arXiv subjects

Chunqiu Li

Publications and source records attributed to Chunqiu Li.

8 recordsLinked to original sources

arXiv2026

Dynamic Bifurcation of Nonautonomous Evolution Equations: Invariant Manifold Methods

In this paper, we use the global invariant manifold and the reduced singular cohomology groups method, which is different from those in the literature, to study the dynamic bifurcation from infinity of the nonautonomous evolution equation in terms of invariant sets by applying the Conley index theory (due to Rybakowski). We first establish a nonautonomous global invariant manifold for the abstract equation, which allows us to reduce the original system to this finite-dimensional manifold. Then, a homotopy between the reduced equation and a product flow is constructed. Finally, by considering the reduced singular cohomology theory of the Conley index, we establish our main theorems on dynamic bifurcations from infinity for this nonautonomous equation. As an example, a nonautonomous parabolic equation on unbounded domains is considered. Some new detailed results on bifurcations from infinity of the parabolic equation under an appropriate Landesman-Lazer type condition are proved, including the existence of a nonautonomous Morse decomposition, and improving the earlier works in the literature.

math.DS↗

arXiv2024

Convergence of bi-spatial pullback random attractors and stochastic Liouville type equations for nonautonomous stochastic p-Laplacian lattice system

We consider convergence properties of the long-term behaviors with respect to the coefficient of the stochastic term for a nonautonomous stochastic $p$-Laplacian lattice equation with multiplicative noise. First, the upper semi-continuity of pullback random $(\ell^2,\ell^q)$-attractor is proved for each $q\in[1,+\infty)$. Then, a convergence result of the time-dependent invariant sample Borel probability measures is obtained in $\ell^2$. Next, we show that the invariant sample measures satisfy a stochastic Liouville type equation and a termwise convergence of the stochastic Liouville type equations is verified. Furthermore, each family of the invariant sample measures is turned out to be a sample statistical solution, which hence also fulfills a convergence consequence.

math.PR↗

arXiv2024

Invariant sample measures and sample statistical solutions for nonautonomous stochastic lattice Cahn-Hilliard equation with nonlinear noise

We consider a stochastic lattice Cahn-Hilliard equation with nonautonomous nonlinear noise. First, we prove the existence of pullback random attractors in $\ell^2$ for the generated nonautonomous random dynamical system. Then, we construct the time-dependent invariant sample Borel probability measures based on the pullback random attractor. Moreover, we develop a general stochastic Liouville type equation for nonautonomous random dynamical systems and show that the invariant sample measures obtained satisfy the stochastic Liouville type equation. At last, we define a new kind of statistical solution -- sample statistical solution corresponding to the invariant sample measures and show that each family of invariant sample measures is a sample statistical solution.

math.PR↗

arXiv2023

Global martingale and pathwise solutions and infinite regularity of invariant measures for a stochastic modified Swift-Hohenberg equation

We consider a 2D stochastic modified Swift-Hohenberg equations with multiplicative noise and periodic boundary. First, we establish the existence of local and global martingale and pathwise solutions in the regular Sobolev space $H^{2m}$ for each $m\geqslant1$. Associated with the unique global pathwise solution, we obtain a Markovian transition semigroup. Then, we show the existence of invariant measures and ergodic invariant measures for this Markovian semigroup on $H^{2m}$. At last, we improve the regularity of the obtained invariant measures to $H^{2(m+1)}$. With appropriate conditions on the diffusion coefficient, we can deduce the infinite regularity of the invariant measures, which was conjectured by Glatt-Holtz \textit{et al.} in their situation.

math.DS↗

arXiv2021

On the forward dynamical behavior of nonautonomous lattice dynamical systems

In this article, we study the forward dynamical behavior of nonautonomous lattice systems. We first construct a family of sets $\{\mathcal{A}_\varepsilon(σ)\}_{σ\in Σ}$ in arbitrary small neighborhood of a global attractor of the skew-product flow generated by a general nonautonomous lattice system, which is forward invariant and uniformly forward attracts any bounded subset of the phase space. Moreover, under some suitable conditions, we further construct a family of sets $\{\mathcal{B}_\varepsilon(σ)\}_{σ\in Σ}$ such that it uniformly forward exponentially attracts bounded subsets of the phase space. As an application, we study the discrete Gray-Scott model in detail and illustrate how to apply our abstract results to some concrete lattice system.

math.DS↗

arXiv2021

A Remark on Attractor Bifurcation

In this paper we present some local dynamic bifurcation results in terms of invariant sets of nonlinear evolution equations. We show that if the trivial solution is an isolated invariant set of the system at the critical value $λ=λ_0$, then either there exists a one-sided neighborhood $I^-$ of $λ_0$ such that for each $λ\in I^-$, the system bifurcates from the trivial solution to an isolated nonempty compact invariant set $K_λ$ with $0\not\in K_λ$, or there is a one-sided neighborhood $I^+$ of $λ_0$ such that the system undergoes an attractor bifurcation for $λ\in I^+$ from $(0,λ_0)$. Then we give a modified version of the attractor bifurcation theorem. Finally, we consider the classical Swift-Hohenberg equation and illustrate how to apply our results to a concrete evolution equation.

math.DS↗

arXiv2021

Bifurcation from Infinity of the Schrödinger Equation via Invariant Manifolds

This paper is concerned with the bifurcation from infinity of the nonlinear Schrödinger equation $$-Δu+V(x)u=λu+f(x,u),\hspace{0.4cm} x\in \mathbb{R}^N.$$ We treat this problem in the framework of dynamical systems by considering the corresponding parabolic equation on unbounded domains. Firstly, we establish a global invariant manifold for the parabolic equation on $\mathbb{R}^N$. Then, we restrict the parabolic equation to this invariant manifold, which generates a system of finite dimension. Finally, we use the Conley index theory and the shape theory of attractors to establish some new results on bifurcations from infinity and multiplicity of solutions of the Schrödinger equation under an appropriate Landesman-Lazer type condition.

math.DS↗

arXiv2020

Upper semi-continuity of random attractors and existence of invariant measures for nonlocal stochastic Swift-Hohenberg equation with multiplicative noise

In this paper, we mainly study the long-time dynamical behaviors of 2D nonlocal stochastic Swift-Hohenberg equations with multiplicative noise from two perspectives. Firstly, by adopting the analytic semigroup theory, we prove the upper semi-continuity of random attractors in the Sobolev space $H_0^2(U)$, as the coefficient of the multiplicative noise approaches zero. Then, we extend the classical "stochastic Gronwall's lemma", making it more convenient in applications. Based on this improvement, we are allowed to use the analytic semigroup theory to establish the existence of ergodic invariant measures.

math.PR↗