interaction of diffusion and boundary conditions on considere le systeme d equations paraboliques u t d u f u x d u n e x u 0 x ou u r n r n n 3 est un ouvert borne de frontiere lisse d diag d 1 d n e diag e 1 e n chaque d j 0 est constant e j r est continu e j 0 j 1 2 n et e 0 est constant pour f e fixe on etudie comment le flot defini ci dessus depend des parametres d

spectral convergence and nonlinear dynamics of reaction diffusion equations under perturbations of the domain abstract in this paper we obtain the continuity of attractors for semilinear parabolic problems with neumann boundary conditions relatively to perturbations of the domain we show that if the perturbations on the domain are such that the convergence of eigenvalues and eigenfunctions of the neumann laplacian is granted then we obtain the upper semicontinuity of the attractors if moreover every equilibrium of the unperturbed problem is hyperbolic we also obtain the continuity of attractors we also give necessary and sufficient conditions for the spectral convergence of neumann problems under perturbations of the domain

continuity of the dynamics in a localized large diffusion problem with nonlinear boundary conditions abstract this paper is concerned with singular perturbations in parabolic problems subjected to nonlinear neumann boundary conditions we consider the case for which the diffusion coefficient blows up in a subregion 0 which is interior to the physical domain r n we prove under natural assumptions that the associated attractors behave continuously as the diffusion coefficient blows up locally uniformly in 0 and converges uniformly to a continuous and positive function in 1 0

upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation abstract for a smooth bounded domain r n n 3 and e a real parameter consider the hyperbolic equation eu tt u t u u g in with dirichlet boundary conditions under certain conditions on this equation has a compact attractor a e in h 0 1 l 2 for e 0 the parabolic equation also has a compact attractor which can be naturally embedded into a compact set a 0 in h 0 1 l 2 it is shown that for any neighborhood u of a 0 the set a e u for e small

layer solutions in a half space for boundary reactions r n 1 are denoted by y y1 yn 1 our main goal is to study bounded solutions of 1 1 that are monotone increasing say from 1 to 1 in one of the y variables we call them layer solutions of 1 1 and we study their existence uniqueness symmetry and variational properties as well as their asymptotic behavior the interest in such increasing solutions comes from some models of boundary phase transitions when the nonlinearity f is given by f u sin cu for some constant c problem 1 1 in a half plane is called the peierls nabarro problem and it appears as a model of dislocations in crystals see 21 36 the peierls nabarro problem is also central to the analysis of boundary vortices in the paper 28 which studies a model for soft thin films in micromagnetism recently derived by kohn and slastikov 26 see also 27 our main result theorem 1 2 characterizes the nonlinearities f for which there exists a layer solution of 1 1 in dimension n 2 we prove that the necessary and sufficient condition is that the potential g defined by g f has two and only two absolute minima in the interval 1 1 located at 1 under the additional hypothesis g 1 0 we also establish the uniqueness of a layer solution up to translations in the y variable the proofs of both the necessity and the sufficiency of the condition on g for existence use new ingredients which we develop in this article a first one is a nonlocal estimate as well as a conserved or hamiltonian quantity satisfied by every layer solution in dimension 2 see theorem 1 3 the estimate can be seen as

singular perturbation of domains and the semilinear elliptic equation ii on considere un domaine a perturbation singuliere d 1 d 2 q avec un petit parametre 0 et on etudie le comportement des solutions et leur structure du probleme aux valeurs limites elliptique semilineaire pour quand 0 est petit v f v 0 dans v v 0 sur ou f est une fonction lisse a valeur reelle non lineaire

jacobi matrices and transversality the paper deals with smooth nonlinear ode systems in n f x such that the derivative f x has a matrix representation of jacobi type not necessarily symmetric with positive off diagonal entries a discrete functional is introduced and is discovered to be nonincreasing along the solutions of the associated linear variational system f x t y two families of transversal cones invariant under the flow of that linear system allow us to prove transversality between the stable and unstable manifolds of any two hyperbolic critical points of the given nonlinear system it is also proved that the nonwandering points are critical points a new class of morse smale systems in n is then explicitly constructed

