Nonlinear Partial Functional Differential Equations: Existence and Stability

Nonlinea Pa ial Func ional Di e en ial Equa ions: Exis ence and S abili y Tom´ as Ca aballo Dp o. Ecuaciones Di e enciales y An´alisis Num´e ico. Uni e sidad de Se illa. Apdo. Co eos 1160. 41080-SEVILLA. Spain. e-mail: [email p o ec ed] Dedica ed o Pepi, my wi e. Abs ac Exis ence and uniqueness o solu ions o a class o nonlinea unc ional di e en ial equa- ions in Hilbe spaces a e es ablished. Su icien condi ions which gua an ee he ans e ence o exponen ial s abili y om pa ial di e en ial equa ions o pa ial unc ional di e en ial equa ions a e s udied. The s abili y esul s de i ed a e also applied o o dina y di e en ial equa ions wi h he edi a y cha ac e is ics. Keywo ds: Pa ial di e en ial equa ion; pa ial unc ional di e en ial equa ion; exponen ial s abili y. AMS 2000 Classi ica ions: 35R10. 1 In oduc ion The s udy o unc ional di e en ial equa ions is mo i a ed by he ac ha when one wan s o model some e olu ion phenomena a ising in Physics, Biology, Enginee ing, e c., some he edi a y cha ac e is ics such as a e e ec , ime lag and ime delay can appea in he a iables. Typical examples a ise om he esea ches o ma e ials wi h e mal memo y, biochemical eac ions, popula ion models, e c. (see, o ins ance, Hale and Lunel [8], Ruess [16]-[17], Webb [20], Wu [23] and he e e ences he ein). On he o he hand, one impo an and in e es ing p oblem in he analysis o unc ional di e en ial equa ions is he s abili y, he heo y o which has been g ea ly de eloped o e he las yea s. As is well known, in he case wi hou any he edi a y ea u es, Lyapuno ’s echnique is a ailable o ob ain su icien condi ions o he s abili y o solu ions o (pa ial) di e en ial equa ions. Howe e , in he case o di e en ial equa ions wi h he edi a y p ope ies, o ins ance, e en in he case o cons an ime delays, Lyapuno ’s me hod becomes di icul o apply e ec i ely 1 as K aso skii [12] poin ed ou . The main eason is ha i is much mo e di icul (o e en impossible in some cases) o cons uc p ope Lyapuno unc ions (o unc ionals) o unc ional di e en ial equa ions han o hose wi hou any he edi a y cha ac e is ics. As a consequence, a compa ison echnique has been de eloped by a ious au ho s such as K aso skii [12] and Mao [14] (among o he s). Le us illus a e his poin in mo e de ail. Conside he ollowing one-dimensional delay di e en ial equa ion dx( ) d = ( , x( ), x( −h)), > 0,(1) whe e h > 0, o equi alen ly, dx( ) d = ( , x( ), x( )) + [ ( , x( ), x( −h)) − ( , x( ), x( ))] (2) Clea ly i h > 0 is small enough, he pe u ba ion e m ( , x( ), x( −h)) − ( , x( ), x( )) could be expec ed o be so small ha he pe u bed equa ion (2) would beha e asymp o ically as equa ion dx( ) d = ( , x( ), x( )).(3) In pa icula , i can be p o ed ha , unde some ci cums ances, exponen ial s abili y is ans- e ed om he nondelay equa ion (3) o he delay one (1) i he cons an ime lag h > 0 appea ing in he p oblem is su icien ly small (see, e.g. [14]). So, in o de o ind ou whe he he unc ional equa ion (1) is exponen ially s able, one can check he exponen ial s abili y o he equa ion (2) and hen compu e whe he he ime lag h > 0 is su icien ly small. Ne e heless, i is wo h poin ing ou ha his kind o esul s can be somewha es ic i e o many p ac ical applica ions. In ac , he si ua ion u ns ou o be a he complica ed when one conside s he gene al unc ional di e en ial equa ions, e en he usual delay di e en ial sys- ems. To his espec , we should men ion ha in a wide a ie y o p oblems, he his o y o he phenomenon has a decisi e in luence on he u u e beha iou o he sys em, and, in some cases, no only a sho pe iod o he pas has