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Random attractors for stochastic retarded reaction-diffusion equations with multiplicative white noise on unbounded domains

  • Xiaoyao Jia EMAIL logo , Xiaoquan Ding and Juanjuan Gao
Published/Copyright: August 3, 2018
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Open Mathematics
From the journal Open Mathematics Volume 16 Issue 1

Abstract

In this paper we investigate the stochastic retarded reaction-diffusion equations with multiplicative white noise on unbounded domain ℝn (n ≥ 2). We first transform the retarded reaction-diffusion equations into the deterministic reaction-diffusion 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-off technique to obtain the pullback asymptotic compactness.

Keywords: Stochastic reaction-diffusion system; Random attractor; Time-delay; Multiplicative white noise
MSC 2010: 35B40; 35B41; 35K45; 35K57

1 Introduction

In this paper we investigate a class of stochastic partial differential equations, which are widely used in quantum field theory, statistical mechanics and financial 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 differential equations. Hence, it is very important to study the properties of retarded partial differential equations.

In this paper, we consider the asymptotic behavior of the solutions to the following stochastic retarded reaction-diffusion equation with multiplicative noise in the whole space ℝn (n ≥ 2):

du+(λu−Δu)dt=(F(x,u(t,x))+G(x,u(t−h,x))+g(x))dt+ϵu∘dw.(1)

Here λ is a positive constant; h > 0 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 defined on ℝn; w is a two-sided real-valued Wiener process on a probability space (Ω, 𝓕, ℙ), with

Ω={ω∈C(R,R):ω(0)=0},

the Borel σ-algebra 𝓕 on Ω is generated by the compact open topology [2] and ℙ is the corresponding Wiener measure on 𝓕; ϵ is a positive parameter; ∘ denotes the Stratonovich sense in the stochastic term. We identify ω(t) with wt(ω), i.e. wt(ω) = w(t, ω) = ω (t), t ∈ ℝ.

One of the most important problems for stochastic differential equations is to study the asymptotic behavior of the solutions. The asymptotic behavior of random dynamical systems can be described by its random attractors. The existence of attractors for partial differential equations has been studied by many authors, see [3, 12, 16, 17, 20, 21, 22, 23, 24, 25, 26, 27, 29, 31, 32, 34] and the references therein. However, to the best of our knowledge, the studies of attractors for retarded partial differential equations are very few, especially for the stochastic retarded partial differential equations. The attractors for the deterministic retarded partial differential equations were investigated in [6, 7, 9, 33, 35], and random attractors for stochastic retarded partial differential equations were studied in [13, 14, 30, 36]. In [13], the authors studied the existence of random attractor for stochastic retarded reaction-diffusion equations with additive noise, while in [14, 30, 36], the authors studied the random attractors for stochastic retarded lattice dynamical systems. In this paper, we will study the existence of the random attractors for stochastic retarded reaction-diffusion equations with multiplicative white noise on unbounded domain ℝn. In the bounded domain case, the compactness of random absorbing set can be obtained by Sobolev embeddings theory. However, in the unbounded domain case, we can not get compactness in this way, because Sobolev embeddings are not compact. Hence, the cut-off technique will be used to obtain the compactness. This technique has been used by many authors (see [13, 21, 26, 27]).

The remainder of the paper is organized as follows. In section 2, we recall an important theorem which will be used to obtain the existence of random attractor, and transform the stochastic retarded partial differential equations to random ones by Ornstein-Uhlenbeck process. We will show that for each ω ∈ Ω, the random equation has a unique solution. In Section 3, we get some useful estimates for the solution of the random equation. At last, we get the existence of the random attractor.

2 Preliminaries and Random Dynamical Systems

In this section we introduce some notations and recall some basic knowledge about random attractors for random dynamical systems (RDS). The reader can refer to [2, 3, 10, 11, 17] for more details.

First, we introduce some notations which will be used in the following. Let ∥⋅∥and (⋅, ⋅) be the norm and inner product in L2(ℝn). For fixed h > 0, let 𝔖 = C ([−h, 0], L2(ℝn)), and let the norm of 𝔖 be ∥f∥𝔖 = supt∈[−h,0] ∥f(t)∥. Then, it is easy to see that 𝔖 is a Banach space. Let (X, ∥⋅∥X) be a Banach space with Borel σ−algebra 𝓑(X), and let (Ω, 𝓕,ℙ) be a probability space. For a ≤ b, t ∈ [a, b]and continuous function u ∈ C([a−h, b], L2(ℝd)), define ut(s) = u(t + s) for s ∈ [−h, 0]. From the definition, one can see that, for t ∈ [a, b], ut ∈ 𝔖.

Let (X, ∥⋅∥X) be a Banach space with Borel σ−algebra 𝓑(X), and let (Ω, 𝓕,ℙ) be a probability space.

Definition 2.1

(Ω, 𝓕, ℙ, (θt)t∈ℝ) is called a metric dynamical system, ifθ:ℝ × Ω ↦ Ωis (𝓑(ℝ) × 𝓕, 𝓕)− measurable, θ0is the identity onΩ, θs+t = θt ∘ θsfor alls, t ∈ ℝ andθt ℙ = ℙ for allt ∈ ℝ.

Definition 2.2

A stochastic processϕis called a continuous random dynamical system (RDS) over (Ω, 𝓕, ℙ, (θt)t∈ℝ), ifϕis (𝓑([0, +∞)) × 𝓕 × 𝓑(X), 𝓑(X))−neasurable, and for allω ∈ Ω,

  1. the mappingϕ(t, ω, ⋅) : X ↦ X, x ↦ ϕ(t, ω, x) is continuous for everyt ≥ 0;

  2. ϕ(0, ω, ⋅) is the identity onX;

  3. ϕ(s + t, ω, ⋅) = ϕ(t, θsω, ⋅) ∘ ϕ(s, ω, ⋅) for alls, t > 0.

Definition 2.3

  1. A random set {B(ω)}ω∈ΩofXis called bounded if there existsx0 ∈ Xand a random variabler(ω) > 0, such that

    B(ω)⊂{x∈X:||x−x0||X≤r(ω),x0∈X},forallω∈Ω.

  2. A random set {B(ω)}ω∈Ωis called a compact random set ifB(ω) is compact for allω ∈ Ω.

  3. A random set {B(ω)}ω∈Ω ⊂ Xis called tempered with respect to (θt)t∈ℝ, if for ℙ − a.e.ω ∈ Ω,

    limt→+∞e−ytsupx∈B(θ−tω)||x||X=0forally>0.(2)

  4. A random variabler(ω) is said to be tempered with respect to (θt)t∈ℝ, if for ℙ − a.e.ω ∈ Ω,

    limt→+∞e−yt|r(θ−tω)|=0forally>0.(3)

In the following, we assume that ϕ is a continuous RDS on X over (Ω, 𝓕, ℙ, (θt)t∈ℝ), and 𝒟 is a collection of random subsets of X.

Definition 2.4

Let {K(ω)}ω∈Ω ∈ 𝒟. Then {K(ω)}ω∈Ωis called a random absorbing set forϕin 𝒟 if for every {B(ω)} ∈ 𝒟 and ℙ − a.e. ω ∈ Ω, there existsT(B, ω) > 0, such that

ϕ(t,θ−tω,B(θ−tω))⊆K(ω)forallt≥T(B,ω).(4)

Definition 2.5

A random set {A(ω)}ω∈ΩofXis called a 𝒟-random attractor (or 𝒟-pullback attractor) forϕ, if the following conditions are satisfied:

  1. {A(ω)}ω∈Ωis compact, andω ↦ d(x, A(ω)) is measurable for everyx ∈ X;

  2. {A(ω)}ω∈Ωis invariant, that is,

    ϕ(t,ω,A(ω))=A(θtω)∀t≥0.(5)

  3. {A(ω)}ω∈Ωattracts every set in 𝒟, that is, for every {B(ω)}ω∈Ω ∈ 𝒟, ω ∈ Ω,

    limt→+∞dist(ϕ(t,θ−tω,B(θ−tω)),A(ω))=0,(6)

    where dist(Y,Z)=supy∈Yinfz∈Z||y−z||Xis the Hausdorff semi-metrix(Y ⊆ X, Z ⊆ X).

Definition 2.6

ϕis said to be 𝒟-pullback asymptotically compact inX, if for allB ∈ 𝒟 and ℙ − a.e. ω ∈ Ω, {ϕ(tn,θ−tnω,xn)}i=1∞has a convergent subsequence inX, astn → ∞, andxn ∈ B(θ−tnω).

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 RDSϕin 𝒟 andϕis 𝒟-pullback asymptotically compact inX. Thenϕhas a unique 𝒟-random attractor {A(ω)}ω∈Ωwhich is given by

A(ω)=∩τ≥0∪t≥τϕ(t,θ−tω,K(θ−tω))¯.(7)

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-diffusion 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-diffusion equation on ℝn with the multiplicative noise:

du+(λu−Δu)dt=(F(x,u(t,x))+G(x,u(t−h,x))+g(x))dt+ϵu∘dw,x∈Rn,t>0,(8)

with the initial condition

u(t,x)=u0(t,x),x∈Rn,t∈[−h,0].(9)

Here g is a given function in L2 (ℝn), and F, G are continuous functions satisfying the following conditions:

  1. F:ℝn × ℝ → ℝ is a continuous function such that for all x ∈ ℝn and s ∈ ℝ,

    F(x,s)s≤−α1|s|p+β1(x);(10)
    |F(x,s)|≤α2|s|p−1+β2(x);(11)
    ∂∂sF(x,s)≤α3;(12)
    |∂∂xF(x,s)|≤β3(x).(13)

    Here α1, α2, α3 are positive constants, p > 2, β1(x), β2(x), β3(x) are nonnegative functions on ℝn, such that β1(x) ∈ L1(ℝn), and β2(x), β3(x) ∈ L2(ℝn).

