arXiv subjects

Wenxian Shen

Publications and source records attributed to Wenxian Shen.

At least 73 records · Page 4Linked to original sources

arXiv2015

Transition Fronts in Time Heterogeneous and Random Media of Ignition Type

The current paper is devoted to the investigation of wave propagation phenomenon in reaction-diffusion equations with ignition type nonlinearity in time heterogeneous and random media. It is proven that such equations in time heterogeneous media admit transition fronts or generalized traveling wave solutions with time dependent profiles and that such equations in time random media admit generalized traveling wave solutions with random profiles. Important properties of generalized traveling wave solutions, including the boundedness of propagation speeds and the uniform decaying estimates of the propagation fronts, are also obtained.

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arXiv2015

Transition fronts in nonlocal Fisher-KPP equations in time heterogeneous media

The present paper is devoted to the study of transition fronts of nonlocal Fisher-KPP equations in time heterogeneous media. We first construct transition fronts with prescribed interface location functions, which are natural generalizations of front location functions in homogeneous media. It it shown that these transition fronts have exact decaying rates as the space variable tends to infinity. Then, by the general results on space regularity of transition fronts of nonlocal evolution equations proven in the authors' earlier work (\cite{ShSh14-4}), these transition fronts are continuously differentiable in space. We show that their space partial derivatives have exact decaying rates as the space variable tends to infinity. Finally, we study the asymptotic stability of transition fronts. It is shown that transition fronts attract those solutions whose initial data decays as fast as transition fronts near infinity and essentially above zero near negative infinity.

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arXiv2015

Regularity of transition fronts in nonlocal dispersal evolution equations

It is known that solutions of nonlocal dispersal evolution equations do not become smoother in space as time elapses. This lack of space regularity would cause a lot of difficulties in studying transition fronts in nonlocal equations. In the present paper, we establish some general criteria concerning space regularity of transition fronts in nonlocal dispersal evolution equations with a large class of nonlinearities, which allows the applicability of various techniques for reaction-diffusion equations to nonlocal equation, and hence serves as an initial and fundamental step for further studying various qualitative properties of transition fronts such as stability and uniqueness. We also prove the existence of continuously differentiable and increasing interface location functions, which give a better characterization of the propagation of transition fronts and are of great technical importance.

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arXiv2015

Transition fronts in nonlocal equations with time heterogeneous ignition nonlinearity

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.

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arXiv2015

Diffusive KPP Equations with Free Boundaries in Time Almost Periodic Environments: I. Spreading and Vanishing Dichotomy

In this series of papers, we investigate the spreading and vanishing dynamics of time almost periodic diffusive KPP equations with free boundaries. Such equations are used to characterize the spreading of a new species in time almost periodic environments with free boundaries representing the spreading fronts. In this first part, we show that a spreading-vanishing dichotomy occurs for such free boundary problems, that is, the species either successfully spreads to all the new environment and stabilizes at a time almost periodic positive solution, or it fails to establish and dies out eventually. The results of this part extend the existing results on spreading-vanishing dichotomy for time and space independent, or time periodic and space independent, or time independent and space periodic diffusive KPP equations with free boundaries. The extension is nontrivial and is ever done for the first time.

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arXiv2015

Diffusive KPP Equations with Free Boundaries in Time Almost Periodic Environments: II. Spreading Speeds and Semi-Wave

In this series of papers, we investigate the spreading and vanishing dynamics of time almost periodic diffusive KPP equations with free boundaries. Such equations are used to characterize the spreading of a new species in time almost periodic environments with free boundaries representing the spreading fronts. In the first part of the series, we showed that a spreading-vanishing dichotomy occurs for such free boundary problems (see [16]). In this second part of the series, we investigate the spreading speeds of such free boundary problems in the case that the spreading occurs. We first prove the existence of a unique time almost periodic semi-wave solution associated to such a free boundary problem. Using the semi-wave solution, we then prove that the free boundary problem has a unique spreading speed.