o be aken in o accoun , bu a la ge one. So i seems a he unna u al o look o esul s which hold only o small alues o he de ia ing a gumen s. In his wo k, we shall ca y ou an in es iga ion in his di ec ion. One o he main aims o his pape is o gi e su icien condi ions (which, in pa icula , may con ain he co esponding esul s in ini e dimension) in o de o ans e he exponen ial s abili y o pa ial di e en ial equa ions o pa ial unc ional di e en ial equa ions. The p oblem we a e e e ing o is de o ed o he conside a ion o an in ini e dimensional e sion o (1) in which has he ollowing o m: ( , x, y) = A( , x) + 1( , y), wi h he amily o (non-linea ) ope a o s A( , ·) sa is ying some kinds o coe ci i y condi ions (see Sec ion 3) as well as 1sa is ying Lipschi z con inuous ones. We would also like o men ion 2 ha , in some sense, a sui able coe ci i y condi ion implies he (exponen ial) s abili y o solu ions in nondelay cases. In addi ion o his, we will be able o assu e exponen ial s abili y o a g ea numbe o ini e dimensional unc ional di e en ial equa ions whe e he esul s in [14] only gua an ee his kind o s abili y o cons an and su icien ly small delays. In Sec ion 2, we begin wi h some p elimina y esul s. Sec ion 3 is de o ed o es ablish some esul s on he exis ence and uniqueness o solu ions o a class o pa ial unc ional di e en ial equa ions in a a ia ional con ex . Some esul s on exponen ial s abili y a e s udied in Sec ion 4. Finally, se e al examples a e gi en in Sec ion 5 o illus a e he heo y de i ed in he p eceding sec ions. 2 P elimina ies Fi s o all, we in oduce he amewo k in which ou analysis is going o be ca ied ou . Le V be a sepa able Banach space and Hbe a eal sepa able Hilbe one such ha V ,→H≡H0,→V0, whe e V0is he dual o Vand he injec ions a e con inuous and dense. We deno e by k · k ,|·| and k·k∗ he no ms in V,Hand V0 espec i ely; by h·,·i he duali y p oduc be ween V0, V , and by (·,·) he scala p oduc in H. Le us ake h≥0, p≥1 and T > 0, le CH=C(−h, 0; H) be he space o all con inuous unc ions om [−h, 0] in o Hwi h sup-no m kψkCH= sup−h≤s≤0|ψ(s)|,ψ∈CH, simila ly, le CV=C(−h, 0; V), Lp V=Lp(−h, 0; V) and Lp H=Lp(−h, 0; H). Gi en a unc ion x(·)∈ Lp(−h, T;V)∩C(−h, T;H), we associa e wi h an Lp V∩CH- alued unc ion x , ≥0, by se ing x (s) = x( +s), s∈[−h, 0]. The i s pu pose o his pape is o es ablish exis ence and uniqueness esul s o a class o nonlinea pa ial unc ional di e en ial equa ions o he o m    dx( ) d =A( , x( ))+ ( , x ), ∈[0, T], x( ) = ψ( ), ∈[−h, 0], (4) whe e, in gene al, he ope a o s a e assumed o be nonlinea . In ac , we a e in e es ed in he case in which A( , ·) : V→V0is a amily o nonlinea mono one and coe ci e ope a o s and ( , ·) : X→His Lipschi z con inuous, whe e Xwill deno e CH, CV, Lp Ho Lp V. On he one hand, i is wo h poin ing ou ha , in many applica ions, Ausually deno es a pa ial di e en ial ope a o (linea o nonlinea ), while uses o be a i s o de pa ial di e en ial one. On he o he , we wan o men ion ha he p oblem o exis ence o solu ions o (4) and i s s abili y ha e been p e iously analyzed by T a is and Webb in [18] o linea au onomous ope a o A gene a ing a s ongly con inuous semig oup, and in [19] when Ais he gene a o o an analy ic semig oup by using he ools o he semig oup heo y o ope a o s. Also Webb [21] conside s a simila p oblem o au onomous and acc e i e ope a o Aby he same heo y o (linea o no ) semig oups o ope a o s. Mo eo e , he e exi s a wide li e a u e on he exis ence o di e en 3 classes o solu ions (s ong, mild, in eg al, e c.) o unc ional di e en ial equa ions e en in he mo e gene al con ex o di e en ial inclusions. I is well wo h eading he wo k by