almost periodic linear differential equations with non separated solutions a celebrated result by favard states that for certain almost periodic linear differential systems the existence of a bounded solution implies the existence of an almost periodic solution a key assumption in this result is the separation among bounded solutions here we prove a theorem of anti favard type if there are bounded solutions which are non separated in a strong sense sometimes almost periodic solutions do not exist strongly non separated solutions appear when the associated homogeneous system has homoclinic solutions this point of view unifies two fascinating examples by zhikov levitan and johnson for the scalar case our construction uses the ideas of zhikov levitan together with the theory of characters in topological groups

asymptotic solutions of the becker doring equations with size dependent rate constants we find the large time asymptotic behaviour for a number of physically interesting cases of the becker doring equations allowing both the forward and the backward rates to depend on cluster size in a power law fashion we consider in detail the constant monomer form of the equations in the special case where the powers are equal since the structure of the large time asymptotic behaviour is then richest we then turn to cases in which aggregation and fragmentation have different exponents examining both the fragmentation and coagulation dominated cases again under constant monomer conditions

the structure of the bounded trajectories set of a scalar convex differential equation the present paper describes the topological and ergodic structure of the set of bounded trajectories of the flow defined by a scalar convex differential equation we characterize the minimal subsets the ergodic measures concentrated on them and study the longtime behaviour of the bounded trajectories in terms of the lyapunov exponents of the linearized equations in particular we obtain conditions that guarantee the existence of almost periodic almost automorphic and recurrent solutions

the becker doring equations at large times and their connection with the lsw theory of coarsening a systematic procedure is given for obtaining the asymptotic late time behavior of the becker doring equations describing the time evolution of a population of clusters of particles in lowest order of approximation the distribution of the sizes of the largest clusters satisfies the equations of the lifshitz slyozov wagner theory of coarsening

non parametric radially symmetric mean curvature flow with a free boundary we study the mean curvature flow of radially symmetric graphs with prescribed contact angle on a fixed smooth hypersurface in euclidean space in this paper we treat two distinct problems the first problem has a free neumann boundary only while the second has two disjoint boundaries a free neumann boundary and a fixed dirichlet height we separate the two problems and prove that under certain initial conditions we have either long time existence followed by convergence to a minimal surface or finite maximal time of existence at the end of which the graphs develop a curvature singularity we also give a rate of convergence for the singularity

asymptotic behavior of solutions to the coagulation fragmentation equations ii weak fragmentation the discrete coagulation fragmentation equations are a model for the kinetics of cluster growth in which clusters can coagulate via binary interactions to form larger clusters or fragment to form smaller ones the assumptions made on the fragmentation coefficients have the physical interpretation that surface effects are important our results on the asymptotic behavior of solutions generalize the corresponding results of ball carr and penrose for the becker doring equation

flow by mean curvature of convex surfaces into spheres the motion of surfaces by their mean curvature has been studied by brakke 1 from the viewpoint of geometric measure theory other authors investigated the corresponding nonparametric problem 2 5 9 a reason for this interest is that evolutionary surfaces of prescribed mean curvature model the behavior of grain boundaries in annealing pure metal in this paper we take a more classical point of view consider a compact uniformly convex w dimensional surface m mo without boundary which is smoothly imbedded in r let mo be represented locally by a diffeomorphism

non parametric mean curvature evolution with boundary conditions soit un domaine borne de r n n 2 de frontiere lisse et soit a l operateur de surface minimale donne par a d i a i p a i p i 1 p 2 1 2 on considere des surfaces dont la vitesse en direction de leur normale unite est egale a la courbure moyenne on etudie l equation d evolution qui en resulte u 1 du 2 1 2 au 0 dans 0 t

the becker doring equations with exponentially size dependent rate coefficients this paper is concerned with an analysis of the becker doring equations which lie at the heart of a number of descriptions of non equilibrium phase transitions and related complex dynamical processes the becker doring theory describes growth and fragmentation in terms of stepwise addition or removal of single particles to or from clusters of similar particles and has been applied to a wide range of problems of physicochemical and biological interest within recent years here we consider the case where the aggregation and fragmentation rates depend exponentially on cluster size these choices of rate coefficients at least qualitatively correspond to physically realistic molecular clustering scenarios such as occur in for example simulations of simple fluids new similarity solutions for the constant monomer becker doring system are identified and shown to be generic in the case of aggregation and fragmentation rates that depend exponentially on cluster size

nonmonotone equations with large almost periodic forcing terms abstract we consider the scalar differential equation u f u c h t where f u is a jumping nonlinearity and h t is an almost periodic function while c is a real parameter deciding the size of the forcing term the main result is that if h t does not vanish too much in some suitable sense then the equation admits a unique almost periodic solution for large values of the parameter c the class of the h t s to which the result applies is studied in detail it includes all the nontrivial trigonometric polynomials and is generic in the baire sense

