偏微分方程的吸引子

Abstract. In this paper the reaction-diffusion systems and the damped wave systems with non-linear terms of gradient form are studied ,and the conditions for making them to be gradient sys-tems are given. So the structure of attractors is clear and simple in some sense by the known re-sult. Some examples are given. Also the reaction-diffusion systems with general nonlinear termsare discussed.     

Exact solutions for logistic reaction–diffusion equations in biology

Reaction–diffusion equations with a nonlinear source have been widely used to model various systems, with particular application to biology. Here, we provide a solution technique for these types of equations in N-dimensions. The nonclassical symmetry method leads to a single relationship between the nonlinear diffusion coefficient and the nonlinear reaction term; the subsequent solutions for the Kirchhoff variable are exponential in time (either growth or decay) and satisfy the linear Helmholtz equation in space. Example solutions are given in two dimensions for particular parameter sets for both quadratic and cubic reaction terms.     

一类弱耦合动力系统的参数识别问题

研究一类弱耦合反应－扩散动力系统的参数识别问题。通过构造上下解，证明了反应－扩散方程组解的存在惟一性；给出了求解参数识别问题的最优化系，从而可以选取适当的梯度法或者共轭梯度法，实现对系统参数的识别。     

A Dissipation-Preserving Integrator for Damped Oscillatory Hamiltonian Systems

In this paper, based on discrete gradient, a dissipation-preserving integrator for weakly dissipative perturbations of oscillatory Hamiltonian system is established. The solution of this system is a damped nonlinear oscillator. Basically, lots of nonlinear oscillatory mechanical systems including frictional forces lend themselves to this approach. The new integrator gives a discrete analogue of the dissipation property of the original system. Meanwhile, since the integrator is based on the variation-of-constants formula for oscillatory systems, it preserves the oscillatory structure of the system. Some properties of the new integrator are derived. The convergence is analyzed for the implicit iterations based on the discrete gradient integrator, and it turns out that the convergence of the implicit iterations based on the new integrator is independent of $\|M\|$, where $M$ governs the main oscillation of the system and usually $\|M\|\gg1$. This significant property shows that a larger stepsize can be chosen for the new schemes than that for the traditional discrete gradient integrators when applied to the oscillatory Hamiltonian system. Numerical experiments are carried out to show the effectiveness and efficiency of the new integrator in comparison with the traditional discrete gradient methods in the scientific literature.     

Fast-diffusion limit with large noise for systems of stochastic reaction-diffusion equations

We consider a class of a stochastic reaction-diffusion equations with additive noise. In the limit of fast diffusion, one can approximate solutions of the stochastic reaction–diffusion equations by the solution of a suitable system of ordinary differential equation only describing the reactions, but due to nonlinear interaction of large diffusion and fluctuations in the limit new effective reaction terms appear. We focus on systems with polynomial nonlinearities and illustrate the result by applying it to a predator-prey system and a cubic auto-catalytic reaction between two chemicals.     

On existence of periodic solutions for gradient systems with applications

In this paper we prove the existence of periodic solutions for gradient systems in finite and infinite dimensional spaces. The techniques of the proofs are based on the application of a global inverse functions theorem, the Schäefer fixed point theorem and the Faedou–Galerkin method. We apply our results in order to solve nonlinear reaction–diffusion equations with Dirichlet and Neumann boundary conditions.     

Nonlinear diffusion and viral spread through the leaf of a plant

The spread of a virus through the leaf of a plant is both spatially and temporally causal in that the present status depends on the past and the spatial spread is compactly supported and progresses outwards. Such spatial spread is known to occur for certain nonlinear diffusion processes. The first compactly supported solution for nonlinear diffusion equations appears to be that of Pattle published in 1959. In that paper, no explanation is given as to how the solution was derived. Here, we show how the solution can be derived using Lie symmetry analysis. This lays a foundation for exploring the behavior of other choices for nonlinear diffusion and exploring the addition of reaction terms which do not eliminate the compactly supported structure. The implications associated with using the reaction–diffusion equation to model the spatial–temporal spread of a virus through the leaf of a plant are discussed.     

高维格上时滞反应扩散方程的行波解

本文利用Schauder不动点定理和上、下解技术,研究了高维格上时滞反应扩散方程组当非线性项满足拟单调条件、指数拟单调条件、部分拟单调条件以及部分指数拟单调条件时行波解的存在性.     

Second-order characteristic schemes in time and age for a nonlinear age-structured population model

In this paper, we analyze two new second-order characteristic schemes in time and age for an age-structured population model with nonlinear diffusion and reaction. By using the characteristic difference to approximate the transport term and the average along the characteristics to treat the nonlinear spatial diffusion and reaction terms, an implicit second-order characteristic scheme is proposed. To compute the nonlinear approximation system, an explicit second-order characteristic scheme in time and age is further proposed by using the extrapolation technique. The global existence and uniqueness of the solution of the nonlinear approximation scheme are established by using the theory of variation methods, Schauder’s fixed point theorem, and the technique of prior estimates. The optimal error estimates of second order in time and age are strictly proved for both the implicit and the explicit characteristic schemes. Numerical examples are given to illustrate the performance of the methods.     

