a semi linear elliptic equation in a strip arising in a two dimensional flame propagation model on montre l existence d une solution stationnaire pour un probleme a 2 dimensions pour l equation elliptique semilineaire de la forme u v y u x g u dans la bande s x y r 2 x r 0 0

stability of strong travelling waves for a non local time delayed reaction diffusion equation the paper is concerned with a non local time delayed reaction diffusion equation we prove the time asymptotic stability of a travelling wavefront without a smallness assumption on its wavelength i e the so called strong wavefront by means of the technical weighted energy method when the initial perturbation around the wave is small the exponential convergent rate is also given selection of a suitable weight plays a key role in the proof

quenching of flames by fluid advection we consider a simple scalar reaction advection diffusion equation with ignitiontype nonlinearity and discuss the following question what kinds of velocity profiles are capable of quenching any given flame provided the velocity s amplitude is adequately large even for shear flows the answer turns out to be surprisingly subtle if the velocity profile changes in space so that it is nowhere identically constant or if it is identically constant only in a region of small measure then the flow can quench any initial data but if the velocity profile is identically constant in a sizable region then the ensuing flow is incapable of quenching large enough flames no matter how much larger the amplitude of this velocity is the constancy region must be wider across than a couple of laminar propagating front widths the proof uses a linear pde associated to the nonlinear problem and quenching follows when the pde is hypoelliptic the techniques used allow the derivation of new nearly optimal bounds on the speed of traveling wave solutions c 2001 john wiley sons inc

stability of numerical methods for delay differential equations abstract consider the following delay differential equation dde y t y t y t t t t 0 with the initial condition y t for t t 0 where and are such that 0 1 0 2 has a unique solution y t the author gives sufficient conditions for the asymptotic stability of the equation 0 1 for which he introduces new definitions of numerical stability the approach is reminiscent of that from the nonlinear stiff ordinary differential equation ode field he investigates stability properties of the class of one point collocation rules in particular the backward euler method turns out to be stable with respect to all the given definitions

multiplicity of positive periodic solutions to superlinear repulsive singular equations abstract in this paper we study positive periodic solutions to the repulsive singular perturbations of the hill equations it is proved that such a perturbation problem has at least two positive periodic solutions when the anti maximum principle holds for the hill operator and the perturbation is superlinear at infinity the proof relies on a nonlinear alternative of leray schauder type and on krasnoselskii fixed point theorem on compression and expansion of cones

speed up of combustion fronts in shear flows this paper is concerned with the analysis of speed up of reaction diffusion advection traveling fronts in infinite cylinders with periodic boundary conditions the advection is a shear flow with a large amplitude and the reaction is nonnegative with either positive or zero ignition temperature the unique or minimal speeds of the traveling fronts are proved to be asymptotically linear in the flow amplitude as the latter goes to infinity solving an open problem from berestycki nonlinear pdes in condensed matter and reactive flows kluwer doordrecht 2003 the asymptotic growth rate is characterized explicitly as the unique or minimal speed of traveling fronts for a limiting degenerate problem and the convergence of the regular traveling fronts to the degenerate ones is proved for positive ignition temperatures under an additional hormander type condition on the flow

pulsating type entire solutions of monostable reaction advection diffusion equations in periodic excitable media abstract we establish the existence of pulsating type entire solutions of reaction advection diffusion equations with monostable nonlinearities in a periodic framework here the nonlinearities include the classic kpp case the pulsating type entire solutions are defined in the whole space and for all time t r by studying a pulsating traveling front connecting a constant unstable stationary state to a stable stationary state which is allowed to be a positive function we proved that there exist pulsating type entire solutions behaving as two pulsating traveling fronts coming from both directions and approaching each other the key techniques are to characterize the asymptotic behavior of the solutions as t in terms of appropriate subsolutions and supersolutions

entire solutions of reaction advection diffusion equations with bistable nonlinearity in cylinders this paper deals with entire solutions and the interaction of traveling wave fronts of bistable reaction advection diffusion equation with infinite cylinders assume that the equation admits three equilibria two stable equilibria 0 and 1 and an unstable equilibrium it is well known that there are different wave fronts connecting any two of those three equilibria by considering a combination of any two of those different traveling wave fronts and constructing appropriate subsolutions and supersolutions we establish three different types of entire solutions finally we analyze a model for shear flows in cylinders to illustrate our main results

