Nonlinear Analysis: Real World Applications 12 (2011) 2811–2821 Contents lists available at ScienceDirect Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa Random attractors for stochastic semi-linear degenerate parabolic equations Meihua Yang a,∗ , P.E Kloeden b a School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, China b Institute of Mathematics, Goethe University, Frankfurt am Main, 60054, Germany article info Article history: Received 27 January 2011 Accepted 11 April 2011 Keywords: Stochastic degenerate parabolic equation Random dynamical system Random attractor abstract The existence of a random attractor is established for a class of stochastic semi-linear degenerate parabolic equations with the leading term of the form div(σ (x)∇ u) and additive spatially distributed temporal noise The nonlinearity is dissipative for large values of the state without restriction on the growth order of the polynomial, while the spatial domain is either bounded or unbounded © 2011 Elsevier Ltd All rights reserved Introduction The theory of nonautonomous attractors for deterministic dissipative systems is now quite well established [1–4] Typically, such systems are generated by a parabolic or hyperbolic evolution equation with a nondegenerate leading term The degenerate case is technically more complicated, and much less has been done, even in the autonomous context [5–7] The situation is similar for stochastic evolution equations These are intrinsically nonautnomous, and they generate a random dynamical system, for which the appropriate attractor concept is a random attractor [8–14] Recently, Röckner and his coauthors [15–17] made significant progress with stochastic porous media equations In this paper, we consider a different class of stochastic semi-linear degenerate parabolic equations with a different kind of degeneracy condition, specifically with a leading term of the form div(σ (x)∇ u) as well as an additive spatially distributed temporal noise The deterministic version of these equations has been investigated by Anh and his coworkers [5,18,19], who also established the existence of nonautonomous attractors when the forcing terms are nonautonomous Their results provide the functional analytic framework for our work here We formulate the equations in the next section, and outline the required background material on functional analytical results in Section and on random dynamical systems in Section Then, we formulate our main result on the existence of a random attractor in Section and, finally, in Section 6, establish the required dissipativity estimates on the solutions The equations We investigate the following stochastic semi-linear degenerate parabolic equation with variable, nonnegative coefficients defined on an arbitrary domain (bounded or unbounded) DN ⊂ RN with N ≥ 2: m − hj dwj , du + [−div(σ (x)∇ u) + λu + f (u)]dt = u(x, 0) = u0 (x), x ∈ DN , u(x, t )|∂ DN = 0, t ≥ 0, ∗ x ∈ DN , t ≥ 0, j =1 Corresponding author Tel.: +86 18971660733; fax: +49 69 798 28846 E-mail addresses: yangmeih@gmail.com (M Yang), kloeden@math.uni-frankfurt.de (P.E Kloeden) 1468-1218/$ – see front matter © 2011 Elsevier Ltd All rights reserved doi:10.1016/j.nonrwa.2011.04.007 (2.1) 2812 M Yang, P.E Kloeden / Nonlinear Analysis: Real World Applications 12 (2011) 2811–2821 where λ > 0, and the nonlinear term f ∈ C (R, R) satisfies the following assumptions: f (0) = 0, f ′ (s) ≥ −l for any s ∈ R, (2.2) with positive constant l (1) The case when DN is bounded: α2 |s|p − β2 ≥ f (s)s ≥ α1 |s|p − β1 for all s ∈ R, (2.3) with positive constants α1 , α2 , β1 and β2 (2) The case when DN is unbounded: f (x, s)s ≥ α1 |s|p − k1 (x), |f (x, s)| ≤ α2 |s|p−1 + k2 (x), (2.4) with positive constants α1 and α2 , where k1 ∈ L1 (DN ) ∩ L∞ (DN ) and k2 ∈ L2 (DN ) ∩ Lq (DN ), with p + q = The degeneracy of problem (2.1) is considered in the sense that the measurable, nonnegative diffusion coefficient σ (x) is allowed to have at most a finite number of essential zeros As in [18], we assume that the function σ : DN → R+ ∪ {0} satisfies the following assumptions: (Hα ) σ ∈ L1loc (DN ) and, for some α ∈ (0, 2), lim infx→z |x − z |−α σ (x) > for every z ∈ DN , when the domain DN is bounded; ∞ (Hα,β ) σ satisfies condition Hα and lim inf|x|→∞ |x|−β σ (x) > for some β > 2, when the domain DN is unbounded The assumptions (Hα ) and (Hα,β ) imply, first, that the set of zeros is finite and, second, that the function σ could be nonsmooth, e.g., cannot be of class C if α ∈ (0, 2) and cannot have bounded derivatives if α ∈ (0, 1) See [5–7,18,19] for more details Function spaces and operators We recall from [5,7,18] some basic concepts and properties about function spaces that we will use later Let N ≥ and α ∈ (0, 2), and define ∈ (2, ∞), if N = 2, α 2∗α = 2N 2N ∈ 2, , N −2+α N −2 (3.1) if N ≥ The exponent 2∗α has the role of the critical exponent in the classical Sobolev embeddings The following lemma is the generalized version of a Poincaré inequality [20, Corollary 2.6]; see also [6,7,18] Lemma 3.1 Let DN be a bounded (respectively, unbounded) domain in RN , N ≥ 2, and assume that condition Hα (respectively, ∞ Hα,β ) is satisfied Then, there exists a constant c > such that ∫ |u|2 dx ≤ c ∫ DN σ (x)|∇ u|2 dx for every u ∈ C0∞ (DN , R) (3.2) DN In the case of a bounded domain, inequality (3.2) holds for α ∈ (0, 2]; however, the case α = can be considered as a ‘‘critical case’’ Moreover, condition Hα is optimal in the sense that for α > there exist functions such that (3.2) is not satisfied (see [7,18,20]) Note that, in the case of an unbounded domain, inequality (3.2) generally does not hold if β ≤ in ∞ Hα,β (see also [7,18,20,21]) 1,2 The natural energy space for problem (2.1) involves the space D0 (DN , σ ) defined as the closure of C0∞ with respect to the norm, ∫ ‖u‖D1,2 (D N ,σ ) σ (x)|∇ u|2 dx := 21 (3.3) DN 1,2 The space D0 (DN , σ ) is a Hilbert space with respect to the scalar product, (u, v)σ := ∫ σ (x)∇ u∇v dx (3.4) DN 1,2 The following two lemmata refer to the continuous and compact inclusions of D0 (DN , σ ) [20, Propositions 3.3–3.5]; see also [7,18] M Yang, P.E Kloeden / Nonlinear Analysis: Real World Applications 12 (2011) 2811–2821 2813 Lemma 3.2 Assume that DN is a bounded domain in RN , N ≥ 2, and that σ satisfies (Hα ) Then, the following embeddings hold: ,2 ∗ (i) D0 (DN , σ ) ↩→ L2α (DN ) continuously, ,2 (ii) D0 (DN , σ ) ↩→ Lp (DN ) compactly if p ∈ [1, 2∗α ) ∞ Lemma 3.3 Assume that DN is an unbounded domain in RN , N ≥ 2, and that σ satisfies (Hα,β ) Then, the following embeddings hold: ,2 (i) D0 (DN , σ ) ↩→ Lp (DN ) continuously for every p ∈ [2∗β , 2∗α ], ,2 (ii) D0 (DN , σ ) ↩→ Lp (DN ) compactly if p ∈ (2∗β , 2∗α ), where 