Numerical Solution of Some Nonlocal, Nonlinear Dispersive Wave Equations

Summary. We use a spectral method to solve numerically two nonlocal, nonlinear, dispersive, integrable wave equations, the Benjamin-Ono and the Intermediate Long Wave equations. The proposed numerical method is able to capture well the dynamics of the solutions; we use it to investigate the behaviour of solitary wave solutions of the equations with special attention to those, among the properties usually connected with integrability, for which there is at present no analytic proof. Thus we study in particular the resolution property of arbitrary initial profiles into sequences of solitary waves for both equations and clean interaction of Benjamin-Ono solitary waves. We also verify numerically that the behaviour of the solution of the Intermediate Long Wave equation as the model parameter tends to the infinite depth limit is the one predicted by the theory. Received October 28, 1997; revised February 11, 1999; accepted April 7, 1999     

Limit behaviour of focusing solutions to nonlinear diffusions

The paper is concerned with the behaviour of focusing solutions to nonlinear diffusion problems. These solutions describe the movement of a flow filling a hole and have consequences for the qualitative theory of degenerate nonlinear parabolic equations. The general equation under study is theso-called doubly nonlinear diffusion equation a²with parameters m > 0 and p > 1 such that m(p - 1) > 1 so that the finite propagation property holds and free boundaries occur. Well-known particular cases are the Porous Medium Equation and the evolutionary p-

Laplacian Equation. We study the behaviour of the families of selfsimilar focusing solutions as the parameters m and p tend to their limiting values and identify the limit problems these limits solve. In the case m(p - 1) -+ 1 we find as appropriate asymptotic problems a family of Hamilton-Jacobi equations. When we let m + ffi we obtain in the limit the Hele-Shaw problem. When p + cc we

obtain linear travelling waves with arbitrary speed, solutions of a certain ∞-Laplacian evolution problem.     

Global existence and asymptotic behaviour for systems of nonlinear hyperbolic equations

We consider the initial-boundary value problem for a class of nonlinear hyperbolic equations system in a bounded domain. Using the potential well theory, the existence of global solutions is investigated. We also established the asymptotic behaviour of global solutions as t?→?+?∞ by applying the multiplier method.     

Linear and nonlinear heat equations in $L^q_\delta$ spaces and universal bounds for global solutions

We develop a theory of both linear and nonlinear heat equations in the weighted Lebesgue spaces , where is the distance to the boundary. In particular, we prove an optimal estimate for the heat semigroup, and we establish sharp results on local existence-uniqueness and local nonexistence of solutions for semilinear heat equations with initial values in those spaces. This theory enables us to obtain new types of results concerning positive global solutions of superlinear parabolic problems. Namely, under certain assumptions, we prove that any global solution is uniformly bounded for by a universal constant, independent of the initial data. In all previous results, the bounds for global solutions were depending on the initial data. Received March 15, 2000 / Accepted October 18, 2000 / Published online February 5, 2001     

Lyapunov-type inequalities for nonlinear systems

In this paper, by using elementary analysis, we establish some new Lyapunov-type inequalities for nonlinear systems of differential equations, special cases of which contain the well-known equations such as Emden-Fowler-type and half-linear equations. The inequalities obtained here can be used as handy tools in the study of qualitative behaviour of solutions of the associated equations.     

Center manifolds for periodic functional differential equations of mixed type

We study the behaviour of solutions to nonlinear functional differential equations of mixed type (MFDEs), that remain sufficiently close to a prescribed periodic solution. Under a discreteness condition on the Floquet spectrum, we show that all such solutions can be captured on a finite dimensional invariant center manifold, that inherits the smoothness of the nonlinearity. This generalizes the results that were obtained previously in [H.J. Hupkes, S.M. Verduyn Lunel, Center manifold theory for functional differential equations of mixed type, J. Dynam. Differential Equations 19 (2007) 497-560] for bifurcations around equilibrium solutions to MFDEs.     