Ruess [16] (see also [17]), whe e we can ind a desc ip ion o some o he di e en echniques used o handle wi h his ques ion, in addi ion o a la ge lis o e e ences conce ning hese me hods. Among o he s, le us men ion, o he uni alued case, he me hod o lines (c . Ka sa os [9]), he Gale kin app oxima ions (c . Ka sa os and Pa o [10]), he Ka o app oximan s (c . Ka sa os and Pa o [11]), e c. Ex ensions o he mul i alued amewo k can be ound in se e al wo ks included in he e e ences in [16]. Howe e , on he one hand, he case in ol ing unbounded (linea o no ) ope a o s (i.e. when is a pa ial di e en ial ope a o o i s o de ), despi e o i s impo ance in applica ions, has only been ea ed in some speci ic and pa icula si ua ions (see, o ins ance, Fi zgibbon [7], Aizico ici [1], C andall e al. [5]) and no always in a sys ema ic way; on he o he hand, wha is missed in he li e a u e is a gene al ea men o his p oblem om a a ia ional poin o iew (see A ola [2]-[3] o some pa icula linea and nonlinea pa ial di e en ial equa ions wi h delays). Consequen ly, in his pape , we shall i s es ablish some exis ence esul s by a a ia ional ype o a gumen simila o ha one ca ied ou by Lions [13] o a case wi hou delays, bu subjec o necessa y changes o make ou scheme go h ough when ( , ·) : Lp H→H. Then we will ea he mo e gene al case wi h ( , ·) : Lp V→Hby using a Gale kin app oxima ion echnique. 3 Exis ence and uniqueness o solu ions Le A( , ·) : V→V0be a amily o (nonlinea ) ope a o s de ined a.e. . ( o almos e e y ) and p≥2. Assume he ollowing hypo heses: Coe ci i y: ∃α > 0, λ, ν ∈Rsuch ha : −2hA( , x), xi+λ|x|2+ν≥αkxkp,∀x∈V , a.e. .; (5) Mono onici y: −2hA( , x)−A( , y), x −yi+λ|x−y|2≥0,∀x, y ∈V , a.e. .; (6) Boundedness: ∃γ > 0 : kA( , x)k∗≤γkxkp−1,∀x∈V , a.e. .; (7) Hemicon inui y: θ∈R→ hA( , x +θy), zi ∈ Ris con inuous ∀x, y, z ∈V , a.e. .; (8) Measu abili y: ∈(0, T)→A( , x)∈V0is Lebesgue-measu able ∀x∈V , a.e. . (9) 4 Le ( , ·) : L2 H→Hbe a amily o nonlinea ope a o s de ined a.e. ., and sa is y he ollowing condi ions: ∃c ≥0 : sup 0≤ ≤T | ( , 0)| ≤ c <+∞; (10) ∃k1=k1(h)>0 : | ( , η)− ( , ξ)| ≤ k1kη−ξkCH,∀η, ξ ∈CH,a.e. .; (11) ∈(0, T)7→ ( , η)∈His Lebesgue-measu able ∀η∈L2 H.(12) Gi en an ini ial alue ψ∈Lp(−h, 0; V)∩C(−h, 0; H), he i s objec i e in his Sec ion is, unde he condi ions desc ibed abo e, o ind a unique unc ion x(·)∈Lp(−h, T;V)∩C(−h, T;H) such ha (x( ) = ψ(0) + R 0[A(s, x(s))+ (s, xs)]d , ∈[0, T], x( ) = ψ( ), ∈[−h, 0],(13) whe e he i s equali y is unde s ood in V0. We will e e o his solu ion as he a ia ional solu ion o (4). Rema k 1 (1) Fi s , we no ice ha , al hough he esul s can be p o ed o p > 1, he in e es ing si ua ions in he applica ions appea when p≥2. Because o his, we con en ou sel es wi h he analysis o he case p≥2. (2) We ha e ixed he ini ial da a in he space Lp(−h, 0; V)∩C(−h, 0; H)jus only o ensu e ha he solu ion belongs o Lp(−h, T ;V)∩C(−h, T;H).Howe e , we can o cou se ake ψ∈Lp Vand a alue x0∈Hins ead o ψ(0) in he p oblem. In his case, he a gumen we will use p o ides a solu ion in he space Lp(−h, T;V)∩C(0, T;H). (3) Al hough i is possible o ex end he esul s de i ed he e o mo e gene al sy ems in ol ing coe icien s such ha ( , x( ), x ), we es ic ou sel es o his mo e simple si ua ion in o de o a oid unnecessa y echnicali ies. Now we shall p o e ha he e exis s a mos one solu ion o (13). This esul will be deduced mainly om (6) and he ene gy equali y. Theo em 2 Assume he p eceding hypo heses hold. Then, he e exis s a mos one solu ion o (13) in Lp(−h, T;V)∩C(−h, T;H). P oo . Suppose ha x, y ∈Lp(−h, T;V)∩C(−h, T;H) a e wo solu ions o (13). Then, aking in o accoun (6), we ob ain |x( )−y( )|2= 2 R 