Dynamics for a non-autonomous reaction diffusion model with the fractional diffusion
Wen Tan Chunyou Sun
Discrete & Continuous Dynamical Systems - A 2017, 37(12): 6035-6067 doi: 10.3934/dcds.2017260

In this paper, we study the dynamics of a non-autonomous reaction diffusion model with the fractional diffusion on the whole space. We firstly prove the existence of a $(L^2,L^2)$ pullback $\mathscr{D}_μ$ -attractor of this model. Then we show that the pullback $\mathscr{D}_μ$ -attractor attract the $\mathscr{D}_μ$ class (especially all $L^2$ -bounded set) in $L^{2+δ}$-norm for any $δ∈[0,∞)$. Moreover, the solution of the model is shown to be continuous in $H^s$ with respect to initial data under a slightly stronger condition on external forcing term. As an application, we prove that the $(L^2,L^2)$ pullback $\mathscr{D}_{μ}$-attractor indeed attract the class of $\mathscr{D}_{μ}$ in $H^s$ -norm, and thus the existence of a $(L^2, H^s)$ pullback $\mathscr{D}_μ$ -attractor is obtained.

keywords: Non-autonomous fractional diffusion Tail estimate $(L^2, L^{2+δ})$ pullback attraction $(L^2, H^s)$ pullback attractors
Dynamics for the complex Ginzburg-Landau equation on non-cylindrical domains I: The diffeomorphism case
Feng Zhou Chunyou Sun
Discrete & Continuous Dynamical Systems - B 2016, 21(10): 3767-3792 doi: 10.3934/dcdsb.2016120
The purpose of this article is to analyze the dynamics of the following complex Ginzburg-Landau equation \begin{align*} \partial_{t}u-(\lambda+i\alpha)\Delta u+(\kappa+i\beta)|u|^{p-2}u-\gamma u=f(t) \end{align*} on non-cylindrical domains, which are obtained by diffeomorphic transformation of a bounded base domain, without any upper restriction on $p>2$, only with some restriction on $\beta/\kappa$. We establish the existence and uniqueness of strong and weak solutions as well as some energy inequalities for this equation on variable domains. Moreover the existence of a $\mathscr{D}$-pullback attractor is established for the process generated by the weak solutions under a slightly weaker condition that the measure of the spatial domains in the past is uniformly bounded above.
keywords: Complex Ginzburg-Landau equation $\mathscr{D}$-pullback asymptotically compact non-autonomous $\mathscr{D}$-pullback attractor. non-cylindrical domains
Attractors for A sup-cubic weakly damped wave equation in $ \mathbb{R}^{3} $
Xinyu Mei Chunyou Sun
Discrete & Continuous Dynamical Systems - B 2017, 22(11): 1-27 doi: 10.3934/dcdsb.2019053

In this paper, the dynamical behavior of weakly damped wave equations with a sup-cubic nonlinearity is considered in locally uniform spaces. We first prove the global well-posedness of the Shatah-Struwe solutions, then we obtain the existence of the $ \big(H_{lu}^{1}(\mathbb{R}^{3})\times L_{lu}^{2}(\mathbb{R}^{3}),H_{\rho}^{1}(\mathbb{R}^{3})\times L_{\rho}^{2}(\mathbb{R}^{3})\big) $-global attractor for the Shatah-Struwe solutions semigroup of this equation. The results are crucially based on the recent extension of Strichartz estimates to the case of bounded domains.

keywords: Wave equation locally uniform spaces global attractor Shatah-Struwe solution contractive function
Non-autonomous wave dynamics with memory --- asymptotic regularity and uniform attractor
Chunyou Sun Daomin Cao Jinqiao Duan
Discrete & Continuous Dynamical Systems - B 2008, 9(3&4, May): 743-761 doi: 10.3934/dcdsb.2008.9.743
The dynamical behavior of non-autonomous strongly damped wave-type evolutionary equations with linear memory, critical nonlinearity, and time-dependent external forcing is investigated. The time-dependent external forcing is assumed to be only translation-bounded, instead of translation-compact. First, the asymptotic regularity of solutions is proved, and then the existence of the compact uniform attractor together with its structure and regularity is obtained.
keywords: Non-autonomous systems with memory critical exponent uniform attractor. asymptotic regularity wave equations
Invariant measures for complex-valued dissipative dynamical systems and applications
Xin Li Wenxian Shen Chunyou Sun
Discrete & Continuous Dynamical Systems - B 2017, 22(6): 2427-2446 doi: 10.3934/dcdsb.2017124

