Existence and Uniqueness of Solutions for Non-Linear Stochastic Partial Differential Equations

Existence and uniqueness of solutions for Non–Linear Stochastic Partial Differential Equations TOM´ AS CARABALLO GARRIDO Departamento de An´alisis Matem´atico, Universidad de Sevilla, Apartado de Correos 1.160. 41080-SEVILLA, Spain. ABSTRACT: We state some results on existence and uniqueness for the solution of non linear stochastic PDEs with deviating arguments. In fact, we consider the equation dx(t)+(A(t, x(t)) + B(t, x(τ(t))) + f(t)) dt = (C(t, x(ρ(t))) + g(t)) dwt, where A(t, .), B(t, .)and C(t, .)are suitable families of non linear operators in Hilbert spaces, wtis a Hilbert valued Wiener process, and τ , ρ are functions of delay. If Asatisfies a coercivity condition and a monotonicity hypothesis, and if B , C are Lipschitz continuous, we prove that there exists a unique solution of an initial value problem for the precedent equation. Some examples of interest for the applications are given to illustrate the results. 1. Introduction The study of stochastic PDE’s has greatly developed over the last years. Stochastic PDE’s are used in modelling physical phenomena [5], population biology [7], filtering [13], etc. The main aim of this paper is to study this type of equation 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(rho(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. 1 When there are not delays ( τ(t) = ρ(t) = t), the equation (1) has been studied: in the case B=C= 0, for Anon linear, in Bensoussan [2] and Curtain [6], and for some type of non linear operators A, in Bensoussan–Temam [3,4] and Marcus [10]; 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 [12]. In the case with deviating arguments, Real [14,15] 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 (see Lions [9]): the monotonicity method. Pardoux [12] also used an adaptation of that method for another type of non linear monotone equations: when B= 0 and without delays. In Section 2 we shall state the problem and the notation we are going to use. Uniqueness of solution will be proved, in Section 3, using Itˆo’s formula. In Section 4, we state the existence results. Some extensions of the results are given in Section 5. Finally, we illustrate our theory with several important examples appearing in the applications. 2. Statement of the problem The theory of stochastic integrals in Hilbert spaces is well developed (see [8], [12], for example). The variational method we are going to use, forces us to work with the classical pair of real separable Hilbert spaces V , H satisfying V ,→H(injection continuous and dense). We identify Hwith its dual space, and denote by V0the dual of V. Then, we have V ,→H≡H0,→V0. 