  2. G : ℝn × ℝ → ℝ is a continuous function such that for all x ∈ ℝn and s1, s2 ∈ ℝ,

    |G(x,s1)−G(x,s2)|≤α4|s1−s2|;(14)
    |G(x,s)|≤β4(x)|s|+β5(x).(15)

    Here α4 is a positive constant, β4(x), β5(x) are nonnegative functions on ℝn such that β4(x) ∈ L2pp−2(ℝn) ∩ L∞(ℝn), β5(x) ∈ L2(ℝn).

Example 2.8

Forx ∈ ℝn, s, s1, s2 ∈ ℝ andp > 2, let

F(x,s)=−α1|s|p−1sgn(s);G(x,s)=α4e−x2s.

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 defined in Section 1. Let

θtω(⋅)=ω(⋅+t)−ω(t),t∈R.(16)

Then (Ω, 𝔉, ℙ, (θt)t∈ℝ) is an ergodic metric dynamical system. Since the probability space (Ω,𝔉,ℙ) is canonical, one has

w(t,ω)=ω(t),w(t,θsω)=w(t+s,ω)−w(s,ω).(17)

To study the random attractor for problem (8)-(9), we first transform that system into a deterministic system with random parameter. Let

z(θtω)≡−μ∫−∞0eμs(θtω)(s)ds,t∈R.

Then, one has that z(θtω) is the Ornstein-Uhlenbeck process and solves the following equation (see [15] for details):

dz+μzdt=dw(t).(18)

Moreover, the random variable z(θtω) is tempered, and z(θtω) is ℙ − a.e. continuous. It follows from Proposition 4.3.3 [2] that there exists a tempered function r(ω) > 0 such that

|z(ω)|+|z(ω)|2≤r(ω),(19)

where r(ω) satisfies, for α > 0 and for ℙ − a.e. ω ∈ Ω,

r(θtω)≤eα2|t|r(ω),t∈R.(20)

Then it follows that for α > 0 and for ℙ − a.e. ω ∈ Ω,

|z(θtω)|+|z(θtω)|2≤eα2|t|r(ω),t∈R,(21)
|z(θt−hω)|+|z(θt−hω)|2≤eα2|t−h|r(ω)≤eα2heα2|t|r(ω),t∈R.(22)

By [9], z(θtω) has the following properties:

limt→±∞z(θtω)t=0,(23)
limt→±∞1|t|∫−t0|z(θsω)|ds=E|z|,(24)
E|z|≤E|z|2=12μ.(25)

It follows from (23) and (25) that there exists a positive constant μ > 0 small enough, such that for t > 0 large enough,

2ϵμ∫−t0|z(θsω)|ds≤λ4t;2ϵ|z(θtω)|≤λ8t.(26)

In this paper we consider the weak solutions of (8)-(9) and (30)-(31).

Definition 2.9

For anyu0 ∈ 𝔖, letu : [0,∞) × Ω → L2 (ℝn). Suppose thatu(⋅, ω, u0) : [0, ∞) → L2 (ℝn) is continuous and∂u∂tis measurable. Then we say thatuis a weak solution of (8)-(9), ifu(t, ω, u0) = u0(t, ω), t ∈ [−h, 0], ω ∈ Ω, andu ∈ L2 ([0, T], H1(ℝn)) ∩ Lp([0, T], Lp(ℝn)), for anyT > 0, and for anyϕ ∈ C0∞ (ℝn)

(u(T),ϕ)−(u0(0),ϕ)+∫0T(u(t),λϕ−Δϕ)dt=∫0T(F(⋅,u(t))+G(⋅,u(t−h))+g,ϕ)dt+ϵ∫0T(ϕ,u∘dw),(27)

ℙ − a.e. ω ∈ Ω.

Suppose that u is a weak solution of (8)-(9). Let v(t) = e−ϵz(θtω)u(t), v0(t) = e−ϵz(θtω)u0(t) and ψ = eϵz(θtω)ϕ. Then one has that

(u(T),ϕ)−(u0(0),ϕ)=∫0T(∂u∂t,ϕ)dt=∫0T(∂∂t(eϵz(θtω)v(t)),ϕ)dt=∫0T(∂∂tv(t),ψ)dt+ϵ∫0T(ψ,v∘dz)=(v(T),ψ)−(v0(0),ψ)+ϵ∫0T(ψ,v∘dw)−μϵ∫0T(ψ,z(θtω)v)dt.(28)

In the last step of (28) we use (18). Hence, it follows from (28) and the definition of weak solution that

(v(T),ψ)−(v0(0),ψ)+∫0T(v(t),λψ−Δψ)dt=∫0Te−ϵz(θtω)(F(⋅,eϵz(θtω)v(t))+G(⋅,eϵz(θt−hω)v(t−h))+g,ψ)dt+μϵ∫0T(ψ,z(θtω)v)dt.(29)

Then function v satisfying (29) is said to be the weak solution of the following equation:

∂v∂t+λv−Δv=e−ϵz(θtω)F(x,eϵz(θtω)v(t,x))+e−ϵz(θtω)G(x,eϵz(θt−hω)v(t−h,x))+e−ϵz(θtω)g(x)+ϵμz(θtω)v,(30)

with the initial condition

v0(t,x)=e−ϵz(θtω)u0(t,x),x∈Rn,t∈[−h,0].(31)

Equation (30)-(31) can be seen as a deterministic partial differential equation with random coefficients.

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 domain O ⊂ ℝn, (30)-(31) has a unique weak solution v(⋅, ω, v0) ∈ C([0, ∞), L2(O)) ∩ Lloc2 ([0, ∞), H01 (O)) ∩ Llocp ([0, ∞), Lp(O)), 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 ∈ L2(ℝn) is the unique weak solution of (30)-(31) on ℝn. Then, u(t) = eϵz(θtω)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≥0 over (Ω, 𝓕, ℙ, (θt)t∈ℝ), with

Φ(t,θ−tω,B(θ−tω))=vt(⋅,ω,v0),ω∈Ω,v0∈S,(32)

and (8)-(9) generates a continuous random dynamical system (Ψ(t))t≥0 over (Ω, 𝓕, ℙ, (θt)t∈ℝ), with

Ψ(t,θ−tω,B(θ−tω))=ut(⋅,ω,u0),ω∈Ω,u0∈S.(33)

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≥0.

3 Uniform estimates of solutions

In this section we prove the existence of the random attractor for random dynamical system (Φ(t))t≥0. We first give some useful estimates on the mild solutions of equation (30)-(31).

Lemma 3.1

Random dynamical systemΦhas a random absorbing set {K(ω)}ω∈Ωin 𝒟, that is, for any {B(ω)}ω∈Ω ∈ 𝒟 and ℙ − a.e.ω ∈ Ω, there isTB(ω) > 0, such thatΦ(t, θ−tω, B(θ−tω)) ⊂ K(ω) for allt ≥ TB(ω).

Proof

Taking the inner product of (30) with v, we get that

12ddt||v||2+λ||v||2+||∇v||2=e−ϵz(θtω)∫RnF(x,eϵz(θtω)v(t,x))vdx+e−ϵz(θtω)∫RnG(x,eϵz(θt−hω)v(t−h,x))vdx+e−ϵz(θtω)∫Rng(x)vdx+ϵμz(θtω)||v||2.(34)

We now estimate each term on the right hand side of (34). For the first term, by condition (A1) and Young inequality we have that

e−ϵz(θtω)∫RnF(x,eϵz(θtω)v(t,x))vdx≤e−2ϵz(θtω)∫Rn−α1|eϵz(θtω)v|p+β1(x)dx=−α1e(p−2)ϵz(θtω)||v||Lpp+e−2ϵz(θtω)||β1||L1.(35)

Note that by Young inequality, we have that for all a, b, c > 0, α, β, y > 1, and 1α+1β+1y=1,

abc≤aαα+bββ+cyy.(36)

For the second term on the right hand side of (34), by condition (A2) and (36), we obtain that

e−ϵz(θtω)∫RnG(x,eϵz(θt−hω)v(t−h,x))vdx≤e−ϵz(θtω)∫Rnβ4(x)|eϵz(θt−hω)v(t−h,x)v|dx+e−ϵz(θtω)∫Rnβ5(x)|v|dx=∫Rn(ep−2pϵz(θtω)|v|)|v(t−h,x)|(e2−2ppϵz(θtω)+ϵz(θt−hω)β4(x))dx+e−ϵz(θtω)∫Rnβ5(x)|v|dx≤α12e(p−2)ϵz(θtω)||v||Lpp+λ4e−λ2h||v(t−h,x)||2+c1e4−4pp−2ϵz(θtω)+2pp−2ϵz(θt−hω)||β4||2pp−2+λ4||v||2+1λe−2ϵz(θtω)||β5||2,(37)

where c1 is a positive constant depending on α, λ and h. For the third term on the right hand side of (34), by Young inequality,

e−ϵz(θtω)∫Rng(x)vdx≤1λe−2ϵz(θtω)||g||2+λ4||v||2.(38)

Then it follows from (34) - (38) that, for all t ≥ 0

ddt||v||2+2||∇v||2+α1e(p−2)ϵz(θtω)||v||Lpp≤(2ϵμ|z(θtω)|−λ)||v||2+λ2e−λ2h||v(t−h,x)||2+c2e−2ϵz(θtω)+c3e4−4pp−2ϵz(θtω)+2pp−2ϵz(θt−hω)=(2ϵμ|z(θtω)|−λ2)||v||2+λ2(e−λ2h||v(t−h,x)||2−||v||2)+c2e−2ϵz(θtω)+c3e4−4pp−2ϵz(θtω)+2pp−2ϵz(θt−hω),(39)

with c2=2||β1||L1+2λ||β5||2+2λ||g||2,c3=2c1||β4||2pp−2. Applying Gronwall inequality, we obtain that, for all t ≥ 0,

||v(t)||2+2∫0teλ2(s−t)+2ϵμ∫st|z(θτω)|dτ||∇v||2ds+α1∫0teλ2(s−t)+2ϵμ∫st|z(θτω)|dτ+(p−2)ϵz(θsω)||v||Lppds≤e−λ2t+2ϵμ∫0t|z(θτω)|dτ||v0||2+λ2∫0teλ2(s−t)+2ϵμ∫st|z(θτω)|dτ(e−λ2h||v(s−h)||2−||v||2)ds+c2∫0teλ2(s−t)+2ϵμ∫st|z(θτω)|dτ−2ϵz(θtω)ds+c3∫0teλ2(s−t)+2ϵμ∫st|z(θτω)|dτ+4−4pp−2ϵz(θsω)+2pp−2ϵz(θs−hω)ds.(40)

