, and Juanjuan Gao Random attractors for stochastic retarded reaction-di usion equations with multiplicative white noise on unbounded domains

Abstract: In this paper we investigate the stochastic retarded reaction-di usion equationswithmultiplicative white noise on unbounded domain Rn (n ≥ 2). We rst transform the retarded reaction-di usion equations into the deterministic reaction-di usion equations with random parameter by Ornstein-Uhlenbeck process. Next, we show the original equations generate the random dynamical systems, and prove the existence of random attractors by conjugation relation between two random dynamical systems. In this process, we use the cut-o technique to obtain the pullback asymptotic compactness.


Introduction
In this paper we investigate a class of stochastic partial di erential equations, which are widely used in quantum eld theory, statistical mechanics and nancial mathematics. It is known that there are many dynamical systems, depending on both current and historical states. They are referred to as time-decay dynamical systems and can be described by retarded partial di erential equations. Hence, it is very important to study the properties of retarded partial di erential equations.
In this paper, we consider the asymptotic behavior of the solutions to the following stochastic retarded reaction-di usion equation with multiplicative noise in the whole space R n (n ≥ ): Here λ is a positive constant; h > is the delay time of the system; F and G are given functions satisfying certain conditions which will be given in Section 2; g is a given function de ned on R n ; w is a two-sided real-valued Wiener process on a probability space (Ω, F, P), with ||x|| X = for all γ > . (2) (4) A random variable r(ω) is said to be tempered with respect to (θ t ) t∈R , if for P − a.e. ω ∈ Ω, lim t→+∞ e −γt |r(θ −t ω)| = for all γ > .
At the end of this section, we refer to [3,17] for the existence of random attractor for continuous RDS.
Proposition 2.7. Let {K(ω)} ω∈Ω ∈ D be a random absorbing set for the continuous RDS ϕ in D and ϕ is Dpullback asymptotically compact in X. Then ϕ has a unique D-random attractor {A(ω)} ω∈Ω which is given by In the rest of this paper we will assume that D is the collection of all tempered random subsets of S and we will prove that the stochastic retarded reaction-di usion equation has a D-pullback random attractor.
In the remaining part of this section we show that there is a continuous random dynamical system generated by the following stochastic retarded reaction-di usion equation on R n with the multiplicative noise: with the initial condition Here g is a given function in L (R n ), and F, G are continuous functions satisfying the following conditions: (A1) F : R n × R → R is a continuous function such that for all x ∈ R n and s ∈ R, Here α , α , α are positive constants, p > , β (x), β (x), β (x) are nonnegative functions on R n , such that β (x) ∈ L (R n ), and β (x), β (x) ∈ L (R n ). (A2) G : R n × R → R is a continuous function such that for all x ∈ R n and s , s ∈ R, |G Here α is a positive constant, Example 2.8. For x ∈ R n , s, s , s ∈ R and p > , let It is easy to check that the functions F and G in Example 2.8 satisfy Condition (A1) and (A2).
In what follows we consider the probability space (Ω, F , P) which is de ned in Section 1. Let Then (Ω, F , P, (θ t ) t∈R ) is an ergodic metric dynamical system. Since the probability space (Ω, F , P) is canonical, one has To study the random attractor for problem (8)-(9), we rst transform that system into a deterministic system with random parameter. Let Then, one has that z(θ t ω) is the Ornstein-Uhlenbeck process and solves the following equation (see [15] for details): Moreover, the random variable z(θ t ω) is tempered, and z(θ t ω) is P-a.e. continuous. It follows from Proposition 4.3.3 [2] that there exists a tempered function r(ω) > such that where r(ω) satis es, for α > and for P-a.e. ω ∈ Ω, Then it follows that for α > and for P-a.e. ω ∈ Ω, By [9], z(θ t ω) has the following properties: It follows from (23) and (25) that there exists a positive constant µ > small enough, such that for t > large enough, In this paper we consider the weak solutions of (8)-(9) and (30)- (31).
Suppose that u is a weak solution of (8)- (9). Let v(t) = e −ϵz(θt ω) u(t), v (t) = e −ϵz(θt ω) u (t) and ψ = e ϵz(θt ω) ϕ. Then one has that In the last step of (28) we use (18). Hence, it follows from (28) and the de nition of weak solution that Then function v satisfying (29) is said to be the weak solution of the following equation: with the initial condition Equation (30)- (31) can be seen as a deterministic partial di erential equation with random coe cients.
We use a similar method as in [4] to obtain the existence of weak solution to equation (30)- (31). First, using the Galerkin method as in [28], we can get that, under the condition (A1) and (A2), for any bounded , for P-a.e. ω ∈ Ω. We can take the domain to be a family of balls, and the radius of balls tend to ∞. Then, one can get that v ∈ L (R n ) is the unique weak solution of (30)-(31) on R n . Then, u(t) = e ϵz(θt ω) v(t) is a unique weak solution to (8)- (9). Similar as Theorem 12 [13], we can show that (30)-(31) generates a continuous random dynamical system (Φ(t)) t≥ over (Ω, F, P, (θ t ) t∈R ), with and (8)-(9) generates a continuous random dynamical system (Ψ(t)) t≥ over (Ω, F, P, (θ t ) t∈R ), with Notice that two dynamical systems are conjugate to each other. Therefore, in the following sections, we only consider the existence of random attractor of (Φ(t)) t≥ .