stability of fronts for a kpp system ii the critical case abstract we study a system of n 1 coupled semilinear parabolic equations on the real line which depends on a small parameter and reduces to the scalar kolmogorov petrovsky piscounov kpp equation when 0 under appropriate scaling the system has a family of traveling fronts parametrized by their speed when 2 as in the scalar kpp case the case of critical speed 2 say is investigated and it is shown that the system inherits some crucial properties of the kpp equation when is small in particular the asymptotic stability of the front in a local and semiglobal sense first we describe the properties of the front and then apply functional arguments to prove its local stability in an adequate weighted sobolev space moreover the decay rate of the perturbations is shown to be polynomial in time finally we show a semiglobal stability property of the front which also is inherited from the scalar kpp equation

on existence of positive solutions and bounded oscillations for neutral difference equations abstract a criterion for the existence of a positive solution for the first order difference equation xn cxn m pnxn k 0 pn 0 is established results are also obtained for the oscillation and nonoscillation of solutions of a second order difference equation 2 xn cxn m pnxn k where the pn are either of constant sign or are oscillatory some of the results are sharp

two periodic solutions of neutral difference systems depending on two parameters in this paper we establish existence multiplicity and nonexistence of periodic solutions for a class of first order neutral difference systems our approach is based on a fixed point theorem in cones as well as some analysis techniques

stability analysis of numerical schemes for stochastic differential equations stochastic differential equations sdes represent physical phenomena dominated by stochastic processes as for deterministic ordinary differential equations odes various numerical schemes are proposed for sdes in this paper we study the stability of numerical schemes for scalar sdes with respect to the mean square norm which we call ms stability we will show some figures of the ms stability domain or regions for some numerical schemes and present numerical results which confirm it this notion is an extension of absolute stability in numerical methods for odes

convergence to travelling waves for solutions of a class of semilinear parabolic equations abstract in this paper we investigate the long time behaviour of the solutions of a class of semilinear parabolic equations in an infinite cylinder such equations arise in various fields such as biology or flame propagation theory we prove that the solutions whose initial data together with being increasing along the axis of the cylinder satisfy certain growth conditions at its ends will converge exponentially to travelling front solutions

front propagation into unstable states universal algebraic convergence towards uniformly translating pulled fronts fronts that start from a local perturbation and propagate into a linearly unstable state come in two classes pulled fronts and pushed fronts the term pulled front expresses that these fronts are pulled along by the spreading of linear perturbations about the unstable state accordingly their asymptotic speedv equals the spreading speed of perturbations whose dynamics is governed by the equations linearized about the unstable state the central result of this paper is the analysis of the convergence of asymptotically uniformly traveling pulled fronts towardsv we show that when such fronts evolve from sufficiently steep initial conditions which initially decay faster than e x forx 1 t hey have auniversal relaxation behavioras timet 1 the velocity of a pulled front always relaxes algebraically like v t d v 3 2t c 3 p d d 2 t 3 2 c o 1 t 2 the parametersv and d are determined through a saddle point analysis from the equation of motion linearized about the unstable invaded state this front velocity is independent of the precise value of the front amplitude which one tracks to measure the front position the interior of the front is essentially slaved to the leading edge and develops universally as x t d8v t x rt dt 0 v t 0 c o 1 t 2 where8v x vt is a uniformly translating front solution with velocity v v our result which can be viewed as a general center manifold result for pulled front propagation is derived in detail for the well known nonlinear diffusion equation of type t d 2 x c 3 where the invaded unstable state is d 0 even for this simple case the subdominant t 3 2 term extends an earlier result of bramson our analysis is then generalized to more general sets of partial differential equations with higher spatial or temporal derivatives to pdes with memory kernels and also to difference equations such as those that occur in numerical finite difference codes our universal result for pulled fronts thus implies independence i of the level curve which is used to track the front position ii of the precise nonlinearities iii of the precise form of the linear operators in the dynamical equation and iv of the precise initial conditions as long as they are sufficiently steep the only remainders of the explicit form of the dynamical equation are the nonlinear solutions 8v and the three saddle point parameters v and d as our simulations confirm all our analytical predictions in every detail it can be concluded that we have a complete analytical understanding of the propagation mechanism and relaxation behavior of pulled fronts if they are uniformly translating for t 1 an immediate consequence of the slow algebraic relaxation is that the standard moving boundary approximation breaks down for weakly curved pulled fronts in two or three dimensions in addition to our main result for pulled fronts we also discuss the propagation and convergence of fronts emerging from