0hA(s, x(s)) −A(s, y(s)), x(s)−y(s)ids +2 R 0( (s, xs)− (s, ys), x(s)−y(s))ds ≤λR 0|x(s)−y(s)|2ds +2 R 0| (s, xs)− (s, ys)||x(s)−y(s)|ds. 5 Now, i ollows om (11) ha o any ∈[0, T] sup 0≤s≤ |x(s)−y(s)|2≤(|λ|+ 1) Z 0 |x(s)−y(s)|2ds (14) +k2 1Z 0 ||x(s)−y(s)||2 CHds (15) On he o he hand, since x(s) = y(s) o s≤0, we easily ge R 0||x(s)−y(s)||2 CHds =R 0sup −h≤ ≤0 |xs( )−ys( )|2ds =R 0sup −h≤ ≤0 |x(s+ )−y(s+ )|2ds ≤R 0sup 0≤ ≤s |x( )−y( )|2ds. (16) Thus, i ollows om (14)–(16) sup 0≤s≤ |x(s)−y(s)|2≤2h|λ|+1+k2 1iZ 0 sup 0≤ ≤s |x( )−y( )|2ds, ∀ ∈[0, T], and G onwall’s lemma ob iously implies uniqueness. Rema k 3 Obse e ha i we assume he ollowing mono onici y hypo hesis Fo all ξ, η ∈Lp(−h, T;V)wi h ξ0=η0i holds −2hA( , ξ( ))+ ( , ξ )−A( , η( )) − ( , η ), ξ( )−η( )i +λ|ξ( )−η( )|2≥0, ∈[0, T],(17) ins ead o (6), uniqueness is also easily deduced. Indeed, no ice ha in his case, (17) implies |x( )−y( )|2≤λZ 0 |x(s)−y(s)|2ds ∀ ∈[0, T], o a bi a y wo solu ions x, y o he p oblem. Now, be o e p o ing ou i s exis ence esul , we shall s a e a heo em on exis ence and uniqueness o solu ions o e olu ion equa ions. Theo em 4 Assume (5)–(9) hold wi h λ= 0.Then, gi en 1∈Lp0(0, T;V0)(wi h 1 p+1 p0= 1) and x0∈H, he e exis s a unique unc ion x∈Lp(0, T;V)∩C(0, T;H)such ha x( ) = x0+Z 0 [A(s, x(s))+ 1(s)]ds, o all ∈[0, T]. In addi ion o his, he ollowing ene gy equali y holds: |x( )|2=|x0|2+ 2 Z 0 hA(s, x(s))+ 1(s), x(s)ids, ∈[0, T]. 6 P oo . See Lions [13] (Theo em 1.2, page 162). Theo em 5 Assume ha (5)–(9), (11) and (12) hold. Then, o each ini ial da um ψ∈ Lp(−h, 0; V)∩C(−h, 0; H), he e exis s a unique solu ion o he p oblem (13) in Lp(−h, T;V)∩ C(−h, T;H). P oo . Uniqueness ollows om Theo em 2. Fo he exis ence, we conside he equa ions (x1( ) = ψ(0) + R 0A(s, x1(s)) −λ 2x1(s)ds, ∈[0, T], x1( ) = ψ( ), ∈[−h, 0],(18)        xn+1( ) = ψ(0) + R 0A(s, xn+1(s)) −λ 2xn+1(s)ds +λ 2R 0xn(s)ds +R 0 (s, xn s)ds, ∈[0, T],∀n≥1, xn+1( ) = ψ( ), ∈[−h, 0],∀n≥1. (19) By i ue o (5)–(9), he amily A1( , .) : V→V0de ined as A1( , x) = A( , x)−(λ/2)x , sa is ies assump ions in Theo em 4. Consequen ly, (18) has a unique solu ion x1∈Lp(−h, T;V)∩ C(−h, T;H).We no e ha , om (11) i ollows ha he mapping ∈(0, T)7→ ( , x1 )∈H belongs o L2(0, T;H).Consequen ly, bea ing hese ema ks in mind, we can use Theo em 4 and ge ha he e exis s a unique unc ion x2∈Lp(−h, T;V)∩C(−h, T;H) , which is he solu ion o (19) o n= 1 . By ecu ence, we ob ain a sequence o solu ions o (18)–(19), {xn}n≥1⊂ Lp(−h, T;V)∩C(−h, T;H).Now, we wan o p o e ha he sequence {xn}con e ges o a unc ion xin Lp(−h, T;V)∩C(−h, T;H) , which will be he solu ion o (13). Fo his end, we shall i s p o e he ollowing lemmas. Lemma 6 {xn}n≥1is a Cauchy sequence in C(−h, T;H). P oo . Indeed, i ollows o n≥2 |xn+1( )−xn( )|2= 2 R 0hA(xn+1)−A(xn), xn+1 −xnids −λR 0|xn+1 −xn|2ds +λR 0(xn+1 −xn, xn−xn−1)ds +2 R 0( (xn s)− (xn−1 s), xn+1 −xn)ds, (20) whe e, o sho we deno e, xn:= xn(s), A(xn):=A(s, xn(s)) and (xn):= (s, xn s).Now, i is easy o deduce om (6) sup 0≤θ≤ |xn+1(θ)−xn(θ)|2≤ |λ|R 0|xn+1 −xn||xn−xn−1|ds +2 R 0| (xn s)− (xn−1 s)||xn+1 −xn|ds. (21) 7 Fi s ly, |λ|R 0|xn+1 −xn||xn−xn−1|ds ≤1 4sup 0≤θ≤ |xn+1(θ)−xn(θ)|2 +λ2TR 0sup 0≤θ≤s |xn(θ)−xn−1(θ)|2ds, (22) and no icing ha xn(s) = xn−1(s) o −h≤s≤0, 2R 0| (xn s)− (xn−1 s)||xn+1 −xn|ds ≤1 4TR 0|xn+1 −xn|2ds + 4k2 1TR 0||xn s−xn−1 s||2 CHds ≤1 4sup 0≤θ≤ |xn+1(θ)−xn(θ)|2+ 4k2 1TR 0sup 0≤θ≤s |xn+1(θ)−xn(θ)|2ds. (23) I we se ϕn( ) = sup0≤θ≤ |xn+1(θ)−xn(θ)|2,i hen ollows om (21)–(23) ha he e exis s a posi i e cons an k > 0 such ha ϕn( )≤kZ 0 ϕn−1(s)ds . (24) By i e a ion om (24), we ge ϕn( )≤kn−2Tn−1 (n−2)! ϕ2(T),∀n≥2,∀ ∈[0, T].