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

Non–Linear Stochastic Partial Differential Equations with delays: Existence and uniqueness of solutions Tom´as Caraballo Garrido Dpto. de An´alisis Matem´atico. Facultad de Matem´aticas (Universidad de Sevilla). Apartado de correos 1.160. 41080–Sevilla Clasificaci´on A.M.S.: 60H, 35K. 1. Introduction The main aim of this paper is to study stochastic PDE’s with delay terms. In fact, we prove existence and uniqueness of solution (in Itˆo’s sense) for a rather general type of stochastic PDEs with non linear monotone operators and with delays. We deal with the following stochastic parabolic equation: (1) ½dx(t)+[A(t, x(t)) + B(t, x(τ(t))) + f(t)] dt = [C(t, x(ρ(t))) + g(t)] dwt, t > 0 x(0) = x0, where A(t, .), B(t, .), C(t, .) are families of operators in Hilbert spaces, non linear eventually, and satisfying a monotonicity condition; wtis a Hilbert valued Wiener process, and τ , ρ are delay functions. When there are not delays ( τ(t) = ρ(t) = 0 ), the equation (1) has been studied: in the case B=C= 0, for Anon linear, in Bensoussan [2] and Curtain [5], and for some type of non linear operators A, in Bensoussan–Temam [3] and Marcus [7]; in the case C6= 0 , B = 0, for linear Aand C, in Balakrishann [1], for linear Aand non linear Cin Dawson [6], and for non linear monotone Aand Lipschitz continuous Cin Pardoux [8]. In the case with deviating arguments, Real [9] studies a rather general case when all of the operators are linear and there exists a term which is a non continuous martingale. However, we have not found in the literature the case we are going to analyze here. We will adapt to our problem one of the most important method for solving non linear PDEs: the monotonicity method. Pardoux [8] also used an adaptation of that method for another type of non linear monotone equations: when B= 0 and without delays. 2. Statement of the problem and the main results The theory of stochastic integrals in Hilbert spaces is well developed (see [8], for example). We consider the classical pair of real separable Hilbert spaces V , H satisfying V ,→H (injection continuous and dense). We will denote by k.k,|.|and k.k∗the norms in V,Hand V0respectively; by h., .ithe duality product between V0, V , and by (.,.) the scalar product in H. Let us fix T > 0 and, let wtbe a Wiener process defined on the complete probability space (Ω,F, P) and taking values in the separable Hilbert space K, with incremental covariance operator W. Let (Ft)t≥0be the σ-algebra generated by {ws,0≤s≤t}, then wtis a martingale relative to (Ft)t≥0. As an abuse of notation, we also use |.|for the norm in the linear continuous operator space L(K, H). We denote by Ip(0, T;V),for p > 1,the space of V–valued processes (x(t))t∈[0,T ](we will write x(t) for short) measurable (from [0, T ]×Ω in V), and satisfying: i) x(t) is Ft−measurable a.e. in t(in the sequel, we will write a.e.t.) ii) ERT 0|xt|pdt < +∞. For short, we shall write L2(Ω; C(−h, T;H)) instead of L2(Ω,F, dP ;C(−h, T;H)) . Let A(t, .) : V→V0be a family of non linear operators defined a.e.t., and let p > 1.We make the following hypotheses: (a.1) Coercivity: ∃α > 0, λ ∈R: 2hA(t, x), xi+λ|x|2≥αkxkp,∀x∈V , a.e.t. (a.2) Monotonicity: 2hA(t, x)−A(t, y), x −yi+λ|x−y|2≥0,∀x, y ∈V , a.e.t. (a.3) Boundedness: ∃β > 0 : kA(t, x)k∗≤βkxkp−1,∀x∈V , a.e.t. (a.4) Hemicontinuity: θ∈R→ hA(t, x +θy), zi ∈ Ris continuous ∀x, y, z ∈V , a.e.t. (a.5) Measurability: t∈(0, T)→A(t, x)∈V0is Lebesgue −measurable ∀x∈V , a.e.t. Let