h i g h l i g h t s • A new method of constructing sequences of Lyapunov functions and anti-Lyapu... more h i g h l i g h t s • A new method of constructing sequences of Lyapunov functions and anti-Lyapunov functions is developed. • Gaussian-like tails estimates and quantitative concentration of stationary measures are obtained. • Quantitative stabilization/de-stabilization of local attractors/repellers are studied. • Upper bounds for the stationary differential entropy and entropy-dimension inequalities are established.
Journal of Dynamics and Differential Equations, 2018
The present paper is devoted to the investigation of long-time behaviors of global probability so... more The present paper is devoted to the investigation of long-time behaviors of global probability solutions of Fokker-Planck equations with rough coefficients. In particular, we prove the convergence of probability solutions under a Lyapunov condition in terms of the Markov semigroup associated to the stationary one. A generalization of earlier results on the existence and uniqueness of global probability solutions is also given.
Discrete & Continuous Dynamical Systems - A, 2016
The present paper is devoted to the study of transition fronts in nonlocal reaction-diffusion equ... more The present paper is devoted to the study of transition fronts in nonlocal reaction-diffusion equations with time heterogeneous nonlinearity of ignition type. It is proven that such an equation admits space monotone transition fronts with finite speed and space regularity in the sense of uniform Lipschitz continuity. Our approach is first constructing a sequence of approximating front-like solutions and then proving that the approximating solutions converge to a transition front. We take advantage of the idea of modified interface location, which allows us to characterize the finite speed of approximating solutions in the absence of space regularity, and leads directly to uniform exponential decaying estimates. Contents 23 Appendix A. Comparison principles 23 References 24
This paper is devoted to the study of the asymptotic dynamics of a class of coupled second order ... more This paper is devoted to the study of the asymptotic dynamics of a class of coupled second order oscillators driven by white noises. It is shown that any system of such coupled oscillators with positive damping and coupling coefficients possesses a global random attractor. Moreover, when the damping and the coupling coefficients are sufficiently large, the global random attractor is a one-dimensional random horizontal curve regardless of the strength of the noises, and the system has a rotation number, which implies that the oscillators in the system tend to oscillate with the same frequency eventually and therefore the so-called frequency locking is successful. The results obtained in this paper generalize many existing results on the asymptotic dynamics for a single second order noisy oscillator to systems of coupled second order noisy oscillators. They show that coupled damped second order oscillators with large damping have similar asymptotic dynamics as the limiting coupled fir...
This paper is devoted to the study of the asymptotic dynamics of the stochastic damped sine-Gordo... more This paper is devoted to the study of the asymptotic dynamics of the stochastic damped sine-Gordon equation with homogeneous Neumann boundary condition. It is shown that for any positive damping and diffusion coefficients, the equation possesses a random attractor, and when the damping and diffusion coefficients are sufficiently large, the random attractor is a onedimensional random horizontal curve regardless of the strength of noise. Hence its dynamics is not chaotic. It is also shown that the equation has a rotation number provided that the damping and diffusion coefficients are sufficiently large, which implies that the solutions tend to oscillate with the same frequency eventually and the so called frequency locking is successful.
To understand the effects that the climate change has on the evolution of species as well as the ... more To understand the effects that the climate change has on the evolution of species as well as the genetic consequences, we analyze an integrodifference equation (IDE) models for a reproducing and dispersing population in a spatio-temporal heterogeneous environment described by a shifting climate envelope. Our analysis on the IDE focuses on the persistence criterion, travelling wave solutions, and the inside dynamics. First, the persistence criterion, characterizing the global dynamics of the IDE, is established in terms of the basic reproduction number. In the case of persistence, a unique travelling wave is found to govern the global dynamics. The effects of the size and the shifting speed of the climate envelope on the basic reproduction number, and hence, on the persistence criterion, are also investigated. In particular, the critical domain size and the critical shifting speed are found in certain cases. Numerical simulations are performed to complement the theoretical results. I...
h i g h l i g h t s • A new method of constructing sequences of Lyapunov functions and anti-Lyapu... more h i g h l i g h t s • A new method of constructing sequences of Lyapunov functions and anti-Lyapunov functions is developed. • Gaussian-like tails estimates and quantitative concentration of stationary measures are obtained. • Quantitative stabilization/de-stabilization of local attractors/repellers are studied. • Upper bounds for the stationary differential entropy and entropy-dimension inequalities are established.
