PROC
Asymptotical dynamics of the modified Schnackenberg equations
Yuncheng You
Conference Publications 2009, 2009(Special): 857-868 doi: 10.3934/proc.2009.2009.857
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.
keywords: absorbing set Schnackenberg equation asymptotical compactness global dynamics global attractor
DCDS
Random attractor of stochastic Brusselator system with multiplicative noise
Junyi Tu Yuncheng You
Discrete & Continuous Dynamical Systems - A 2016, 36(5): 2757-2779 doi: 10.3934/dcds.2016.36.2757
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.
keywords: Random attractor stochastic Brusselator system pullback asymptotic compactness flattening property. multiplicative noise
CPAA
Asymptotic dynamics of reversible cubic autocatalytic reaction-diffusion systems
Yuncheng You
Communications on Pure & Applied Analysis 2011, 10(5): 1415-1445 doi: 10.3934/cpaa.2011.10.1415
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.
keywords: cubic autocatalysis Reversible reaction-diffusion system Gray-Scott equations exponential attractor. global attractor asymptotic dynamics
DCDS-S
Asymptotical dynamics of Selkov equations
Yuncheng You
Discrete & Continuous Dynamical Systems - S 2009, 2(1): 193-219 doi: 10.3934/dcdss.2009.2.193
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.
keywords: global attractor absorbing set asymptotic compactness asymptotical dynamics fractal dimension. Selkov equation
DCDS
Preface
Yuncheng You
Discrete & Continuous Dynamical Systems - A 2014, 34(1): i-iii doi: 10.3934/dcds.2014.34.1i
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
Random attractors and robustness for stochastic reversible reaction-diffusion systems
Yuncheng You
Discrete & Continuous Dynamical Systems - A 2014, 34(1): 301-333 doi: 10.3934/dcds.2014.34.301
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.
keywords: random robustness. asymptotic dynamics pullback uniform dissipation stochastic reaction-diffusion system Random attractor
CPAA
Global attractor of the Gray-Scott equations
Yuncheng You
Communications on Pure & Applied Analysis 2008, 7(4): 947-970 doi: 10.3934/cpaa.2008.7.947
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.
keywords: global attractor fractal dimension. global dynamics asymptotic compactness Gray-Scott equation absorbing set
DCDS-S
Random attractor for stochastic reversible Schnackenberg equations
Yuncheng You
Discrete & Continuous Dynamical Systems - S 2014, 7(6): 1347-1362 doi: 10.3934/dcdss.2014.7.1347
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.
keywords: Random attractor pullback absorbing property. stochastic Schnackenberg equations asymptotic dynamics
PROC
Pullback uniform dissipativity of stochastic reversible Schnackenberg equations
Yuncheng You
Conference Publications 2015, 2015(special): 1134-1142 doi: 10.3934/proc.2015.1134
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.
keywords: pullback absorbing set Reaction-diffusion system random attractor. pullback uniform dissipativity stochastic Schnackenberg equations
DCDS-B
Dynamics of three-component reversible Gray-Scott model
Yuncheng You
Discrete & Continuous Dynamical Systems - B 2010, 14(4): 1671-1688 doi: 10.3934/dcdsb.2010.14.1671
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.
keywords: asymptotic compactness. Reversible Gray-Scott system global attractor asymptotic dynamics absorbing set

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