(25) The e o e, sup 0≤θ≤T |xn+1(θ)−xn(θ)|2≤kn−2Tn−2 (n−2)! ϕ2(T),∀n≥2.(26) Ob iously, since xn+1(θ) = xn(θ) o θ∈[−h, 0], (26) implies ha {xn}is a Cauchy sequence in C(−h, T;H). Lemma 7 The sequence {xn}is bounded in Lp(−h, T;V). P oo . Indeed, o n≥2 we immedia ely ob ain |xn(T)|2= 2 RT 0hA(xn), xnids −λRT 0|xn|2ds +|ψ(0)|2+ 2 RT 0( (xn−1), xn)ds +λRT 0(xn−1, xn)ds. (27) The e o e, −2RT 0hA(xn), xnids +λRT 0|xn|2ds ≤ |ψ(0)|2+ 2 RT 0| (xn−1)||xn|ds +|λ|RT 0|xn−1||xn|ds. (28) Since {xn}con e ges in C(−h, T;H), i will be bounded in his space. Now, i is no di icul o check ha he e exis s a posi i e cons an k0>0 such ha he igh -hand side o (28) is 8 bounded by his cons an . Fo ins ance, we will es ima e one o hose e ms. Fi s ly, we obse e ha RT 0||xn−1 s||2 CH≤RT 0sup −h≤θ≤s |xn−1(θ)|2ds Nex , (10) and (11) imply 2RT 0| (xn−1)||xn|ds ≤RT 0| (xn−1)|2+|xn|2ds ≤Tk1||xn−1||C(−h,T;H)+c 2+T||xn||2 C(0,T;H),(29) which, in addi ion o (28) and (5), leads o αZT 0 kxn(s)kpds ≤ −2ZT 0 hA(xn), xnids +λZT 0 |xn|2ds +νT ≤k0, and he lemma is p o ed. Now in o de o comple e he p oo o he heo em, we shall p o e ha he limi o he sequence {xn}is a solu ion o (13). Fi s ly, obse e ha Lemma 6 implies ha he e exis s x∈C(−h, T;H) such ha xn→x in C(−h, T;H). Now, hanks o (11), we ha e (xn)→ (x) (in L∞(0, T;H)). On he o he hand, by i ue o Lemma 7, {xn}has a subsequence which con e ges weakly in Lp(−h, T;V). Bu , since xn→xin C(−h, T;H), we can assu e ha xn→xweakly in Lp(−h, T;V) (in he sequel, we will deno e his by xn* x in Lp(−h, T;V)). Ne e heless, i ollows om (7) ha {A(xn)}is bounded in Lp0(0, T ;V0) (wi h p0such ha (1/p) + (1/p0) = 1), since ZT 0 kA( , xn( ))kp/(p−1) ∗d ≤γZT 0 kxn( )kpd ≤γk0/α. The e o e, om each subsequence o {A(xn)}, we can ge ano he subsequence weakly con e gen in Lp0(0, T;V0). Now, as i is easy o see ha all he limi s o di e en subsequences coincide, we inally ge ha A(xn)* in Lp0(0, T;V0). In conclusion, we ha e p o ed: xn→xin C(0, T;H),(30) (xn)→ (x) in L∞(0, T;H),(31) xn* x in Lp(−h, T;V),(32) A(xn)* in Lp0(0, T;V0).(33) Finally, by he i ue o (30)-(33), we can ake limi s in (19) and ob ain x( ) = ψ(0) + Z 0 (s)ds +Z 0 (s, xs)ds. (34) 9 4 STABILITY OF SOLUTIONS In his sec ion we shall show ha unde sui able condi ions exponen ial s abili y can be ans- e ed om equa ions wi hou ime lags o hose wi h ime lag ones. Since we a e mainly in e es ed in exponen ial s abili y p oblems, we will assume he e exis s x∈L2(−h, T;V)∩ C(−h, T;H),∀T > 0,which is he a ia ional solu ion o he ollowing p oblem:    dx( ) d =A( , x( ))+ ( , x ) > 0, x( ) = ψ( ), ∈[−h, 0]. (61) In o he wo ds, x( ) sa is ies he ollowing in eg al equa ion (in V0): (x( ) = ψ(0) + R 0[A(s, x(s))+ (s, xs)]ds, ≥0, x( ) = ψ( ), ∈[−h, 0].(62) In pa icula , we suppose in his sec ion ha all condi ions in Sec ion 3 hold so ha he e exis s a unique solu ion o he unc ional di e en ial equa ion (61). Fi s o all, we in es iga e he case wi hou he edi a y cha ac e is ics. In o he wo ds, conside Eq. (61) wi h h= 0 and hus k1>0 in (11) does no depend on h. Then Eq. (61) educes o      dx( ) d =A( , x( ))+ ( , x( )), ≥0, x(0) = x0. (63) I i is possible o know he exis ence o a Lyapuno unc ion, we could p o e exponen ial s abili y o solu ions. Indeed, assume he e exis ∈C2(H;R+) and posi i e cons an s ci,1≤i≤4 , such ha 0(x)∈V o all x∈Vand c1|x|2≤ (x)≤c2|x|2,L (x)≤ −c3 (x),| 0(x)| ≤ c4|x|, o all x∈V, whe e Lis he associa ed Lyapuno ope a o de ined as L (x) = hA( , x) + ( , x), 0(x)i,∀x∈V. We can ge o he unc ion ec3 (x), and x∈H ec3 (x( )) = (x(0)) + c3R 0ec3s (x(s))ds +R 0ec3shA(s, x(s))+ (s, x(s)), 0(x(s))ids. Obse ing ha L (x)≤ −c3 (x) , we ha e (x( )) ≤e−c3 (x(0)) ,∀ ≥0, and om he assump ions on , we easily deduce ha |x( )|2≤c2 c1 e−c3 |x(0)|2,∀ ≥0. 