a dual to lyapunov s stability theorem lyapunov s second theorem is a standard tool for stability analysis of ordinary differential equations here we introduce a theorem which can be viewed as a dual to lyapunov s result from existence of a scalar function satisfying certain inequalities it follows that almost all trajectories of the system tend to zero the scalar function has a physical interpretation as the stationary density of a substance that is generated in all points of the state space and flows along the system trajectories if the stationary density is bounded everywhere except at a singularity in the origin then almost all trajectories tend towards the origin the weaker notion of stability allows for applications also in situations where lyapunov s theorem cannot be used moreover the new criterion has a striking convexity property related to control synthesis

nesting inertial manifolds for reaction and diffusion equations with large diffusivity we study the asymptotic behaviour in large diffusivity of inertial manifolds governing the long time dynamics of a semilinear evolution system of reaction and diffusion equations a priori we review both local and global dynamics of the system in scales of banach spaces of hilbert type and we prove the existence of a universal compact attractor for the equations extensions yield the existence of a family of nesting inertial manifolds dependent on the diffusion of the system of equations it is introduced an upper semicontinuity notion in large diffusivity for inertial manifolds the limit inertial manifold whose dimension is strictly less than those of the infinite dimensional system of semilinear evolution equations is obtained

interaction of localized large diffusion and boundary conditions abstract in this paper we derive the limiting pdes for parabolic problems where localized large diffusion touches the boundary of the domain hence the limit problem reflects the interaction of large diffusion and boundary conditions the limit problem is a pde coupled with a system of odes through some nonlocal terms we study the well posedness and dissipativity of the nonstandard limit problem as well as the continuity of the dynamics as diffusion becomes large

convergence of binomial tree methods for european american path dependent options the binomial tree method first proposed by cox ross and rubinstein journal of financial economics 7 1979 pp 229 263 is one of the most popular approaches to pricing options by introducing an additional path dependent variable such methods can be readily extended to the valuation of path dependent options in this paper using numerical analysis and the notion of viscosity solutions we present a unifying theoretical framework to show the uniform convergence of binomial tree methods for european american path dependent options including arithmetic average options geometric average options and lookback options

a nonlinear parabolic equation with varying domain abstract this document considers a boundary value problem where the basic problem is to understand how new equilibrium solutions may appear for a given nonlinear function f as the domain omega is changed these new solutions being bifurcations from known equilibrium solutions for a given cubic function f an intuitive discussion of this phenomenon was given by hale indicating how stable non constant equilibria appear as secondary rather than primary bifurcations additional keywords continuity perturbations reprints

attractors of nonlinear dynamical systems with a weakly monotone measure abstract this paper is devoted to the study of attractive sets for dynamical systems in a metric space with a measure it is assumed that the measure of a set of points in the phase space is increasing along the flow we prove that an invariant set is an attractor for almost all initial conditions under some extra assumptions for a system of autonomous ordinary differential equations we present attractivity conditions in terms of the divergence with a density function unlike previous results in the literature our approach allows the use of a wider class of density functions if the divergence vanishes on a set of positive measure as an example the attitude stabilization problem of a rigid body is solved by using an affine feedback control for the kinematic equations in terms of quaternions

well posedness and long time behavior for p laplacian equation with nonlinear boundary condition abstract in this paper we study the homogeneous nonlinear boundary value problem for the p laplacian equation u t p u a x t u 2 u b x t u 2 u h x t we prove the existence of weak solutions which is global or local in time in dependence on the relation between the exponent of nonlinear part in boundary value and p boundedness of weak solution is proved we established conditions of uniqueness we prove also the properties of extinction in a finite time finite speed propagation and waiting time lastly by using the energy method we obtain sufficient conditions that the solutions of this problem with non positive initial energy blow up in finite time

dynamics in dumbbell domains iii continuity of attractors in this paper we conclude the analysis started in 3 and continued in 4 con cerning the behavior of the asymptotic dynamics of a dissipative reactions di usion equation in a dumbbell domain as the channel shrinks to a line segment in 3 we have established an appropriate functional analytic framework to address this problem and we have shown the continuity of the set of equilibria in 4 we have analyzed the behavior of the limiting problem in this paper we show that the attractors are upper semicontinuous and moreover if all equilibria of the limiting problem are hyperbolic then they are lower semicontinuous and therefore continuous the continuity is obtained in lp and h1 norms