the evolution of reaction diffusion waves in a class of scalar reaction diffusion equations algebraic decay rates in this paper we consider an initial boundary value problem for the generalized fisher kolmogorov petrovskii piscounov fkpp equation of order m 1 with the non negative initial data being of o x at large x dimensionless distance where 1 m 1 the corresponding problem when 1 m 1 having been considered in detail by needham and barnes nonlinearity 12 1999 41 58 using matched asymptotic expansions we are able to obtain the complete structure of the solution to the initial boundary value problem for large t time which exhibits the formation of a permanent form reaction diffusion travelling wave structure this being in contrast to the case when 1 m 1 where the large t time structure exhibits the formation of an accelerating phase wave

nonplanar traveling fronts in reaction diffusion equations with combustion and degenerate fisher kpp nonlinearities abstract this paper is concerned with nonplanar traveling fronts in reaction diffusion equations with combustion nonlinearity and degenerate fisher kpp nonlinearity our study contains two parts in the first part we establish the existence of traveling fronts of pyramidal shape in r 3 and in the second part we establish the existence and stability of v shaped traveling fronts in r 2

stability of travelling fronts in a model for flame propagation part i linear analysis we investigate the stability of travelling wave solutions of the multidimensional thermodiffusive model for flame propagation with unit lewis number this model consists in a system of two nonlinear parabolic equations posed in an infinite cylinder with neumann boundary conditions in this paper we prove that every travelling wave solution of this model is linearly stable our tools are exponential decay estimates for solutions of elliptic equations in a cylinder and the maximum principle for parabolic equations

periodic solutions of linear and quasilinear neutral functional differential equations abstract in this paper we will use the fourier expansions to generalize hale s alternative results for the existence of periodic solutions of linear retarded functional differential equations to neutral equations using some a priori estimation and the leray schauder degree theory we will obtain some existence theorems of periodic solutions for a class of quasilinear neutral equations the results also generalize the works of hatvani and krisztin on retarded equations

multiple solutions of some semilinear elliptic equations in slender cylindrical domains we analyze the semilinear problem ul ul 0 in 0 2l ul 0 at 0 2l where rn 1 is a bounded smooth domain and is a given nonlinearity in the limit l the solutions of that problem are related to those of the n 1 dimensional problem v v 0 in v 0 at in particular necessary and sufficient conditions on a given solution of the latter v are given for the former to have a solution ul such that ul x1 x2 xn v x2 xn as l in most part of the cylinder 0 2l related bifurcation problems and some generalizations are also considered

existence of traveling waves of invasion for ginzburg landau type problems in infinite cylinders we study a class of systems of reaction diffusion equations in infinite cylinders which arise within the context of ginzburg landau theories and describe the kinetics of phase transformation in second order or weakly first order phase transitions with non conserved order parameters we use a variational characterization to study the existence of a special class of traveling wave solutions which are characterized by a fast exponential decay in the direction of propagation our main result is a simple verifiable criterion for existence of these traveling waves under the very general assumptions of non linearities we also prove boundedness regularity and some other properties of the obtained solutions as well as several sufficient conditions for existence or non existence of such traveling waves and give rigorous upper and lower bounds for their speed in addition we prove that the speed of the obtained solutions gives a sharp upper bound for the propagation speed of a class of disturbances which are initially sufficiently localized we give a sample application of our results using a computer assisted approach

exponential mean square stability of numerical methods for systems of stochastic differential equations this paper is concerned with exponential mean square stability of the classical stochastic theta method and the so called split step theta method for stochastic systems first we consider linear autonomous systems under a sufficient and necessary condition for exponential mean square stability of the exact solution it is proved that the two classes of theta methods with q 0 5 are exponentially mean square stable for all positive step sizes and the methods with q 0 5 still unconditionally preserves the exponential mean square stability of the underlying systems but the stochastic theta method does not have this property finally we consider stochastic differential equations with jumps some similar results are derived

the evolution of travelling waves in the weakly hyperbolic generalized fisher model in this paper we consider an initial boundary value problem for a weakly hyperbolic generalized fisher equation of order p 0 we show that for each p 0 the large time structure of the solution involves the evolution of a propagating wavefront however we demonstrate that the case p 1 is a bifurcation point in the sense that for p 1 the wavefront is of reaction diffusion type while for 0 p

convergence and stability of two classes of theta milstein schemes for stochastic differential equations abstract this paper examines convergence and stability of the two classes of theta milstein schemes for stochastic differential equations sdes with non global lipschitz continuous coefficients the split step theta milstein sstm scheme and the stochastic theta milstein stm scheme for 1 2 1 this paper concludes that the two classes of theta milstein schemes converge strongly to the exact solution with the order 1 for 0 1 2 under the additional linear growth condition for the drift coefficient these two classes of the theta milstein schemes are also strongly convergent with the standard order this paper also investigates exponential mean square stability of these two classes of the theta milstein schemes for 1 2 1 these two theta milstein schemes can share the exponential mean square stability of the exact solution for 0 1 2 similar to the convergence under the additional linear growth condition these two theta milstein schemes can also reproduce the exponential mean square stability of the exact solution