16 Consequen ly, wha we ha e p o ed is ha e e y solu ion o (61) con e ges exponen ially o ze o, e en i ze o is no a s a iona y solu ion. In pa icula , i ( , 0) = 0 o all ≥0,ou esul implies ha he i ial solu ion o (61) is globally exponen ially s able. Al hough, as we ha e men ioned be o e, he cons uc ion o Lyapuno unc ions is no , in gene al, a i ial p oblem, he e exis s a condi ion which makes (x) = |x|2become a na u al Lyapuno unc ion. We a e e e ing o he ollowing hypo hesis (H): he e exis s a posi i e cons an γ > 0 such ha 2hA( , x) + ( , x), xi ≤ −γ|x|2,∀x∈V (obse e ha on his occasion L (x) = 2hA( , x) + ( , x), xi ≤ −γ|x|2). Rema k 10 In wha ollows, we assume ha ( , 0) = 0, o all ≥0since we a e in e es ed in analysing he s abili y o he i ial solu ion o ou p oblem. To his espec , i is wo h men ioning ha in a a ie y o p ac ical si ua ions, he ollowing assump ion (H)0(which seems easie o check) implies (H): (H)0: The e exis s a posi i e cons an bα > 0such ha −2hA( , x), xi ≥ bα|x|2,∀x∈Vand −bα+ 2k1<0, whe e k1is he nonnega i e cons an in (11). Indeed, no e ha 2hA( , x) + ( , x), xi≤−bα|x|2+ 2( ( , x), x) ≤(−bα+ 2k1)|x|2 and deno ing γ=bα−2k1, assump ion (H) ollows. Now, we shall show ha he same hypo heses as abo e (mainly (11) and (H)0) imply expo- nen ial s abili y o he i ial solu ion o he unc ional di e en ial equa ion (61). Howe e , i is pa icula ly wo h poin ing ou ha on his occasion he cons an s k1is gene ally dependen on he ime lag cons an h > 0. This ac simply means ha , in o de o ob ain exponen ial s abili y, he ime lag mus be su icien ly small. Howe e , as will be shown by he examples in he inal sec ion (see also Theo em ??), on some occasions such as he ime delay case, he cons an k1could be independen on h > 0 so ha he s abili y holds ue o any h > 0. Fo ou ends, le us i s ly s udy some s abili y c i e ia o he unc ional di e en ial equa ion (61) by using a Razumikhin ype a gumen . Theo em 11 Assume ha ope a o s Aand a e con inuous wi h espec o ime , and ha he e exis s a posi i e cons an λ > 0such ha o all ≥0, 2hA( , φ(0)) + ( , φ), φ(0)i<−λ|φ(0)|2(64) 17 p o ided φ={φ(s) : −h≤s≤0} ∈ C(−h, 0; V)sa is ying kφk2 CH≤eλh|φ(0)|2.(65) Then, he e exis s a posi i e cons an K≥1such ha o all ψ∈C(−h, 0; V) o which he solu ion o (61) co esponding o he ini ial alue ψ, deno ed by x( , ψ),belongs o C(−h, T;V) o all T > 0, sa is ies |x( , ψ)|2≤K||ψ||2 CH·e−λ , o all ≥0.(66) P oo . Suppose he asse ion does no hold. Then, o any K≥1 he e exis s an ini ial da a ψsuch ha (66) is no ue. Le us ix a K≥1 and ake he co esponding ψ. Thus, we can a i m ha he e exis s a ρ≥0 such ha eλ |x( ;ψ)|2≤eλρ|x(ρ;ψ)|2=K||ψ||2 CH,(67) o all 0 ≤ ≤ρ, and he e is a sequence { k}k≥1in R+such ha k↓ρ, as k→ ∞, and eλ k|x( k;ψ)|2> eλρ|x(ρ;ψ)|2.(68) On he o he hand, by i ue o (67) we deduce |x(ρ+θ;ψ)|2≤eλ(ρ− )|x(ρ;ψ)|2≤eλh|x(ρ;ψ)|2, o all −h≤θ≤0, which, in iew o he assump ions (64)-(65), immedia ely implies ha 2hA(ρ, x(ρ))+ (ρ, xρ), x(ρ;ψ)i<−λ|x(ρ)|2. By he con inui y o he solu ion and he unc ions Aand , we see ha o some su icien ly small h > 0, 2hA( , x( ))+ ( , x ), x( ;ψ)i<−λ|x( )|2, o all ∈[ρ, ρ +h]. Thus, o all su icien ly small h > 0, eλ(ρ+h)|x(ρ+h;ψ)|2−eλρ|x(ρ;ψ)|2 =Rρ+h ρeλ λ|x( ;ψ)|2+ 2 hA( , x( ;ψ))+ ( , x ), x( ;ψ)id  ≤0. Howe e , his con adic s (68), so he esul (66) mus be ue. Theo em 12 Suppose he assump ions in Sec ion 3 o exis ence and uniqueness o solu ions o (13) hold. In addi ion o (11) and (H)0, assume ha ope a o s Aand a e con inuous wi h espec o o ime . Then, he e exis λ > 0and K≥1such ha o all ψ∈C(−h, 0; V) o which he solu ion o (13) belongs o C(−h, T ;V) o all T > 0, i.e. x(·;ψ)∈C(−h, T;V),i ollows |x( ;ψ)|2≤K· ||ψ||2 CH·e−λ , o all ≥0. 