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This paper is concerned with the convergence of invariant measures in the Wasserstein sense for fractional stochastic reaction–diffusion equations defined on unbounded domains as the noise intensity approaches zero. Based on uniform estimates of solutions, we prove the family of invariant measures of the stochastic equations converges to the invariant measure of the corresponding deterministic equations in terms of the Wasserstein metric. We also provide the rate of such convergence. [ABSTRACT FROM AUTHOR]

This article is concerned with the asymptotic behavior of solutions of a wide class of non-autonomous, non-local, fractional, stochastic p -Laplacian equations driven by multiplicative white noise. The time-dependent nonlinear drift term of the equation has a polynomial growth of arbitrary order in its third component which is allowed to be greater than the exponent p. We first employ the Faedo–Galerkin method to prove the well-posedness of the equation in an appropriate Hilbert space. We then establish the existence, uniqueness and periodicity of tempered pullback random attractors for the equations. The upper semi-continuity of these attractors is also derived as the density of noise tends to zero. The results of this paper are new even when the stochastic equation reduces to the deterministic one. [ABSTRACT FROM AUTHOR]

We first apply the Galerkin method to prove the existence and uniqueness of solutions for a class of non-autonomous fractional reaction–diffusion equations driven by multiplicative noise. We then establish the existence and uniqueness of tempered pullback random attractors for the equations in an appropriate Hilbert space. The upper semicontinuity of random attractors is also obtained when the intensity of noise approaches zero. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study the existence, stability and bifurcation of random complete and periodic solutions for stochastic parabolic equations with multiplicative noise. We first prove the existence and uniqueness of tempered random attractors for the stochastic equations and characterize the structures of the attractors by random complete solutions. We then examine the existence and stability of random complete quasi-solutions and establish the relations of these solutions and the structures of tempered attractors. When the stochastic equations are incorporated with periodic forcing, we obtain the existence and stability of random periodic solutions. For the stochastic Chafee–Infante equation, we further establish the multiplicity and stochastic bifurcation of complete and periodic solutions. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the asymptotic behavior of solutions of a stochastic nonlinear wave equation with dispersive and dissipative terms defined on an unbounded domain. It is proved that the random dynamical system generated by the equation has a random attractor in a Sobolev space. To overcome the difficulty caused by the non-compactness of Sobolev embeddings on unbounded domains, a cut-off method and a decomposition trick are combined to prove the asymptotic compactness of the solutions. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the dynamics of an infinite-dimensional gradient system under small almost periodic perturbations. Under the assumption that the original autonomous system has a global attractor given as the union of unstable manifolds of a finite number of hyperbolic equilibrium solutions, we prove that the perturbed non-autonomous system has exactly the same number of almost periodic solutions. As a consequence, the pullback attractor of the perturbed system is given by the union of unstable manifolds of these finitely many almost periodic solutions. An application of the result to the Chafee–Infante equation is discussed. [Copyright &y& Elsevier]

Abstract: We study the asymptotic behavior of solutions of a class of singularly perturbed non-autonomous parabolic equations defined on with unbounded external terms. We first prove the existence of a pullback attractor for the perturbed equation in and then establish the upper semicontinuity of these attractors as small perturbations approach zero. The uniform estimates on the tails of solutions are used to overcome the difficulty caused by the non-compactness of Sobolev embeddings on unbounded domains. [Copyright &y& Elsevier]

Abstract: We investigate the asymptotic behavior of solutions of a class of non-autonomous degenerate parabolic equations on with unbounded external forcing terms. The existence of a pullback global attractor is proved in for any . [Copyright &y& Elsevier]

Abstract: The existence of a random attractor for the stochastic FitzHugh–Nagumo system defined on an unbounded domain is established. The pullback asymptotic compactness of the stochastic system is proved by uniform estimates on solutions for large space and time variables. These estimates are obtained by a cut-off technique. [Copyright &y& Elsevier]

Abstract: The existence of a pullback attractor is established for the singularly perturbed FitzHugh–Nagumo system defined on the entire space when external terms are unbounded in a phase space. The pullback asymptotic compactness of the system is proved by using uniform a priori estimates for far-field values of solutions. Although the limiting system has no global attractor, we show that the pullback attractors for the perturbed system with bounded external terms are uniformly bounded, and hence do not blow up as a small parameter approaches zero. [Copyright &y& Elsevier]

Abstract: We prove the existence of a global attractor for the Newton–Boussinesq equation defined in a two-dimensional channel. The asymptotic compactness of the equation is derived by the uniform estimates on the tails of solutions. We also establish the regularity of the global attractor. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the continuity of strong solutions of the Reaction–Diffusion equation with respect to initial conditions. It is proved that the solutions of the equation are continuous in with respect to initial data provided the nonlinearity satisfies a dissipative condition with polynomial growth order . [Copyright &y& Elsevier]

We investigate the asymptotic behavior of solutions of the p -Laplace equation driven simultaneously by non-autonomous deterministic forcing and multiplicative white noise on R n . We show the tails of solutions of the equation are uniformly small outside a bounded domain, which is used to derive asymptotic compactness of solution operators in L 2 ( R n ) by overcoming the non-compactness of Sobolev embeddings on unbounded domains. We then prove existence and uniqueness of random attractors and further establish upper semicontinuity of attractors as the intensity of noise approaches zero. [ABSTRACT FROM AUTHOR]

Natural anthocyanins are safer and nutritious as compared to synthetic pigments; however, their stability is poor. They can produce spontaneous copigmentation with organic acids, leading to the improvement of colour stability, albeit slowly. Box–Behnken experimental design was used to elucidate the mechanism of copigmentation between Vitis amurensis Rupr anthocyanins (0.1 mg/mL) and organic acids (0.87 mg/mL, ferulic acid: d -gluconic acid:caffeic acid:vanillic acid = 1.5:2.5:2.5:0.5, w/w/w/w) promoted by high hydrostatic pressure (HHP; 300 MPa, 2 min). The copigmentation effect and antioxidant activity of anthocyanins were also evaluated. The structure of anthocyanins was analysed using ultraviolet–visible spectroscopy, Fourier-transform infrared spectroscopy, high-performance liquid chromatography, and mass spectrometry. The results of HHP copigmentation showed that the following anthocyanins were newly formed—delphinidin-3- O -catechol, petunidin-3- O -catechol, delphinidin-4-vinyl-catechol, petunidin-3- O -guaiacol, malvidin-4-vinyl-guaiacol, cyanidin-3- O -(6″- O -caffeoyl)-glucoside, peonidin-3- O -(6″- O -caffeoyl)-glucoside, delphinidin-3- O -(6″- O -caffeoyl)-glucoside, malvidin-3- O -glucoside-4-vinyl-guaiacol, and malvidin-3- O -(6″- O -feruloyl)-glucoside—owing to appropriate modifications that increased the copigmentation rate (R = 42.12%), photo-thermal stability (R > 45%), and potential antioxidant activities expressed in vivo ( p  < 0.01 vs. Model Group). [ABSTRACT FROM AUTHOR]