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≥0and we have the following representation of wt: wt= ∞ X i=1 βi tei, where (ei) is an orthonormal set of eigenvectors of W,βi tare mutually independent real Wiener processes with incremental covariance λi>0, W ei=λieiand tr W=P∞ i=1 λi(tr denotes the trace of an operator, see [8], [12], [13]). 2 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 < +∞. It is easy to check that Ip(0, T;V) is a closed subspace of Lp(Ω×[0, T],F ⊗B([0, T]), dP ⊗dt;V), where by B([0, T]) we denote the Borel σ–algebra. For short, we shall write L2(Ω; C(−h, T;H)) instead of L2(Ω,F, dP ;C(−h, T;H)) ,where C(−h, T;H) denotes the space of continuous functions from [−h, T] to 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 ∈Rsuch that: 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: ∃k1such that |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: ∃k2such that |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 3 (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 ]. Remark 2.1. We observe that, if x∈L2(0, T ;H) then, by (b.1)–(b.3), B(x)∈L2(0, T;H) , where B(x)(t) = B(t, x(t)) .Moreover, x∈L2(0, T;H)→B(x)∈L2(0, T ;H) is continuous, and so, measurable. Since x∈H→B(t, x)∈His continuous a.e.t., it follows that, if x(t) is an H–valued stochastic process and Ft–adapted, then B(t, x(t)) also is. In addition, if x∈ L2(Ω ×(0, T); H),then B(x)∈L2(Ω ×(0, T ); H) too. Finally, if xnis a bounded sequence in L2(Ω ×(0, T); H), B(xn) also is bounded. Similar observations are deduced from (c.1)–(c.3) for C:L2(0, T ;H)→L2(0, T;L(K, H)) defined by C(x)(t) = C(t, x(t)) . These remarks and the measurability of ρand τimply that the integrals appearing in (P C) are well defined. 3. Uniqueness of solution In this Section we will prove that there exists at most one solution of (PC). We will obtain this result from (a.2) and Itˆo’s formula (see [8], [13] for that formula). Theorem 3.1 Assume the hypotheses in Section 2. Then, there exists at most one solution of (P C)in Ip(0, T;V)∩L2(Ω; C(0, T ;H)) . Proof. Suppose that x, y ∈Ip(0, T;V)∩L2(Ω; C(0, T ;H)) are solutions of (PC). Then, applying Itˆo’s formula, we obtain E|x(t)−y(t)|2 (3.1) =−2EZt 0 hA(s, x(s)) −A(s, y(s)), x(s)−y(s)ids −2EZt 0 (B(s, x(τ(s))) −B(s, y(τ(s))), x(s)−y(s)) ds +EZt 0 tr £(C(s, x(ρ(s))) −C(s, y(ρ(s)))) W(C(s, x(ρ(s))) −C(s, y(ρ(s))))∗¤ds. 4 Now, by putting z(t) = x(t)−y(t) and using conditions (a.2), (b.2), (c.2), it follows E|z(t)|2≤λE Zt 0 |z(s)|2ds + 2k1EZt 0 |z(τ(s))||z(s)|ds(3.2) +k2 2tr(W)EZt 0 |z(ρ(s))|2ds . We are going to estimate the terms on the right-hand side of (3.2). First, λE Zt 0 |z(s)|2ds ≤ |λ|Zt 0 sup r∈[0,s] E|z(r)|2ds.