Random attractor of stochastic partly dissipative systems perturbed by Lévy noise

The current paper is devoted to random dynamics of stochastic partly dissipative systems perturbed by Lévy noise. By the technique of dissipative in probability and multivalued random dynamical systems (MRDS), the existences of random attractor for MRDS generated by the stochastically perturbed partly dissipative systems are provided, both the weaker restrictions and stronger restrictions on the coefficients of Lévy noise respectively.

Mathematics Subject Classification: 35R60; 60H15; 35K57.

Global attractors play an important role in the study of asymptotic behavior of various nonlinear systems. There is a great amount of works toward the global attractors for dissipative autonomous as well as nonautonomous and random equations, see [1-7], etc. Dynamical systems driven by non-Gaussian processes, such as Lévy processes, have attracted a lot of attention recently. Stochastic differential equations driven by Lévy processes have been summarized in [8]. Peszat and Zabczyk [9] have presented a basic framework for partial differential equations driven by Lévy processes, which extended several results known for stochastic partial differential equations (SPDEs) driven by Wiener processes. For more works on SPDEs driven by Lévy processes, see [9] and references therein.

Recently, the authors in [6] developed the new frame of the random attractor for infinite dimensional systems, in which, the solution is not necessarily required to be unique, but the corresponding multivalued random dynamical systems (MRDS) is dissipative with probability one. Kapustyan et al. [10] follows the idea of the one in [6], and study the random attractor of reaction diffusion systems perturbed by càdlàg process.

where a > 0, Q ⊂ Rⁿ is a bounded domain with smooth boundary, f, g ∈ C(R), h ∈ L²(Q), f and g satisfy some appropriate assumptions, and η(t, ω) is a stochastic càdlag process with right continuity and left limits.

Global attractors for deterministic FitzHugh-Nagumo systems and partly dissipative deterministic reaction diffusion systems have been investigated in [11] for the bounded domain case and in [7] for the unbounded domain case. Regarding the stochastic FitzHugh-Nagumo system as well as the following general partly dissipative random reaction diffusion system

where D ⊂ Rⁿ is a smooth bounded domain, h : Ω × D × R → R, f : Ω × D × R² → R and g : Ω × D × R → R are measurable and satisfy some appropriate hypotheses. The authors in [5] show the existence of random attractor for the general random partly dissipative reaction diffusion equation (1.2), and provide the Hausdorff dimension of random attractor of stochastic FitzHugh-Nagumo systems, [12] shows the existence of pullback attractor of the non-autonomous FitzHugh-Nagumo systems on unbounded domains. The author in [13] study the stochastic partly FitzHugh-Nagumo systems driven by Gaussian white noise, and show the existence of random attractor by uniform estimates on solution for large space and time variable via a cut-off technique.

Motivated by Kapustyan et al. [10], we consider the following partly dissipative reaction diffusion systems perturbed by Lévy noise.

where d > 0,L(t, ω) is a stochastic Lévy process which trajectories are right-continuous and have left limits. Functions h, f, g, k₁, k₂ and σ are twice continuously differentiable in all variables and satisfy

(H1) c₁|u|^p - c₃ ≤ h(x, u)u ≤ c₂|u|^p + c₃, p > 2.

(H2) , 0 < p₁ < p - 1.

(H3) δ(x) ≥ δ > 0.

(H4) |g'(u)| ≤ c₅, , i = 1, . . ., n, where δ_i > 0, i = 1, . . ., 5.

(H5) , i = 1, . . ., n, where δ₆ > 0.

(H6) there exist some positive constants a₁, a₂, b₁ and b₂ such that

In current paper, we study the long time behavior of stochastic partly dissipative equations (1.3) perturbed by Lévy noise. By the technique of dissipative in probability and MRDS, the existences of random attractor for MRDS generated by the perturbed partly dissipative systems (1.3) are provided, both the weaker restrictions and stronger restrictions on the coefficients of Lévy noise, respectively.