Journal of Dynamics and Differential Equations, 2018
The present paper is devoted to the investigation of long-time behaviors of global probability so... more The present paper is devoted to the investigation of long-time behaviors of global probability solutions of Fokker-Planck equations with rough coefficients. In particular, we prove the convergence of probability solutions under a Lyapunov condition in terms of the Markov semigroup associated to the stationary one. A generalization of earlier results on the existence and uniqueness of global probability solutions is also given.
Discrete & Continuous Dynamical Systems - A, 2016
The present paper is devoted to the study of transition fronts in nonlocal reaction-diffusion equ... more The present paper is devoted to the study of transition fronts in nonlocal reaction-diffusion equations with time heterogeneous nonlinearity of ignition type. It is proven that such an equation admits space monotone transition fronts with finite speed and space regularity in the sense of uniform Lipschitz continuity. Our approach is first constructing a sequence of approximating front-like solutions and then proving that the approximating solutions converge to a transition front. We take advantage of the idea of modified interface location, which allows us to characterize the finite speed of approximating solutions in the absence of space regularity, and leads directly to uniform exponential decaying estimates. Contents 23 Appendix A. Comparison principles 23 References 24
This paper is devoted to the study of the asymptotic dynamics of a class of coupled second order ... more This paper is devoted to the study of the asymptotic dynamics of a class of coupled second order oscillators driven by white noises. It is shown that any system of such coupled oscillators with positive damping and coupling coefficients possesses a global random attractor. Moreover, when the damping and the coupling coefficients are sufficiently large, the global random attractor is a one-dimensional random horizontal curve regardless of the strength of the noises, and the system has a rotation number, which implies that the oscillators in the system tend to oscillate with the same frequency eventually and therefore the so-called frequency locking is successful. The results obtained in this paper generalize many existing results on the asymptotic dynamics for a single second order noisy oscillator to systems of coupled second order noisy oscillators. They show that coupled damped second order oscillators with large damping have similar asymptotic dynamics as the limiting coupled fir...
This paper is devoted to the study of the asymptotic dynamics of the stochastic damped sine-Gordo... more This paper is devoted to the study of the asymptotic dynamics of the stochastic damped sine-Gordon equation with homogeneous Neumann boundary condition. It is shown that for any positive damping and diffusion coefficients, the equation possesses a random attractor, and when the damping and diffusion coefficients are sufficiently large, the random attractor is a onedimensional random horizontal curve regardless of the strength of noise. Hence its dynamics is not chaotic. It is also shown that the equation has a rotation number provided that the damping and diffusion coefficients are sufficiently large, which implies that the solutions tend to oscillate with the same frequency eventually and the so called frequency locking is successful.
To understand the effects that the climate change has on the evolution of species as well as the ... more To understand the effects that the climate change has on the evolution of species as well as the genetic consequences, we analyze an integrodifference equation (IDE) models for a reproducing and dispersing population in a spatio-temporal heterogeneous environment described by a shifting climate envelope. Our analysis on the IDE focuses on the persistence criterion, travelling wave solutions, and the inside dynamics. First, the persistence criterion, characterizing the global dynamics of the IDE, is established in terms of the basic reproduction number. In the case of persistence, a unique travelling wave is found to govern the global dynamics. The effects of the size and the shifting speed of the climate envelope on the basic reproduction number, and hence, on the persistence criterion, are also investigated. In particular, the critical domain size and the critical shifting speed are found in certain cases. Numerical simulations are performed to complement the theoretical results. I...
Uploads
Papers by Zhongwei Shen