positive periodic solutions of functional differential equations abstract we consider the existence multiplicity and nonexistence of positive periodic solutions for the periodic equation x t a t g x x t b t f x t t where a b c r 0 are periodic 0 a t dt 0 0 b t dt 0 f g c 0 0 and f u 0 for u 0 g x is bounded t is a continuous periodic function define f 0 lim u 0 f u u f lim u f u u i0 number of zeros in the set f 0 f and i number of infinities in the set f 0 f we show that the equation has i0 or i positive periodic solution s for sufficiently large or small 0 respectively

numerical methods for nonlinear stochastic differential equations with jumps we present and analyse two implicit methods for ito stochastic differential equations sdes with poisson driven jumps the first method ssbe is a split step extension of the backward euler method the second method cssbe arises from the introduction of a compensated martingale form of the poisson process we show that both methods are amenable to rigorous analysis when a one sided lipschitz condition rather than a more restrictive global lipschitz condition holds for the drift our analysis covers strong convergence and nonlinear stability we prove that both methods give strong convergence when the drift coefficient is one sided lipschitz and the diffusion and jump coefficients are globally lipschitz on the way to proving these results we show that a compensated form of the euler maruyama method converges strongly when the sde coefficients satisfy a local lipschitz condition and the pth moment of the exact and numerical solution are bounded for some p 2 under our assumptions both ssbe and cssbe give well defined unique solutions for sufficiently small stepsizes and ssbe has the advantage that the restriction is independent of the jump intensity we also study the ability of the methods to reproduce exponential mean square stability in the case where the drift has a negative one sided lipschitz constant this work extends the deterministic nonlinear stability theory in numerical analysis we find that ssbe preserves stability under a stepsize constraint that is independent of the initial data cssbe satisfies an even stronger condition and gives a generalization of b stability finally we specialize to a linear test problem and show that cssbe has a natural extension of deterministic a stability the difference in stability properties of the ssbe and cssbe methods emphasizes that the addition of a jump term has a significant effect that cannot be deduced directly from the non jump literature

how travelling waves attract the solutions of kpp type equations we consider in this paper a general reaction diffusion equation of the kpp kolmogorov petrovskii piskunov type posed on an infinite cylinder such a model will have a family of pulsating waves of constant speed larger than a critical speed c the family of all supercritical waves attract a large class of initial data and we try to understand how we describe in this paper the fate of an initial datum trapped between two supercritical waves of the same velocity the solution will converge to a whole set of translates of the same wave and we identify the convergence dynamics as that of an effective drift around which an effective diffusion process occurs in several nontrivial particular cases we are able to describe the dynamics by an effective equation

strong and weak divergence in finite time of euler s method for stochastic differential equations with non globally lipschitz continuous coefficients the stochastic euler scheme is known to converge to the exact solution of a stochastic differential equation sde with globally lipschitz continuous drift and diffusion coefficients recent results extend this convergence to coefficients that grow at most linearly for superlinearly growing coefficients finite time convergence in the strong mean square sense remains in this article we answer this question to the negative and prove for a large class of sdes with non globally lipschitz continuous coefficients that euler s approximation converges neither in the strong mean square sense nor in the numerically weak sense to the exact solution at a finite time point even worse the difference of the exact solution and of the numerical approximation at a finite time point diverges to infinity in the strong mean square sense and in the numerically weak sense

spatial decay and stability of traveling fronts for degenerate fisher type equations in cylinder abstract this paper is concerned with the spatial decay and asymptotic stability of multidimensional traveling fronts for the degenerate fisher type equation in an unbounded cylinder firstly by applying the moving plane argument and the generalized center manifold theorem we obtain the uniqueness and exponential decay of the traveling front with the critical speed and the non exponential decay of traveling fronts with non critical speeds especially for the p degree fisher equation we get the precise algebraic decaying rates and the higher order expansion of traveling fronts with non critical speeds secondly by applying the spectral analysis and sub super solution method we prove the nonlinear exponential stability of all traveling fronts in some exponentially weighted spaces and the lyapunov stability of traveling fronts with non critical speeds in some polynomially weighted spaces finally by combining the sub super solution method with the nonlinear stability results we get the asymptotic behavior and the asymptotic spreading speeds of the solution for more general initial data which are proved to be determined by the spatial decay of the initial data at one end

