Attractors for differential equations with variable delays

A ac o s o Di e en ial Equa ions wi h Va iable Delays Tom´as Ca aballo, Jos´e A. Langa 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-mail: ca aball@nume .us.es ; [email p o ec ed] and James C. Robinson Ma hema ics Ins i u e. Uni e si y o Wa wick. Co en y, CV4 7AL. UK E-mail: [email p o ec ed]a wick.ac.uk Using he ela i ely new concep o a pullback a ac o , we p esen some esul s on he exis ence o a ac o s o di e en ial equa ions wi h a iable delay. We gi e a a ie y o examples o which ou esul applies. 1. INTRODUCTION The heo y o global a ac o s o au onomous sys ems as de eloped by Hale in [7] owes much o examples a ising in he s udy o e a ded unc ional di e en ial equa ions [8] ( o sligh ly di e en app oaches see Babin and Vishik [1], Ladyzhenskaya [12], o Temam [15]). Al hough he classical heo y can be ex ended in a ela i ely s aigh o wa d manne o deal wi h ime-pe iodic equa ions, gene al non-au onomous equa ions such as ˙x( ) = F( , x( ), x( −ρ( ))) (1) all ou side i s scope. Recen ly, a heo y o ‘pullback a ac o s’ has been de eloped (see sec ion 2) which allows many o he ideas o he au onomous heo y o be ex ended o deal wi h such examples. Howe e , un il now his has only been applied o o dina y and pa ial di e en ial equa ions. I is ou in en ion he e o show how pullback a ac o s can be used o in es iga e he beha iou o non-au onomous delay equa ions. In pa icu- la , we a e able o compa e he dynamics o sys ems o o dina y di e en ial equa ions wi h ha o he same sys em wi h a small delay, and show ha hese a e ‘close’ in some global sense. 1 2TOM´ AS CARABALLO, JOS´ E A. LANGA AND JAMES C. ROBINSON 2. DELAY DIFFERENTIAL EQUATIONS AS DYNAMICAL SYSTEMS We ake as ou canonical example o a non-au onomous delay equa ion a sys em wi h one, ime- a ying delay, ρ( ) whe e ρ:R→[0, h] is a con- inuous unc ion and h > 0, d d x( ) = F( , x( ), x( −ρ( ))) xs=ψ, ψ ∈ C.(2) The ini ial condi ion xsis speci ied in C, he space C0([−h, 0]; Rn) o con in- uous unc ions om [−h, 0] in o Rn, and, o a unc ion x∈C0([−h, T]; Rn), he no a ion xsdeno es he unc ion in Cgi en by xs(θ) = x(s+θ) o all θ∈[−h, 0] (and so makes sense o any 0 ≤s≤T). This equa ion can be w i en in a mo e gene al amewo k, which allows one o conside a la ge se o p oblems in a uni ied way. Ra he han make he delay explici , we w i e ( , x ) = F( , x( ), x( −ρ( ))), and so can ew i e (2) as ˙x( ) = ( , x )xs=ψ, ψ ∈ C.