Uniform estimates of solutions
In this section we prove the existence of the random attractor for random dynamical system (Φ(t)) t≥ . We rst give some useful estimates on the mild solutions of equation (30)- (31).
Proof. Taking the inner product of (30) with v, we get that We now estimate each term on the right hand side of (34). For the rst term, by condition (A1) and Young inequality we have that Note that by Young inequality, we have that for all a, b, c > , α, β, γ > , and α + β + γ = , For the second term on the right hand side of (34), by condition (A2) and (36), we obtain that where c is a positive constant depending on α, λ and h. For the third term on the right hand side of (34), by Young inequality, Then it follows from (34) -(38) that, for all t ≥ Hence, it follows from (41) that Then, (40) and (42) imply that and for t ∈ [ , −σ], Due to (44) and (45) Replacing ω by θ −t ω in (46), we get that for all t ≥ By (26), one has that for any It follows from (26) Similarly, we can get that Set Then, there exists a T B (ω) > , such that for all t > T B (ω), To show ρ (ω) is tempered, we need only to prove that for any γ > small enough, the following holds lim t→+∞ e −γt ρ (θ −t ω) = . (53) Using (26) In view of (54) and (55), we nd that ρ (ω) is tempered. This ends the proof.
In Lemma 3.1, we show that {K(ω)} ω∈Ω is a random absorbing set for Φ. To prove Φ has a random attractor, we need to show that Φ is D-pullback asymptotically compact. Therefore, we should obtain some estimates for ∇v.
In another aspect, for s ∈ [t, t + ], It follows from (58) and (59) that Replacing ω by θ ω in the last inequality we obtain that This ends the proof.
Proof. Taking the inner product of (30) with −∆v, we get that We now estimate each term on the right-hand side of (64 For the second term, it follows from condition (A2) and Young inequality that −e −ϵz(θt ω) The third term is bounded by −e −ϵz(θt ω) By (64) -(67), we nd that for all t ≥ , with c = ||β || + ||g|| + ||β || . Let T B (ω) be the positive constant de ned in Lemma 3.1, take t ≥ T B (ω) and s ∈ [t, t + ]. Integrating (68) over [s, t + ], we get that Integrating the above inequality with respect to s over [t, t + ], we get that Replacing ω by θ −t− ω in (70), we obtain that It follows from (56) that for all t ≥ T B (ω) + h, It follows from (71) -(74) that, for all t ≥ T B (ω) + h Then we have that for all t ≥ T B (ω) + h + , Replacing ω by θ −t ω in the last inequality, we have that Using (76), we get that By (52), we can get that It follows from (76), (78)-(81) that This ends the proof.
Proof. Let ρ be a smooth function de ned on R + such that ≤ ρ(s) ≤ for all s ∈ R + , and Then there exists a positive constant c such that |ρ ′ (s)| ≤ c for all s ∈ R + . Taking the inner product of (30) with ρ |x| k v, we get that We now estimate each term in (85). For the third term on the left hand-side of (85), we have that and By (86) and (87), one has that For the rst term on the right-hand side of (85) by condition (A1), we have For the second term on the right-hand side of (85) we have that By (36), the rst term on the right-hand side of (90) is bounded by and the second term on the right-hand side of (90) is bounded by For the third term on the right-hand side of (85) we have that It follows from (85) -(93) that Let T B (ω) be the positive constant de ned in Lemma 3.1. Set T (B, ω) ≥ T B (ω) + h + . Applying Gronwall inequality to (94), we obtain that for t ≥ T , Replacing ω by θ −t ω, we obtain from (97) that for all t ≥ T (B, ω), Now we estimate each term on the right hand side of (98). We rst replace t by T and replace ω by θ −t ω in (46), we have that Therefore, for the rst term on the right hand side of (98), We now estimate the terms in (100) as follows. For the rst term, by (26), we nd that for t ≥ T and v (θ −t ω) ∈ B(θ −t ω), For the second term, by (26) lim Similarly, we can get the following estimate for the third term on the right-hand side of (100): It follows from (100) -(103) that which implies that for any ϵ > , there is a T = T (B, ω, ϵ) > T such that e λh e λ (T −t)+ ϵµ t T |z(θτ−t ω)|dτ ||v T (θ −t (ω), v (θ −t (ω)))|| S ≤ ϵ .
Next, we use Ascoli theorem to show that Φ is D pullback asymptotically compact.