It is necessary to point out that, comparing with the main result for (1.1) in [10], the two difficulties we need to tackle are the measurability and the asymptotic compactness of the solutions operators of stochastic partly dissipative systems (1.3). By the similar arguments in [4], the measurability for stochastic partly dissipative systems perturbed by Lévy noise, so we will use the technique of dissipative in probability and MRDS, to show the existences of random attractor for MRDS generated by the stochastically perturbed partly dissipative systems (1.3).

The rest of the paper is organized as follows. In Section 2, we present some definitions and theorems about MRDS. Section 3 is devoted to the study of the existence of random attractor of partly dissipative random reaction diffusion systems perturbed by Lévy noise in terms of dissipative in probability, the weaker restrictions and stronger restrictions on the coefficients of Lévy noise, respectively.

In this section, we recall the definitions of multivalued random dynamical systems and random attractors, the reader is referred to [6,10] for details.

Definition 2.1. [9]Let E be a Banach space, and let X = (X(t), t ≥ 0) be a E valued stochastic process defined on a probability space . It is called as a Lévy process if

(L1) X(0) = 0, a.s.;

(L2) X has independent and stationary increments; and

(L3) X is stochastically continuous, i.e. for all δ > 0 and for all s ≥ 0

We can choose the sample space Ω = D(R) of Lévy process as

It is easy to check that Ω = D(R) is the Skorokhod metric space with Borel σ-algebra Φ.

P is θ_t-invariant probability measure. The matric ρ can be defined by

where

Λ = {λ(⋅)| : λ(⋅) : [-i, i] → [-i, i], λ(-i) = -i, λ(i) = i, λ(⋅) is continuous and monotonically increasing function}.

Remark 2.1. It follows from [9]that Lévy process η(t, ω) is a stochastic càdlàg process with trajectories without discontinuities of the second kind, that is to say, the sample path is right continuous with left limits.

In this paper, we just use the càdlàg property of the Lévy process, and L(t, ω) = ω(t) = π(θ_tω), where π : Ω → R, and π(ω) = ω(0).

Let C(X) be the family of all nonempty closed subset of X, β(X) be the family of all nonempty and bounded sunset of X. In the sequel, we will introduce the definition of MRDS.

Definition 2.2. [10]A multivalued mapping G : R₊ × Ω × X → C(X) is called as a MRDS if

(1) the mapping (t, x) → G(t, ω)x is measurable for all x ∈ X

(2) G(0, ω)x = x and G(t + s, ω)x ⊂ G(t, θ_sω)G(s, ω)x for all t, s ∈ R₊, x ∈ X, ω ∈ Ω.

Definition 2.3. [10]A measurable set A(ω) is called a random attractor for a MRDS G if for P-almost all ω ∈ Ω,

(1) A(θ_tω) ⊂ G(t, ω)A(ω) for all t ∈ R₊;

(2) for all B ∈ β(X), dist(G(t, θ_-tω))B, A(ω)) → 0, t → +∞;

(3) A(ω) is a compact set in X.

The following hypotheses is developed by Kapustyan et al. [10], which are key tools to show the existence of random attractor for MRDS generated by stochastic equations (1.3).

(G1) the mapping is measurable for all B ∈ β (X)

(G2) for all ϵ > 0, there exists R = R(ϵ) such that for all B ∈ β (X), there exists T = T (B, R, ϵ) for which

As the authors pointed out in the paper [10] that, the condition (G1) is used to show the measurability of the map, and the condition (G2) is used to show the MRDS is dissipative in probability.

For a given , define

Then, consists of the limit of all convergent sequences {ξ_n}, where , and t_n → ∞.