(3) In wha ollows we concen a e on his o m o he equa ion, assuming ha :R×C → Rnis con inuous and ‘a bounded map’ (i.e. maps bounded se s in o bounded se s). We no e he e ha his o mula ion immedia ely includes examples o he han he single, ime- a ying delay o (2). Fo example, he in eg o- di e en ial equa ion (see Kuang [11] o mo e de ails) ˙x( ) = Z0 −h g( , s, x( +s)) ds also i s in o his amewo k, al hough we do no de elop his heo y he e. I is known (Hale [7]) ha o any (s, ψ)∈R×C he e exis s a unique so- lu ion x( ;s, ψ) o (3) de ined on [s−h, αs,ψ). We assume ha αs,ψ = +∞, o all s∈R,since we a e in e es ed in long- ime beha iou o solu ions. We de ine a solu ion ope a o φ( , s) which gi es he solu ion (in C) a ime when xs=ψ, ia φ( , s)ψ=x (·;s, ψ).(4) ATTRACTORS FOR DELAY DIFFERENTIAL EQUATIONS 3 3. PULLBACK ATTRACTORS We now discuss he heo y o pullback a ac o s, as de eloped in Kloe- den and S onie [9], Kloeden and Schmal uss [10], and C auel e al. [5]). As is clea abo e, in he case o non-au onomous di e en ial equa ions he ini- ial ime is jus as impo an as he inal ime, and he classical semig oup p ope y o au onomous dynamical sys ems is no longe a ailable. Ins ead o a amily o one ime-dependen maps S( ) we need o use a wo-pa ame e p ocess φ( , s), as in oduced abo e in (4) (c . Sell [14]); φ( , s)ψdeno es he solu ion a ime which was equal o ψa ime s. The semig oup p ope y is eplaced by he p ocess composi ion p ope y φ( , s)φ(s, ) = φ( , ) o all ≥s≥ , and, ob iously, he ini ial condi ion implies φ(s, s) =Id. As wi h he semi- g oup composi ion S( )S(s) = S( +s), his jus exp esses he uniqueness o solu ions. [I is possible o p esen he heo y wi hin he mo e gene al amewo k o cocycle dynamical sys ems. In his case he second componen o φis iewed as an elemen o some pa ame e space J, so ha he solu ion can be w i en as φ( , p)ψ, and a shi map θ :J→Jis de ined so ha he p ocess composi ion becomes he cocycle p ope y, φ( +τ, p) = φ( , θτp)φ(τ, p). We do no pu sue his app oach he e, bu no e ha i has p o ed ex emely ui ul, pa icula ly in he case o andom dynamical sys ems. Fo a ious examples using his gene al se ing, see Kloeden and Schmal uss [10], o Sell [14]. Fo his eason, pullback a ac o s a e o en e e ed o as ‘cocycle a ac o s’]. As in he s anda d heo y o a ac o s, we seek an in a ian a ac ing se . Howe e , since he equa ion is non-au onomous his se also depends on ime. De ini ion 3.1. Le φbe a p ocess on a comple e me ic space X. A amily o compac se s {A( )} ∈Ris said o be a (global) pullback a ac o o φi , o all s∈R, i sa is ies i) φ( , s)A(s) = A( ) o all ≥s, and ii) lims→∞ dis (φ( , −s)D, A( ))=0, o allboundedsubse sDo X. In he de ini ion, dis (A, B) is he Hausdo semidis ance be ween Aand B, de ined as dis (A, B) = sup a∈A in b∈Bd(a, b), o A, B ⊆X. 4TOM´ AS CARABALLO, JOS´ E A. LANGA AND JAMES C. ROBINSON P ope y i) is a gene aliza ion o he in a iance p ope y o au onomous dynamical sys ems. The pullback a ac ing p ope y ii) conside s he s a e o he sys em a ime when he ini ial ime −sgoes o −∞ (c . Chepyzho and Vishik [3]) The no ion o an a ac o is closely ela ed o ha o an abso bing se . De ini ion 