Asymptotic Compactness Of Solutions Research Articles

Well-posedness and long term behavior of supercritical wave equations driven by nonlinear colored noise on [formula omitted

This paper is concerned with the well-posedness and long term behavior of the non-autonomous random wave equations driven by nonlinear colored noise on Rn with n≤5. The drift nonlinearity has a supercritical growth exponent (n+2)/(n−2). We first prove the existence and uniqueness of solutions in the energy space by showing the non-concentration of energy via the Morawetz identity and the uniform Strichartz estimates. We then prove the existence and uniqueness of tempered pullback random attractors of the non-autonomous random dynamical system associated with the equation. The asymptotic compactness of solutions is obtained by the idea of energy equation due to Ball and the uniform tail-ends estimates in order to circumvent the difficulty caused by the lack of compactness of Sobolev embeddings on unbounded domains.

Long Term Behavior for a Class of Stochastic Delay Lattice Systems inXρSpace

In this paper, we focus on the asymptotic behavior of solutions to stochastic delay lattice equations with additive noise and deterministic forcing. We first show the existence of a continuous random dynamical system for the equations. Then we investigate the pullback asymptotical compactness of solutions as well as the existence and uniqueness of tempered random attractor inXρspace. Finally, ergodicity of the systems is achieved.

RANDOM ATTRACTORS FOR NON-AUTONOMOUS FRACTIONAL STOCHASTIC GINZBURG-LANDAU EQUATIONS ON UNBOUNDED DOMAINS

This paper deals with the dynamical behavior of solutions for non-autonomous stochastic fractional Ginzburg-Landau equations driven by additive noise with $ \alpha\in(0,1) $. We prove the existence and uniqueness of tempered pullback random attractors for the equations in $ L^{2}({\bf{R}}^{3}) $. In addition, we also obtain the upper semicontinuity of random attractors when the intensity of noise approaches zero. The main difficulty here is the noncompactness of Sobolev embeddings on unbounded domains. To solve this, we establish the pullback asymptotic compactness of solutions in $ L^{2}({\bf{R}}^{3}) $ by the tail-estimates of solutions.

Tempered random attractors of a non-autonomous non-local fractional equation driven by multiplicative white noise

In this paper, we prove the existence and regularity of a tempered pullback attractor for a non-autonomous stochastic fractional equation involving an abstract non-local operator defined aswhere is the kernel of which satisfies the general fractional-type condition of order and is a ball in centered at x with radius ε. The asymptotical compactness of solutions in is demonstrated by the spectral decomposition and truncation technique, where is a Hilbert space with and As a special example, we obtain the existence and regularity of tempered pullback attractor of stochastic fractional reaction-diffusion equation.

Upper semicontinuity of attractors of stochastic delay reaction-diffusion equations in the delay

A system of stochastic delayed reaction-diffusion equations with multiplicative noise and deterministic non-autonomous forcing is considered. We first prove the existence and uniqueness of a bi-spatial pullback attractor for these equations when the initial space is C−ρ,0,L2O and the terminate space is C−ρ,0,H01O. The asymptotic compactness of solutions in C−ρ,0,H01O is established by combining “positive and negative truncations” technique and some new estimates on solutions. Then the periodicity of the random attractors is proved when the stochastic delay equations are forced by periodic functions. Finally, upper semicontinuity of the global random attractors in the delay is obtained as the length of time delay approaches zero.

Random dynamics of stochastic p-Laplacian equations on [formula omitted] with an unbounded additive noise

This article is concerned with the random dynamics of solutions of the non-autonomous parabolic p-Laplacian equations on RN driven by an unbounded additive noise, where the nonlinearity has a (p,q)-growth exponents including the general polynomial-type functions. Firstly by using an inductive method, we establish the higher-order integrability of difference of solutions near the initial time for arbitrary space dimension. Secondly based on this very estimate, the asymptotical compactness of solutions in Lp(RN)∩Lq(RN) for any p,q≥2 is proved by a new technique instead of the usual truncation estimate method, and then we obtain the existence of pullback attractor in Lp(RN)∩Lq(RN) for p-Laplacian equations with an unbounded additive noise. Thirdly we prove that the obtained L2-pullback attractor is attracting and translation bounded in Lδ(RN) topology for any δ∈[2,∞) and arbitrary space dimension N,N≥1.

Multivalued non-autonomous random dynamical systems for wave equations without uniqueness

This paper deals with the multivalued non-autonomous random dynamical system generated by the non-autonomous stochastic wave equations on unbounded domains, which has a non-Lipschitz nonlinearity with critical exponent in the three dimensional case. We introduce the concept of weak upper semicontinuity of multivalued functions and use such continuity to prove the measurability of multivalued functions from a metric space to a separable Banach space. By this approach, we show the measurability of pullback attractors of the multivalued random dynamical system of the wave equations regardless of the completeness of the underlying probability space. The asymptotic compactness of solutions is proved by the method of energy equations, and the difficulty caused by the non-compactness of Sobolev embeddings on $\mathbb{R}^n$ is overcome by the uniform estimates on the tails of solutions.

Random attractors for stochastic two-compartment Gray-Scott equations with a multiplicative noise

Abstract In this paper, we consider the existence of a pullback attractor for the random dynamical system generated by stochastic two-compartment Gray-Scott equation for a multiplicative noise with the homogeneous Neumann boundary condition on a bounded domain of space dimension n ≤ 3. We first show that the stochastic Gray-Scott equation generates a random dynamical system by transforming this stochastic equation into a random one. We also show that the existence of a random attractor for the stochastic equation follows from the conjugation relation between systems. Then, we prove pullback asymptotical compactness of solutions through the uniform estimate on the solutions. Finally, we obtain the existence of a pullback attractor.

Regularity of random attractors for stochastic reaction-diffusion equations on unbounded domains

The existence of a unique random attractors in [Formula: see text] for a stochastic reaction-diffusion equation with time-dependent external forces is proved. Due to the presence of both random and non-autonomous deterministic terms, we use a new theory of random attractors which is introduced in [B. Wang, J. Differential Equations 253 (2012) 1544–1583] instead of the usual one. The asymptotic compactness of solutions in [Formula: see text] is established by combining “tail estimate” technique and some new estimates on solutions. This work improves some recent results about the regularity of random attractors for stochastic reaction-diffusion equations.

Pullback𝒟-Attractor of Nonautonomous Three-Component Reversible Gray-Scott System on Unbounded Domains

The long time behavior of solutions of the nonautonomous three-components reversible Gray-Scott system defined on the entire spaceℝnis studied when the external forcing terms are unbounded in a phase space. The existence of a pullback global attractor for the equation is established inL2ℝn3andH1ℝn3, respectively. The pullback asymptotic compactness of solutions is proved by using uniform estimates on the tails of solutions on unbounded domains.

Random attractors for partly dissipative stochastic lattice dynamical systems1

In this paper, we consider the long term behavior for the stochastic lattice dynamical systems with some partly dissipative nonlinear term in l 2 × l 2. The main purpose of this paper is to establish the existence of a compact global random attractor. The uniqueness and existence is first proved for the solution of an infinite dimensional random dynamical system, and a priori estimate is obtained on the solutions. The existence of a random absorbing set is then discussed for the systems, and an estimate on tails of the solutions is derived when the time is large enough, which ensures the asymptotic compactness of solutions. Finally, the global random attractor is proved to exist within the set of tempered random bounded sets rather than all bounded deterministic sets, i.e. the stochastic lattice system has a global random attractor in l 2 × l 2.