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

continuity of attractors for parabolic problems with localized large diffusion abstract in this paper we study the continuity of asymptotics of semilinear parabolic problems of the form u t div p x u u f u in a bounded smooth domain r n with dirichlet boundary conditions when the diffusion coefficient p becomes large in a subregion 0 which is interior to the physical domain we prove under suitable assumptions that the family of attractors behave upper and lower semicontinuously as the diffusion blows up in 0

upper semicontinuity for attractors of parabolic problems with localized large diffusion and nonlinear boundary conditions the motivations to study the problem considered in this paper come from the theory of composite materials where the heat diffusion properties can change from one part of the domain to another mathematically this leads to a nonlinear second order parabolic equation for which the diffusion coefficient becomes large in a subdomain 0 the equation is supplemented by a nonlinear boundary condition on and an initial condition the authors determine the form of the limit problem the so called shadow system which involves an evolution equation for the averages of the density over 0 the main results include global in time existence of solutions and upper semicontinuity of the associated global attractors when the system approaches the shadow system

continuity of the attractors in a singular problem arising in composite materials abstract in this paper we consider the second order parabolic problem pe u t e p e x u x e x c x u e f u e in 0 1 u e n b x u e g u e for x 0 1 where the diffusion coefficient becomes large in a subset which is interior to the interval 0 1 as e 0 we prove the existence of invariant manifolds for pe and using the transversality of invariant manifolds of hyperbolic points obtained in v l carbone j g ruas filho transversality of stable and unstable manifolds for parabolic problems arising in composite materials j math anal appl 303 2005 220 241 we prove that equation pe and its limit problem are topologically equivalent on the attractors

a mathematical modeling for the lookback option with jump diffusion using binomial tree method abstract the binomial tree method btm first proposed by cox et al 1979 4 in diffusion models and extended by amin 1993 9 to jump diffusion models is one of the most popular approaches to pricing options in this paper we present a binomial tree method for lookback options in jump diffusion models and show its equivalence to certain explicit difference scheme we also prove the existence and convergence of the optimal exercise boundary in the binomial tree approximation to american lookback options and give the terminal value of the genuine exercise boundary further numerical simulations are performed to illustrate the theoretical results

upper semicontinuity of attractors and synchronization abstract in this paper we prove that diffusively coupled abstract semilinear parabolic systems synchronize we apply the abstract results obtained to a class of ordinary differential equations and to reaction diffusion problems the technique consists of proving that the attractors for the coupled differential equations are upper semicontinuous with respect to the attractor of a limiting problem explicitly exhibited in the diagonal

parabolic problems with nonlinear boundary conditions and critical nonlinearities we prove existence uniqueness and regularity of solutions for heat equations with nonlinear boundary conditions we study these problems with initial data in l q omega w 1 w q omega 1 q infinity or measures and with critically growing non linearities

density functions for navigation function based systems in this paper we present a scheme for constructing density functions for systems that are almost globally asymptotically stable i e systems for which all trajectories converge to an equilibrium except for a set of measure zero using navigation functions nfs although recently proven converse theorems guarantee the existence of density functions for such systems such results are only existential and the construction of a density function for almost globally asymptotically stable systems remains a challenging task we show that for a specific class of dynamical systems that are defined based on an nf a density function can be easily derived from the system s underlying nf

bifurcations caused by perturbing the domain in an elliptic equation we consider the neumann problem u u up on a continuous family of bounded domains which approach as 0 a set 0 with two connected components and analyze it as a bifurcation problem with two parameters and the bifurcation diagrams and the qualitative properties of the bifurcation sets for p odd and p even are obtained and the relations between them are studied by considering the problem u u au2 u3 for different values of a

stability in the almost everywhere sense a linear transfer operator approach abstract the problem of almost everywhere stability of a nonlinear autonomous ordinary differential equation is studied using a linear transfer operator framework the infinitesimal generator of a linear transfer operator perron frobenius is used to provide stability conditions of an autonomous ordinary differential equation it is shown that almost everywhere uniform stability of a nonlinear differential equation is equivalent to the existence of a non negative solution for a steady state advection type linear partial differential equation we refer to this non negative solution verifying almost everywhere global stability as lyapunov density a numerical method using finite element techniques is used for the computation of lyapunov density

global attractor for sectorial evolutionary equation an abstract differential equation with sectorial operator is shown to generate a dissipative semigroup whenever an appropriate introductory estimate for solutions is known based on hale s theory of dissipative systems such an a priori estimate is then shown to be sufficient for existence of a global attractor examples contained in the final part of the paper illustrate presented ideas 1996 academic press inc 1 introductory notes