3.2. {B( )} ∈Ris said o be abso bing wi h espec o he p ocess φi , o all ∈Rand all D⊂Xbounded, he e exis s TD( )>0 such ha o all τ≥TD( ) φ( , −τ)D⊂B( ). Indeed, jus as in he au onomous case, he exis ence o compac abso b- ing se s is he c ucial p ope y in o de o ob ain pullback a ac o s. Fo he ollowing esul see C auel and Flandoli [4] o Schmal uss [13]. Theo em 3.1. Le φ( , s)be a wo-pa ame e p ocess, and suppose φ( , s) : X→Xis con inuous o all ≥s. I he e exis s a amily o compac ab- so bing se s {B( )} ∈R, hen he e exis s a pullback a ac o {A( )} ∈R, and A( )⊂B( ) o all ∈R. Fu he mo e, A( ) = S SS D⊂X boundedΛD( ),whe e ΛD( ) = n∈[ s≥n φ(s, −s)D. 4. ATTRACTORS FOR NON-AUTONOMOUS DELAY DIFFERENTIAL EQUATIONS We showed abo e how o de ine he p ocess associa ed wi h he solu ion o he delay di e en ial equa ion ˙x( ) = ( , x )xs=ψ(5) ia φ( , s)ψ=x (·;s, ψ). ATTRACTORS FOR DELAY DIFFERENTIAL EQUATIONS 5 We now p o e a simple gene al esul on he exis ence o pullback a - ac o s o his p oblem. P oo ha he condi ion o he heo em holds is signi ican ly mo e one ous han he p oo o he heo em i sel . Theo em 4.1. Suppose ha φ( , s)maps bounded se s in o bounded se s, and ha he e exis s a amily {B( 0)} ∈Ro bounded abso bing se s o φ. Then he e exis s a pullback a ac o o p oblem (5). P oo . Using heo em 3.1 i su ices o p o e ha he e exis s a amily o compac abso bing se s o φ. Fo each 0∈R, de ine K( 0) = φ( 0, 0−h)B( 0−h). K( 0) is clea ly abso bing, since o any bounded D⊂ C we ha e, o ≥TD( 0) + h, (he e TD( 0) deno es he abso p ion ime co esponding o he amily {B( 0)} ∈R) φ( 0, 0− )D=φ( 0, 0−h)φ( 0−h, ( 0−h)−( −h))D ⊂φ( 0, 0−h)B( 0−h) = K( 0). Also, K( 0) is bounded, since φmaps bounded se s in o bounded se s. Finally, K( 0) is a compac subse o C. This ollows using he A zel`a- Ascoli heo em, since we ha e jus shown ha K( 0) is bounded, and he equicon inui y ollows since, o ψ∈B( 0−h) and θ∈[−h, 0],  d dθφ( 0, 0−h)ψ(θ)= d dθx( 0+θ; 0−h, ψ)=| ( 0+θ, x 0+θ(·; 0−h, ψ)| which is bounded, using he assump ion on . 4.1. The case o s ong dissipa i i y In his sec ion we suppose a dissipa i e p ope y o he nonlinea e m o he di e en ial equa ion which will lead us o he exis ence o a uni o m (o e ∈R) bounded abso bing se o he p ocess φand hence a pullback a ac o . We will suppose in his sec ion ha o some α > 0, β≥0 h ( , ψ), ψ(0)i≤−α|ψ(0)|2+β o all ψ∈Φ(h)C(6) whe e h·,·i deno es he scala p oduc in Rnand Φ(h)C={χ∈ C :χ=φ(s+h, s)ψ, some s∈R, ψ ∈ C}. (No e ha Φ(h)Cis essen ially he se o all hose unc ions in Cwhich a e ealisable as solu ions o he equa ion a e a ime h). 