Notice that

∫0teλ2s−2ϵμ∫0s|z(θτω)|dτe−λ2h||v(s−h)||2ds=∫−ht−heλ2(s+h)−2ϵμ∫0s+h|z(θτω)|dτe−λ2h||v(s)||2ds=∫−h0eλ2s−2ϵμ∫0s+h|z(θτω)|dτ||v(s)||2ds+∫0t−heλ2s−2ϵμ∫0s+h|z(θτω)|dτ||v(s)||2ds≤h||v0||S+∫0t−heλ2s−2ϵμ∫0s|z(θτω)|dτ||v(s)||2ds≤h||v0||S+∫0teλ2s−2ϵμ∫0s|z(θτω)|dτ||v(s)||2ds.(41)

Hence, it follows from (41) that

λ2∫0teλ2(s−t)+2ϵμ∫st|z(θtω)|dτ(e−λ2h||v(s−h)||2−||v||2)ds=λ2e−λ2t+2ϵμ∫0t|z(θtω)|dτ∫0teλ2s−2ϵμ∫0s|z(θtω)|dτe−λ2h||v(s−h)||2ds−λ2e−λ2t+2ϵμ∫0t|z(θtω)|dτ∫0teλ2s−2ϵμ∫0s|z(θtω)|dτ||v||2ds≤λh2e−λ2t+2ϵμ∫0t|z(θtω)|dτ||v0||S.(42)

Then, (40) and (42) imply that

||v(t)||2+2∫0teλ2(s−t)+2ϵμ∫st|z(θtω)|dτ||∇v||2ds+α1∫0teλ2(s−t)+2ϵμ∫st|z(θtω)|dτ+(p−2)ϵz(θsω)||v||Lppds≤(1+λh2)e−λ2t+2ϵμ∫0t|z(θtω)|dτ||v0||S2+c2∫0teλ2(s−t)+2ϵμ∫st|z(θtω)|dτ−2ϵz(θtω)ds+c3∫0teλ2(s−t)+2ϵμ∫st|z(θtω)|dτ+4−4pp−2ϵz(θsω)+2pp−2ϵz(θs−hω)ds.(43)

For fixed σ ∈ [−h, 0], we have that, for t ≥ −σ,

||v(t+σ)||2≤(1+λh2)e−λ2(t+σ)+2ϵμ∫0t+σ|z(θτω)|dτ||v0||S2+c2∫0t+σeλ2(s−t−σ)+2ϵμ∫st+σ|z(θτω)|dτ−2ϵz(θsω)ds+c3∫0t+σeλ2(s−t−σ)+2ϵμ∫st+σ|z(θτω)|dτ+4−4pp−2ϵz(θsω)+2pp−2ϵz(θs−hω)ds≤(1+λh2)eλ2(h−t)+2ϵμ∫0t|z(θτω)|dτ||v0||S2+c2eλh2∫0teλ2(s−t)+2ϵμ∫st|z(θτω)|dτ−2ϵz(θsω)ds+c3eλh2∫0teλ2(s−t)+2ϵμ∫st|z(θτω)|dτ+4−4pp−2ϵz(θsω)+2pp−2ϵz(θs−hω)ds.(44)

and for t ∈ [0, −σ],

||v(t+σ)||2≤||v0||S2≤(1+λh2)eλ2(h−t)+2ϵμ∫0t|z(θτω)|dτ||v0||S2+c2eλh2∫0teλ2(s−t)+2ϵμ∫st|z(θτω)|dτ−2ϵz(θsω)ds+c3eλh2∫0teλ2(s−t)+2ϵμ∫st|z(θτω)|dτ+4−4pp−2ϵz(θsω)+2pp−2ϵz(θs−hω)ds.(45)

Due to (44) and (45), we find that for all t ≥ 0

||vt||S2≤(1+λh2)eλ2(h−t)+2ϵμ∫0t|z(θτω)|dτ||v0||S2+c2eλh2∫0teλ2(s−t)+2ϵμ∫st|z(θτω)|dτ−2ϵz(θsω)ds+c3eλh2∫0teλ2(s−t)+2ϵμ∫st|z(θτω)|dτ+4−4pp−2ϵz(θsω)+2pp−2ϵz(θs−hω)ds.(46)

Replacing ω by θ−tω in (46), we get that for all t ≥ 0

||vt(θ−tω,v0(θ−tω))||S2≤(1+λh2)eλ2(h−t)+2ϵμ∫0t|z(θτ−tω)|dτ||v0(θ−tω)||S2+c2eλh2∫0teλ2(s−t)+2ϵμ∫st|z(θτ−tω)|dτ−2ϵz(θs−tω)ds+c3eλh2∫0teλ2(s−t)+2ϵμ∫st|z(θτ−tω)|dτ+4−4pp−2ϵz(θs−tω)+2pp−2ϵz(θs−t−hω)ds.(47)

By (26), one has that for any v0(θ−tω) ∈ B(θ−tω),

limt→+∞(1+λh2)eλ2(h−t)+2ϵμ∫0t|z(θτ−tω)|dτ||v0||S2=limt→+∞(1+λh2)eλ2(h−t)+2ϵμ∫−t0|z(θτω)|dτ||v0||S2≤limt→+∞(1+λh2)eλ2(h−t)+λ4t||v0||S2=0.(48)

It follows from (26) that for any v0(θ−tω) ∈ B(θ−tω),

c2eλh2∫0teλ2(s−t)+2ϵμ∫st|z(θτ−tω)|dτ−2ϵz(θs−tω)ds=c2eλh2∫−t0eλ2s+2ϵμ∫s0|z(θτω)|dτ−2ϵz(θsω)ds≤c2eλh2∫−∞0eλ4s−2ϵz(θtω)ds≤c2eλh2∫−∞0eλ2s−λ4s−λ8sds<+∞.(49)

Similarly, we can get that

c3eλh2∫0teλ2(s−t)+2ϵμ∫st|z(θτ−tω)|dτ+4−4pp−2ϵz(θs−tω)+2pp−2ϵz(θs−t−hω)ds=c3eλh2∫−t0eλ2s+2ϵμ∫s0|z(θτω)|dτ+4−4pp−2ϵz(θsω)+2pp−2ϵz(θs−hω)ds≤c3eλh2∫−∞0eλ4s+4−4pp−2ϵz(θsω)+2pp−2ϵz(θs−hω)ds<+∞.(50)

Set

ρ12(ω)=1+c2eλh2∫−∞0eλ4s−2ϵz(θsω)ds+c3eλh2∫−∞0eλ4s+4−4pp−2ϵz(θsω)+2pp−2ϵz(θs−hω)ds.(51)

Then, there exists a TB(ω) > 0, such that for all t > TB(ω),

||vt(θ−tω,v0(θ−tω))||S2≤ρ12(ω).(52)

To show ρ12 (ω) is tempered, we need only to prove that for any y > 0 small enough, the following holds

limt→+∞e−ytρ12(θ−tω)=0.(53)

Using (26) again, one has that for y < λ small enough,

limt→+∞e−yt∫−∞0eλ4s−2ϵz(θs−tω)ds≤limt→+∞e−yt∫−∞0ey4s−2ϵz(θs−tω)ds=limt→+∞e−yt∫−∞tey4(s+t)−2ϵz(θsω)ds=limt→+∞e−34yt(∫−∞0ey4s−2ϵz(θsω)ds+∫0tey4s−2ϵz(θsω)ds)≤limt→+∞e−34yt(∫−∞0ey4s−y8sds+∫0tey4s+y8sds)=0.(54)

Similarly, we can get that

limt→+∞e−yt∫−∞0eλ4s+4−4pp−2ϵz(θs−tω)+2pp−2ϵz(θs−t−hω)ds=0.(55)

In view of (54) and (55), we find that ρ12 (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.