2∗β = 2N N −2+β ( 0, Ts is an unbounded self-adjoint operator in L2 (DN ) whose domain Dom(Ts ) is a dense subset in L2 (DN ) The operator Ts is strictly positive and injective Also, Dom(Ts ) endowed with scalar product (u, v)Dom(Ts ) := (Ts u, Ts v)L2 (DN ) is a Hilbert space We write V2s = Dom(Ts ) and have the following identifications: Dom T− ,2 = D− (DN , σ ), the dual of D0 (DN , σ ), as well as Dom(T ) = L (DN ) and Dom(T ) = ,2 D0 (DN , σ ) Moreover, the injection V 2s1 ⊂ V2s2 for s1 > s2 is compact and dense ,2 Furthermore, we define Dp (T) := u ∈ D0 (DN , σ ) : Tu ∈ Lp (DN ) Random dynamical systems We recall some basic concepts related to random attractors for stochastic dynamical systems; for more details, see [2,8, 10,11,24] Let (X , ‖ · ‖X ) be a separable Hilbert space with Borel σ -algebra B (X ), and let (Ω , F , P) be a probability space Definition 4.1 (Ω , F , P, (θt )t ∈R ) is called a metric dynamical system if θ : R × Ω → Ω is (B (R) × F , F )-measurable, θ0 is the identity on Ω , θs+t = θt ◦ θs for all s, t ∈ R, and θt (P) = P for all t ∈ R Definition 4.2 A random dynamical system (RDS) (θ , φ) consists of a metric dynamical system (Ω , F , P, (θt )t ∈R ) and a cocycle mapping φ : R+ × Ω × X → X , which is (B (R+ ) × F × B (X ), B (X ))-measurable and satisfies the following properties: φ(0, ω, x) = x (initial condition), φ(s, θt ω, φ(t , ω, x)) = φ(s + t , ω, x) (cocycle property), for all s, t ∈ R+ , x ∈ X , and ω ∈ Ω It is called a continuous RDS if φ is continuous with respect to x for each t ≥ and ω ∈ Ω 2814 M Yang, P.E Kloeden / Nonlinear Analysis: Real World Applications 12 (2011) 2811–2821 To study the asymptotic behavior of the RDS determined by Eq (2.1), we first need to recall some concepts and properties An (F , B (R)) measurable function R : Ω → R is said to be tempered with respect to (θt )t ∈R if lim e−γ t R(θ−t ω) = t →∞ for P-almost all ω ∈ Ω and all γ > A set-valued map B : Ω → 2X is called a random closed set if B(ω) is nonempty and closed for each ω ∈ Ω and the mapping ω → d(x, B(ω)) is (F , B (R)) measurable for all x ∈ X A random bounded set B := {B(ω)}ω∈Ω of X is said to be tempered with respect to (θt )t ∈R if R(ω) := diam(B(ω)) := supx,y∈B d(x, y) is a tempered random variable Let D be the collection of all tempered random sets in X We will only consider the class D of tempered random sets in this paper Definition 4.3 A random set A := {A(ω)}ω∈Ω ∈ X is called a D-random attractor (or D-pullback attractor) for an RDS (θ, φ) on a Polish space (X , d) if A is a random compact set, i.e., A(ω) is nonempty and compact for P-almost all ω ∈ Ω and ω → d(x, A(ω)) is measurable for every x ∈ X ; A is a φ -invariant, i.e., φ(t , ω, A(ω)) = A(θt ω), for all t ≥ and P-almost all ω ∈ Ω ; A attracts all tempered random sets D = {D(ω)}ω∈Ω ∈ D in the sense that lim dist (φ(t , θ−t ω, D(θ−t ω)), A(ω)) = 0, t →∞ P-almost all ω ∈ Ω When the RDS is continuous, the existence of a random compact absorbing set is a sufficient condition for the existence of a random attractor [9–11,13,25] Definition 4.4 A random closed set B = {B(ω)}ω∈Ω ∈ D is called a D-random absorbing set for an RDS (θ , φ) if, for any D = {D(ω)}ω∈Ω ∈ D and P-almost all ω ∈ Ω , there exists a TD (ω) > such that φ(t , θ−t ω, D(θ−t ω)) ⊂ B(ω) for all t ≥ TD (ω) (4.1) Theorem 4.5 Let (θ , φ) be a continuous RDS on