina ies on he heo y o pullbak a a o s Now, we will eall he main p oin s om he heo y o pullbak a a o s whih will b e needed o p o e ou ob je i e (see [1℄ and [2℄ o mo e de ails). Le us onside a p o ess (also alled a wo-pa ame e semig oup) U on a me i spae X , i.e., a amily {U( , τ); −∞ < τ ≤ < +∞} o on inuous mappings U( , τ) : X→X , suh ha U(τ, τ)x=x, and U( , τ) = U( , )U( , τ) o all τ≤ ≤ . (7) Supp ose ha D is a nonemp y lass 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ni ion 1 The p oess U(·,·) is said o be pul lbak D -asymp o ial ly ompa i o any ∈R , any b D∈ D, any sequene τn→ −∞, and any sequene xn∈D(τn) , he sequene {U( , τn)xn} is ela i ely ompa (i.e. p e-ompa ) in X . Deni ion 2 I is said ha b B∈ D is pul lbak D -abso bing o he p oess U(·,·) i o any ∈R and any b D∈ D , he e exis s a τ0( , b D)≤ suh ha U( , τ)D(τ)⊂B( ) o al l τ≤τ0( , b D). Deni ion 3 The amily b A={A( ); ∈R} ⊂ P(X) is said o be a pul lbak D -a a o o U(·,·) i 1. A( ) is ompa o al l ∈R , 2. b A is pul lbak D -a a ing, i.e., lim τ→−∞dis (U( , τ)D(τ), A( )) = 0, o al l b D∈ D , and al l ∈R , 3. b A is in a ian , i.e., U( , τ)A(τ) = A( ), o −∞ < τ ≤ < +∞. 12 M. Anguiano We ha e he ollowing esul (see [2℄ o mo e de ails). Theo em 2 Suppose ha he p oess U(·,·) is pul lbak D -asymp o ial ly ompa and ha b B∈ D is a amily o pul lbak D -abso bing se s o U(·,·) . Then, he amily b A={A( ); ∈R} ⊂ P(X) dened by A( ) = Λ( b B, ), ∈ R, whe e o eah b D∈ D Λ( b D, ) = s≤  [ τ≤s U( , τ)D(τ) , is a pul lbak D -a a o o U(·,·) whih sa ises in addi ion ha A( ) = Sb D∈D Λ( b D, ), o ∈R. Fu hemo e, b A is minimal in he sense ha i b C={C( ); ∈R} ⊂ P(X) is a amily o losed se s suh ha limτ→−∞ dis (U( , τ)B(τ), C( )) = 0, hen A( )⊂C( ) . 4 Exis ene o he pullbak a a o Now, we an p o e ou main esul in his pap e . Fi s , we need a on inui y esul whih is es ablished in he nex subse ion. 4.1 Weak Con inui y Assume ha he un ion ∈C(R) sa ises (4) and (5), and ha h∈ L2 loc(R;H−1(Ω)) . Thanks o Theo em 1, we an dene a p o ess {U( , τ) , τ≤ } in L2(Ω) , as U( , τ)uτ=u( ;τ, uτ)∀uτ∈L2(Ω) ,∀τ≤ . (8) F om he uniqueness o solu ion o p oblem (3), i ollows ha (8) denes a p o ess in L2(Ω) . In addi ion, i an b e p o ed ha he p o ess dened by (8) is on inuous in L2(Ω) . Mo eo e , U is weakly on inuous, and mo e exa ly he ollowing esul holds ue. We will deno e by  ⇀  he weak on e gene in he o esp onding india ed spae, while  →  will deno e he s ong on e gene, as usual. P op osi ion 3 Le {uτn} ⊂ L2(Ω) be a sequene on e ging weakly in L2(Ω) o an elemen uτ∈L2(Ω) . Then, o al l T > τ , i ol lows U( , τ)uτn⇀ U ( , τ)uτ in L2(Ω) ∀ ≥τ , (9) U(·, τ)uτn⇀ U (·, τ)uτ in L2(τ, T ;H1 0(Ω)), (10) U(·, τ)uτn⇀ U (·, τ)uτ in Lp(τ, T ;Lp(Ω)) , (11) (U(·, τ)uτn)⇀ (U(·, τ)uτ) in Lp′(τ, T ;Lp′(Ω)). (12) I Ω is a bounded se , hen U(·, τ)uτn−→ U(·, τ)uτ in L2(τ, T ;L2(Ω)) . (13) A ea ion-diusion equa ion in some unb ounded domains 13 P oo . This esul may be p o ed in muh he same way as Theo em 1, and using simila a gumen s o [11℄.  