the singularly perturbed domain and the characterization for the eigenfunctions with neumann boundary condition on considere un domaine mobile 0 parametre qui degenere partiellement quand 0 et on donne une caracterisation elaboree des comportements des fonctions propres du laplacien avec la condition limite de neumann

dynamics in dumbbell domains i continuity of the set of equilibria we analyze the dynamics of a reaction diffusion equation with homogeneous neumann boundary conditions in a dumbbell domain we provide an appropriate functional setting to treat this problem and as a first step we show in this paper the continuity of the set of equilibria and of its linear unstable manifolds

extremal equilibria for reaction diffusion equations in bounded domains and applications we show the existence of two special equilibria the extremal ones for a wide class of reaction diffusion equations in bounded domains with several boundary conditions including non linear ones they give bounds for the asymptotic dynamics and so for the attractor some results on the existence and or uniqueness of positive solutions are also obtained as a consequence several well known results on the existence and or uniqueness of solutions for elliptic equations are revisited in a unified way obtaining in addition information on the dynamics of the associated parabolic problem finally we ilustrate the use of the general results by applying them to the case of logistic equations in fact we obtain a detailed picture of the positive dynamics depending on the parameters appearing in the equation

on the stability and stabilization of quaternion equilibria of rigid bodies we study attitude control of rigid bodies on quaternion coordinates under three mathematically different perspectives depending on how the system dynamics are assumed to evolve in the first case we suppose that one equilibrium point is chosen a priori and a continuous controller is used under the assumption that the rigid body always spins in the same direction in the second case we relax the assumption that the sense of rotation is constant finally a third scenario is considered in which hybrid switching control is used to choose the direction in which to spin that is both equilibria are continuously considered with regard to less energy consumption it is showed that each of three scenarios must be treated in a different theoretical setting a comparative study in simulations is also provided

transversality of stable and unstable manifolds for parabolic problems arising in composite materials abstract in this paper we study one dimensional parabolic problems that arise from composite materials we show that the eigenvalues and eigenfunctions of the associated linear unbounded operator have the sturm liouville property and the nonincrease of the lap number along the solutions these results are used to show that the stable and unstable manifolds of equilibrium points are transversal

optimal control on the l infty norm of a diffusion process stochastic control problems are considered where the cost to be minimized is either a running maximum of the state variable or more generally a running maximum of a function of the state variable and the control in both cases it is proved that the value function which must be defined on an augmented state space to take care of the non markovian feature of the running maximum is the unique viscosity solution of the associated bellman equation which turns out to be in the second case a variational inequality with an oblique derivative boundary condition most of this work consists of proving the convergence of lp approximations and this is done by purely partial differential equation pde methods

patterns in parabolic problems with nonlinear boundary conditions we obtain existence of asymptotically stable nonconstant equilibrium solutions for semilinear parabolic equations with nonlinear boundary conditions on small domains connected by thin channels we prove the convergence of eigenvalues and eigenfunctions of the laplace operator in such domains this information is used to show that the asymptotic dynamics of the heat equation in this domain is equivalent to the asymptotic dynamics of a system of two ordinary differential equations diffusively weakly coupled the main tools employed are the invariant manifold theory and a uniform trace theorem

large time dynamics of a nonlinear spring mass damper model abstract in this paper we consider a nonlinear strongly damped wave equation as a model for a controlled spring mass damper system and give some results concerning its large time behaviour it can be seen that the infinite dimensional system admits a two dimensional attracting manifold where the equation is well represented by a classical nonlinear oscillations ode which can be exhibited explicitly in contrast to other papers this one applies invariant manifold theory to a problem whose linear part is not self adjoint

asymptotic properties in parabolic problems dominated by a p laplacian operator with localized large diffusion abstract this paper is concerned with upper semicontinuity of the family of attractors associated with nonlinear reaction diffusion equations with principal part governed by a degenerate p laplacian in which the diffusion d blows up in localized regions inside the domain

the dynamics of a one dimensional parabolic problem versus the dynamics of its discretization abstract in this paper we prove that the spatial discretization of a one dimensional system of parabolic equations with suitably small step size contains exactly the same asymptotic dynamics as the continuous problem