一类非线性系统解的有界性

本文研究了一类非线性系统解的正向有界性.运用常微分方程定性理论的方法,获得了这类方程所有解有界的充分条件和必要条件.推广和改进了已有文献中的结果.     

Stability analysis of models of cell production systems

We investigate the qualitative behaviour of the models of cell production systems, in the form of systems of nonlinear delay differential equations. Considered are three general models of a system involving the subpopulations of stem cells, precursor cells and mature cells, with different configurations of regulation feedbacks. The models correspond basically to the blood cell production process; however, other applications are possible. First, the simplified version (describable by ordinary differential equations) is considered. Fairly complete characterization of the trajectories is possible in this case, using the Lyapunov functions and phase plane techniques. Next, for the general models, the stability of equations linearized around the equilibria is investigated. Certain results can be obtained here, using both exact methods and numerical procedures based on an original lemma on the zeros of exponential polynomials. Then global properties (boundedness, attractivity, etc.) are examined for the nonlinear, delay case using a range of methods: Lyapunov functionals, Razumikhin functions and direct estimates on solutions. Certain special cases of our models reduce to previous literature models of blood production. Results of our analysis enable to exclude these configurations of regulation feedbacks which yield model behaviour not compatible with biological and medical observations. Techniques developed in this paper are applicable to a wide range of possible models of cell production systems.     

Floquet theory and stability of nonlinear integro-differential equations

Summary One of the classical topics in the qualitative theory of differential equations is the Floquet theory. It provides a means to represent solutions and helps in particular for stability analysis. In this paper first we shall study Floquet theory for integro-differential equations (IDE), and then employ it to address stability problems for linear and nonlinear equations.     

Spatial estimates for an equation with a delay term

In this note, we investigate the spatial behaviour of the solutions for a theory for the heat conduction with a delay term. We obtain an alternative of the Phragmen–Lindelof type. That is the solutions either decay in a exponential way or blow-up at infinity in a exponential way. We also describe how to obtain an upper bound for the amplitude term. It is worth noting that this is the first contribution on spatial behaviour for partial differential equations involving a delay term. We use the energy arguments to obtain our main results. The main point of the contribution is the use of a suitable weighted energy function.     

Existence and uniqueness of solutions to singular Cahn–Hilliard equations with nonlinear viscosity terms and dynamic boundary conditions

We prove global existence and uniqueness of solutions to a Cahn–Hilliard system with nonlinear viscosity terms and nonlinear dynamic boundary conditions. The problem is highly nonlinear, characterized by four nonlinearities and two separate diffusive terms, all acting in the interior of the domain or on its boundary. Through a suitable approximation of the problem based on abstract theory of doubly nonlinear evolution equations, existence and uniqueness of solutions are proved using compactness and monotonicity arguments. The asymptotic behaviour of the solutions as the diffusion operator on the boundary vanishes is also shown.     

The similarity forms and invariant solutions of two-layer shallow-water equations

In this study symmetry group properties and general similarity forms of the two-layer shallow-water equations are discussed by Lie group theory. We represent that Lie group theory can be used as an effective approach for investigation of the self-similar solutions for the shallow-water equations with variable inflow as the generalization of dimensional analysis that was used so far for a regular approach in the literature. We also represent that the results obtained by dimensional analysis are just a special case of the results obtained by Lie group theory and it is possible to obtain the new similarity forms and the new variable inflow functions for the study of gravity currents in two-layer flow under shallow-water approximations based on Lie group theory. The symmetry groups of the system of nonlinear partial differential equations are found and the corresponding similarity and reduced forms are obtained. Some similarity solutions of the reduced equations are investigated. It is shown that reduced equations and similarity forms of the system depend on the group parameters. We show that an analytic similarity solution for the system of equations can be found for some special values of them. For other values of the group parameters, the similarity solutions of the two-layer shallow-water equations representing the gravity currents with a variable inflow are found by the numeric integration.     