6TOM´ AS CARABALLO, JOS´ E A. LANGA AND JAMES C. ROBINSON Al hough his seems s ange a a i s iew, no e ha (6) is a consequence o a mo e na u al se o assump ions in a ious pa icula examples. Indeed, i we conside (2) wi h F:Rn→Rnuni o mly bounded and uni o mly con inuous, i.e., o some k≥0 and some unc ion ω:R+→R+, |F(x)| ≤ kand |F(x)−F(y)| ≤ ω(|x−y|), and dissipa i e in a simila sense o (6), so ha , o some α0>0 and β0≥0 hF(x), xi≤−α0|x|2+β0, we eco e (6). Obse e ha , in his case, we a e assuming ha ( , x ) = F(x( −ρ( )) o , mo e gene ally, ( , ψ) = F(ψ(−ρ( )), o all ψ∈ C, ∈R. Indeed, we ha e hF(x( −ρ( )), x( )i ≤ hF(x( )), x( )i+hF(x( −ρ( )) −F(x( )), x( )i ≤ −α0|x( )|2+β0+|x( )| |F(x( −ρ( )) −F(x( ))| ≤ −α0|x( )|2+β0+|x( )|ω(|x( −ρ( )) −x( )|) ≤ −α0|x( )|2+β0+|x( )|ω(kh) ≤ −α|x( )|2+β o all ≥h, since hen x( ) is a solu ion o (2). We now show ha (6) ensu es he exis ence o a pullback a ac o . Theo em 4.2. Suppose ha (6) holds. Then he e exis s a amily o bounded abso bing se s {B( 0)} 0∈R o (3), and hus we can conclude he exis ence o a pullback a ac o o his p oblem. P oo . We will p o e mo e han he exis ence o a amily o bounded abso bing se s: in ac , he e exis s a uni o m (in 0) bounded abso bing se o (3). Indeed, gi en D⊂ C bounded, he e exis s d≥0 such ha o all ψ∈D, kψkC≤d, i.e. kψkC= sup θ∈[−h,0] |ψ(θ)| ≤ d. Take now ψ∈Dand conside |φ( 0, 0− )ψ|= sup θ∈[−h,0] |x( 0+θ; 0− , ψ)| = sup τ∈[ 0−h, 0] |x(τ; 0− , ψ)|. Le us w i e x(τ) = x(τ; 0− , ψ), τ∈[ 0− , 0]. Then, mul iplying (5) by x(τ) we ge d dτ|x(τ)|2= 2 hx(τ), (τ, xτ)i ≤ 2β−2α|x(τ)|2, ATTRACTORS FOR DELAY DIFFERENTIAL EQUATIONS 7 o all τ≥ 0− . Then, by G onwall’s lemma |x(τ)|2≤ |x( 0− )|2e−2α(τ− 0+ )+β α(1 −e−2α(τ− 0+ )) ≤ |ψ|2e−2α(τ− 0+ )+β α ≤ |ψ|2e−2α(θ+ )+β α ≤de2αhe−2α +β α. Thus we ob ain sup θ∈[−h,0] |x( 0+θ)|2≤de2αhe−2α +β α≤1 + β α i we ake ≥1 2αlog(de2αh) = TD. No e ha his ime TDdoes no depend on 0. 4.2. A mo e gene al case In he p e ious sec ion we conside ed di e en ial equa ions which only depended on he delay e m, and had no explici dependence o he cu en s a e (in o he wo ds, ( , x ) = F(x( ), x( −ρ( ))) = F(x( −ρ( )))). Howe e , a dependence on bo h he cu en and e a ded s a e is mo e usual in applica ions, and o en he equa ion can be in e p e ed as a pe u ba ion o an o dina y di e en ial equa ion. In his sec ion we will assume ha Fcan be w i en as he ollowing sum: F(x( ), x( −ρ( ))) = F0(x( )) + F1(x( −ρ( ))). In his si ua ion, we shall show ha i a dissipa i e hypo hesis o he e m F0holds, hen he assump ions on he o he e m can be elaxed. Le us assume ha F0:Rn→Rnis a con inuous unc ions sa is ying he dissipa i e assump ion as abo e: hF0(x), xi ≤ −α0|x|2+β0, o all x∈Rn.