Theorem 2.1. ( [10], Theorem 1)

Let a mapping x → G(t, ω)x be upper semicontinuous and compact-valued for all t ∈ R₊ and ω ∈ Ω, let conditions (G1) and (G2) hold for a MRDS G and let the set G(t, ω)B_R be precompact in X for all ω ∈ Ω, t > 0 and R > 0. Then,

is a random attractor for G. Therefore, the attractor is unique, it is a minimal set among closed attracting sets, and it is a maximal set among compact, measurable, semiinvariant sets.

In this section, we will prove the solution of equation (1.3) can generate a MRDS, and the MRDS posses a random attractor.

Lemma 3.1. Under the assumptions (H1)-(H5), there exists the positive constants m₁, m₂, δ₃, c_a, c_r such that the solution ((u(t), v(t)) of the equations (1.3) satisfy the following estimates.

Proof. Multiplying the first equation of system (1.3) by u and the second equation of system (1.3) by v, Integrating them over D, we have

It follows from (H6) and Young inequality that there exist two constants a > 0 and r > 0 such that

Due to (H4), there exists a constant c₇ > 0 such that

We deduce from (H1) to (H3) that

Notice that

Let q = max{p₁ + 1, 2}, then there exists a constant δ₂ > 0 such that

Thus,

Hence

Let m₁ = max{2a₁ + r, 2a₂ + a}, m₂ = min {δ, 2dλ₁}, where λ₁ > 0 is the first eigenvalue of -Δ in . By the inequality (3.1), we have

Integrating (3.2) on the interval [s, t] (t ≥ s ≥ 0), we obtain that

Using the Gronwall's Lemma,

for all t > 0. The proof of Lemma 3.1 is completed.

Lemma 3.2. Let be an arbitrary sequence of solution of equation (1.1), where ω_n → ω₀ in Ω, weakly in L² × L², and t_n → t₀ > 0. Then,

in L² × L² at least along some subsequence where is the solution of equation (1.3).

Proof. Let T > 0. It follows from (3.1) that the sequence {((u_n, v_n))} is bounded in By the similar argument of Proposition 1.1 that there exists a subsequence , v(⋅)) in L²(0, T, L²(D)) × L²(0, T, L²(D)), (u_n(t, ω_n), strongly in L²(D) × L²(D)) for almost all t ∈ (0, T ) and weakly in L²(D) × L²(D)) uniformly in t ∈ [0, T].

Let

and denote

It follows from (3.3) that

for all t ≥ s, t, s ∈ [0, T ], and J_n(t, ω) → J(t, ω₀) for almost all t ∈ (0, T). It is easy to check that J_n(t, ω_n) → J(t, ω₀) uniformly on an arbitrary interval [a, b] ⊂ (0, T). The convergence ω_n → ω₀ implies that L_n(⋅) → L₀(⋅) in L¹(0, t) and L_n(⋅), n ≥ 1 are bounded in L^∞(0, t). Then

Hence,

for all t_n ≥ t₀ > 0.

Hence

Since (u_n(t_n), v_n(t_n)) → (u(t₀), v(t₀)) weakly in L²(D) × L²(D). The converse inequality also holds and therefore

Moreover, (u_n(t_n), v_n(t_n)) converges strongly to (u(t₀), v(t₀)) in L²(D) × L²(D). The proof of Lemma 3.2 is completed.

Lemma 3.3. Let Ω be a metric space and Φ be a Borel σ -algebra. Assume that a multivalued mapping G : R₊ × Ω × X → C(X) satisfies the following condition: if x_n → x weakly in X as t_n → t₀ > 0, ω_n → ω in Ω and y_n ∈ G(t_n, ω_n)x_n then y_n → y₀ ∈ G(t₀, ω₀)x₀ in X for some subsequence. Then, assumption G₁ holds for the mapping G.

Proof. The proof is similar to the argument of Lemma 4 in [10], and is omitted here.

Lemma 3.4. For (u₀, v₀) ∈ L²(D) × L²(D), there exists an unique solution (u, v) ∈ C(0, ∞; L²(D) × L²(D)) of the equations (1.3) satisfying . The mapping defined by

is a multivalued random dynamical system generated by equation (1.3).