Lemma 3.2

There exists a tempered random variableρ2(ω) > 0 such that for any {B(ω)}ω∈Ω ∈ 𝒟 and v0(ω) ∈ B(ω), there exists a TB(ω) > 0 such that the solutionvof (30)-(31) satisfies, for ℙ-a.e. ω ∈ Ω, for allt ≥ TB(Ω),

∫tt+1||∇v(s,θ−tω,v0(θ−tω))||2ds≤ρ2(ω).(56)

Proof

By (43), we have that for all t ≥ 0

2∫0teλ2(s−t)+2ϵμ∫st|z(θtω)|dτ||∇v||2ds≤(1+λh2)e−λ2t+2ϵμ∫0t|z(θtω)|dτ||v0||S2+        c2∫0teλ2(s−t)+2ϵμ∫st|z(θtω)|dτ−2ϵz(θtω)ds+c3∫0teλ2(s−t)+2ϵμ∫st|z(θtω)|dτ+4−4pp−2ϵz(θsω)+2pp−2ϵz(θs−hω)ds.(57)

Let TB(ω) be the positive constant defined in Lemma 4.1. Replacing ω by θ−tω in (57), by (48) - (50), one has that for all t ≥ TB(ω),

2∫0teλ2(s−t)+2ϵμ∫st|z(θτ−tω)|dτ||∇v(s,θ−tω,v0(θ−tω))||2ds≤(1+λh2)e−λ2t+2ϵμ∫0t|z(θτ−tω)|dτ||v0(θ−tω)||S2+c2∫0teλ2(s−t)+2ϵμ∫st|z(θτ−tω)|dτ−2ϵz(θs−tω)ds                          +c3∫0teλ2(s−t)+2ϵμ∫st|z(θτ−tω)|dτ+4−4pp−2ϵz(θs−tω)+2pp−2ϵz(θs−t−hω)ds                          ≤ρ12(ω).(58)

In another aspect, for s ∈ [t, t + 1],

∫0t+1eλ2(s−t−1)+2ϵμ∫st+1|z(θτ−t−1ω)|dτ||∇v(s,θ−t−1ω,v0(θ−t−1ω))||2ds≥∫tt+1eλ2(s−t−1)+2ϵμ∫st+1|z(θτ−t−1ω)|dτ||∇v(s,θ−t−1ω,v0(θ−t−1ω))||2ds≥e−λ2−2ϵμmax−1≤τ≤0|z(θτω)|∫tt+1||∇v(s,θ−t−1ω,v0(θ−t−1ω))||2ds.(59)

It follows from (58) and (59) that

∫tt+1||∇v(s,θ−t−1ω,v0(θ−t−1ω))||2ds≤eλ2+2ϵμmax−1≤τ≤0|z(θτω)|ρ12(ω).(60)

Replacing ω by θ1ω in the last inequality we obtain that

∫tt+1||∇v(s,θ−tω,v0(θ−tω))||2ds≤eλ2+2ϵμmax−1≤τ≤0|z(θτω)|ρ12(ω)≡ρ2(ω).(61)

This ends the proof. □

Lemma 3.3

There exists a tempered random variableρ3(ω) > 0 such that for any {B(ω)}ω∈Ω ∈ 𝒟 andv0(ω) ∈ B(ω), there exists aTB(ω) > 0 such that the solutionvof (30)-(31) satisfies, for ℙ-a.e. ω ∈ Ω, for all t ≥ TB(ω) + h + 1, andσ1, σ2 ∈ [−h, 0],

∇v(t,θ−tω,v0(θ−tω))2≤ρ3(ω),(62)
∫t+σ1t+σ2||Δv(s,θ−tω,v0(θ−tω))||2ds≤ρ4(ω).(63)

Proof

Taking the inner product of (30) with −Δv, we get that

12ddt||∇v||2+λ||∇v||2+||Δv||2=−e−ϵz(θtω)∫RnF(x,eϵz(θtω)v(t,x))Δvdx−        e−ϵz(θtω)∫RnG(x,eϵz(θt−hω)v(t−h,x))Δvdx−e−ϵz(θtω)∫Rng(x)Δvdx+ϵμz(θtω)||∇v||2.(64)

We now estimate each term on the right-hand side of (64). For the first term, by Young inequality and condition (A1), we have

−e−ϵz(θtω)∫RnF(x,eϵz(θtω)v(t,x))Δvdx=e−ϵz(θtω)∫Rn∂F∂x(x,eϵz(θtω)v)⋅∇vdx+∫Rn∂F∂s(x,eϵz(θtω)v)|∇v|2dx≤e−ϵz(θtω)||β3|| ||∇v||+α3||∇v||2≤(1+α3)||∇v||2+14e−2ϵz(θtω)||β3||2.(65)

For the second term, it follows from condition (A2) and Young inequality that

−e−ϵz(θtω)∫RnG(x,eϵz(θt−hω)v(t−h,x))Δvdx≤e−ϵz(θtω)∫Rneϵz(θt−hω)β4(x)|v(t−h,x)|+β5(x)Δvdx≤18||Δv||2+2e−2ϵz(θtω)+2ϵz(θt−hω)∫Rnβ42(x)|v(t−h,x)|2dx+18||Δv||2+2e−2ϵz(θtω)∫Rnβ52(x)dx≤14||Δv||2+2e−2ϵz(θtω)+2ϵz(θt−hω)||β4||L∞2||v(t−h)||2+2e−2ϵz(θtω)||β5||2.(66)

The third term is bounded by

−e−ϵz(θtω)∫Rng(x)Δvdx≤14||Δv||2+e−2ϵz(θtω)||g||2.(67)

By (64) - (67), we find that for all t ≥ 0,

ddt||∇v||2+||Δv||2≤2(1+α3)||∇v||2+c5e−2ϵz(θtω)+4e−2ϵz(θtω)+2ϵz(θt−hω)||β4||L∞2||v(t−h)||2,(68)

with c5=4||β5||2+2||g||2+12||β3||2. Let TB(ω) be the positive constant defined in Lemma 3.1, take t ≥ TB(ω) and s ∈ [t, t + 1]. Integrating (68) over [s, t + 1], we get that

||∇v(t+1,ω,v0(ω))||2≤||∇v(s,ω,v0(ω))||2+2(1+α3)∫st+1||∇v(τ,ω,v0(ω))||2dτ         +c5∫st+1e−2ϵz(θτω)dτ+4||β4||L∞2∫st+1e−2ϵz(θτω)+2ϵz(θτ−hω)||v(τ−h,ω,v0(ω))||2dτ.(69)

Integrating the above inequality with respect to s over [t, t + 1], we get that

||∇v(t+1,ω,v0(ω))||2≤2(2+α3)∫tt+1||∇v(τ,ω,v0(ω))||2dτ         +c5∫tt+1e−2ϵz(θτω)dτ+4||β4||L∞2∫tt+1e−2ϵz(θτω)+2ϵz(θτ−hω)||v(τ−h,ω,v0(ω))||2dτ.(70)

Replacing ω by θ−t−1ω in (70), we obtain that

||∇v(t+1,θ−t−1ω,v0(θ−t−1ω))||2≤2(2+α3)∫tt+1||∇v(τ,θ−t−1ω,v0(θ−t−1ω))||2dτ     +c5∫tt+1e−2ϵz(θτ−t−1ω)dτ+4||β4||L∞2∫tt+1e−2ϵz(θτ−t−1ω)+2ϵz(θτ−t−h−1ω)||v(τ−h,θ−t−1ω,v0(θ−t−1ω))||2dτ.(71)

It follows from (56) that for all t ≥ TB(ω) + h,

2(2+α3)∫tt+1||∇v(τ,θ−t−1ω,v0(θ−t−1ω))||2dτ≤2(2+α3)ρ2(θ−1ω),(72)
∫tt+1e−2ϵz(θτ−t−1ω)dτ≤e2ϵmax−1≤τ≤0|z(θτω)|,(73)

and

∫tt+1e−2ϵz(θτ−t−1ω)+2ϵz(θτ−t−h−1ω)||∇v(τ−h,θ−t−1ω,v0(θ−t−1ω))||2dτ≤e2ϵmax−1≤τ≤0|z(θτω)|+|z(θτ−hω)|∫t−ht−h+1||∇v(τ,θ−t−1ω,v0(θ−t−1ω))||2dτ=e2ϵmax−1≤τ≤0|z(θτω)|+|z(θτ−hω)|∫t−ht−h+1||∇v(τ,θ−t+h−1(θ−hω),v0(θ−t+h−1(θ−hω))||2dτ≤e2ϵmax−1≤τ≤0|z(θτω)|+|z(θτ−hω)|ρ2(θ−hω).(74)

It follows from (71) - (74) that, for all t ≥ TB(ω) + h

||∇v(t+1,θ−t−1ω,v0(θ−t−1ω))||2≤2(2+α3)ρ2(θ−1ω)+c5e2ϵmax−1≤τ≤0|z(θτω)|                             +4||β4||L∞2e2ϵmax−1≤τ≤0|z(θτω)|+|z(θτ−hω)|ρ2(θ−hω)≡ρ3(ω).(75)

Then we have that for all t ≥ TB(ω) + h +1,

||∇v(t,θ−tω,v0(θ−tω))||2≤ρ3(ω).(76)

Let t ≥ h, −h ≤ σ1 ≤ σ2 ≤ 0. Integrating (68) over [t + σ1, t + σ2], we obtain that

||∇v(t+σ2,ω,v0(ω))||2+∫t+σ1t+σ2||Δv(τ,ω,v0(ω))||2dτ         ≤||∇v(t+σ1,ω,v0(ω))||2+2(1+α3)∫t+σ1t+σ2||∇v(τ,ω,v0(ω))||2dτ         +c5∫t+σ1t+σ2e−2ϵz(θτω)dτ+4||β4||L∞2∫t+σ1t+σ2e−2ϵz(θτω)+2ϵz(θτ−hω)||v(τ−h,ω,v0(ω))||2dτ.(77)

Replacing ω by θ−tω in the last inequality, we have that

||∇v(t+σ2,θ−tω,v0(θ−tω))||2+∫t+σ1t+σ2||Δv(τ,θ−tω,v0(θ−tω))||2dτ         ≤||∇v(t+σ1,θ−tω,v0(θ−tω))||2+2(1+α3)∫t+σ1t+σ2||∇v(τ,θ−tω,v0(θ−tω))||2dτ         +c5∫t+σ1t+σ2e−2ϵz(θτ−tω)dτ+4||β4||L∞2∫t+σ1t+σ2e−2ϵz(θτ−tω)+2ϵz(θτ−t−hω)||v(τ−h,θ−tω,v0(θ−tω))||2dτ.(78)

Using (76), we get that

∫t+σ1t+σ2||∇v(τ,θ−tω,v0(θ−tω))||2dτ=∫t+σ1t+σ2||∇v(τ,θ−τ(θτ−tω),v0(θ−τ(θτ−tω)))||2dτ≤hmax−h≤τ≤0ρ3(θτω)).(79)