(7) Fi s ly, i we suppose ha F1:Rn→Rnis a con inuous and bounded unc ion, i.e. he e exis s k≥0 such ha |F1(x)| ≤ k, o all x∈Rn, 8TOM´ AS CARABALLO, JOS´ E A. LANGA AND JAMES C. ROBINSON hen i is easy o p o e ha (6) holds. Indeed, o e e y ψ∈Φ(h)Cand a ixed ε<α0, h ( , ψ), ψ(0)i=hF0(ψ(0)), ψ(0)i+hF1(ψ(−ρ( ))), ψ(0)i ≤ −α0|ψ(0)|2+β0+k|ψ(0)| ≤ −(α0−ε)|ψ(0)|2+β0+k2 4ε. Secondly, i is s ill possible o weaken his boundedness on F1,al hough now i is necessa y o assume mo e egula i y o he delay unc ion. Ins ead o p o ing ha (6) holds, we p o e he exis ence o a amily o bounded abso bing se s di ec ly. Theo em 4.3. Assume F0sa is ies (7). Assume ha F1is sublinea , i.e. he e exis s k > 0such ha |F1(x)|2≤k2(1 + |x|2), o all x∈Rn, and suppose ha he delay unc ion ρis con inuously di e en iable wi h ρ0( )≤ρ∗<1.Then, i k2< α2 0(1 −ρ∗), he e exis s a amily o bounded abso bing se s, {B( 0)} 0∈R o (3), and consequen ly he e exis s a pullback a ac o o his p oblem. P oo . Choose a posi i e λ(small enough) and ano he posi i e εwhich will be ixed la e . As in he las heo em, le us w i e x(τ) = x(τ; 0− , ψ), τ∈[ 0− , 0], o ψin a gi en bounded se D⊂ C,i.e. kψkC≤d, o all ψ∈D. Then, i ollows d dτeλτ |x(τ)|2=λeλτ |x(τ)|2+ 2eλτ hx(τ), (τ, xτ)i =λeλτ |x(τ)|2+ 2eλτ hx(τ), F0(x(τ))i +2eλτ hx(τ), F1(x(τ−ρ(τ)))i ≤(λ−2α0)eλτ |x(τ)|2+ 2β0eλτ +εeλτ |x(τ)|2+eλτ ε−1|F1(x(τ−ρ(τ)))|2 ≤(λ−2α0+ε)eλτ |x(τ)|2+ (2β0+k2ε−1)eλτ +k2ε−1eλτ |x(τ−ρ(τ))|2. By in eg a ion on he in e al [ 0− , τ], eλτ |x(τ)|2−eλ( 0− )|x( 0− )|2≤2β0+k2ε−1 λeλτ −eλ( 0− ) +(λ−2α0+ε)Rτ 0− eλs|x(s)|2ds +k2 εRτ 0− eλs|x(s−ρ(s))|2ds. (8) ATTRACTORS FOR DELAY DIFFERENTIAL EQUATIONS 9 E alua ing he e m con aining he delay unc ion by making he change o a iable s−ρ(s) = uin he in eg al, we ob ain Rτ 0− eλs|x(s−ρ(s))|2ds ≤1 1−ρ∗Rτ 0− −heλu+λh|x(u)|2du ≤eλh 1−ρ∗hR 0− 0− −heλu|x(u)|2du +Rτ 0− eλu|x(u)|2dui ≤eλh 1−ρ∗hR 0− 0− −heλu|ψ(u)|2du +Rτ 0− eλu|x(u)|2dui ≤eλh 1−ρ∗Rτ 0− eλu|x(u)|2du +d2eλh λ(1−ρ∗)eλ( 0− )−eλ( 0− −h), and, consequen ly, eλτ |x(τ)|2≤eλ( 0− )d2+2β0+k2ε−1λ−1eλτ −eλ( 0− ) +d2eλhk2ε−1 λ(1−ρ∗)eλ( 0− )−eλ( 0− −h) +hλ−2α0+ε+eλhk2ε−1 (1−ρ∗)iRτ 0− eλs|x(s)|2ds. Now, aking ε=α0and no icing ha o λsmall enough we can assu e ha λ−2α0+ε+eλhk2ε−1 (1−ρ∗)is nega i e, i immedia ely ollows ha |x(τ)|2≤d2h1 + eλh k2ε−1 λ(1−ρ∗)ieλ( 0− −τ)+2β0+k2ε−1λ−1, and se ing τ= 0+θ, o θ∈[−h, 0], |x( 0+θ)|2≤d21 + eλhk2ε−1 λ(1 −ρ∗)e−λ( +θ)+2β0+k2ε−1λ−1, and, hus sup θ∈[−h,0] |x( 0+θ)|2≤d2h1 + eλh k2ε−1 λ(1−ρ∗)ie−λ +λh +2β0+k2ε−1λ−1 ≤1 + 2β0+k2ε−1λ−1 i ≥TD=λ−1log d21 + eλhk2ε−1 λ(1 −ρ∗)eλh. 16 TOM´ AS CARABALLO, JOS´ E A. LANGA AND JAMES C. ROBINSON Gi en an ini ial condi ion xs=ψ, wi h kψkC0≤M, conside d d (x( )−y( )) = F(x( )) −F(y( −ρ( ))). Taking he inne p oduc wi h x( )−y( ) gi es 1 2 d d |x( )−y( )|2=hF(x( )) −F(y( )), x( )−y( )i +hF(y( )) −F(y( −ρ( ))), x( )−y( )i ≤L|x( )−y( )|2+L|y( )−y( −ρ( ))||x( )−y( )| ≤2L|x( )−y( )|2+2M0≤ ≤ kL|ρ( )| ≥ ≤2L|x( )−y( )|2+2M0≤ ≤ kL ≥. On [0, ] we can deduce ha |x( )−y( )|2≤(eL −1)2M L, and so, in pa icula , |x( )−y( )|2≤(eL −1)2M L o all ∈[0, ]. Now, s a ing om =we ha e |x( )−y( )|2≤1 L[2M(eL −1) + kL]eL( −), and, he e o e, |x( )−y( )|2≤C(, ), whe e C(, )→0 as ↓0+uni o mly on bounded ime in e als. In pa icula , i ollows ha sup s∈R kφ( +s, s)ψ− S( )ψkC0→0 o all ≥0.