Proof. By the same proof as that of Property 1.1 in [11] and the classical Galerkin approximates, noting that k₁(u)ω(t) and k₂(v)ω(t) are right continuous with left limit with respect to t, we can show that the equation (1.3) admits at least a solution (u(t, ω, u₀, v₀), v(t, ω, u₀, v₀)) ∈ C(0, ∞; L²(D) × L²(D)) for every ω ∈ Ω. Moreover, . Similar to the Property 4 in [6], it can be verified the family of mappings {G(t, ω)_t≥0 generates a MRDS.

Lemma 3.5. Assume that (H1)-(H6) hold, and Levy noise ω(t) satisfies the following conditions

(HY0) If for all ϵ > 0, α > 0, there exists T = T (ϵ) > 0 such that Lévy noise ω(t) satisfies

(HY1) If for all ϵ > 0, there exists a positive constant D > 0 such that Lévy noise ω(t) satisfies

Then, the MRDS G generated by equation (1.3) satisfies the condition G₂.

Proof. It follows from Lemma 3.1 that there exist random variable t(ω) ∈ [T₁, T₂] and initial value x₀(ω) ∈ B_r such that the solution of equation (1.3) reaches the superium at t(ω) for all T₂ > T₁ > 0 and any ω ∈ Ω, that is,

Notes that is Φ-measurable, therefore, (u(t(ω), θ_-t(ω)ω)x₀(ω), and v(t(ω), θ_-t(ω))ω)) are also Φ-measurable.

Fixed ϵ > 0, then for any arbitrary N > 0, T and for t ∈ [T, T + N], denote

Define

Choosing T = T (r) such that

Then,

It follows from (HY0) that there exists a positive constant random variant T₁ = T₁(ω) such that for t ≥ T₁(ϵ),

Thus, there exists a random set A₂ ⊂ Ω such that , and

for all ω ∈ Ω\A₂.

Thus, there exists T₂ = T₂(ϵ, r) ≥ T₁ + T(r) such that

for all t(ω) ∈ [T₂, T₂ + N].

It follows from (3.1) that

The definitions A₁ and A₂ imply that

where A and B are some positive constants, and f_ϵ : ω → R is a measurable and P -almost everywhere bounded function. Hence there exists a real number R₁ = R₁(ϵ) and a random set A₃ ⊂ Ω such that

for all ω ∈ A₃. Therefore,

It follows from (HY1) that there exists^: a positive constant D = D(ϵ) such that

for all t > 0. Choose R = R(ϵ) such that

Then, for all ϵ > 0, there exists R = R(ϵ) > 0 for which, whatever B_r is a ball with radius r, there exists T = T (ϵ, R, r) such that

Notice that L^N ⊂ L^N+1, and let , then P (L) < ϵ, and

which implies that G2 holds. Thus, the proof of Lemma (3.5) is completed.

Theorem 3.1. Assume that conditions (H1)-(H6) hold, and the Lévy noise satisfy the (HY0)- (HY1). Then, the MRDS G generated by the systems (1.3) admits a random attractors.

Proof. It follows from Lemma 3.4 that the equations (1.3) generates a MRDS G. Lemma 3.3 and 3.5 imply that the MRDS G satisfies the conditions G₁ and G₂. Then, the existence random attractor is established by theorem (2.1). Thus, the proof is completed.

In the sequel, we are going to show the existence of random attractor for the equation (1.3) with weaker restrictions on the functions k₁(u) and k₂(v) instead of the stronger restrictions imposed on it, that is to say, k₁(u) and k₂(v) are assumed to be the functions

for γ < p - 1.