a Polish space (X , d) Suppose that there exists a closed random absorbing set B = {B(ω)}ω∈Ω ∈ D and that φ is D-pullback asymptotically compact in X Then, φ has a unique D-random attractor A = {A(ω)}ω∈Ω , which is unique in the class of tempered random sets, with component subsets given by A(ω) = φ(t , θ−t ω, B(θ−t ω)), ω ∈ Ω τ ≥0 t ≥τ Existence of a D-pullback attractor In this paper, we will prove that the stochastic degenerate parabolic equation in DN generates a continuous random dynamical system, which has a D-random attractor in L2 (DN ) m − hj dwj , du + [Au + λu + f (u)]dt = u(x, 0) = u0 (x), x ∈ DN , u(x, t )|∂ DN = 0, t ≥ 0, x ∈ DN , t ≥ j =1 (5.1) where Au := −div(σ (x)∇ u) and h1 , , hm ∈ Lp (DN ) ∩ Dom(A) ∩ Dp (A), for some p ≥ (given in (2.2)) and the w1 , , wm are pairwise independent two-sided real-valued Wiener processes on the following probability space We consider the canonical probability space (Ω , F , P), where Ω = ω = (ω1 , ω2 , , ωm ) ∈ C (R, Rm ) : ω(0) = and F is Borel σ -algebra induced by the compact open topology of Ω , while P is the corresponding Wiener measure on (Ω , F ) Then, we identify ω with W (t , ω) = (w1 (t ), w2 (t ), , wm (t )) = ω(t ) for t ∈ R Finally, define the time shift by θt ω(·) = ω(· + t ) − ω(t ), ω ∈ Ω , t ∈ R Then, (Ω , F , P, (θt )t ∈R ) is a metric dynamical system M Yang, P.E Kloeden / Nonlinear Analysis: Real World Applications 12 (2011) 2811–2821 2815 We now associate a continuous random dynamical system with stochastic degenerate parabolic equation over (Ω , F , P, (θt )t ∈R ) To this end, we need to convert the stochastic equation with an additive noise term into a pathwise random partial differential equation Given j = 1, , m, consider the stochastic stationary solution of the one-dimensional Ornstein–Uhlenbeck equation dzj + λzj dt = dwj (t ) (5.2) For this solution, the random variable |zj (ωj )| is tempered, and zj ((θt ω)j ) is P-almost sure continuous Therefore, there exists a tempered function r (ω) > such that m − |zj (ωj )|2 + |zj (ωj )|p ≤ r (ω), (5.3) j=1 where r (ω) satisfies, P-almost surely, λ r (θt ω) ≤ e |t | r (ω), t ∈ R (5.4) Then, it follows from (5.3)–(5.4) that, P-almost surely, m − λ |zj ((θt ω)j )|2 + |zj ((θt ω)j )|p ≤ e |t | r (ω), t ∈ R (5.5) j=1 Putting z (θt ω) := dz + λz dt = ∑m m − j=1 hj zj ((θt ω)j ), by (5.2), we see that hj dwj j =1 The existence of a solution to the stochastic degenerate parabolic equation (5.1) follows from [26] We show that problem (5.1) generates a random dynamical system Let v(t ) := u(t ) − z (θt ω), where u is a solution of (5.1) Then, v satisfies ∂v + Av + λv + f (v + z (θt ω)) = −Az (θt ω) (5.6) ∂t 1,2 We know from [18] that −A is the infinitesimal generator of a C -semigroup e−At on D0 (DN , σ ), and it is easy to check 1,2 1,2 from assumptions (2.2)–(2.4) that the function f (·, ω) : D0 (DN , σ ) → D0 (DN , σ ) is locally Lipschitz continuous Thus, the classical semigroup theory on the local existence and uniqueness of the solutions of evolution differential equations in [27] applies here pathwise Then, by the Galerkin method, one can show that, for all v0 ∈ L2 (DN ), P-almost surely, (5.6) 1,2 has a unique and continuous solution v(·, ω, v0 ) ∈ C ([0, ∞), L2 (DN )) ∩ L2 ((0, T ); D0 (DN , σ )) ∩ Lp ((0, T ); Lp (DN )) with the initial value v(0, ω, v0 ) = v0 in L (DN ) for every