(3.3) Using (ρ.τ) we get 2EZt 0 |z(τ(s))||z(s)|ds ≤EZt 0 |z(τ(s))|2ds +EZt 0 |z(s)|2ds(3.4) ≤2Zt 0 sup r∈[0,s] E|z(r)|2ds , EZt 0 |z(ρ(s))|2ds ≤Zt 0 sup r∈[0,s] E|z(r)|2ds .(3.5) Consequently, (3.2) −(3.5) yield (3.6) sup r∈[0,t] E|z(r)|2≤£|λ|+ 2k1+ 2k2 2tr (W)¤Zt 0 sup r∈[0,s] E|z(r)|2ds . Finally, Gronwall’s Lemma implies (3.7) sup r∈[0,t] E|z(r)|2= 0 ,∀t∈[0, T]. Obviously, uniqueness follows from (3.7). 4. Existence of solution First, we state a theorem on existence and uniqueness of solution for a stochastic evolution equation, and an energy equality. Next, we will prove the existence of solution of (PC) using this result. Theorem 4.1 Assume the hypotheses in Section 2, with λ= 0 . Then, there exists a unique process x∈Ip(0, T;V)∩L2(Ω; C(0, T ;H)) such that x(t) + Zt 0 [A(s, x(s)) + f(s)] ds =x0+Mt, P −a.s. , ∀t∈[0, T], 5 where Mtis a H–valued continuous, square integrable Ft–martingale. The solution also verifies the following energy equality: |x(t)|2+ 2 Zt 0 hA(s, x(s)), x(s)ids + 2 Zt 0 (f(s), x(s)) ds(4.1) =|x0|2+ 2 Zt 0 (x(s), dMs) + trhhMiit, P −a.s. , ∀t∈[0, T], where hhMiitdenotes the quadratic variation of Mt(see M´etivier and Pellaumail [11]). Proof. See [11], [15] and the references given there. Remark 4.1. We observe that, in our situation (see the proof of theorem 4.2), the martingale Mt will be Rt 0g(s)dwsand hence, the energy equality yields E|x(t)|2+ 2EZt 0 hA(s, x(s)), x(s)ids + 2EZt 0 (f(s), x(s)) ds(4.2) =E|x0|2+EZt 0 tr(g(s)Wg(s)∗)ds , P −a.s. , ∀t∈[0, T]. In Pardoux [12,13] and Ichikawa [8] we can find a rather general Itˆo’s formula. Now, using a Picard’s scheme, we can prove the existence of solution for the problem (PC). Theorem 4.2 Assume the conditions in Section 2. Then, there exists a unique solution of (PC)in Ip(0, T;V)∩L2(Ω; C(0, T ;H)) . Proof. Uniqueness holds from theorem 3.1. For the existence, we consider the equations x1(t) + Zt 0·A(s, x1(s)) + λ 2x1(s)¸ds +Zt 0 f(s)ds(4.3) =x0+Zt 0 g(s)dws xn+1(t) + Zt 0·A(s, xn+1(s)) + λ 2xn+1(s)¸ds(4.4) +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, ... By (a.1)–(a.5), the family A1(t, .) : V→V0defined by A1(t, x) = A(t, x) + (λ/2)x , satisfies the assumptions in theorem 4.1. Consequently, (4.3) has a unique solution x1∈Ip(0, T;V)∩ L2(Ω; C(0, T;H)) . We note that, from (b.2),(c.2) and the measurability of the functions ρ , τ , it follows: 6 i) The map (t, ω)∈(0, T )×Ω→B(t, x1(τ(t)) ∈Hbelongs to I2(0, T;H). ii) The map (t, ω)∈(0, T)×Ω→C(t, x1(ρ(t)) ∈Hbelongs to