4.2 The exis ene o he global pullbak a a o Le Rλ1 b e he se o all un ions :R→(0,+∞) suh ha lim →−∞eλ1 2( ) = 0 , and deno e by Dλ1 he lass o all amilies b D={D( ) : ∈R} ⊂ P(L2(Ω) ) suh ha D( )⊂B(0, b D( )) , o some b D∈ Rλ1 , whe e B(0, b D( )) deno es he losed ball in L2(Ω) en e ed a ze o wi h adius b D( ) . Now, we an p o e he ollowing esul . Theo em 4 Suppose ha Ω sa ises (2), and suppose ha ∈C(R) sa ises (4) and (5) wi h l= 0 . Le h∈L2 loc(R;H−1(Ω)) be suh ha Z −∞ eλ1skh(s)k2 H−1(Ω) ds < +∞ ∀ ∈R . Then, he e exis s a unique global pul lbak Dλ1 -a a o o he p oess U , whih belongs o Dλ1 , and is dened by (8). P oo . We only gi e he main ideas o he p o o . Le τ∈R , and uτ∈L2(Ω) b e xed, and deno e u( ) = u( ;τ, uτ) = U( , τ)uτ∀ ≥τ . Le b D∈ Dλ1 b e gi en. Taking in o aoun (2), (4), he ene gy equali y and in eg a ing b e ween τ and , |U( , τ)uτ|2≤e−λ1 Z −∞ eλ1skh(s)k2 H−1(Ω) ds +eλ1(τ− ) 2 D(τ) , (14) o all uτ∈D(τ) and o all ≥τ . Deno e by Rλ1( ) he nonnega i e numbe gi en o eah ∈R by R2 λ1( ) = e−λ1 Z −∞ eλ1skh(s)k2 H−1(Ω) ds + 1 . (15) Obse e ha Rλ1∈ Rλ1 . Now, onside he amily b Bλ1 o losed balls in L2(Ω) , b Bλ1={Bλ1( ) : ∈R} , dened by Bλ1( ) =  ∈L2(Ω) : | | ≤ Rλ1( ) . I is s aigh o wa d o hek ha b Bλ1∈ Dλ1, and mo eo e , by (14), he amily b Bλ1 is pullbak Dλ1 -abso bing o he p o ess U . Ao ding o Theo em 2, o nish he p o o o he heo em we only ha e o p o e ha U is pullbak Dλ1 -asymp o ially ompa . 14 M. Anguiano Le us x b D∈ Dλ1 , a sequene τn→ −∞ , a sequene uτn∈D(τn) , and ∈R . We ha e o p o e ha om he sequene {U( , τn)uτn} we an ex a a subsequene ha on e ges in L2(Ω) . As he amily b Bλ1 is pullbak Dλ1 -abso bing, by a diagonal p o edu e, i is no diul o onlude ha he e exis a subsequene τn′, uτn′⊂ {(τn, uτn)} , and a sequene {wk;k≥0} ⊂ L2(Ω) suh ha o all k≥0 , and wk∈ Bλ1( −k) , U( −k, τn′)uτn′⇀ wk in L2(Ω) , (16) and |w0| ≤ lim in n′→∞ U( , τn′)uτn′ . (17) I we now p o e ha also lim sup n′→∞ U( , τn′)uτn′≤ |w0| , (18) hen we will ha e lim n′→∞U( , τn′)uτn′=|w0| . And his, oge he wi h he weak on e gene, will imply he s ong on e gene in L2(Ω) o U( , τn′)uτn′ o w0 . In o de o p o e (18), onside [u] := |∇u|2−λ1 2|u|2−h (u), ui . Taking in o aoun (2), (4), he ene gy equali y and in eg a ing b e ween τ and , i is immedia e ha o all k≥0 and all τn′≤ −k , U( , τn′)uτn′2 (19) =U( −k, τn′)uτn′2e−λ1k + 2 Z −k eλ1(s− )h(s), U(s, −k)U( −k, τn′)uτn′ds −2Z −k eλ1(s− )U(s, −k)U( −k, τn′)uτn′ds . Now we will p o e ha Z −k eλ1(s− )[U(s, −k)wk]ds (20) ≤lim in n′→∞ Z −k eλ1(s− )U(s, −k)U( −k, τn′)uτn′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 , A ea ion-diusion equa ion in some unb ounded domains 15 and J(2) k( ) = −Z −k eλ1(s− )h ( ), ids , o all ∈L2( −k, ;H1 0(Ω)) ∩Lp( −k, ;Lp(Ω)) . We also ob ain om (16) and using P op osi ion 3 lim in n′→∞ (J(1) k(U(·, −k)U( −k, τn′)uτn′) ≥J(1) k(U(·, −k)wk) . (21) Using (5) wi h l= 0 , om (16) and P op osi ion 3 we easily ob ain lim n′→∞in J(2) k(U(·, −k)U( −k, τn′)uτn′) ≥J(2) k(U(·, −k)wk) . The e o e (20) is easily ob ained om he las inequali y and (21). Then, aking in o aoun ha he amily b Bλ1 is pullbak Dλ1 -abso bing, om (16), using P op osi ion 3 and hanks o (19) and (20), we ob ain lim n′→∞sup U( , τn′)uτn′2 ≤R2 λ1( −k)e−λ1k+|w0|2 , o all k≥1. Taking in o aoun (15), we easily ob ain lim n′→∞sup U( , τn′)uτn′2≤ |w0|2 .  Aknowledgemen s. This wo k has b een pa ially supp o ed by Jun a de Andaluía unde p o je P07-FQM-02468. Re e enes [1℄ T. Ca aballo, G. 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