(20) 3. This ollows immedia ely om (15), since he p oo o he exis ence o an abso bing se in Rnunde his condi ion is simple. The e is hen a global a ac o A ⊂Rn, and we can de ine Aas in (19). 4. Finally, no e ha he adius o he abso bing se Bin heo em 4.2 depends on αand β, and using he calcula ions om sec ion 4.1.1 i ollows ATTRACTORS FOR DELAY DIFFERENTIAL EQUATIONS 17 ha αand βcan be aken uni o m o e ∈(0, 0]. I hen ollows ha he compac abso bing se in heo em 4.2 is gi en by φ( 0, 0−0)B. I ollows om (20) ha o e e y ∈(0, 0] his is a subse o a ixed compac se K. Since A( )⊂B( ) (see heo em 3.1) he inal condi ion is sa is ied, and an applica ion o heo em 6.1 gi es he esul as s a ed. Rema k 6.1. No e ha one could also compa e he a ac o s by con- side ing he subse s o Rn A( ) = {y∈Rn:y=x( ), ∈[−, 0], x ∈A( )}. I hen ollows ha dis (A( ),A)→0 as →0, whe e now he dis ance is measu ed in Rn. 7. CONCLUSIONS By using he cocycle a ac o we ha e ex ended he classical ea men o a ac o s o delay di e en ial equa ions o he gene al nonau onomous case, and in pa icula we ha e eco e ed esul s on pe iodic equa ions. Fu he mo e, by using he uppe semicon inui y esul om Ca ballo and Langa [2], we ha e shown ha he in oduc ion o a small delay has li le e ec on he asymp o ic dynamics. Finally, we no e ha hese ideas should be applicable o a wide class o equa ions. In pa icula , equa ions wi h dis ibu ed delays, such as ˙x( ) = Z0 −h ( , s, x( +s)) ds, and e en equa ions wi h in ini e delays, such as ˙x( ) = Z0 −∞ ( , s, x( +s)) ds. Fo he second o hese he phase space needs o be chosen much mo e ca e ully han abo e, and we ha e no p esen ed esul s o his sys em he e o a oid oo much no a ion. Fo de ails o he s anda d heo y see Kuang [11]. 18 TOM´ AS CARABALLO, JOS´ E A. LANGA AND JAMES C. ROBINSON ACKNOWLEDGMENT This wo k has been pa ially suppo ed by P oyec o DGICYT PB98-1134. JCR is a Royal Socie y Uni e si y Resea ch Fellow, and would like o hank he Socie y o all hei suppo . He would also like o hank EDAN o hei hospi ali y, and Ibe d ola o hei gene osi y. REFERENCES 1. Babin A. V., Vishik M. I., A ac o s o E olu ion Equa ions. Ams e dam, No h Holland, (1992) 2. Ca aballo T., Langa J. A., On he uppe semicon inui y o cocycle a ac o s o nonau onomous and s ochas ic di e en ial equa ions. Submi ed. 3. Chepyzho V., Vishik M., A Hausdo dimension es ima e o ke nel sec ions o non- au onomous e olu ion equa ions, Indiana Uni . Ma h. J. 42, No. 3 (1993), 1057-1076. 4. C auel H., Flandoli F., A ac o s o andom dynamical sys ems, P obabili y Theo y and Rela ed Fields 100 (1994), 365-393. 5. C auel H., Debussche A., Flandoli F., Random a ac o s, J. Dyn. Di . 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