I2(0, T;L(K, H)) ,and so, R· 0C(s, x1(ρ(s))) dwsis a continuous and square integrable Ft–martingale. Again, by these remarks, we can apply theorem 4.1 and we get that there exists a unique process x2∈Ip(0, T;V)∩L2(Ω; C(0, T;H)) , which is solution of (4.4) for n= 1 . By recurrence, we obtain a sequence of solutions for (4.3) −(4.4) , {xn}n≥1⊂Ip(0, T;V)∩L2(Ω; C(0, T;H)) . In the sequel, we shall prove that the sequence {xn}is convergent to a process in Ip(0, T;V)∩ L2(Ω; C(0, T;H)) , and this process is the solution of (PC). We shall split this proof in four steps. STEP 1.– {xn}is a Cauchy sequence in L2(Ω; C(0, T;H)). Indeed, for n > 1, it follows from Itˆo’s formula for the process xn+1(t)−xn(t), |xn+1(t)−xn(t)|2+ 2 Zt 0 hA(xn+1)−A(xn), xn+1 −xnids(4.5) +λZt 0 |xn+1 −xn|2ds + 2 Zt 0¡B(xn τ)−B(xn−1 τ), xn+1 −xn¢ds =λZt 0¡xn+1 −xn, xn−xn−1¢ds + 2 Zt 0¡xn+1 −xn,(C(xn ρ)−C(xn−1 ρ)) dws¢ +Zt 0 tr h¡C(xn ρ)−C(xn−1 ρ)¢W¡C(xn ρ)−C(xn−1 ρ)¢∗ids , where, by definition: xn:= xn(s), A(xn) := A(s, xn(s)) , B(xn τ) := B(s, xn(τ(s))) and C(xn ρ) := C(s, xn(ρ(s))) . ¿From (a.2), |xn+1(t)−xn(t)|2≤|λ|Zt 0 |xn+1 −xn||xn−xn−1|ds(4.6) + 2 ¯¯¯¯Zt 0¡xn+1 −xn,(C(xn ρ)−C(xn−1 ρ)) dws¢¯¯¯¯ +Zt 0¯¯¯tr h¡C(xn ρ)−C(xn−1 ρ)¢W¡C(xn ρ)−C(xn−1 ρ)¢∗i¯¯¯ds + 2 Zt 0 |B(xn τ)−B(xn−1 τ)||xn+1 −xn|ds . Consequently, (4.6) yields E·sup 0≤θ≤t |xn+1(θ)−xn(θ)|2¸≤ |λ|EZt 0 |xn+1 −xn||xn−xn−1|ds(4.7) + tr(W)EZt 0 |C(xn ρ)−C(xn−1 ρ)|2ds + 2EZt 0 |B(xn τ)−B(xn−1 τ)||xn+1 −xn|ds + 2E·sup 0≤θ≤t¯¯¯¯Zt 0¡xn+1 −xn,(C(xn ρ)−C(xn−1 ρ)) dws¢¯¯¯¯¸. Now, we estimate the terms on the right-hand side of (4.7), and we apply the inequality 2ab ≤a2 l2+l2b2, a, b ∈R, l > 0, 7 for suitable l. |λ|EZt 0 |xn+1 −xn||xn−xn−1|ds(4.8) ≤1 4E·sup 0≤θ≤t |xn+1(θ)−xn(θ)|2¸+λ2TZt 0 E·sup 0≤θ≤s |xn(θ)−xn−1(θ)|2¸ds . tr(W)EZt 0 |C(xn ρ)−C(xn−1 ρ)|2ds(4.9) ≤tr(W)k2 2Zt 0 E·sup 0≤θ≤s |xn(θ)−xn−1(θ)|2¸ds . 2EZt 0 |B(xn τ)−B(xn−1 τ)||xn+1 −xn|ds(4.10) ≤1 4TEZt 0 |xn+1 −xn|2ds + 4k2 1TE Zt 0 |xn(τ(s)) −xn−1(τ(s))|2ds ≤1 4E·sup 0≤θ≤t |xn+1(θ)−xn(θ)|2¸+ 4k2 1TZt 0 E·sup 0≤θ≤s |xn(θ)−xn−1(θ)|2¸ds . Burkholder–Davis–Gundy’s inequality implies 2E·sup 0≤θ≤t¯¯¯¯Zt 0¡xn+1 −xn,(C(xn ρ)−C(xn−1 ρ)) dws¢¯¯¯¯¸ (4.11) ≤6tr (W)E·µsup 0≤θ≤t |xn+1(θ)−xn(θ)|2¶Zt 0 |C(xn ρ)−C(xn−1 ρ)|2ds¸1/2 ≤1 4E·sup 0≤θ≤t |xn+1(θ)−xn(θ)|2¸+ 36k2 2tr (W)Zt 0 E·sup 0≤θ≤s |xn(θ)−xn−1(θ)|2¸ds . If we set (4.12) ϕn(t) = E·sup 0≤θ≤t |xn+1(θ)−xn(θ)|2¸, then, (4.7) −(4.11) yield (4.13) ϕn(t)≤3 4ϕn(t) + (λ2T+k2 2tr(W) + 4k2 1T+ 36k2 2)Zt 0 ϕn−1(s)ds , and so, there exists k > 0 such that (4.14) ϕn(t)≤kZt 0 ϕn−1(s)ds . By iteration from (4.14), we get (4.15) ϕn(t)≤kn−1Tn−1 (n−1)! ϕ1(T),∀n > 1,∀t∈[0, T]. Therefore, (4.16) E·sup 0≤θ≤T |xn+1(θ)−xn(θ)|2¸≤kn−1Tn−1 (n−1)! ϕ1(T),∀n > 1. 