Finite difference reaction diffusion equations with nonlinear boundary conditions

This article is concerned with numerical solutions of finite difference systems of reaction diffusion equations with nonlinear internal and boundary reaction functions. The nonlinear reaction functions are of general form and the finite difference systems are for both time-dependent and steady-state problems. For each problem a unified system of nonlinear equations is treated by the method of upper and lower solutions and its associated monotone iterations. This method leads to a monotone iterative scheme for the computation of numerical solutions as well as an existence-comparison theorem for the corresponding finite difference system. Special attention is given to the dynamical property of the time-dependent solution in relation to the steady-state solutions. Application is given to a heat-conduction problem where a nonlinear radiation boundary condition obeying the Boltzmann law of cooling is considered. This application demonstrates a bifurcation property of two steady-state solutions, and determines the dynamic behavior of the time-dependent solution. Numerical results for the heat-conduction problem, including a test problem with known analytical solution, are presented to illustrate the various theoretical conclusions. © 1995 John Wiley & Sons, Inc.     

一个非线性扩散系统解的存在性及线性系统的最优控制

讨论关于生物种群的一个非线性扩散系统和线性系统的一些问题,得到了非线性扩散系统弱解的存在性;线性系统最优繁殖率的存在性和关于边界扰动的最优解的存在唯一性。     

Approximate analytical solution of two-dimensional space-time fractional diffusion equation

This work presents an iterative scheme for the numerical solution of the space-time fractional two-dimensional advection–reaction–diffusion equation applying homotopy perturbation with Laplace transform using Caputo fractional-order derivatives. The solution obtained is beneficial and significant to analyze the modeling of superdiffusive systems and subdiffusive system, anomalous diffusion, transport process in porous media. This iterative technique presents the combination of homotopy perturbation technique, and Laplace transforms with He's polynomials, which can further be applied to numerous linear/nonlinear two-dimensional fractional models to computes the approximate analytical solution. In the present method, the nonlinearity can be tackle by He's polynomials. The salient features of the present scientific work are the pictorial presentations of the approximate numerical solution of the two-dimensional fractional advection–reaction–diffusion equation for different particular cases of fractional order and showcasing of the damping effect of reaction terms on the nature of probability density function of the considered two-dimensional nonlinear mathematical models for various situations.     

SOME ASYMPTOTIC INFERENCE IN QUASI-LIKELIHOOD NONLINEAR MODELS:A GEOMETRIC APPROACH

§ 1　IntroductionIt is well known that quasi-likelihood models introduced by Wedderburn[1 ] greatlywiden the scope of generalized linear models by using a much weaker assumption in whichonly the firstand second moments ofresponse vector Yare needed to replace the full distri-butional assumption about Y in the models.It has drawn considerable attention in recentliterature(e.g.see[2～ 6] and so on) .However,little work has been done on the issuefrom a geometric viewpoint.The purpose of this p…     

Gain-Scheduled Worst-Case Control on Nonlinear Stochastic Systems Subject to Actuator Saturation and Unknown Information

In this paper, we propose a method for designing continuous gain-scheduled worst-case controller for a class of stochastic nonlinear systems under actuator saturation and unknown information. The stochastic nonlinear system under study is governed by a finite-state Markov process, but with partially known jump rate from one mode to another. Initially, a gradient linearization procedure is applied to describe such nonlinear systems by several model-based linear systems. Next, by investigating a convex hull set, the actuator saturation is transferred into several linear controllers. Moreover, worst-case controllers are established for each linear model in terms of linear matrix inequalities. Finally, a continuous gain-scheduled approach is employed to design continuous nonlinear controllers for the whole nonlinear jump system. A numerical example is given to illustrate the effectiveness of the developed techniques.     

A Variational Analysis for the Moving Finite Element Method for Gradient Flows

By using the Onsager principle as an approximation tool, we give a novel derivation for the moving finite element method for gradient flow equations. We show that the discretized problem has the same energy dissipation structure as the continuous one. This enables us to do numerical analysis for the stationary solution of a nonlinear reaction diffusion equation using the approximation theory of free-knot piecewise polynomials. We show that under certain conditions the solution obtained by the moving finite element method converges to a local minimizer of the total energy when time goes to infinity. The global minimizer, once it is detected by the discrete scheme, approximates the continuous stationary solution in optimal order. Numerical examples for a linear diffusion equation and a nonlinear Allen-Cahn equation are given to verify the analytical results.     

Blow-up estimates for system of heat equations coupled via nonlinear boundary flux

This paper deals with a system of heat equations coupled via nonlinear boundary flux. The precise blow-up rate estimates are established together with the blow-up set. It is observed that there is some quantitative relationship regarding the blow-up properties between the heat system with coupled nonlinear boundary flux terms and the corresponding reaction–diffusion system with the same nonlinear terms as the source.     

具有扩散的Lotka-Volterra系统的持续性

利用最大值原理结合上、下解的方法，讨论了一类具有扩散的竞争－捕食的Lotka-Volterra系统静态解的存在性与持续生存。     

Blow-up analysis for a nonlinear diffusion system

This paper deals with a nonlinear diffusion system coupled via nonlinear reaction terms of power type. As results of interactions among the multi-nonlinearities in the system described by six exponents, global boundedness and blow-up criteria of positive solutions are determined.     

The attractor for a nonlinear reaction‐diffusion system in an unbounded domain

In this paper the quasi‐linear second‐order parabolic systems of reaction‐diffusion type in an unbounded domain are considered. Our aim is to study the long‐time behavior of parabolic systems for which the nonlinearity depends explicitly on the gradient of the unknown functions. To this end we give a systematic study of given parabolic systems and their attractors in weighted Sobolev spaces. Dependence of the Hausdorff dimension of attractors on the weight of the Sobolev spaces is considered. © 2001 John Wiley & Sons, Inc.