In this work, we extend the classical real-valued framework to deal with complex-valued dissipative dynamical systems. With our new complex-valued framework and using generalized complex Banach limits, we construct invariant measures for continuous complex semigroups possessing global attractors. In particular, for any given complex Banach limit and initial data $u_{0}$, we construct a unique complex invariant measure $\mu$ on a metric space which is acted by a continuous semigroup $\{S(t)\}_{t\geq 0}$ possessing a global attractor $\mathcal{A}$. Moreover, it is shown that the support of $\mu$ is not only contained in global attractor $\mathcal{A}$ but also in $\omega(u_{0})$. Next, the structure of the measure $\mu$ is studied. It is shown that both the real and imaginary parts of a complex invariant measure are invariant signed measures and that both the positive and negative variations of a signed measure are invariant measures. Finally, we illustrate the main results of this article on the model examples of a complex Ginzburg-Landau equation and a nonlinear Schrödinger equation and construct complex invariant measures for these two complex-valued equations.

keywords: Complex-valued dynamical systems complex invariant measure global attractor Ginzburg-Landau equation nonlinear Schrödinger equation
Dynamics for a non-autonomous degenerate parabolic equation in $\mathfrak{D}_{0}^{1}(\Omega, \sigma)$
Xin Li Chunyou Sun Na Zhang
Discrete & Continuous Dynamical Systems - A 2016, 36(12): 7063-7079 doi: 10.3934/dcds.2016108
In this paper, we study the dynamics of a non-autonomous semilinear degenerate parabolic equation $u_{t}-div(\sigma(x)\nabla u)+ f(u)=g(x, t)$ defined on a bounded domain $\Omega\subset \mathbb{R}^N$ with smooth boundary. We first establish a Nash-Moser-Alikakos type a priori estimate for the difference of solutions near the initial time; Then we prove that the solution process $U(t,\tau)$ is continuous from $L^{2}(\Omega)$ to $\mathscr{D}_{0}^{1}(\Omega, \sigma)$ w.r.t. initial data; And finally show that the known $(L^{2}(\Omega), L^{2}(\Omega))$ pullback $\mathscr{D}_{\lambda}$-attractor indeed can attract in $\mathscr{D}_{0}^{1}(\Omega, \sigma)$-norm. Any differentiability on the forcing term is not required.
keywords: pullback attractor. asymptotic behavior degenerate parabolic equation Nash-Moser-Alikakos type a priori estimate Non-autonomous
Dynamics for the damped wave equations on time-dependent domains
Feng Zhou Chunyou Sun Xin Li
Discrete & Continuous Dynamical Systems - B 2018, 23(4): 1645-1674 doi: 10.3934/dcdsb.2018068

We consider the asymptotic dynamics of a damped wave equations on a time-dependent domains with homogeneous Dirichlet boundary condition, the nonlinearity is allowed to have a cubic growth rate which is referred to as the critical exponent. To this end, we establish the existence and uniqueness of strong and weak solutions satisfying energy inequality under the assumption that the spatial domains $\mathcal{O}_{t}$ in $\mathbb{R}^{3}$ are obtained from a bounded base domain $\mathcal{O}$ by a $C^{3}$-diffeomorphism $r(·, t)$. Furthermore, we establish the pullback attractor under a slightly weaker assumption that the measure of the spatial domains are uniformly bounded above.

keywords: Non-autonomous dynamical systems wave equation time-dependent domain critical exponent pullback attractor
Robust exponential attractors for non-autonomous equations with memory
Peter E. Kloeden José Real Chunyou Sun
Communications on Pure & Applied Analysis 2011, 10(3): 885-915 doi: 10.3934/cpaa.2011.10.885
The aim of this paper is to consider the robustness of exponential attractors for non-autonomous dynamical systems with line memory which is expressed through convolution integrals. Some properties useful for dealing with the memory term for non-autonomous case are presented. Then, we illustrate the abstract results by applying them to the non-autonomous strongly damped wave equations with linear memory and critical nonlinearity.
keywords: wave equations. robust exponential attractor Non-autonomous systems with memory

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