Non-Linear Stochastic Partial Differential Equations with Delays: Existence and Uniqueness of Solutions

Non–Linea S ochas ic Pa ial Di e en ial Equa ions wi h delays: Exis ence and uniqueness o solu ions Tom´as Ca aballo Ga ido Dp o. de An´alisis Ma em´a ico. Facul ad de Ma em´a icas (Uni e sidad de Se illa). Apa ado de co eos 1.160. 41080–Se illa Clasi icaci´on A.M.S.: 60H, 35K. 1. In oduc ion The main aim o his pape is o s udy s ochas ic PDE’s wi h delay e ms. In ac , we p o e exis ence and uniqueness o solu ion (in I ˆo’s sense) o a a he gene al ype o s ochas ic PDEs wi h non linea mono one ope a o s and wi h delays. We deal wi h he ollowing s ochas ic pa abolic equa ion: (1) ½dx( )+[A( , x( )) + B( , x(τ( ))) + ( )] d = [C( , x(ρ( ))) + g( )] dw , > 0 x(0) = x0, whe e A( , .), B( , .), C( , .) a e amilies o ope a o s in Hilbe spaces, non linea e en ually, and sa is ying a mono onici y condi ion; w is a Hilbe alued Wiene p ocess, and τ , ρ a e delay unc ions. When he e a e no delays ( τ( ) = ρ( ) = 0 ), he equa ion (1) has been s udied: in he case B=C= 0, o Anon linea , in Bensoussan [2] and Cu ain [5], and o some ype o non linea ope a o s A, in Bensoussan–Temam [3] and Ma cus [7]; in he case C6= 0 , B = 0, o linea Aand C, in Balak ishann [1], o linea Aand non linea Cin Dawson [6], and o non linea mono one Aand Lipschi z con inuous Cin Pa doux [8]. In he case wi h de ia ing a gumen s, Real [9] s udies a a he gene al case when all o he ope a o s a e linea and he e exis s a e m which is a non con inuous ma ingale. Howe e , we ha e no ound in he li e a u e he case we a e going o analyze he e. We will adap o ou p oblem one o he mos impo an me hod o sol ing non linea PDEs: he mono onici y me hod. Pa doux [8] also used an adap a ion o ha me hod o ano he ype o non linea mono one equa ions: when B= 0 and wi hou delays. 2. S a emen o he p oblem and he main esul s The heo y o s ochas ic in eg als in Hilbe spaces is well de eloped (see [8], o example). We conside he classical pai o eal sepa able Hilbe spaces V , H sa is ying V ,→H (injec ion con inuous and dense). We will 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 ix T > 0 and, le w be a Wiene p ocess de ined on he comple e p obabili y space (Ω,F, P) and aking alues in he sepa able Hilbe space K, wi h inc emen al co a iance ope a o W. Le (F ) ≥0be he σ-algeb a gene a ed by {ws,0≤s≤ }, hen w is a ma ingale ela i e o (F ) ≥0. As an abuse o no a ion, we also use |.| o he no m in he linea con inuous ope a o space L(K, H). We deno e by Ip(0, T;V), o p > 1, he space o V– alued p ocesses (x( )) ∈[0,T ](we will w i e x( ) o sho ) measu able ( om [0, T ]×Ω in V), and sa is ying: i) x( ) is F −measu able a.e. in (in he sequel, we will w i e a.e. .) ii) ERT 0|x |pd < +∞. Fo sho , we shall w i e L2(Ω; C(−h, T;H)) ins ead o L2(Ω,F, dP ;C(−h, T;H)) . Le A( , .) : V→V0be a amily o non linea ope a o s de ined a.e. ., and le p > 1.We make he ollowing hypo heses: (a.1) Coe ci i y: ∃α > 0, λ ∈R: 2hA( , x), xi+λ|x|2≥αkxkp,∀x∈V , a.e. . (a.2) Mono onici y: 2hA( , x)−A( , y), x −yi+λ|x−y|2≥0,∀x, y ∈V , a.e. . (a.3) Boundedness: ∃β > 0 : kA( , x)k∗≤βkxkp−1,∀x∈V , a.e. . (a.4) Hemicon inui y: θ∈R→ hA( , x +θy), zi ∈ Ris con inuous ∀x, y, z ∈V , a.e. . (a.5) Measu abili y: ∈(0, T)→A( , x)∈V0is Lebesgue −measu able ∀x∈V , a.e. . Le B( , .) : H→Hbe a amily o ope a o s de