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arXiv2015

Spreading Speeds and Linear Determinacy for Two Species Competition Systems with Nonlocal Dispersal in Periodic Habitats

The current paper is concerned with the existence of spreading speeds and linear determinacy for two species competition systems with nonlocal dispersal in time and space periodic habitats. The notion of spreading speed intervals for such a system is first introduced via the natural features of spreading speeds. The existence and lower bounds of spreading speed intervals are then established. When the periodic dependence of the habitat is only on the time variable, the existence of a single spreading speed is proved. It also shows that, under certain conditions, the spreading speed interval in any direction is a singleton, and, moreover, the linear determinacy holds.

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arXiv2015

Approximations of Random Dispersal Operators/Equations by Nonlocal Dispersal Operators/Equations

This paper is concerned with the approximations of random dispersal operators/equations by nonlocal dispersal operators/equations. It first proves that the solutions of properly rescaled nonlocal dispersal initial-boundary value problems converge to the solutions of the corresponding random dispersal initial-boundary value problems. Next, it proves that the principal spectrum points of nonlocal dispersal operators with properly rescaled kernels converge to the principal eigenvalues of the corresponding random dispersal operators. Finally, it proves that the unique positive time periodic solutions of nonlocal dispersal KPP equations with properly rescaled kernels converge to the unique positive time periodic solutions of the corresponding random dispersal KPP equations.

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arXiv2015

Stability, Uniqueness and Recurrence of Generalized Traveling Waves in Time Heterogeneous Media of Ignition Type

The present paper is devoted to the study of stability, uniqueness and recurrence of generalized traveling waves of reaction-diffusion equations in time heterogeneous media of ignition type, whose existence has been proven by the authors of the present paper in a previous work. It is first shown that generalized traveling waves exponentially attract wave-like initial data. Next, properties of generalized traveling waves, such as space monotonicity and exponential decay ahead of interface, are obtained. Uniqueness up to space translations of generalized traveling waves is then proven. Finally, it is shown that the wave profile of the unique generalized traveling wave is of the same recurrence as the media. In particular, if the media is time almost periodic, then so is the wave profile of the unique generalized traveling wave.

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arXiv2015

Regularity and stability of transition fronts in nonlocal equations with time heterogeneous ignition nonlinearity

The present paper is devoted to the investigation of various properties of transition fronts in nonlocal equations in heterogeneous media of ignition type, whose existence has been established by the authors of the present paper in a previous work. It is first shown that the transition front is continuously differentiable in space with uniformly bounded and uniformly Lipschitz continuous space partial derivative. This is the first time that regularity of transition fronts in nonlocal equations is ever studied. It is then shown that the transition front is uniformly steep. Finally, asymptotic stability, in the sense of exponentially attracting front like initial data, of the transition front is studied.

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arXiv2014

Principal Lyapunov exponents and principal Floquet spaces of positive random dynamical systems. II. Finite-dimensional systems

This is the second part in a series of papers concerned with principal Lyapunov exponents and principal Floquet subspaces of positive random dynamical systems in ordered Banach spaces. The current part focuses on applications of general theory, developed in the authors' paper "Principal Lyapunov exponents and principal Floquet spaces of positive random dynamical systems. I. General theory," Trans. Amer. Math. Soc., in press, to positive random dynamical systems on finite-dimensional ordered Banach spaces. It is shown under some quite general assumptions that measurable linear skew-product semidynamical systems generated by measurable families of positive matrices and by strongly cooperative or type-K strongly monotone systems of linear ordinary differential equations admit measurable families of generalized principal Floquet subspaces, generalized principal Lyapunov exponents, and generalized exponential separations.

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arXiv2014

Principal Lyapunov exponents and principal Floquet spaces of positive random dynamical systems. I. General theory

This is the first of a series of papers concerned with principal Lyapunov exponents and principal Floquet subspaces of positive random dynamical systems in ordered Banach spaces. It focuses on the development of general theory. First, the notions of generalized principal Floquet subspaces, generalized principal Lyapunov exponents, and generalized exponential separations for general positive random dynamical systems in ordered Banach spaces are introduced, which extend the classical notions of principal Floquet subspaces, principal Lyapunov exponents, and exponential separations for strongly positive deterministic systems in strongly ordered Banach to general positive random dynamical systems in ordered Banach spaces. Under some quite general assumptions, it is then shown that a positive random dynamical system in an ordered Banach space admits a family of generalized principal Floquet subspaces, a generalized principal Lyapunov exponent, and a generalized exponential separation. We will consider in the forthcoming parts applications of the general theory developed in this part to positive random dynamical systems arising from a variety of random mappings and differential equations, including random Leslie matrix models, random cooperative systems of ordinary differential equations, and random parabolic equations.