metastable states for the becker doring cluster equations the becker doring equations in whichc l t can represent the concentration ofl particle clusters or droplets in say a condensing vapour at timet are begin array 20 c dc l t mathord left vphantom dc l t dt right kern nulldelimiterspace dt j l 1 t j l t l 2 3 end array with j l t a l c 1 t c l t b l 1 c l 1 t and eitherc1 const case a or rho sum limits 1 infty lc l const case b the equilibrium solutions arec l q l z l where q l prod limits 2 l a r 1 mathord left vphantom a r 1 b r right kern nulldelimiterspace b r the density of the saturated vapour defined as rho s sum limits 1 infty lq l z s l wherez s is the radius of convergence of the series is assumed finite it is proved here that subject to some further plausible conditions on the kinetic coefficientsa l andb l there is a class of metastable solutions of the equations withc1 z s small and positive which take an exponentially long time to decay to their asymptotic steady states an exponentially long time means one that increases more rapidly than any negative power of the given value ofc1 z s or in caseb s as the latter tends to zero the main ingredients in the proof are i a time independent upper bound on the solution of the kinetic equations this upper bound is a steady state solution of case a of the equations of the type used in the becker doring theory of nucleation and ii an upper bound on the total concentration of particles in clusters greater than a certain critical size which with suitable initial conditions remains exponentially small until the time becomes exponentially large

asymptotically dichotomic almost periodic differential equations abstract consider a non linear differential equation in r n which asymptotically behaves as a linear equation admitting an exponential dichotomy we wonder if almost periodic solutions exist when we add to the equation an almost periodic forcing term large enough and not vanishing too much a positive answer has been given in 3 for the scalar case n 1 and our aim is to extend that result to higher dimensions we discover that the extension seems to be driven by a new ingredient namely the type of the exponential dichotomy besides the pure stable types the mixed hyperbolic type is now possible and leads to a weaker than expected extension an example shows that a stronger extension cannot be obtained by the same method the approach is blended and mixes methods of differential equations and functional analysis especially when estimating norm and spectral radius of some crucial positive but non compact linear integral operators

asymptotic solutions of the becker doring equations we describe the solution of various becker doring models first we analyse equilibrium and steady state solutions together with the large time behaviour of solutions for the case of constant monomer concentration then we solve a weak fragmentation constant mass case by using matched asymptotic expansions the methods are applied both to the full becker doring equations and to a coarse grained or contracted system comparison of the results show good qualitative agreement all the phenomena present in the full model are reproduced in the contracted system and no anomalous effects are introduced however the quantitative agreement depends strongly on the choice of parameter values in the contracted system

the becker doring cluster equations basic properties and asymptotic behaviour of solutions existence and uniqueness results are established for solutions to the becker doring cluster equations the density is shown to be a conserved quantity under hypotheses applying to a model of a quenched binary alloy the asymptotic behaviour of solutions with rapidly decaying initial data is determined denoting the set of equilibrium solutions byc 0 s the principal result is that if the initial density 0 s then the solution converges strongly toc o while if 0 s the solution converges weak toc s in the latter case the excess density 0 s corresponds to the formation of larger and larger clusters i e condensation the main tools for studying the asymptotic behaviour are the use of a lyapunov function with desirable continuity properties obtained from a known lyapunov function by the addition of a special multiple of the density and a maximum principle for solutions

the becker doring equations and the lifshitz slyozov theory of coarsening in this paper the relation between the kinetic set of becker doring bd equations and the classical lifshitz slyozov ls theory of coarsening is studied a model that resembles the ls theory but keeps some of the nucleation effects is derived for this model a solution is described that shows how the kinetic effects explain the particular solution selected in the ls theory by means of a renormalization procedure a discrete group of transformations is shown to play an important role in describing the structure of the solution near the critical size of the ls theory

renormalization theoretic analysis of non equilibrium phase transitions i the becker doring equations with power law rate coefficients we study in detail the application of renormalization theory to models of cluster aggregation and fragmentation of relevance to nucleation and growth processes we investigate the becker d oring equations originally formulated to describe and analyse non equilibrium phase transitions and more recently generalized to describe a wide range of physicochemical problems in this paper we analyse how the systematic coarse graining renormalization of the becker d oring system of equations affects the aggregation and fragmentation rate coefficients we consider the case of power law size dependent cluster rate coefficients which we show lead to only three classes of system that require analysis coagulation dominated systems fragmentation dominated systems and those where coagulation and fragmentation are exactly balanced we analyse the late time asymptotics associated with each class