By (52), we can get that

4||β4||L∞2∫t+σ1t+σ2e−2ϵz(θτ−tω)+2ϵz(θτ−t−hω)||v(τ−h,θ−tω,v0(θ−tω))||2dτ≤4||β4||L∞2max−2h≤τ≤−hρ12(θτω)∫t+σ1t+σ2e−2ϵz(θτ−tω)+2ϵz(θτ−t−hω)dτ=4h||β4||L∞2max−2h≤τ≤−hρ12(θτω)e2ϵmax−h≤τ≤0|z(θτω)|+|z(θτ−hω)|.(80)

Notice that

∫t+σ1t+σ2e−2ϵz(θτ−tω)dτ=∫σ1σ2e−2ϵz(θτω)dτ≤he2ϵmax−h≤τ≤0|z(θτω)|.(81)

It follows from (76), (78)-(81) that

∫t+σ1t+σ2||Δv(τ,θ−tω,v0(θ−tω))||2dτ≤(1+2h(1+α3))max−h≤τ≤0ρ3(θτω)+c5he2ϵmax−h≤τ≤0|z(θτω)|+       4h||β4||L∞2max−2h≤τ≤−hρ12(θτω)e2ϵmax−h≤τ≤0|z(θτω)|+|z(θτ−hω)|≡ρ4(ω).(82)

This ends the proof. □

Lemma 3.4

LetB(ω) ∈ 𝒟 andv0(ω) ∈ B(ω). Then for anyϵ1 > 0 and ℙ-a.e. ω ∈ Ω, there exists a T∗ = T∗(B, ω, ϵ1) > 0, and R∗ = R∗ (ω, ϵ1) > 0 such that the solution (30)-(31), satisfies for allt ≥ T∗,

sups∈[−h,0]∫|x|≥R∗|vt(s,θ−tω,v0(θ−tω))|dx≤ϵ1.(83)

Proof

Let ρ be a smooth function defined on ℝ+ such that 0 ≤ ρ(s) ≤ 1 for all s ∈ ℝ+, and

ρ(s)=0,     0≤s≤1,1,     s≥2.(84)

Then there exists a positive constant c0 such that |ρ′(s)| ≤ c0 for all s ∈ ℝ+. Taking the inner product of (30) with ρ|x|2k2v, we get that

12ddt∫Rnρ|x|2k2|v|2dx+λ∫Rnρ|x|2k2|v|2dx−∫Rnρ|x|2k2vΔvdx=e−ϵz(θtω)∫RnF(x,eϵz(θtω)v)ρ|x|2k2vdx+e−ϵz(θtω)∫RnG(x,eϵz(θt−hω)v(t−h))ρ|x|2k2vdx                        +e−ϵz(θtω)∫Rng(x)ρ|x|2k2vdx+ϵμz(θtω)∫Rnρ|x|2k2|v|2dx.(85)

We now estimate each term in (85). For the third term on the left hand-side of (85), we have that

−∫Rnρ|x|2k2vΔvdx=∫Rnρ|x|2k2|∇v|2dx+∫Rnρ′|x|2k2v2xk2⋅∇vdx  =∫Rnρ|x|2k2|∇v|2dx+∫k≤|x|≤2kρ′|x|2k2v2xk2⋅∇vdx,(86)

and

∫k≤|x|≤2kρ′|x|2k2v2xk2⋅∇vdx≤22k∫k≤|x|≤2k|ρ′|x|2k2| |v| |∇v|dx≤4c0k∫Rn|v| |∇v|dx≤2c0k||v||2+||∇v||2.(87)

By (86) and (87), one has that

−∫Rnρ|x|2k2vΔvdx≥∫Rnρ|x|2k2|∇v|2dx−2c0k||v||2+||∇v||2.(88)

For the first term on the right-hand side of (85) by condition (A1), we have

e−ϵz(θtω)∫RnF(x,eϵz(θtω)v)ρ|x|2k2vdx=e−2ϵz(θtω)∫RnF(x,eϵz(θtω)v)ρ|x|2k2eϵz(θtω)vdx≤−α1e(p−2)ϵz(θtω)∫Rnρ|x|2k2|v|pdx+e−2ϵz(θtω)∫Rnρ|x|2k2β1(x)dx.(89)

For the second term on the right-hand side of (85) we have that

e−ϵz(θtω)∫RnG(x,eϵz(θt−hω)v(t−h))ρ|x|2k2vdx≤e−ϵz(θtω)∫Rnβ4(x)eϵz(θt−hω)v(t−h)ρ|x|2k2|v|dx+e−ϵz(θtω)∫Rnβ5(x)ρ|x|2k2|v|dx.(90)

By (36), the first term on the right-hand side of (90) is bounded by

e−ϵz(θtω)∫Rnβ4(x)eϵz(θt−hω)v(t−h)ρ|x|2k2|v|dx=∫Rne−p−2pϵz(θtω)|v|ρ1p|x|2k2 ρ12|x|2k2|v(t−h)|ρp−22p|x|2k2e2−2ppϵz(θtω)+ϵz(θt−hω)β4(x)dx≤α1∫Rne(p−2)ϵz(θtω)ρ|x|2k2|v|pdx+λ4e−λ2h∫Rnρ|x|2k2|v(t−h)|2dx                    +c6∫Rnρ|x|2k2e4−4pp−2ϵz(θtω)+2pp−2ϵz(θt−hω)β4(x)2pp−2dx,(91)

and the second term on the right-hand side of (90) is bounded by

e−ϵz(θtω)∫Rnβ5(x)ρ|x|2k2|v|dx≤λ4∫Rnρ|x|2k2|v|2dx+1λe−2ϵz(θtω)∫Rnρ|x|2k2β52(x)dx.(92)

For the third term on the right-hand side of (85) we have that

e−ϵz(θtω)∫Rng(x)ρ|x|2k2vdx≤λ4∫Rnρ|x|2k2|v|2dx+1λe−2ϵz(θtω)∫Rnρ|x|2k2g2(x)dx.(93)

It follows from (85) - (93) that

ddt∫Rnρ|x|2k2|v|2dx≤2ϵμ|z(θtω)|−λ2∫Rnρ|x|2k2|v|2dx              +λ2∫Rnρ|x|2k2e−λ2h|v(t−h)|2−|v|2dx+4c0k||v||2+||∇v||2              +2e−2ϵz(θtω)∫Rnρ|x|2k2β1(x)+1λβ52(x)+1λg2(x)dx              +2c6∫Rnρ|x|2k2e4−4pp−2ϵz(θtω)+2pp−2ϵz(θt−hω)β42pp−2(x)dx.(94)

Let TB(ω) be the positive constant defined in Lemma 3.1. Set T1(B, ω) ≥ TB(ω) + h +1. Applying Gronwall inequality to (94), we obtain that for t ≥ T1,

∫Rnρ|x|2k2|v|2dx≤eλ2(T1−t)+2ϵμ∫T1t|z(θτω)|dτ∫Rnρ|x|2k2|v(T1)|2dx          +λ2∫T1teλ2(s−t)+2ϵμ∫st|z(θτω)|dτ∫Rnρ|x|2k2e−λ2h|v(t−h)|2−|v|2dx ds          +4c0k∫T1teλ2(s−t)+2ϵμ∫st|z(θτω)|dτ||v||2+||∇v||2ds          +2∫T1teλ2(s−t)+2ϵμ∫st|z(θτω)|dτ−2ϵz(θsω)∫Rnρ|x|2k2β1(x)+1λβ52(x)+1λg2(x)dx ds          +2c6∫T1teλ2(s−t)+2ϵμ∫st|z(θτω)|dτ+4−4pp−2ϵz(θsω)+2pp−2ϵz(θs−hω)∫Rnρ|x|2k2β42pp−2(x)dx ds.(95)

If we take t ≥ T1 + h, then, by (95), we can get that for all σ ∈ [−h, 0],

∫Rnρ|x|2k2|v(t+σ)|2dx≤eλ2(T1−t−σ)+2ϵμ∫T1t+σ|z(θτω)|dτ∫Rnρ|x|2k2|v(T1)|2dx          +λ2∫T1t+σeλ2(s−t−σ)+2ϵμ∫st+σ|z(θτω)|dτ∫Rnρ|x|2k2e−λ2h|v(s−h)|2−|v|2dx ds          +4c0k∫T1t+σeλ2(s−t−σ)+2ϵμ∫st+σ|z(θτω)|dτ||v||2+||∇v||2ds          +2∫T1t+σeλ2(s−t−σ)+2ϵμ∫st+σ|z(θτω)|dτ−2ϵz(θsω)∫Rnρ|x|2k2β1(x)+1λβ52(x)+1λg2(x)dx ds          +2c6∫T1t+σeλ2(s−t−σ)+2ϵμ∫st+σ|z(θτω)|dτ+4−4pp−2ϵz(θsω)+2pp−2ϵz(θs−hω)∫Rnρ|x|2k2β42pp−2(x)dx ds.(96)

It follows that for all t ≥ T1 + h,

sup−h≤σ≤0∫Rnρ|x|2k2|v(t+σ)|2dx≤eλ2(T1+h−t)+2ϵμ∫T1t|z(θτω)|dτ||v(T1)||S2   +λ2eλh2∫T1teλ2(s−t)+2ϵμ∫st|z(θτω)|dτ∫Rnρ|x|2k2e−λ2h|v(s−h)|2−|v|2dx ds   +4c0keλh2∫T1teλ2(s−t)+2ϵμ∫st|z(θτω)|dτ||v||2+||∇v||2ds   +2eλh2∫T1teλ2(s−t)+2ϵμ∫st|z(θτω)|dτ−2ϵz(θsω)∫Rnρ|x|2k2β1(x)+1λβ52(x)+1λg2(x)dx ds   +2c1eλh2∫T1teλ2(s−t)+2ϵμ∫st|z(θτω)|dτ+4−4pp−2ϵz(θsω)+2pp−2ϵz(θs−hω)∫Rnρ|x|2k2β42pp−2(x)dx ds.(97)