18 P oo . As 0 <bα−2k1, we can ake a su icien ly small posi i e λsuch ha 0 <bα−2k1eλh. Now i ≥0 and φ={φ(s) : −h≤s≤0} ∈ C([−h, 0]; V) sa is ies kφkCH≤eλh|φ(0)|, hen Assump ion (11) and condi ion (H)0imply 2hA( , φ(0)) + ( , φ), φ(0)i ≤ −bα+ 2k1eλh|φ(0)|2.(69) The e o e, in iew o Theo em 11 he p oo is comple e. Rema k 13 In he ini e dimensional case, ha is, when V=H=Rn, Theo em 12 gua an ees exponen ial s abili y o he solu ions o (61) i (H)0is ul illed, whe e (H)0can be ew i en now as (H)00 The e exis s a posi i e cons an bαsuch ha −2xTA( , x)≥bα|x|2,∀x∈Rnand −bα+ 2k1<0, whe e xTdeno es he anspose o x. No ice ha his is he same condi ion deduced by Hale and Lunel [8], so ou assump ion becomes a na u al ex ension o he ini e-dimensional esul s. Fu he mo e, unde his condi ion he esul s in Mao [14] only ensu e exponen ial s abili y i he delay unc ion he e is −hwi h hsu icien ly small. Rema k 14 Obse e ha he s abili y esul jus p o ed equi es some egula i y condi ions on he p oblem and some addi ional con inui y assump ions on he ope a o s. Ne e heless, o some pa icula cases, be e esul s could be ob ained. As an example, we shall p o e below an exponen ial s abili y esul o he case o a pa ial di e en ial equa ion wi h a iable delay. To his espec , le us conside F:H→Hglobally Lipschi z wi h cons an bk1and F(0) = 0, ω:R+→[0, h] con inuously di e en iable wi h ω0( )≤0 o all ≥0,and de ine ( , φ) = F(φ(−ω( )), o φ∈C(−h, 0; H).This si ua ion co esponds o he p oblem (dx( ) d =A( , x( ))+F(x( −ω( )), > 0, x( ) = ψ( ), ∈[−h, 0].(70) Theo em 15 In he p eceding si ua ion, assume ha −2hA( , x), xi ≥ bα|x|2,∀x∈Vwi h −bα+ 2bk1<0. Then, he e exis λ > 0, K ≥1such ha o e e y ψ∈CH, he solu ion o (70), deno ed again x( , ψ),sa is ies |x( ;ψ)|2≤K· ||ψ||2 CH·e−λ , o all ≥0. 19 P oo . Le us deno e ρ( ) = −ω( ), ≥0.Then, ρis a con inuously di e en iable unc ion wi h ρ0( )≥1.This immedia ely implies ha ρ−1( )≤ +ρ−1(0) = +k. Now, we can choose λ > 0 such ha λ−bα+bk1(1 + eλk)<0,and, by se ing x( ) = x( ;ψ),i ollows ha eλ |x( )|2≤ |ψ(0)|2+Z 0 eλs |x(s)|2ds + 2 Z 0 eλs hA(s, x(s)), x(s)ids + 2 Z 0 eλs (F(x(ρ(s)), x(s)) ds ≤ ||ψ||2 CH+ (λ−bα)Z 0 eλs |x(s)|2ds + 2bk1Z 0 eλs |x(ρ(s))| |x(s)|ds ≤ ||ψ||2 CH+ (λ−bα+bk1)Z 0 eλs |x(s)|2ds +bk1Z 0 eλs |x(ρ(s))|2ds, and e alua ing he las e m by using he change o a iables u=ρ(s) in he in eg al, i holds Z 0 eλs |x(ρ(s))|2ds ≤Z −h eλu+λk |x(u)|2du ≤Z0 −h eλu+λk |ψ(u)|2du +Z 0 eλu+λk |x(u)|2du ≤heλk||ψ||2 CH+eλk Z 0 eλu |x(u)|2du. Thus, we ha e eλ |x( )|2≤(1 + bk1heλk)||ψ||2 CH+ (λ−bα+bk1+bk1eλk)Z 0 eλs |x(s)|2ds, and he e o e he p oo is comple e. 5 Examples Now, we a e going o apply he esul s p o ed in he p e ious sec ions o ob ain s abili y o some unc ional di e en ial equa ions. Fi s o all, we conside a gene al si ua ion conce ning he semilinea case p e iously s udied by se e al au ho s (see T a is and Webb [18],[19], Ma in and Smi h [15], Wu [23], among o he s). Example 1. In ou a ia ional se ing, conside a linea ope a o A∈ L(V, V 0) sa is ying he coe ci i y condi ion (5) wi h ν= 0.I we se D(A):={ ∈V:A ∈H}, hen he ope a o A es ic ed o his se D(A) becomes a closed and densely de ined ope a o . Mo eo e , (see 20 Dau ay and Lions [6] o he de ails) Ais he gene a o o a s ongly con inuous semig oup o linea ope a o s in H, deno ed by S( ),which sa is ies: |S( )| ≤ eλ /2,∀ ≥0. Con e sely, i S( ) is a s ongly con inuous semig oup sa is ying, o some ω∈R, |S( )| ≤ eω ,∀ ≥0, hen, i is no di icul o p o e ha i s gene a o Asa is ies hAx, xi ≤ ω|x|2,∀x∈D(A). Now, conside he