convergence of the binomial tree method for american options in a jump diffusion model the paper studies the binomial tree method for american options in a jump diffusion model we employ the theory of viscosity solution to show uniform convergence of the binomial tree method for american options we also prove existence and convergence of the optimal exercise boundary in the binomial tree approximation in addition the terminal value of the optimal exercise boundary is given for american options in jump diffusion models

delay partial differential equations with some large diffusion it is often the case in infinite dimensional dynamical systems that a certain unbounded operator depends upon a parameter v its eigenvalues and eigenfunctions will also depend upon v in many cases see for example l 7 there is a finite number of eigenvalues of the unbounded operator that stay bounded as the parameter varies the remaining eigenvalues blow up at a certain rate in this case one expects the dynamics to be dictated by a system of ordinary differential equations this is indeed the case in several applications and its proof strongly uses the invariant manifold theory in the cases considered in this paper we study the dynamics of infinite dimensional dynamical systems for which a countable number of eigenvalues of the unbounded operator converges to the eigenvalues of an unbounded operator in a lower dimensional domain whereas the remaining blow up this is the case for example in parabolic equations in thin domains studied by hale and raugel s we will not be able to address the general case of thin domains but only the case of a thin straight channel however the property that allows this theory to work is present in the general case that is there is a sequence of eigenvalues that stay bounded and a sequence of eigenvalues that blow up see also arrieta 9 the theory of invariant manifolds will no longer work in this case since it is centered around the existence of backward solutions which cannot be expected in this situation the main tools in this work are a converse theorem on existence of compact attractors and an eigenvalue problem a physical interpretation of a parabolic problem in a thin straight channel is that there is a very high diffusion in one of the directions from which our intuition would say that in that direction spatial changes will not be observed the same interpretation can be given to a damped wave problem in a thin straight channel the wave speed can be interpreted as being very large in one direction this would intuitively imply that there would be no spatial changes for the amplitude of the wave in that direction we also consider the problem called shadow systems where the diffusion is very large for some of the substances in the reaction and not so large for the remaining substances in this

on necessary conditions for almost global stability in the year 2000 the sufficiency of the existence of density functions for the almost global stability of a system was proved in this note we prove the necessity of the existence of density functions for the particular case of global asymptotical stability

numerical analysis on binomial tree methods for a jump diffusion model this paper studies the numerical approximation for an european option pricing model with jump diffusion equivalence of the binomial tree method and an explicit difference scheme is discussed the optimal error estimation of the binomial tree approximation is also given another explicit difference scheme is constructed which has higher accuracy than the binomial tree method numerical results coincide with the theoretical results

large diffusivity and asymptotic behavior in parabolic systems abstract for systems of reaction diffusion equations with neumann boundary conditions it is shown that the solutions are asymptotic to the solutions of an ordinary differential equation if the diffusivity is large the methods apply also to reaction diffusion systems with time delays

an almost global notion of input to state stability a new definition of almost global input to state stability for systems on differentiable manifolds is discussed some of its properties are investigated and specific examples are shown which can be treated by means of dual lyapunov techniques

a morse smale ordinary differential equation in rn abstract in this paper we develop some results on morse smale parabolic differential equation we prove that the limiting problem of morse smale parabolic equations is also morse smale the assumptions for obtaining this result are compact convergence of resolvent operators and the gap condition on eigenvalues as application we exhibit a morse smale ordinary differential equation in r n for each n 2 and a scalar morse smale semilinear equation with localized large diffusion

almost global attraction in planar systems abstract in this work we present a result relating the recent ideas of almost global stability and density functions with the classical poincare bendixson theory for planar systems

well posed p laplacian problems with large diffusion abstract in this work we study the asymptotic behavior of p laplacian parabolic problems of the form u t d p u u p 2 u b u in a bounded smooth domain in r n and under neumann boundary conditions when the diffusion coefficient d becomes large we prove under suitable assumptions that the family of attractors behaves lower and upper semicontinuously as the diffusion increases to infinity

rate of convergence of attractors for singularly perturbed semilinear problems abstract we exhibit a class of singularly perturbed parabolic problems which the asymptotic behavior can be described by a system of ordinary differential equation we estimate the convergence of attractors in the hausdorff metric by rate of convergence of resolvent operators application to spatial homogenization and large diffusion except in a neighborhood of a point will be considered