Replacing ω by θ–tω, we obtain from (97) that for all t ≥ T1(B, ω),

sup−h≤σ≤0∫Rnρ|x|2k2|vt(σ,θ−tω,v0(θ−tω))|2dx≤eλh2eλ2(T1−t)+2ϵμ∫T1t|z(θτ−tω)|dτ||vT1(θ−t(ω),v0(θ−t(ω)))||S2   +λ2eλh2∫T1teλ2(s−t)+2ϵμ∫st|z(θτ−tω)|dτ∫Rnρ|x|2k2e−λ2h|v(s−h,θ−tω,v0(θ−tω))|2−|v(s,θ−tω,v0(θ−tω))|2dx ds   +4c0keλh2∫T1teλ2(s−t)+2ϵμ∫st|z(θτ−tω)|dτ||v(s,θ−tω,v0(θ−tω))||2+||∇v(s,θ−tω,v0(θ−tω))||2ds   +2eλh2∫T1teλ2(s−t)+2ϵμ∫st|z(θτ−tω)|dτ−2ϵz(θs−tω)∫Rnρ|x|2k2β1(x)+1λβ52(x)+1λg2(x)dx ds   +2c1eλh2∫T1teλ2(s−t)+2ϵμ∫st|z(θτ−tω)|dτ+4−4pp−2ϵz(θs−tω)+2pp−2ϵz(θs−t−hω)∫Rnρ|x|2k2β42pp−2(x)dx ds.(98)

Now we estimate each term on the right hand side of (98). We first replace t by T1 and replace ω by θ–tω in (46), we have that

||v(T1,θ−tω,θ−tv0(ω))||S2≤(1+λh2) eλ2(h−T1)+2ϵμ∫0T1|z(θτ−tω)|dτ||v0(θ−tω)||S2+c2eλ2h∫0T1eλ2(s−T1)+2ϵμ∫sT1|z(θτ−tω)|dτ−2ϵz(θs−tω)ds+c3eλ2h∫0T1eλ2(s−T1)+2ϵμ∫sT1|z(θτ−tω)|dτ+4−4pp−2ϵz(θs−tω)+2pp−2ϵz(θs−t−hω)ds.(99)

Therefore, for the first term on the right hand side of (98),

eλh2eλ2(T1−t)+2ϵμ∫T1t|z(θτ−tω)|dτ||vT1(θ−t(ω),v0(θ−t(ω)))||S2≤(1+λh2)eλh e−λ2t+2ϵμ∫0t|z(θτ−tω)|dτ||v0(θ−tω)||S2+c2eλh∫0T1eλ2(s−t)+2ϵμ∫st|z(θτ−tω)|dτ−2ϵz(θs−tω)ds+c3eλh∫0T1eλ2(s−t)+2ϵμ∫st|z(θτ−tω)|dτ+4−4pp−2ϵz(θs−tω)+2pp−2ϵz(θs−t−hω)ds.(100)

We now estimate the terms in (100) as follows. For the first term, by (26), we find that for t ≥ T1 and v0(θ–tω) ∈ B(θ–tω),

limt→+∞(1+λh2)eλh e−λ2t+2ϵμ∫0t|z(θτ−tω)|dτ||v0(θ−tω)||S2=limt→+∞(1+λh2)eλh e−λ2t+2ϵμ∫−t0|z(θτω)|dτ||v0(θ−tω)||S2≤limt→+∞(1+λh2)eλh e−λ2t+λ4t||v0(θ−tω)||S2=0.(101)

For the second term, by (26)

limt→+∞c2eλh∫0T1eλ2(s−t)+2ϵμ∫st|z(θτ−tω)|dτ−2ϵz(θs−tω)ds=limt→+∞c2eλh∫0T1eλ2(s−t)+2ϵμ∫s−t0|z(θτω)|dτ−2ϵz(θs−tω)ds≤limt→+∞c2eλh∫0T1eλ2(s−t)−λ4(s−t)−λ8(s−t)ds=0.(102)

Similarly, we can get the following estimate for the third term on the right-hand side of (100):

limt→+∞c3eλh∫0T1eλ2(s−t)+2ϵμ∫st|z(θτ−tω)|dτ+4−4pp−2ϵz(θs−tω)+2pp−2ϵz(θs−t−hω)ds=0.(103)

It follows from (100) - (103) that

limt→+∞eλh2eλ2(T1−t)+2ϵμ∫T1t|z(θτ−tω)|dτ||vT1(θ−t(ω),v0(θ−t(ω)))||S2=0,(104)

which implies that for any ϵ1 > 0, there is a T2 = T2(B, ω, ϵ) > T1 such that

eλh2eλ2(T1−t)+2ϵμ∫T1t|z(θτ−tω)|dτ||vT1(θ−t(ω),v0(θ−t(ω)))||S2≤ϵ1.(105)

Next, we estimate the second term on the right hand side of (98). Similarly as (42), we can obtain that

λ2eλh2∫T1teλ2(s−t)+2ϵμ∫st|z(θτ−tω)|dτ∫Rnρ|x|2k2e−λ2h|v(s−h,θ−tω,v0(θ−tω))|2−|v(s,θ−tω,v0(θ−tω))|2dx ds≤λ2eλh2eλ2(T1−t)+2ϵμ∫0t|z(θτ−tω)|dτ∫Rnρ|x|2k2sup−h≤σ≤0|vT1(σ,θ−tω,v0(θ−tω))|2dx ds≤λ2eλh2eλ2(T1−t)+2ϵμ∫0t|z(θτ−tω)|dτ||vT1(θ−tω,v0(θ−tω))||S.(106)

By (104) and (106), we know that for any ϵ1 > 0, there exists a T3 = T3(B, ω, ϵ) > T1 such that

λ2eλh2∫T1teλ2(s−t)+2ϵμ∫st|z(θτ−tω)|dτ××∫Rnρ|x|2k2e−λ2h|v(s−h,θ−tω,v0(θ−tω))|2−|v(s,θ−tω,v0(θ−tω))|2dx ds≤ϵ1.(107)

For the third term on the right hand side of (98), by Lemma 3.1 and Lemma 3.3, we obtain that

∫T1teλ2(s−t)+2ϵμ∫st|z(θτ−tω)|dτ||v(s,θ−tω,v0(θ−tω))||2+||∇v(s,θ−tω,v0(θ−tω))||2ds≤∫T1teλ2(s−t)+2ϵμ∫st|z(θτ−tω)|dτρ12(θs−tω)+ρ3(θs−tω)ds=∫T1−t0eλ2s+2ϵμ∫s0|z(θτω)|dτρ12(θsω)+ρ3(θsω)ds≤∫−∞0eλ2s−λ4sρ12(θsω)+ρ3(θsω)ds<∞.(108)

This implies that there exists a T4 = T4(B, ω, ϵ) > T1 and R2 = R2(ω, ϵ1) such that for all t ≥ T4 and k ≥ R2,

4c0keλh2∫T1teλ2(s−t)+2ϵμ∫st|z(θτ−tω)|dτ||v(s,θ−tω,v0(θ−tω))||2+||∇v(s,θ−tω,v0(θ−tω))||2ds≤ϵ1.(109)

Note that β1 ∈ L1(ℝn), β5 ∈ L2(ℝn), g ∈ L2(ℝn). Thus, there is R3 = R3(ω, ϵ1) such that for all k ≥ R3

∫Rnρ|x|2k2β1(x)+1λβ52(x)+1λg2(x)dx≤∫|x|≥kβ1(x)+1λβ52(x)+1λg2(x)dx≤ϵ1.(110)

Then, for the fourth term on the right hand side of (98) we have that

2eλh2∫T1teλ2(s−t)+2ϵμ∫st|z(θτ−tω)|dτ−2ϵz(θs−tω)∫Rnρ|x|2k2β1(x)+1λβ52(x)+1λg2(x)dx ds≤2ϵ1eλh2∫T1teλ2(s−t)+2ϵμ∫st|z(θτ−tω)|dτ−2ϵz(θs−tω)ds=2ϵ1eλh2∫T1−t0eλ2s+2ϵμ∫s0|z(θτω)|dτ−2ϵz(θsω)ds≤2ϵ1eλh2∫−∞0eλ2s−λ4s−λ8sds=16λeλh2ϵ1.(111)

Notice that β4∈L2pp−2(Rn). Thus, there is R4 = R4(ϵ1, ω) such that for all k ≥ R4,

∫Rnρ|x|2k2β42pp−2(x)dx≤∫|x|≥kβ42pp−2(x)dx≤ϵ1.(112)

Then, similarly as (111), there exists a constant c > 0, such that

2c1eλh2∫T1teλ2(s−t)+2ϵμ∫st|z(θτ−tω)|dτ+4−4pp−2ϵz(θs−tω)+2pp−2ϵz(θs−t−hω)∫Rnρ|x|2k2β42pp−2(x)dx ds≤cϵ1.(113)

Let T = max {T1, T2, T3. T4} and R = max {R1, R2, R3}. Then, it follows from (98), (105), (107), (109), (111) and (113) that

sup−h≤σ≤0∫Rnρ|x|2k2|vt(σ,θ−tω,v0(θ−tω))|2dx≤(c+16λeλh2+3)ϵ1,(114)

This ends the proof.□

Lemma 3.5

Let B(ω) ∈ 𝒟 andv0(ω) ∈ B(ω). Then there are tempered random variablesρ5(ω), ρ6(ω), such that the solution of (30)-(31) satisfies for all t ≥ TB(ω) + 2h + 1, andσ, σ1, σ2 ∈ [–h, 0],

||vt(σ,θ−tω,v0(θ−tω))||H1(Rn)2≤ρ5(ω);(115)
||vt(σ1,θ−tω,v0(θ−tω))−vt(σ2,θ−tω,v0(θ−tω))||≤ρ6(ω)|σ1−σ2|1/2.(116)