ined a.e. ., and sa is ying: (b.1) B( , 0) = 0 (b.2) Lipschi z condi ion: ∃k1:|B( , x)−B( , y)| ≤ k1|x−y|,∀x, y ∈H , a.e. . (b.3) Measu abili y: ∈(0, T )→B( , x)∈His Lebesgue–measu able, ∀x∈V . And le C( , .) : H→ L(K, H) be ano he amily de ined a.e. . and e i ying: (c.1) C( , 0) = 0 (c.2) Lipschi z condi ion: ∃k2:|C( , x)−C( , y)| ≤ k2|x−y|,∀x, y ∈H , a.e. . (c.3) Measu abili y: ∈(0, T )→C( , x)∈ L(K, H) is Lebesgue–measu able ∀x∈H . We also conside wo measu able unc ions (o delay) ρ, τ : [0, T ]→[0, T ] , such ha (ρ.τ) 0 ≤ρ( ), τ( )≤ , ∀ ∈[0, T ]. Fo , g we suppose ha ( .g) ∈I2(0, T;H), g ∈I2(0, T ;L(K, H)). And inally, we a e gi en an ini ial alue x0∈L2(Ω,F0, P ;H). Now, we s a e he ollowing p oblem: (PC)   To ind a p ocess x∈Ip(0, T ;V)∩L2(Ω; C(0, T;H)) such ha : x( ) + R 0[A(s, x(s)) + B(s, x(τ(s))) + (s)] ds =x0+R 0[C(s, x(ρ(s))) + g(s)] dws, P −a.s., ∀ ∈[0, T ]. The main esul we p o e is he ollowing heo em Theo em 1 Assume he p eceden condi ions. Then, he e exis s a unique solu ion o (P C)in Ip(0, T ;V)∩L2(Ω; C(0, T;H)) . P oo . (See [4]) Uniqueness ollows om I o’s o mula and G onwall’s inequali y. Fo he exis ence, we conside he equa ions (∗)x1( ) + Z 0·A(s, x1(s)) + λ 2x1(s)¸ds +Z 0 (s)ds =x0+Z 0 g(s)dws xn+1( ) + Z 0·A(s, xn+1(s)) + λ 2xn+1(s)¸ds +Z 0 B(s, xn(τ(s))) ds +Z 0 (s)ds(∗∗) =x0+Z 0 λ 2xn(s)ds +Z 0 C(s, xn(ρ(s))) dws+Z 0 g(s)dws,∀n= 1,2,3, ... and we p o e ha he e exis s a sequence o solu ions o (∗)−(∗∗) , {xn}n≥1⊂Ip(0, T;V)∩ L2(Ω; C(0, T;H)) . Las , we p o e ha he sequence {xn}is con e gen in Ip(0, T ;V)∩L2(Ω; C(0, T;H)) , and he limi p ocess is he solu ion o (P C). Rema k 1.– We obse e ha heo em 1 also holds when Vis a sepa able and e lexi e Banach space wi h V ,→H . Rema k 2.– We no e ha heo em 1 holds when ρ , τ ake nega i e alues. Theo em 2 Assume he hypo heses in heo em 1, bu changing (ρ.τ)by he ollowing: ∃h > 0such ha −h≤τ( ), ρ( )≤ , ∀ ∈[0, T ], and le ψbe a p ocess such ha ψ∈Ip(−h, 0; V)∩L2(Ω; C(−h, 0; H)) (whe e hese spaces a e de ined in he ob ious manne , se ing F =F0,∀ ∈[−h, 0] ). Then, he e exis s a unique p ocess x∈Ip(−h, T ;V)∩L2(Ω; C(−h, T ;H)) such ha , (PC)0   x( ) + R 0[A(s, x(s)) + B(s, x(τ(s))) + (s)] ds =ψ(0) + R 0[C(s, x(ρ(s))) + g(s)] dws, P −a.s., ∀ ∈[0, T ], x( ) = ψ( ), ∈(−h, 0] P oo . See Ca aballo [4] Rema k 3.– Some examples a e gi en in Ca aballo [4] in o de o jus i y he esul s. Re e ences [1] A. Balak ishnan, S ochas ic bilinea pa ial di e en ial equa ions, U.S.–I aly Con e ence on Va iable S uc u e Sys ems, O egon (1974). [2] A. Bensoussan, Fil age op imal des sys emes lin´eai es, Dunod. [3] A. Bensoussan and R. Temam, Equa ions aux d´e i ´ees pa ielles s ochas iques non lin´eai es, Is ael J. Ma h.,11 (1972), 95–129. [4] T. Ca aballo, Exis ence and uniqueness o solu ions o non–linea sx ochas ic PDE’s, o appea in Collec anea Ma hema ica. [5] R. Cu ain, S ochas ic di e en ial equa ions in Hilbe spaces, Ph. D. Thesis, B own Uni e si y (1969). [6] D. Dawson, S ochas ic e olu ion equa ion, Ma h. Biosc.,15 (1972) [7] R. Ma cus, Pa abolic I o equa ions, T ans. Am. Ma h. Soc.,198 (1974), 177–190. [8] E. Pa doux, ´ Equa ions aux D´e i ´ees Pa ielles S ochas iques non Lin´eai es Mono ones, Thesis, Uni e si y o Pa is XI (1975). [9] J. Real, S ochas ic Pa ial Di e en ial Equa ions wi h Delays, S ochas ics 8, 2 (1982-83), 81-102. To appea in Collec anea Ma hema ica