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### Open Access Journals

PROC

The existence of a global attractor in the $L^2$ product phase space for the solution semiflow of the modified Schnackenberg equations with the Dirichlet boundary condition on a bounded domain of space dimension $n\le 3$ is proved. This reaction-diffusion system features two pairs of oppositely-signed nonlinear terms so that the dissipative sign-condition is not satisfied. The proof features two types of rescaling and grouping estimation in showing the absorbing property and the uniform smallness in proving the asymptotical compactness by the approach of a new decomposition.

DCDS

Asymptotic dynamics of stochastic Brusselator system with multiplicative noise is investigated in this work. The existence of random attractor is proved via the exponential transformation of Ornstein-Uhlenbeck process and some challenging estimates. The proof of pullback asymptotic compactness here is more rigorous through the bootstrap pullback estimations than a non-dynamical substitution of Brownian motion by its backward translation. It is also shown that the random attractor has the attracting regularity to be an $(L^2\times L^2,H^1\times H^1)$ random attractor.

CPAA

In this paper we prove the existence of a global attractor, an $(H,E)$ global attractor, and an exponential attractor for the cubic autocatalytic reaction-diffusion systems represented by the reversible Gray-Scott equations. The two pairs of oppositely signed nonlinear terms feature the challenge in conducting various estimates. A new rescaling and grouping estimation method is introduced and combined with the other approaches to achieve the proof of dissipation, asymptotic compactness, and discrete squeezing property in all the stages.

DCDS-S

The existence of a global attractor for the solution semiflow of Selkov equations with Neumann boundary conditions on a bounded domain in space dimension $n\le 3$ is proved. This reaction-diffusion system features the oppositely-signed nonlinear terms so that the dissipative sign-condition is not satisfied. The asymptotical compactness is shown by a new decomposition method. It is also proved that the Hausdorff dimension and fractal dimension of the global attractor are finite.

DCDS

The theory of infinite dimensional and stochastic dynamical systems is a rapidly expanding and vibrant field of mathematics. In the recent three decades it has been highlighted as a core knowledge and an advancing thrust in the qualitative study of complex systems and processes described by evolutionary partial differential equations in many different settings, stochastic differential equations, functional differential equations and lattice differential equations. The central research topics include the invariant and attracting sets, stability and bifurcation of patterns and waves, asymptotic theory of dissipative systems and reduction of dimensions, and more and more problems of nonlocal systems, ill-posed systems, multicomponent and network dynamics, random dynamics and chaotic dynamics.

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DCDS

For a typical stochastic reversible reaction-diffusion system with multiplicative white noise, the trimolecular autocatalytic Gray-Scott system on a three-dimensional bounded domain with random noise perturbation proportional to the state of the system, the existence of a random attractor and its robustness with respect to the reverse reaction rates are proved through sharp and uniform estimates showing the pullback uniform dissipation and the pullback asymptotic compactness.

CPAA

In this work the existence of a global attractor for the solution semiflow of the
Gray-Scott equations with the Neumann boundary conditions on bounded domains
of space dimensions $n\leq 3$ is proved. This reaction-diffusion system does
not have dissipative property inherently due to the oppositely signed nonlinearity.
The asymptotical compactness is shown by a new decomposition method.
It is also proved that the Hausdorff dimension and the fractal dimension of the global attractor are finite.

DCDS-S

Asymptotic dynamics of stochastic reversible Schnackenberg equations with multiplicative white noise, which is a typical trimolecular autocatalytic reaction-diffusion system on a three-dimensional bounded domain with Dirichlet boundary condition, is investigated in this paper. The existence of a random attractor is proved through uniform grouping estimates showing the pullback absorbing property and the pullback asymptotic compactness.

PROC

Asymptotic dynamics of stochastic reversible Schnackenberg equations with multiplicative white noise on a three-dimensional bounded domain is investigated in this paper. The pullback uniform dissipativity in terms of the existence of a common pullback absorbing set with respect to the reverse reaction rate of this typical autocatalytic reaction-diffusion system is proved through decomposed grouping estimates.

DCDS-B

The existence of a global attractor for the solution semiflow of a three-component reversible Gray-Scott system with Neumann boundary condition on a bounded domain of space dimension $n\le 3$ is proved. The methodology features the re-scaling and grouping estimation to overcome the difficulty of non-dissipative coupling of three variables and the coefficient barrier. It is also shown that the global attractor turns out to be an $(H, E)$ global attractor.

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