T ≥ Let u(t , ω, u0 ) = v(t , ω, u0 − z (ω)) + z (θt ω) Then, the process u is the solution of problem (5.1) We now define a mapping φ : R+ × Ω × L2 (DN ) → L2 (DN ) by φ(t , ω, u0 ) = u(t , ω, u0 ) = v(t , ω, u0 − z (ω)) + z (θt ω) (5.7) + for all (t , ω, u0 ) ∈ R × Ω × L (DN ) Then, φ satisfies the conditions in Definition 4.2 and, hence, (θ , φ) is a continuous random dynamical system associated with stochastic degenerate parabolic equation (5.1) In the next section, we establish uniform estimates for the solutions of (5.1) needed to prove the following theorem on the existence of a random attractor for (θ , φ) Theorem 5.1 Assume that (2.2)–(2.4) hold Then, the continuous random dynamical system (θ , φ) generated by (5.1) has a unique D-random attractor in L2 (DN ) belonging to the class D Proof First, notice that φ has a closed random absorbing set B = {B(ω)}ω∈Ω in D by Lemma 6.1 Thanks to the compact embedding (see Remark 3.4), φ is D-pullback compact in L2 (DN ) by Lemma 6.5 Hence, the existence of a unique D-random attractor for the RDS (θ , φ) belonging to the class D follows from Theorem 4.5 ∞ Remark 5.2 In the case of an unbounded domain DN ⊂ RN , N ≥ 2, we assume that σ satisfies the condition (Hα,β ) Then, the operator A := −div(σ (x)∇) has the same properties as in the case of a bounded domain Therefore, one can apply similar methods as for the bounded case, with some small changes in the conditions on f More precisely, we assume that f : DN × R → R is a Carathéodory function and that it satisfies the following assumptions: for all x ∈ RN and s ∈ R, ∂f (x, s) ≥ −l, ∂s ∂f (x, s) ≤ ϕ1 (x) ∂s f (x, s)s ≥ α1 |s|p − ϕ2 (x), |f (x, s)| ≤ α2 |s|p−1 + ϕ3 (x), with positive constants α1 , α2 and l, where ϕ1 ∈ L2 (RN ), ϕ2 ∈ L1 (RN ) ∩ L∞ (RN ), and ϕ3 ∈ L2 (RN ) ∩ Lq (RN ), with See [9,18] for more details p + 1q = 2816 M Yang, P.E Kloeden / Nonlinear Analysis: Real World Applications 12 (2011) 2811–2821 Uniform estimates of solutions In this section, we establish uniform estimates for the solutions of (5.1) by the uniform estimates for the solutions of (5.6) Henceforth, we always assume that D is the collection of all tempered subsets of L2 (DN ) with respect to (Ω , F , P, (θt )t ∈R ) The next lemma shows that the RDS (θ , φ) has a random absorbing set in D Lemma 6.1 Assume that (2.2)–(2.4) hold Then, there exists a random absorbing set B = {B(ω)}ωΩ ∈ D for the RDS (θ , φ) Proof We first derive uniform estimates on v(t ) = u(t ) − z (θt ω) from which the uniform estimates on u(t ) follow immediately Multiplying (5.6) by v and then integrating over DN , we find that ∫ d dt |v|2 dx + ∫ σ (x)|∇v|2 dx + ∫ f (v + z (θt ω))v dx − ∫ Az (θt ω)v dx (6.1) DN DN DN DN DN ∫ λ|v|2 dx = − For the nonlinear term, by (2.3) and (2.4), we obtain ∫ f (v + z (θt ω))v dx = ∫ f (v + z (θt ω))(v + z (θt ω)) dx − ≥ α1 ∫ |u|p dx − β1 mes DN − ∫ ∫ ∫ p α1 |u|p−1 |z (θt ω)| dx − β1 mes DN |u| dx − DN ≥ DN ∫ f (v + z (θt ω))z (θt ω) dx DN DN ≥ α1 f (v + z (θt ω))z (θt ω) dx DN DN DN ∫ ∫ |u|p dx − c1 |z (θt ω)|p dx − β1 mes DN (6.2) DN DN On the other hand, ∫ DN Az (θt ω)v dx ≤ λ ∫ |v|2 dx + DN ∫ 2λ |Az (θt ω)|2 dx, (6.3) DN and it follows from (6.1)–(6.3) that ∫ d dt |v|2 dx + λ ∫ DN |v|2 dx + ∫ DN ∫ |Az (θt ω)| dx + DN Note that z (θt ω) = p1 (θt ω) = c2 ∫ DN ≤ c2 σ (x)|∇v|2 dx + α1 ∫ |u|p dx DN |z (θt ω)| dx + c3 p (6.4) DN ∑m j =1 hj zj ((θt ω)j ) and hj ∈ Lp (DN ) ∩ Dom(A) ∩ Dp (A) Therefore, let m − (|zj (θt ω)|p + |zj (θt ω)|2 ) (6.5) j =1 Then, it follows from (5.5) that, P-almost surely, p1 (θτ ω) ≤ c4 e λ|τ | r (ω) for all τ ∈ R (6.6) Hence, for all t ≥ 0, d dt ∫ |v|2 dx + λ ∫ DN |v|2 dx + DN ∫ σ (x)|∇v|2 dx + α1 DN ∫ |u|p dx ≤ p1 (θt ω) + c3 (6.7) DN Applying Gronwall’s lemma, we find that, for all t ≥ 0, ‖v(t , ω, v0 (ω))‖2L2 (D N) ≤ e−λt ‖v0 (ω)‖2L2 (D ) + N t ∫ eλ(τ −t ) p1 (θτ ω) dτ + c3 λ (6.8) M Yang, P.E Kloeden / Nonlinear Analysis: Real World Applications 12 (2011) 2811–2821 2817 Then, replacing ω by θ−t ω, for all t ≥ 0, we obtain ‖v(t , θ−t ω, v0 (θ−t ω))‖ L2 (DN ) ≤e −λt ‖v0 (θ−t ω)‖ L2 (DN ) t ∫ eλ(s−t ) p1 (θs−t ω) ds + + ≤ e−λt ‖v0 (θ−t ω)‖2L2 (D ) + ∫ N −t ≤ e−λt ‖v0 (θ−t ω)‖2L2 (D ) + c4 ∫ N ≤ e−λt ‖v0 (θ−t ω)‖2L2 (D ) + eλτ p1 (θτ ω) dτ + −t 2c4 N e λτ r (ω) dτ + λ r (ω) + c3 λ c3 λ c3 λ c3 λ (6.9) Recall that φ(t , ω, u0 (ω)) = v(t , ω, u0 (ω) − z (ω)) + z (θt ω) Hence, the above estimate (6.9) implies, for all t ≥ 0, that ≤ 2‖v(t , θ−t ω, u0 (θ−t ω))‖2L2 (D ) + 2‖z (ω)‖2L2 (D ‖φ(t , θ−t ω, u0 (θ−t ω))‖2L2 (D N) N N) c3 ≤ 2e−λt ‖u0 (θ−t ω) − z (θ−t ω)‖2L2 (D ) + c5 r (ω) + + 2‖z (ω)‖2L2 (D ) N N λ c3 −λt 2 ≤ 4e ‖u0 (θ−t ω)‖L2 (D ) + ‖z (θ−t ω)‖L2 (D ) + c5 r (ω) + N N λ + 2‖z (ω)‖2L2 (D ) (6.10) N By assumption, D = {D(ω)}ω∈Ω ∈ D is tempered On the other hand, by definition, ‖z (ω)‖2L2 (D ) is also tempered Therefore, N if u0 (θ−t ω) ∈ D(θ−t ω), then there exists a TD (ω) > such that, for all t ≥ TD (ω), 4e−λt ‖u0 (θ−t ω)‖2L2 (D ) + ‖z (θ−t ω)‖2L2 (D ) N N ≤ c6 r (ω) + c6 , which, along with (6.10), shows that, for all t ≥ TD (ω), ‖φ(t , θ−t ω, u0 (θ−t ω))‖2L2 (D N) ≤ c6 r (ω) + c6 + ‖z (ω)‖2L2 (D ) (6.11) N Given ω ∈ Ω , define B(ω) = u ∈ L2 (DN ) : ‖u‖2L2 (D ) ≤ c6 r (ω) + c6 + ‖z (ω)‖2L2 (D ) N N (6.12) Then, B = {B(ω)}ω∈Ω ∈ D Further, (6.11) implies that B is a random absorbing set for the RDS (θ , φ) in D ,2 We next derive uniform estimates for v in D0 (DN , σ ) and for u in Lp (DN ) Lemma 6.2 Assume that (2.2)–(2.4) hold Let D = {D(ω)}ω∈Ω ∈ D and u0 (ω) ∈ D(ω) Then, for every T1 ≥ 0, the solution u(t , ω, u0 (ω)) of problem (5.1) and the solution v(t , ω, v0 (ω)) of problem (5.6) with v0 (ω) = u0 (ω) − z (ω) satisfy, P-almost surely, t ∫ T1 t ∫ p eλ(s−t ) ‖u(s, θ−t ω, u0 (θ−t ω))‖Lp (D ) ds ≤ e−λt ‖v0 (θ−t ω)‖2L2 (D ) + c (1 + r (ω)), N N (6.13) eλ(s−t ) ‖v(s, θ−t ω, v0 (θ−t ω))‖2 1,2 (6.14) D0 (DN ,σ ) T1 ds ≤ e−λt ‖v0 (θ−t ω)‖2L2 (D ) + c (1 + r (ω)), N for t ≥ T1 Proof First, replacing t by T1 and then replacing ω by θ−t ω in (6.8), the following inequality holds: ‖v(T1 , θ−t ω, v0 (θ−t ω))‖2L2 (D N) ≤ e−λT1 ‖v0 (θ−t ω)‖2L2 (D ) + N T1 ∫ eλ(s−T1 ) p1 (θs−t ω) ds + c Multiplying by eλ1 (T1 −t ) gives eλ(T1 −t ) ‖v(T1 , θ−t ω, v0 (θ−t ω))‖2L2 (D ) ≤ e−λt ‖v0 (θ−t ω)‖2L2 (D ) + N N T1 ∫ eλ(s−t ) p1 (θs−t ω) ds + ceλ(T1 −t ) (6.15) where, from (6.6), T1 ∫ eλ(s−t ) p1 (θs−t ω) ds = ∫ T −t −t eλτ p1 (θτ ω) dτ ≤ λ1 c4 r (ω)e λ(T1 −t ) (6.16) 2818 M Yang, P.E Kloeden / Nonlinear Analysis: Real World Applications 12 (2011) 2811–2821 It follows that eλ(T1 −t ) ‖v(T1 , θ−t ω, v0 (θ−t ω))‖2L2 (D ) ≤ e−λt ‖v0 (θ−t ω)‖2L2 (D ) + c4 r (ω)e λ(T1 −t ) + ceλ(T1 −t ) N N λ (6.17) Note that, from (6.7), for t ≥ T1 , 2‖v(t , ω, v0 (ω))‖2L2 (D ) + N ∫ t eλ(s−t ) ‖v(s, ω, v0 (ω))‖2 1,2 D0 (DN ,σ ) T1 ≤ 2eλ(T1 −t ) ‖v(T1 , ω, v0 (ω))‖2L2 (D ) + t ∫ N + 2α1 ‖u(s, ω, u0 (ω))‖pLp (DN ) eλ(s−t ) p1 (θs ω) ds + 2c3 t ∫ ds eλ(s−t ) ds (6.18) T1 T1 Replacing ω by θ−t ω, we obtain, for all t ≥ T1 , t ∫ e λ(s−t ) ‖v(s, θ−t ω, v0 (θ−t ω))‖ T1 λ(T1 −t ) ≤ 2e 1,2 D0 (DN ,σ ) ‖v(T1 , θ−t ω, v0 (θ−t ω))‖ ∫ ds + 2α1 t p T1 L2 (DN ) t ∫ +2 e eλ(s−t ) ‖u(s, θ−t ω, u0 (θ−t ω))‖Lp (D ) ds N λ(s−t ) p1 (θs−t ω) ds + 2c3 N ∫ eλ(s−t ) ds T1 T1 ≤ 2eλ(T1 −t ) ‖v(T1 , θ−t ω, v0 (θ−t ω))‖2L2 (D ) + t ∫ eλτ p1 (θτ ω) dτ + T1 −t c λ , (6.19) where ∫ eλτ p1 (θτ ω) dτ ≤ c4 r (ω) T1 −t ∫ e λτ dτ ≤ T −t λ c4 r (ω) (6.20) Then, it follows from (6.17) and (6.19) that t ∫ eλ(s−t ) ‖v(s, θ−t ω, v0 (θ−t ω))‖2 1,2 D0 (DN ,σ ) T1 + 2α1 ‖u(s, θ−t ω, u0 (θ−t ω))‖pLp (DN ) ds ≤ e−λt ‖v0 (θ−t ω)‖2L2 (D ) + c (1 + r (ω)) (6.21) N This completes the proof As a special case of Lemma 6.2, we have the following estimates Lemma 6.3 Assume that (2.2)–(2.4) hold Let D = {D(ω)}ω∈Ω ∈ D and u0 (ω) ∈ D(ω) Then, P-almost surely, there exists a TD (ω) > such that the solution u(t , ω, u0 (ω)) of problem (5.1) and the solution v(t , ω, v0 (ω)) of problem (5.6) with v0 (ω) = u0 (ω) − z (ω) satisfy t +1 ∫ t ‖u(s, θ−t −1 ω, u0 (θ−t −1 ω))‖pLp (DN ) ds ≤ c (1 + r (ω)), (6.22) ‖v(s, θ−t −1 ω, v0 (θ−t −1 ω))‖2 1,2 (6.23) t +1 ∫ D0 (DN ,σ ) t ds ≤ c (1 + r (ω)), for all t ≥ TD (ω) Proof First, replacing t by t + and T1 by t in (6.14), we obtain that t +1 ∫ eλ1 (s−t −1) ‖v(s, θ−t −1 ω, v0 (θ−t −1 ω))‖2 1,2 D0 (DN ,σ ) t ds ≤ e−λ1 (t +1) ‖v0 (θ−t −1 ω)‖2L2 (D ) + c (1 + r (ω)) N (6.24) Then, noting that eλ1 (s−t −1) ≥ e−λ1 , for s ∈ [t , t + 1], e−λ1 t +1 ∫ ‖v(s, θ−t −1 ω, v0 (θ−t −1 ω))‖2 1,2 D0 (DN ,σ ) t ds ≤ e−λ1 (t +1) ‖v0 (θ−t −1 ω)‖2L2 (D ) + c (1 + r (ω)) N −λ1 (t +1) ≤ 2e ‖u0 (θ−t −1 ω)‖ L2 (DN ) + ‖z (θ−t −1 ω)‖ L2 (DN ) + c (1 + r (ω)) (6.25) Since ‖u0 (ω)‖2L2 (D ) and ‖z (ω)‖2L2 (D ) are tempered, there is a TD (ω) > such that N N 2e−λ1 (t +1) (‖u0 (θ−t −1 ω)‖2L2 (D ) + ‖z (θ−t −1 ω)‖2L2 (D ) ) ≤ c (1 + r (ω)), N N (6.26) M Yang, P.E Kloeden / Nonlinear Analysis: Real World Applications 12 (2011) 2811–2821 2819 for all t ≥ TD (ω), which, with (6.25), implies that t +1 ∫ ‖v(s, θ−t −1 ω, v0 (θ−t −1 ω))‖2 1,2 D0 (DN ,σ ) t ds ≤ 2eλ1 c (1 + r (ω)), (6.27) for all t ≥ TD (ω) Similarly, we find that t +1 ∫ t ‖u(s, θ−t −1 ω, u0 (θ−t −1 ω))‖pLp (DN ) ds ≤ 2eλ1 c (1 + r (ω)), (6.28) for all t ≥ TD (ω) Lemma 6.4 Assume that (2.2)–(2.4) hold Let D = {D(ω)}ω∈Ω ∈ D and u0 (ω) ∈ D(ω) Then, P-almost surely, there exists a TD (ω) > such that the solution u(t , ω, u0 (ω)) of problem (5.1) satisfies t +1 ∫ ‖u(s, θ−t −1 ω, u0 (θ−t −1 ω))‖2 1,2 D0 (DN ,σ ) t ds ≤ c (1 + r (ω)), (6.29) for all t ≥ TD (ω) Proof Let TD (ω) be as given in Lemma 6.3, and take t ≥ TD (ω) and s ∈ (t , t + 1) We obtain that ‖u(s, θ−t −1 ω, u0 (θ−t −1 ω))‖2 1,2 ≤ 2‖v(s, θ−t −1 ω, v0 (θ−t −1 ω))‖2 1,2 D0 (DN ,σ ) D0 (DN ,σ ) + 2‖z (θs−t −1 ω)‖2 1,2 D0 (DN ,σ ) , (6.30) where ‖z (θs−t −1 ω)‖2 1,2 D0 (DN ,σ ) ≤c m − |zj ((θs−t −1 ω)j )|2 ≤ ce λ1 (t +1−s) r (ω)j ≤ ce λ1 r (ω) (6.31) j =1 Integrating (6.30) with respect to s over (t , t + 1), by Lemma 6.3 and inequality (6.31), we obtain t +1 ∫ ‖u(s, θ−t −1 ω, u0 (θ−t −1 ω))‖2 1,2 D0 (DN ,σ ) t ds ≤ c + cr (ω) (6.32) Then, lemma follows from (6.32) 1,2 Finally, we derive uniform estimates for u in D0 (DN , σ ) Lemma 6.5 Assume that (2.2)–(2.4) hold Let D = {D(ω)}ω∈Ω ∈ D and u0 (ω) ∈ D(ω) Then, P-almost surely, there exists a TD (ω) > such that ‖u(t , θ−t ω, u0 (θ−t ω))‖2 1,2 D0 (DN ,σ ) ≤ c (1 + r (ω)), (6.33) for all t ≥ TD (ω), where c is positive deterministic and r (ω) is the tempered function given in (5.3) Proof Taking the inner product of (5.6) with Av = −div(σ (x)∇v) in L2 (DN ), we obtain d dt ∫ ‖v‖2 1,2 D0 (DN ,σ ) |Av|2 dx + λ + ∫ DN σ (x)|∇v|2 dx = − DN ∫ f (u)Av − DN ∫ Az (θt ω)Av dx (6.34) DN We first estimate the nonlinear term in (6.34) By (2.2)–(2.4), we have ∫ f (u)Av dx = ∫ DN f (u)Au dx − DN ∫ ∫ f (u)Az (θt ω) dx DN f (u)σ (x)|∇ u| dx − ′ = DN ∫ f (u)Az (θt ω) dx, (6.35) DN where, from (2.2), ∫ f ′ (u)σ (x)|∇ u|2 dx ≥ −l DN ∫ σ (x)|∇ u|2 dx, (6.36) DN and, from (2.4), ∫ − DN ∫ f (u)Az (θt ω) dx ≤ α3 |u|p−1 |Az | dx + c DN ∫ |u|p dx + c DN ∫ |Az (θt ω)|p dx, DN (6.37) 2820 M Yang, P.E Kloeden / Nonlinear Analysis: Real World Applications 12 (2011) 2811–2821 and ∫ − DN Az (θt ω)Av dx ≤ ∫ |Av|2 dx + DN ∫ |Az (θt ω)|2 dx (6.38) DN Then, we obtain d dt ∫ ‖v‖2 1,2 D0 (DN ,σ ) |Av|2 dx ≤ l + ∫ DN σ (x)|∇ u|2 dx + c ∫ DN |u|p dx DN ∫ |Az (θt ω)|p dx + +c |Az (θt ω)|2 dx ∫ (6.39) DN DN Let p2 (θt ω) = c ∫ |Az (θt ω)|p dx + ∑m j =1 p2 (θt ω) ≤ k1 (6.40) DN DN Since z (θt ω) = |Az (θt ω)|2 dx ∫ hj zj ((θt ω)j ) and hj ∈ Lp (DN ) ∩ Dom(A) ∩ Dp (A), there are positive constants k1 and k2 such that m − |zj ((θt ω)j )|p + |zj (θt ω)|2 + k2 , (6.41) j =1 which shows that λ p2 (θt ω) ≤ k1 e |t | r (ω) + k2 for all t ∈ R (6.42) Hence, d dt ‖v‖ 1,2 D0 (DN ,σ ) ≤ c ‖ u‖ 1,2 D0 (DN ,σ ) + ‖ u‖ p Lp (DN ) + p2 (θt ω) (6.43) Let TD (ω) be the positive constant in Lemma 6.3 Take t ≥ TD (ω) and s ∈ (t , t + 1), and then integrate (6.43) over (s, t + 1) to obtain ‖v(t + 1, ω, v0 (ω))‖2 1,2 D0 (DN ,σ ) ∫ t +1 ≤ ‖v(s, ω, v0 (ω))‖2 1,2 + p2 (θτ ω) dτ D0 (DN ,σ ) s ∫ t +1 +c ‖u(τ , ω, u0 (ω))‖2 1,2 + ‖u(τ , ω, u0 (ω))‖pLp (DN ) dτ D0 (DN ,σ ) s ≤ ‖v(s, ω, v0 (ω))‖2 1,2 t +1 +c p2 (θτ ω) dτ + D0 (DN ,σ ) ∫ t +1 ∫ t ‖u(τ , ω, u0 (ω))‖ t 1,2 D0 (DN ,σ ) + ‖u(τ , ω, u0 (ω))‖ p Lp (DN ) dτ (6.44) dτ (6.45) Now, integrating the above inequality with respect to s over (t , t + 1), we find that ‖v(t + 1, ω, v0 (ω))‖2 1,2 D0 (DN ,σ ) t +1 ∫ ∫ ‖v(s, ω, v0 (ω))‖2 1,2 ≤ D0 (DN ,σ ) t ∫ t +1 +c ‖u(τ , ω, u0 (ω))‖ t t +1 ds + 1,2 D0 (DN ,σ ) p2 (θτ ω) dτ t + ‖u(τ , ω, u0 (ω))‖ p Lp (DN ) Replacing ω by θ−t −1 ω, we obtain ‖v(t + 1, θ−t −1 ω, v0 (θ−t −1 ω))‖2 1,2 D0 (DN ,σ ) ∫ t +1 ≤ ‖v(s, θ−t −1 ω, v0 (θ−t −1 ω))‖2 1,2 D0 (DN ,σ ) t t +1 ∫ p2 (θτ −t −1 ω) dτ ds + t t +1 ∫ (‖u(τ , θ−t −1 ω, u0 (θ−t −1 ω))‖2 1,2 +c t D0 (DN ,σ ) + ‖u(τ , θ−t −1 ω, u0 (θ−t −1 ω))‖pLp (DN ) ) dτ (6.46) M Yang, P.E Kloeden / Nonlinear Analysis: Real World Applications 12 (2011) 2811–2821 2821 Then, by Lemmas 6.3 and 6.4, it follows from (6.42) and (6.46), for all t ≥ TD (ω), that ‖v(t + 1, θ−t −1 ω, v0 (θ−t −1 ω))‖2 1,2 D0 (DN ,σ ) ≤ c (r (ω) + 1) + ∫ p2 (θs ω) ds −1 ≤ c (r (ω) + 1) + ∫ λ k1 e− s r (ω) + k2 ds −1 ≤ c (r (ω) + 1) (6.47) Finally, by (5.3) and (6.47), we have, for all t ≥ TD (ω), ‖u(t + 1, θ−t −1 ω, u0 (θ−t −1 ω))‖2 1,2 D0 (DN ,σ ) ≤ 2‖v(t + 1, θ−t −1 ω, v0 (θ−t −1 ω))‖2 1,2 D0 (DN ,σ ) + 2‖z (ω)‖2 1,2 D0 (DN ,σ ) ≤ c (r (ω) + 1), (6.48) which completes the proof of the lemma Acknowledgment This work was supported by NSFC Grant 10901063 and The 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