Nonlinear singular sturm-liouville problems and an application to transonic flow through a nozzle

We consider a class of singular Sturm-Liouville problems with a nonlinear convection and a strongly coupling source. Our investigation is motivated by, and then applied to, the study of transonic gas flow through a nozzle. We are interested in such solution properties as the exact number of solutions, the location and shape of boundary and interior layers, and nonlinear stability and instability of solutions when regarded as stationary solutions of the corresponding convective reaction-diffusion equations. Novel elements in our theory include a priori estimate for qualitative behavior of general solutions, a new class of boundary layers for expansion waves, and a local uniqueness analysis for transonic solutions with interior and boundary layers.     

On the preservation of invariants in the simulation of solitary waves in some nonlinear dispersive equations

In this paper we generalize some results in the literature concerning the structure of numerical approximations to solitary wave solutions of some nonlinear, dispersive equations is studied. We prove that those time discretizations with the property of preserving, exactly or approximately up to certain order, some invariants of the problems, have a better propagation of the error and provide a more suitable simulation of the solitary waves. The generalization involves the treatment of nonlocal operators and two different kinds of equations.     

Dynamic Mean Flow and Small-Scale Interaction through Topographic Stress

Summary. The equations describing the mean flow and small-scale interaction of a barotropic flow via topographic stress with layered topography are studied here through the interplay of theory and numerical experiments. Both a viewpoint toward atmosphere—ocean science and one toward chaotic nonlinear dynamics are emphasized. As regards atmosphere—ocean science, we produce prototype topographic blocking patterns without damping or driving, with topographic stress as the only transfer mechanism; these patterns and their chaos bear some qualitative resemblance to those observed in recent laboratory experiments on topographic blocking. As regards nonlinear dynamics, it is established that the equations for mean flow and small-scale interaction with layered anisotropic topography form a novel Hamiltonian system with rich regimes of intrinsic conservative chaos, which include both global and weak homoclinic stochasticity, as well as other regimes with complete integrability involving complex heteroclinic structure. Received August 7, 1997; second revision received December 18, 1997     

Reaction-Diffusion Equations in Homogeneous Media: Existence,Uniqueness and Stability of Travelling Fronts

The goal of this survey is to describe the construction and some qualitative properties of particular global solutions of certain reaction-diffusion equations. These solutions are known as travelling fronts (or travelling waves) and play an important role in the long-time behaviour of the solutions of the parabolic system. We will mainly focus on the existence of travelling wave solutions and their stability. We will also give some standard tools in elliptic and parabolic theory, which are of general interest.     

Numerical treatment of oscillatory functional differential equations

In this paper we are concerned with oscillatory functional differential equations (that is, those equations where all the solutions oscillate) under a numerical approximation. Our interest is in the preservation of qualitative properties of solutions under a numerical discretisation. We give conditions under which an equation is oscillatory, and consider whether the discrete schemes derived using linear ?-methods will also be oscillatory. We conclude with some general theory.     

带强迫项的高阶中立型微分方程的振动性

研究了一类强迫高阶非线性中立型时滞微分方程一切解振动的充分条件,建立了两个振动定理,推广和改进了已有结果.     

On the behaviour of solutions of degenerated nonlinear parabolic equations

The aim of this work is to study the behaviour of solutions of the initial boundary problem for degenerated nonlinear parabolic equations of the second order. The conditions of existence and non-existence solutions are established. Moreover, the behaviour of the solution is studied. We obtain the estimations in terms of characterizing the initial and weight functions on infinity, without a lower bound on the initial function.     

Some alternative results for the nonlinear viscoelasticity equations

In this article, the spatial behaviour of a class of nonlinear viscoelasticity equations is studied. We obtained some alternative results for the solutions under suitable conditions on the nonlinear terms. The main tool used is the weighted energy method. Our results can be viewed as a version of Saint-Venant's principle to the viscoelasticity equations.