Proof

By Lemma 3.1 and Lemma 3.3, we get that for all t ≥ TB(ω) + h + 1, and σ ∈ [–h,0 ],

||vt(σ,θ−tω,v0(θ−tω))||H1(Rn)2=||v(t+σ,θ−tω,v0(θ−tω))||2+||∇v(t+σ,θ−tω,v0(θ−tω))||2≤max−h≤σ≤0ρ12(θσω)+ρ3(θσω)≡ρ5(ω).(117)

For σ1, σ2 ∈ [–h, 0] (assuming σ1 ≤ σ2 for simplicity), one has that,

||vt(σ1,θ−tω,v0(θ−tω))−vt(σ2,θ−tω,v0(θ−tω))||2=||v(t+σ1,θ−tω,v0(θ−tω))−v(t+σ2,θ−tω,v0(θ−tω))||2=||∫t+σ1t+σ2ddtv(s,θ−tω,v0(θ−tω))||2ds≤λ∫t+σ1t+σ2||v(s,θ−tω,v0(θ−tω))||ds+∫t+σ1t+σ2||Δv(s,θ−tω,v0(θ−tω))||ds    +∫t+σ1t+σ2||e−ϵz(θs−tω)F(x,eϵz(θs−tω)v(s,θ−tω,v0(θ−tω)))||ds    +∫t+σ1t+σ2||e−ϵz(θs−tω)G(x,eϵz(θs−t−hω)v(s−h,θ−tω,v0(θ−tω)))||ds    +∫t+σ1t+σ2||e−ϵz(θs−tω)g||ds+ϵμ∫t+σ1t+σ2||z(θs−tω)v(s,θ−tω,v0(θ−tω))||ds.(118)

Now, we estimate each term on the right hand side of (118). By Lemma 3.1, and Lemma 3.3 we find that for all t ≥ TB(ω) + 2h + 1,

∫t+σ1t+σ2||v(s,θ−tω,v0(θ−tω))||ds=∫t+σ1t+σ2||v(s,θ−s(θs−tω),v0(θ−s(θs−tω)))||ds≤max−h≤τ≤0ρ1(θτω)|σ1−σ2|;(119)
∫t+σ1t+σ2||z(θs−tω)v(s,θ−tω,v0(θ−tω))||ds≤max−h≤τ≤0z(θτω)ρ1(θτω)|σ1−σ2|;(120)
∫t+σ1t+σ2||Δv(s,θ−tω,v0(θ−tω))||ds≤∫t+σ1t+σ2||Δv(s,θ−tω,v0(θ−tω))||2ds1/2|σ1−σ2|1/2≤ρ4(ω)|σ1−σ2|1/2;(121)

and

∫t+σ1t+σ2||e−ϵz(θs−tω)g||ds≤max−h≤τ≤0eϵ|z(θτω)|||g|| |σ1−σ2|.(122)

Using conditions (A1) and (A2), we obtain

∫t+σ1t+σ2||e−ϵz(θs−tω)F(x,eϵz(θs−tω)v(s,θ−tω,v0(θ−tω)))||ds≤∫t+σ1t+σ2α2e(p−2)ϵz(θs−tω)||v(s,θ−tω,v0(θ−tω))||p−1+e−ϵz(θs−tω)||β2||ds≤α2max−h≤τ≤0e(p+2)ϵ|z(θτω)|ρ1p−1(θτω)+max−h≤τ≤0eϵ|z(θτω)|||β2|||σ1−σ2|,(123)

and

∫t+σ1t+σ2||e−ϵz(θs−tω)G(x,eϵz(θs−t−hω)v(s−h,θ−tω,v0(θ−tω)))||ds≤max−h≤τ≤0eϵ|z(θτω)|+ϵ|z(θτ−hω)|ρ1(θτ−hω)+||β5|||σ1−σ2|.(124)

The lemma follows from (119) to (124). This ends the proof.□

Next, we use Ascoli theorem to show that Φ is 𝒟 pullback asymptotically compact.

Lemma 3.6

The random dynamical systemΦis 𝒟 pullback asymptotically compact in 𝔖; that is, for ℙ-a.e. ω ∈ Ω, the sequenceΦ(tn,θ−tn,v0(θ−tn))}n=1∞has a convergent subsequence in 𝔖 providedtn → ∞ and v0(θ–tn) ∈ B(θtnω).

Proof

Let Qk be the set of {x ∈ ℝn : |x| < k}. Since tn → ∞, there exists N = N(B, ω) such that tn > TB(ω) + 2h + 1, for all n > N. Using Lemma 3.5, we obtain that for n > N and σ ∈ [–h, 0],

||vtn(σ,θ−tnω,v0(θ−tnω))||H1(Qk)≤ρ5(ω).(125)

This implies that {Φ(tn,θ−tnω,v0(θ−tnω))}n=N∞ is relatively compact in L2(Qk), since the embedding from H1(Qk) to L2(Qk) is compact. From Lemma 3.5, we can also obtain that for all n > N, and σ1, σ2 ∈ [–h, 0],

||vtn(σ1,θ−tnω,v0(θ−tnω))−vtn(σ1,θ−tnω,v0(θ−tnω))||L2(Qk)≤ρ6(ω)|σ1−σ2|1/2.(126)

This means that {Φ(tn,θ−tnω,v0(θ−tnω))}n=N∞ is equicontinuous. By Ascoli Theorem we have that for fixed k∈N,{vtn(σ,θ−tnω,v0(θ−tnω))}n=1∞ is relatively compact in C([–h, 0];L2(Qk)). Therefore, for each k ∈ ℕ, there exists a subsequence {Φ(tni, θ–tniω, v0(θ–tniω))} converges to ηk(⋅, ω) in C([–h, 0]; L2(Qk)). Notice that, for fixed σ ∈ [–h, 0] and ω ∈ Ω, ηk+1(σ, ω) coincides with ηk(σ, ω) on Qk. Hence, for any ϵ > 0, there exists N1 = N1(ω, ϵ) ∈ ℕ, such that for all i > N1,

supσ∈[−h,0]||vtni(σ,θ−tniω,v0(θ−tniω))−η(σ,ω)||L2(Qk)≤ϵ.(127)

By Lemma 3.1, we have that for all σ ∈ [–h, 0], and i > N,

||vtni(σ,θ−tniω,v0(θ−tniω))||L2(Qk)≤supσ∈[−h,0]||vtni(σ,θ−tniω,v0(θ−tniω))||L2(Rn)≤ρ1(ω).(128)

Hence, for all k ∈ ℕ,

||η(σ,ω)||L2(Qk)≤limi→∞||vtni(σ,θ−tniω,v0(θ−tniω))||L2(Qk)≤ρ1(ω).(129)

It follows that,

||η(σ,ω)||L2(Rn)≤supk∈N||η(σ,ω)||L2(Qk)≤ρ1(ω).(130)

Therefore, η(σ, ω) ∈ L2(ℝn). By Lemma 3.4, for every B ∈ 𝒟 and ϵ1 > 0, there exist K = K(ω, ϵ1) and N2 = N2(ω, ϵ1), such that for all i > N2 and k > K,

supσ∈[−h,0]||vtni(σ,θ−tni,v0(θ−tni))||L2(|x|≥k)≤ϵ1.(131)

Notice that for all l > k > K and σ ∈ [–h, 0],

||ξ(σ,ω)||L2(k≤|x|<l)=limi→∞||vtni(σ,θ−tniω,v0(θ−tniω))||L2(k≤|x|<l)≤supσ∈[−h,0],i>N2||vtni(σ,θ−tniω,v0(θ−tniω))||L2(|x|≥k)≤ϵ1.(132)

It follows that

supσ∈[−h,0]||ξ(σ,ω)||L2(|x|≥k)≤supσ∈[−h,0],l>k||ξ(σ,ω)||L2(k≤|x|<l)≤ϵ1.(133)

Then, (113), (127) and (131) imply that for all i > max N1, N2,

supσ∈[−h,0]||vtni(σ,θ−tniω,v0(θ−tniω))−ξ(σ,ω)||≤supσ∈[−h,0]||vtni(σ,θ−tniω,v0(θ−tniω))−ξ(σ,ω)||L2(Qk)+2supσ∈[−h,0]||vtni(σ,θ−tniω,v0(θ−tniω))||L2(|x|≥k) +2supσ∈[−h,0]||ξ(σ,ω)||L2(|x|≥k)≤5ϵ1.(134)

This ends the proof.□

Theorem 3.7

The random dynamical systemΦhas a unique 𝒟-random attractor in 𝔖.

Proof

By Lemma 3.1, we know that Φ has a random absorbing set, and by Lemma 3.6, we obtain that Φ is 𝒟-pullback asymptotically compact in 𝔖. Therefore, by Proposition 2.1, Φ has an unique 𝒟-pullback attractor. This ends the proof.□

  1. Competing interests: The authors declare that there is no conflict of interest regarding the publication of this paper.

Acknowledgement

This work is partially supported by the National Natural Science Foundation of China under Grants 11301153 and 11271110, the Key Programs for Science and Technology of the Education Department of Henan Province under Grand 12A110007, and the Scientific Research Funds of Henan University of Science and Technology.