semilinea p oblem    dx( ) d =Ax( ) + (x ), ≥0, x( ) = ψ( ), ∈[−h, 0]. (71) whe e we assume ha is con inuous, sa is ies he Lipschi z condi ion (11) and (0) = 0. Then, by applying he esul s in [18], we ob ain ha he e exis s a unique mild solu ion o (71) which, in addi ion, is a s ong solu ion (whose exis ence is also ensu ed by ou heo y). On he o he hand, i ω < 0 and k1<−ω, he esul s in [18] gua an ee ha he null solu ion o (71) is exponen ially asymp o ically s able. As we can easily see, he same esul ollows om ou heo y, since on his occasion, (H)0holds wi h bα=−2ω, and he e o e, −bα+ 2k1<0 is equi alen o ω+k1<0. In conclusion, ou heo y gi es a na u al ex ension o he esul s p e iously ob ained in he semilinea case. Example 2. Conside he ollowing semilinea hea equa ion wi h ini e ime lags 1, 2 ( > 1, 2≥0) and wi h µ > 0, α1≥0 d d X( , x) = µ∂2 ∂x2X( , x) + α1Z0 − 1 X( +u, x)h(u)du +α(X( ))X( − 2, x), ≥0, X( , 0) = X( , π) = 0, ≥0, X(s, x) = φ(s, x), φ(·, x)∈C:= C(− , 0; R), φ(s, ·)∈L2(0, π), s ∈[− , 0], x ∈[0, π],kφkC<∞, whe e α:R→R,h: [− 1,0] →Ra e wo bounded, Lipschi z con inuous unc ion wi h |α(x)| ≤ K,|h(u)| ≤ M,x∈R,u∈[− 1,0], M, K > 0. De ine V=H1 0[0, π], H=L2[0, π] wi h he co esponding bounda y condi ions abo e. Le A=∂2 ∂x2wi h he domain D(A) = nu∈L2(0, π),∂u ∂x,∂2u ∂x2∈L2(0, π), u(0) = u(π) = 0o, so i is easy o deduce 2hAu, ui ≤ −2µkuk2, u ∈V. 21 On he o he hand, i is clea ha α1Z0 − 1 X( +u, ·)h(u)du 2 ≤2(α1 1M)2+K2kX k2 C. By a s aigh o wa d compu a ion and applying Theo ems 11, 12 o he abo e equa ion, i µ > 2(α1 1M)2+K2and 2>0 is a bi a y, hen he null solu ion is exponen ially s able. Example 3. Le us now exhibi a inal nonlinea example. Le D= [0,1],2< p < +∞, > 0 and conside he ollowing nonlinea pa ial unc ional di e en ial equa ion          d d u( , x) = ∂ ∂x  ∂u( , x) ∂x  p−2∂u( , x) ∂x !−a(x)u( , x) + ( , u (x)), > 0, x ∈D, x( , x) = ψ( , x), ∈[− , 0], x ∈D, x( , 0) = x( , 1) = 0, > 0, (72) whe e a∈L∞(D), a(x)≥ea > 0, o all x∈D, and ( , φ) = g(φ(−ω( ))),whe e ω:R→[0, ] is a measu able unc ion and g:R→Ris a Lipschi z unc ion wi h cons an L > 0 and sa is ying g(0) = 0.I we se H=L2(D), V =W1,p 0(D) and conside he ope a o Ade ined as hAu, i=−ZD" ∂u(x) ∂x  p−2∂u(x) ∂x ∂ (x) ∂x +a(x)u(x) (x)#dx, ∀u, ∈V. Then i is no di icul o check ha assump ions (5)-(9) hold wi h λ=−2ea, α = 2.Consequen ly, condi ion (H)0holds p o ided −ea+L < 0.So, we ge exponen ial s abili y. 6 Conclusions and inal ema ks We ha e p o ed some esul s on he exis ence and uniqueness o solu ions o a nonlinea pa ial unc ional di e en ial equa ion wi h ini e delay by using a a ia ional app oach. Then, by means o a Razumikhin ype a gumen , we ha e es ablished a s abili y esul in he case o addi ional egula i y o he p oblem. Howe e , no hing has been said in his con ex conce ning he exis ence o solu ions in he case o in ini e delays. Also, a s udy o he mul i alued e sion o ou p oblem could be analyzed and some mo e s abili y esul s should be ob ained. Finally, a compa ison o hese esul s and he ones p e iously p o ed by he e olu ion ope a o app oach seems o be an in e es ing ask which should be done. We plan o in es iga e hese ques ions in some subsequen pape s. Acknowledgmen s. The au ho wishes exp ess his hanks o he anonymous e e ee o help ul sugges ions. He also wan s o hank P o . W. Ruess o he use ul co epondence on he opic o his pape . This wo k has been pa ially suppo ed by DGICYT P ojec (Spain) PB98-1134 and Jun a de Andalucia FQM733. 22 Re e ences [1] S. Aizico ici, On a semilinea Vol e a in eg odi e en ial equa ion, Is ael J. Ma h. 36 (1980), 273-284. [2] M. 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