8 Obviously, (4.16) implies that {xn}is a Cauchy sequence in L2(Ω; C(0, T ;H)) . STEP 2.– The sequence {xn}is bounded in Ip(0, T;V). Indeed, Itˆo’s formula for |xn|2,with n≥2, yields E|xn(T)|2+ 2EZT 0 hA(xn), xnids +λE ZT 0 |xn|2ds(4.17) =E|x0|2−2EZT 0 (B(xn−1 τ), xn)ds −2EZT 0 (f, xn)ds +λE ZT 0 (xn, xn−1)ds +EZT 0 tr h¡C(xn−1 ρ) + g¢W¡C(xn−1 ρ) + g¢∗ids . Therefore, 2EZT 0 hA(xn), xnids +λE ZT 0 |xn|2ds(4.18) ≤E|x0|2+ 2EZT 0 |B(xn−1 τ)||xn|ds + 2EZT 0 |f||xn|ds +|λ|EZT 0 |xn||xn−1|ds + tr (W)EZT 0 |C(xn−1 ρ) + g|2ds . Since {xn}is convergent in L2(Ω; C(0, T;H),it will be bounded in this space. Now, it is not difficult to check that there exists a positive constant, k0,such that the right-hand side of (4.18) is bounded by this constant. As an example, we will estimate one of those terms: 2EZT 0 |B(xn−1 τ)||xn|ds ≤2k1EZT 0 |xn−1(τ(s))||xn(s)|ds ≤k1EZT 0£|xn−1(τ(s))|2+|xn(s)|2¤ds ≤k1EZT 0·sup 0≤θ≤T |xn−1(θ)|2+ sup 0≤θ≤T |xn(θ)|2¸ds ≤Tk1·Eµsup 0≤θ≤T |xn−1(θ)|2¶+Eµsup 0≤θ≤T |xn(θ)|2¶¸ ≤Tk1³kxnk2 L2(Ω;C(0,T ;H)) +kxn−1k2 L2(Ω;C(0,T ;H))´. This fact, (4.18) and (a.1) lead to the following inequalities: (4.19) αZT 0 Ekxn(s)kpds ≤2EZT 0 hA(xn), xnids +λE ZT 0 |xn|2ds ≤k0, and Step 2 is proved. STEP 3.– We can take limits in (4.4). Indeed, from Step 1, xn→x , for some xin L2(Ω; C(0, T ;H)) .Since (b.2) and (c.2) hold, we also have B(xn τ)→B(xτ) (in L2(Ω; L∞(0, T;H)) ), and C(xn ρ)→C(xρ) (in L2(Ω; L∞(0, T;L(K, H))) ). On the other hand, by Step 2, {xn}has a subsequence which is weakly convergent in Ip(0, T;V) . But, since xn→xin L2(Ω; C(0, T;H)) ,we can assure that xn→xweakly in 9 such that du(t, x)−∆u(t, x)dt +u3(t, x)dt +ϕ1(u(τ(t), x)) dt =ϕ2(u(ρ(t), x)) dwt,in (0, T)× O , u(t, x) = ψ(t, x) in (−h, 0) × O , u(t, x) = 0 in (−h, T)×∂O. Example 6.3.– A stochastic non–linear monotone parabolic equation. Let p > 2 . Now, we consider V=W1,p 0(O),and we define A:V→V0by hA(u), vi= N X i=1 ZO¯¯¯¯ ∂u ∂xi¯¯¯¯ p−2∂u ∂xi ∂v ∂xi dx +ZO |u|p−2uv dx , ∀u, v ∈V , and let B , C , ρ , τ , wtas in example 6.2. It is easy to check that (a.1)–(a.5) hold, with α= 1/2, λ = 0 and β=N+ 1 . Consequently, given ψ∈Ip(−h, 0; V)∩L2(Ω; C(−h, 0; H)) and f , g ∈I2(0, T;H) , there exists a unique process uin Ip(−h, T;V)∩L2(Ω; C(−h, T;H)) solution of (PC)0. In other words, du(t, x) = ÃN X i=1 ∂ ∂xiﯯ¯ ∂u(t, x) ∂xi¯¯¯¯ p−2∂u(t, x) ∂xi!dt +|u(t, x)|p−2u(t, x)!dt + (ϕ1(u(τ(t), x)) + f(t, x)) dt + (ϕ2(u(ρ(t), x)) + g(t, x)) dwt,in (0, T)× O , u(t, x) = ψ(t, x),in (−h, 0) × O , u(t, x) = 0 ,in (−h, T)×∂O. References [1] A. 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