B(t, .) : H→Hbe a family of operators defined a.e.t., and satisfying: (b.1) B(t, 0) = 0 (b.2) Lipschitz condition: ∃k1:|B(t, x)−B(t, y)| ≤ k1|x−y|,∀x, y ∈H , a.e.t. (b.3) Measurability: t∈(0, T )→B(t, x)∈His Lebesgue–measurable, ∀x∈V . And let C(t, .) : H→ L(K, H) be another family defined a.e.t. and verifying: (c.1) C(t, 0) = 0 (c.2) Lipschitz condition: ∃k2:|C(t, x)−C(t, y)| ≤ k2|x−y|,∀x, y ∈H , a.e.t. (c.3) Measurability: t∈(0, T )→C(t, x)∈ L(K, H) is Lebesgue–measurable ∀x∈H . We also consider two measurable functions (of delay) ρ, τ : [0, T ]→[0, T ] , such that (ρ.τ) 0 ≤ρ(t), τ(t)≤t , ∀t∈[0, T ]. For f , g we suppose that (f.g) f∈I2(0, T;H), g ∈I2(0, T ;L(K, H)). And finally, we are given an initial value x0∈L2(Ω,F0, P ;H). Now, we state the following problem: (PC)   To find a process x∈Ip(0, T ;V)∩L2(Ω; C(0, T;H)) such that : x(t) + Rt 0[A(s, x(s)) + B(s, x(τ(s))) + f(s)] ds =x0+Rt 0[C(s, x(ρ(s))) + g(s)] dws, P −a.s., ∀t∈[0, T ]. The main result we prove is the following theorem Theorem 1 Assume the precedent conditions. Then, there exists a unique solution of (P C)in Ip(0, T ;V)∩L2(Ω; C(0, T;H)) . Proof. (See [4]) Uniqueness follows from Ito’s formula and Gronwall’s inequality. For the existence, we consider the equations (∗)x1(t) + Zt 0·A(s, x1(s)) + λ 2x1(s)¸ds +Zt 0 f(s)ds =x0+Zt 0 g(s)dws xn+1(t) + Zt 0·A(s, xn+1(s)) + λ 2xn+1(s)¸ds +Zt 0 B(s, xn(τ(s))) ds +Zt 0 f(s)ds(∗∗) =x0+Zt 0 λ 2xn(s)ds +Zt 0 C(s, xn(ρ(s))) dws+Zt 0 g(s)dws,∀n= 1,2,3, ... and we prove that there exists a sequence of solutions for (∗)−(∗∗) , {xn}n≥1⊂Ip(0, T;V)∩ L2(Ω; C(0, T;H)) . Last, we prove that the sequence {xn}is convergent in Ip(0, T ;V)∩L2(Ω; C(0, T;H)) , and the limit process is the solution of (P C). Remark 1.– We observe that theorem 1 also holds when Vis a separable and reflexive Banach space with V ,→H . Remark 2.– We note that theorem 1 holds when ρ , τ take negative values. Theorem 2 Assume the hypotheses in theorem 1, but changing (ρ.τ)by the following: ∃h > 0such that −h≤τ(t), ρ(t)≤t , ∀t∈[0, T ], and let ψbe a process such that ψ∈Ip(−h, 0; V)∩L2(Ω; C(−h, 0; H)) (where these spaces are defined in the obvious manner, setting Ft=F0,∀t∈[−h, 0] ). Then, there exists a unique process x∈Ip(−h, T ;V)∩L2(Ω; C(−h, T ;H)) such that, (PC)0   x(t) + Rt 0[A(s, x(s)) + B(s, x(τ(s))) + f(s)] ds =ψ(0) + Rt 0[C(s, x(ρ(s))) + g(s)] dws, P −a.s., ∀t∈[0, T ], x(t) = ψ(t), t ∈(−h, 0] Proof. See Caraballo [4] Remark 3.– Some examples are given in Caraballo [4] in order to justify the results. References [1] A. Balakrishnan, Stochastic bilinear partial differential equations, U.S.–Italy Conference on Variable Structure Systems, Oregon (1974). [2] A. Bensoussan, Filtrage optimal des systemes lin´eaires, Dunod. [3] A. Bensoussan and R. Temam, Equations aux d´eriv´ees partielles stochastiques non lin´eaires, Israel J. Math.,11 (1972), 95–129. [4] T. Caraballo, Existence and uniqueness of solutions for non–linear sxtochastic PDE’s, to appear in Collectanea Mathematica. [5] R. Curtain, Stochastic differential equations in Hilbert spaces, Ph. D. Thesis, Brown University (1969). [6] D. Dawson, Stochastic evolution equation, Math. Biosc.,15 (1972) [7] R. Marcus, Parabolic Ito equations, Trans. Am. Math. Soc.,198 (1974), 177–190. [8] E. Pardoux, ´ Equations aux D´eriv´ees Partielles Stochastiques non Lin´eaires Monotones, Thesis, University of Paris XI (1975). [9] J. Real, Stochastic Partial Differential Equations with Delays, Stochastics 8, 2 (1982-83), 81-102. To appear in Collectanea Mathematica