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Received: 2017-12-10
Accepted: 2018-06-12
Published Online: 2018-08-03

© 2018 Jia et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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  3. On a viscous two-fluid channel flow including evaporation
  4. Generation of pseudo-random numbers with the use of inverse chaotic transformation
  5. Singular Cauchy problem for the general Euler-Poisson-Darboux equation
  6. Ternary and n-ary f-distributive structures
  7. On the fine Simpson moduli spaces of 1-dimensional sheaves supported on plane quartics
  8. Evaluation of integrals with hypergeometric and logarithmic functions
  9. Bounded solutions of self-adjoint second order linear difference equations with periodic coeffients
  10. Oscillation of first order linear differential equations with several non-monotone delays
  11. Existence and regularity of mild solutions in some interpolation spaces for functional partial differential equations with nonlocal initial conditions
  12. The log-concavity of the q-derangement numbers of type B
  13. Generalized state maps and states on pseudo equality algebras
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  15. Note on group irregularity strength of disconnected graphs
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  24. On algebraic characterization of SSC of the Jahangir’s graph 𝓙n,m
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  30. Restriction conditions on PL(7, 2) codes (3 ≤ |𝓖i| ≤ 7)
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  46. Topological entropy for positively weak measure expansive shadowable maps
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  117. Periodic and subharmonic solutions for a 2nth-order p-Laplacian difference equation containing both advances and retardations
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https://doi.org/10.1515/math-2018-0076
Keywords for this article
Stochastic reaction-diffusion system; Random attractor; Time-delay; Multiplicative white noise
Creative Commons
BY-NC-ND 4.0

Articles in the same Issue

  1. Regular Articles
  2. Algebraic proofs for shallow water bi–Hamiltonian systems for three cocycle of the semi-direct product of Kac–Moody and Virasoro Lie algebras
  3. On a viscous two-fluid channel flow including evaporation
  4. Generation of pseudo-random numbers with the use of inverse chaotic transformation
  5. Singular Cauchy problem for the general Euler-Poisson-Darboux equation
  6. Ternary and n-ary f-distributive structures
  7. On the fine Simpson moduli spaces of 1-dimensional sheaves supported on plane quartics
  8. Evaluation of integrals with hypergeometric and logarithmic functions
  9. Bounded solutions of self-adjoint second order linear difference equations with periodic coeffients
  10. Oscillation of first order linear differential equations with several non-monotone delays
  11. Existence and regularity of mild solutions in some interpolation spaces for functional partial differential equations with nonlocal initial conditions
  12. The log-concavity of the q-derangement numbers of type B
  13. Generalized state maps and states on pseudo equality algebras
  14. Monotone subsequence via ultrapower
  15. Note on group irregularity strength of disconnected graphs
  16. On the security of the Courtois-Finiasz-Sendrier signature
  17. A further study on ordered regular equivalence relations in ordered semihypergroups
  18. On the structure vector field of a real hypersurface in complex quadric
  19. Rank relations between a {0, 1}-matrix and its complement
  20. Lie n superderivations and generalized Lie n superderivations of superalgebras
  21. Time parallelization scheme with an adaptive time step size for solving stiff initial value problems
  22. Stability problems and numerical integration on the Lie group SO(3) × R3 × R3
  23. On some fixed point results for (s, p, α)-contractive mappings in b-metric-like spaces and applications to integral equations
  24. On algebraic characterization of SSC of the Jahangir’s graph 𝓙n,m
  25. A greedy algorithm for interval greedoids
  26. On nonlinear evolution equation of second order in Banach spaces
  27. A primal-dual approach of weak vector equilibrium problems
  28. On new strong versions of Browder type theorems
  29. A Geršgorin-type eigenvalue localization set with n parameters for stochastic matrices
  30. Restriction conditions on PL(7, 2) codes (3 ≤ |𝓖i| ≤ 7)
  31. Singular integrals with variable kernel and fractional differentiation in homogeneous Morrey-Herz-type Hardy spaces with variable exponents
  32. Introduction to disoriented knot theory
  33. Restricted triangulation on circulant graphs
  34. Boundedness control sets for linear systems on Lie groups
  35. Chen’s inequalities for submanifolds in (κ, μ)-contact space form with a semi-symmetric metric connection
  36. Disjointed sum of products by a novel technique of orthogonalizing ORing
  37. A parametric linearizing approach for quadratically inequality constrained quadratic programs
  38. Generalizations of Steffensen’s inequality via the extension of Montgomery identity
  39. Vector fields satisfying the barycenter property
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  43. Recursive interpolating sequences
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  45. Constacyclic codes over 𝔽pm[u1, u2,⋯,uk]/〈 ui2 = ui, uiuj = ujui〉
  46. Topological entropy for positively weak measure expansive shadowable maps
  47. Oscillation and non-oscillation of half-linear differential equations with coeffcients determined by functions having mean values
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  50. A reduced space branch and bound algorithm for a class of sum of ratios problems
  51. Some recurrence formulas for the Hermite polynomials and their squares
  52. A relaxed block splitting preconditioner for complex symmetric indefinite linear systems
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  56. A stochastic differential game of low carbon technology sharing in collaborative innovation system of superior enterprises and inferior enterprises under uncertain environment
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  58. The points and diameters of quantales
  59. Directed colimits of some flatness properties and purity of epimorphisms in S-posets
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  64. On Banach and Kuratowski Theorem, K-Lusin sets and strong sequences
  65. On the boundedness of square function generated by the Bessel differential operator in weighted Lebesque Lp,α spaces
  66. On the different kinds of separability of the space of Borel functions
  67. Curves in the Lorentz-Minkowski plane: elasticae, catenaries and grim-reapers
  68. Functional analysis method for the M/G/1 queueing model with single working vacation
  69. Existence of asymptotically periodic solutions for semilinear evolution equations with nonlocal initial conditions
  70. The existence of solutions to certain type of nonlinear difference-differential equations
  71. Domination in 4-regular Knödel graphs
  72. Stepanov-like pseudo almost periodic functions on time scales and applications to dynamic equations with delay
  73. Algebras of right ample semigroups
  74. Random attractors for stochastic retarded reaction-diffusion equations with multiplicative white noise on unbounded domains
  75. Nontrivial periodic solutions to delay difference equations via Morse theory
  76. A note on the three-way generalization of the Jordan canonical form
  77. On some varieties of ai-semirings satisfying xp+1 ≈ x
  78. Abstract-valued Orlicz spaces of range-varying type
  79. On the recursive properties of one kind hybrid power mean involving two-term exponential sums and Gauss sums
  80. Arithmetic of generalized Dedekind sums and their modularity
  81. Multipreconditioned GMRES for simulating stochastic automata networks
  82. Regularization and error estimates for an inverse heat problem under the conformable derivative
  83. Transitivity of the εm-relation on (m-idempotent) hyperrings
  84. Learning Bayesian networks based on bi-velocity discrete particle swarm optimization with mutation operator
  85. Simultaneous prediction in the generalized linear model
  86. Two asymptotic expansions for gamma function developed by Windschitl’s formula
  87. State maps on semihoops
  88. 𝓜𝓝-convergence and lim-inf𝓜-convergence in partially ordered sets
  89. Stability and convergence of a local discontinuous Galerkin finite element method for the general Lax equation
  90. New topology in residuated lattices
  91. Optimality and duality in set-valued optimization utilizing limit sets
  92. An improved Schwarz Lemma at the boundary
  93. Initial layer problem of the Boussinesq system for Rayleigh-Bénard convection with infinite Prandtl number limit
  94. Toeplitz matrices whose elements are coefficients of Bazilevič functions
  95. Epi-mild normality
  96. Nonlinear elastic beam problems with the parameter near resonance
  97. Orlicz difference bodies
  98. The Picard group of Brauer-Severi varieties
  99. Galoisian and qualitative approaches to linear Polyanin-Zaitsev vector fields
  100. Weak group inverse
  101. Infinite growth of solutions of second order complex differential equation
  102. Semi-Hurewicz-Type properties in ditopological texture spaces
  103. Chaos and bifurcation in the controlled chaotic system
  104. Translatability and translatable semigroups
  105. Sharp bounds for partition dimension of generalized Möbius ladders
  106. Uniqueness theorems for L-functions in the extended Selberg class
  107. An effective algorithm for globally solving quadratic programs using parametric linearization technique
  108. Bounds of Strong EMT Strength for certain Subdivision of Star and Bistar
  109. On categorical aspects of S -quantales
  110. On the algebraicity of coefficients of half-integral weight mock modular forms
  111. Dunkl analogue of Szász-mirakjan operators of blending type
  112. Majorization, “useful” Csiszár divergence and “useful” Zipf-Mandelbrot law
  113. Global stability of a distributed delayed viral model with general incidence rate
  114. Analyzing a generalized pest-natural enemy model with nonlinear impulsive control
  115. Boundary value problems of a discrete generalized beam equation via variational methods
  116. Common fixed point theorem of six self-mappings in Menger spaces using (CLRST) property
  117. Periodic and subharmonic solutions for a 2nth-order p-Laplacian difference equation containing both advances and retardations
  118. Spectrum of free-form Sudoku graphs
  119. Regularity of fuzzy convergence spaces
  120. The well-posedness of solution to a compressible non-Newtonian fluid with self-gravitational potential
  121. On further refinements for Young inequalities
  122. Pretty good state transfer on 1-sum of star graphs
  123. On a conjecture about generalized Q-recurrence
  124. Univariate approximating schemes and their non-tensor product generalization
  125. Multi-term fractional differential equations with nonlocal boundary conditions
  126. Homoclinic and heteroclinic solutions to a hepatitis C evolution model
  127. Regularity of one-sided multilinear fractional maximal functions
  128. Galois connections between sets of paths and closure operators in simple graphs
  129. KGSA: A Gravitational Search Algorithm for Multimodal Optimization based on K-Means Niching Technique and a Novel Elitism Strategy
  130. θ-type Calderón-Zygmund Operators and Commutators in Variable Exponents Herz space
  131. An integral that counts the zeros of a function
  132. On rough sets induced by fuzzy relations approach in semigroups
  133. Computational uncertainty quantification for random non-autonomous second order linear differential equations via adapted gPC: a comparative case study with random Fröbenius method and Monte Carlo simulation
  134. The fourth order strongly noncanonical operators
  135. Topical Issue on Cyber-security Mathematics
  136. Review of Cryptographic Schemes applied to Remote Electronic Voting systems: remaining challenges and the upcoming post-quantum paradigm
  137. Linearity in decimation-based generators: an improved cryptanalysis on the shrinking generator
  138. On dynamic network security: A random decentering algorithm on graphs
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