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arXiv2014

Spreading Speeds and Traveling Waves of Nonlocal Monostable Equations in Time and Space Periodic Habitats

This paper is devoted to the investigation of spatial spreading speeds and traveling wave solutions of monostable evolution equations with nonlocal dispersal in time and space periodic habitats. It has been shown in an earlier work by the first two authors of the current paper that such an equation has a unique time and space periodic positive stable solution $u^*(t,x)$. In this paper, we show that such an equation has a spatial spreading speed $c^*(ξ)$ in the direction of any given unit vector $ξ$. A variational characterization of $c^*(ξ)$ is given. Under the assumption that the nonlocal dispersal operator associated to the linearization of the monostable equation at the trivial solution $0$ has a principal eigenvalue, we also show that the monostable equation has a continuous periodic traveling wave solution connecting $u^*(\cdot,\cdot)$ and $0$ propagating in any given direction of $ξ$ with speed $c>c^*(ξ)$.

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arXiv2014

One-dimensional random attractor and rotation number of the stochastic damped sine-Gordon equation

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 one-dimensional 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.

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arXiv2014

Liouville Type Property and Spreading Speeds of KPP Equations in Periodic Media with Localized Spatial Inhomogeneity

The current paper is devoted to the study of semilinear dispersal evolution equations of the form $$ u_t(t,x)=(\mathcal{A}u)(t,x)+u(t,x)f(t,x,u(t,x)),\quad x\in\mathcal{H}, $$ where $\mathcal{H}=\RR^N$ or $\ZZ^N$, $\mathcal{A}$ is a random dispersal operator or nonlocal dispersal operator in the case $\mathcal{H}=\RR^N$ and is a discrete dispersal operator in the case $\mathcal{H}=\ZZ^N$, and $f$ is periodic in $t$, asymptotically periodic in $x$ (i.e. $f(t,x,u)-f_0(t,x,u)$ converges to 0 as $\|x\|\to\infty$ for some time and space periodic function $f_0(t,x,u)$), and is of KPP type in $u$. It is proved that Liouville type property for such equations holds, that is, time periodic strictly positive solutions are unique. It is also proved that if $u\equiv 0$ is a linearly unstable solution to the time and space periodic limit equation of such an equation, then it has a unique stable time periodic strictly positive solution and has a spatial spreading speed in every direction.

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arXiv2013

On Principal Spectrum Points/Principal Eigenvalues of Nonlocal Dispersal Operators and Applications

This paper is to investigate the dependence of the principal spectrum points of nonlocal dispersal operators on underlying parameters and to consider its applications. In particular, we study the effects of the spatial inhomogeneity, the dispersal rate, and the dispersal distance on the existence of the principal eigenvalues, the magnitude of the principal spectrum points, and the asymptotic behavior of the principal spectrum points of nonlocal dispersal operators with Dirichlet type, Neumann type, and periodic boundary conditions in a unified way. We also discuss the applications of the principal spectral theory of nonlocal dispersal operators to the asymptotic dynamics of two species competition systems with nonlocal dispersal.

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arXiv2012

Asymptotic Dynamics of a Class of Coupled Oscillators Driven by White Noises

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 first order oscillators as the damping goes to infinite and also that coupled damped second order oscillators have similar asymptotic dynamics as their proper space continuous counterparts, which are of great practical importance.

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arXiv2012

Traveling Wave Solutions of Spatially Periodic Nonlocal Monostable Equations

This paper deals with front propagation dynamics of monostable equations with nonlocal dispersal in spatially periodic habitats. In the authors' earlier works, it is shown that a general spatially periodic monostable equation with nonlocal dispersal has a unique spatially periodic positive stationary solution and has a spreading speed in every direction. In this paper, we show that a spatially periodic nonlocal monostable equation with certain spatial homogeneity or small nonlocal dispersal distance has a unique stable periodic traveling wave solutions connecting its unique spatially periodic positive stationary solution and the trivial solution in every direction for all speeds greater than the spreading speed in that direction.

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