Lemma 3.6. Assume that (H1)-(H5) and (3.6) hold. Then there exists the some positive constants m₂, a₅ and b₅ such that the solution (u(t), v(t)) of the equations (1.3) satisfy the following estimates

Proof. By the Young inequality and (3.6), we have

Multiplying the first equation of (1.3) by u and the second one by v, and integrate over D, we obtain

Let m₂ = min{δ, 2dλ₁}, where λ₁ > 0 is the first eigenvalue of -Δ in . Then

Integrating (3.7) on the interval [s, t](t ≥ s ≥ 0), we get

Applying the Gronwall's Lemma to (3.8) gives

for all t > 0. Thus, we complete the proof of Lemma 3.6.

Lemma 3.7. Assume that (H1)-(H5) and (3.6) hold, and Lévy noise ω(t) satisfies the following conditions

(HY2) If for all ϵ > 0, α > 0, there exists T = T (ϵ) > 0 such that Levy noise ω(t) satisfies

(HY3) If for all ϵ > 0, there exists a positive constant D > 0 such that Levy noise ω(t) satisfies

Then, the MRDS G generated by equation (1.3) satisfies the condition G₂.

Proof. The proof is similar to the one of Lemma 3.5, and is omitted here.

Theorem 3.2. Assume that conditions (H1)-(H5) and (3.6) hold, and the Lévy noise satisfy the (HY2)-(HY3). Then, the MRDS G generated by the systems (1.3) admits a random attractors.

Proof. The proof is similar to the one of Theorem 3.1, and is omitted here.

Remark 3.1. From the proof of Theorem 3.1 and Theorem 3.2, the Theorem 3.1 and Theorem 3.2 hold for more general càdlàg process.

The authors declare that they have no competing interests.

This article was supported by the NSF of China (No. 10971225) NSF of Hunan (No. 11JJ3004) and sponsored by SRF for ROCS, SEM, China.

1. Chepyzhov, V, Vishik, M: Attractors for Equations of Mathematical Physics. American Mathematical Society, Providence, Rhode Island (2002)

2. Crauel, H, Flandoli, F: Attractor for random dynamical systems. Probab Theory Relat Fields. 100, 365–393 (1994). Publisher Full Text

3. Crauel, H, Debussche, A, Flandoli, F: Random attractors. J Dyn Differ Equ. 9, 307–341 (1997). Publisher Full Text

4. Huang, J, Shen, W: Pullback attractors for nonautonomous and random parabolic equations on non-Smooth domains. Discret Contin Dyn Syst. 24, 855–882 (2009)

5. Huang, J, Shen, W: Global sttractors for partly dissipative random/stochastic reaction diffusion equations. Int J Evol Equ. 4, 383–412 (2010)

6. Caraballo, T, Langa, JA, Valero, J: Global attractors for multivalued random dynamical systems. Nolinear Anal. 48, 805–829 (2002). Publisher Full Text

7. Rodriguez-Bernal, A, Wang, B: Attractor for partly dissipative reaction diffusion system in Rⁿ. J Math Anal Appl. 252, 790–803 (2000). Publisher Full Text

8. Applebaum, D: Levy Processes and Stochastic Calculus. Cambridge University Press, Cambridge (2009)

9. Peszat, S, Zabczyk, J: Stochastic Partial Differential Equations with Levy noise-Evolution Equation Approach. Cambridge University Press, Cambridge (2009)

10. Kapustyan, OV, Valero, J, Pereguda, OV: Random attractor for the reaction diffusion equation pertubed by a stochastic Cadag process. Theor Probab Math Stat. 73, 57–69 (2006)

11. Marion, M: Finite-dimensional attractor associated with partly dissipative reaction diffusion equations. SIAM J Math Anal. 20, 816–844 (1989). Publisher Full Text

12. Wang, B: Pullback attractor for the non-autonomous FitzHugh-Nagumo systems on un-bounded domains. Nonlinear Anal. 70, 3799–3815 (2009). Publisher Full Text

13. Wang, B: Random attractor for stochastic FitzHugh-Nagumo systems on unbounded domains. Nonlinear Anal. 71, 2811–2828 (2009). Publisher Full Text