periodic solutions of neutral differential equation with multiple deviating arguments in this paper by means of an abstract continuous theorem of k set contractive operator and some analysis techniques the authors study the existence of periodic solution of the nonlinear neutral differential equation with multiple deviating argumentsddt x t cx email protected email protected i 1 email protected i t f x email protected i t email protected j 1 email protected j t g x email protected j t p t a new result is obtained

stability of analytical and numerical solutions of nonlinear stochastic delay differential equations this paper concerns the stability of analytical and numerical solutions of nonlinear stochastic delay differential equations sddes we derive sufficient conditions for the stability contractivity and asymptotic contractivity in mean square of the solutions for nonlinear sddes the results provide a unified theoretical treatment for sddes with constant delay and variable delay including bounded and unbounded variable delays then the stability contractivity and asymptotic contractivity in mean square are investigated for the backward euler method it is shown that the backward euler method preserves the properties of the underlying sddes the main results obtained in this work are different from those of razumikhin type theorems indeed our results hold without the necessity of constructing or finding an appropriate lyapunov functional

sharp thresholds between finite spread and uniform convergence for a reaction diffusion equation with oscillating initial data abstract we investigate the large time dynamics of solutions of multi dimensional reaction diffusion equations with ignition type nonlinearities we consider solutions which are in some sense locally persistent at large time and initial data which asymptotically oscillate around the ignition threshold we show that as time goes to infinity any solution either converges uniformly in space to a constant state or spreads with a finite speed uniformly in all directions furthermore the transition between these two behaviors is sharp with respect to the period vector of the asymptotic profile of the initial data we also show the convergence to planar fronts when the initial data are asymptotically periodic in one direction

stability of the drift implicit and double implicit milstein schemes for nonlinear sdes abstract this paper examines the stability of numerical solutions of nonlinear stochastic differential equations sdes with non global lipschitz continuous coefficients two implicit milstein schemes called drift implicit milstein scheme and double implicit milstein scheme are considered to simulate the underlying sdes it is proved that the schemes can preserve the stability and contractivity in mean square of the underlying systems

periodic solutions of difference equations with general periodicity abstract motivated by 1 we study the existence of periodic solutions of a perturbed continuous difference equation of first order without the assumptions of one periodicity moreover we generalize our results to the case of higher order difference equations we also give an algorithm to estimate the bound of the parameter for which periodic solutions exist

balanced implicit methods for stiff stochastic systems this paper introduces some implicitness in stochastic terms of numerical methods for solving stiff stochastic differential equations and especially a class of fully implicit methods the balanced methods their order of strong convergence is proved numerical experiments compare the stability properties of these schemes with explicit ones

traveling waves for the fitzhugh nagumo system on an infinite channel abstract we are concerned with the traveling wave solutions for the fitzhugh nagumo system on an infinite channel based on a variational formulation in which a non local term depends on a parameter the speed of a traveling wave can be selected out furthermore to show the existence of a traveling wave solution with such a speed we seek a minimizer subject to a constraint in the way of solving the variational problem we apply a truncation technique to the nonlocal term to obtain a minimizer located in a bounded invariant region

a discrete periodic lotka volterra system with delays in this paper we investigate the following discrete periodic lotka volterra system with delays x n 1 x n expb n i 1qai n x n ti j 1mcj n y n li y n 1 y n expr n i 1qdi n x n ki j 1mej n y n si the sufficient and realistic conditions are obtained for the existence of positive periodic solutions and the permanence for the above system

algebraic dichotomies with an application to the stability of riemann solutions of conservation laws abstract recently there has been some interest on the stability of waves where the functions involved grow or decay at an algebraic rate x m in this paper we define the so called algebraic dichotomy that may aid in treating such problems we discuss the basic properties of the algebraic dichotomy methods of detecting it and calculating the power of the weight function we present several examples 1 the bessel equation 2 the n degree fisher type equation 3 hyperbolic conservation laws in similarity coordinates 4 a system of conservation laws with a dafermos type viscous regularization we show that the linearized system generates an analytic semigroup in the space of algebraic decay functions this example motivates our work on algebraic dichotomies

the asymptotic behavior of the solutions of some semilinear elliptic equations in cylindrical domains abstract the bounded solutions of some semilinear elliptic equations in unbounded cylindrical domains are considered their asymptotic behavior as x x coordinate along the axis of the cylinder is analyzed and used to establish uniqueness and monotonicity in the x variable of travelling wavefronts of some parabolic problems

the evolution of travelling waves in fractional order autocatalysis with decay i permanent form travelling waves this paper considers a model isothermal autocatalytic reaction scheme with termination when the reaction orders m n are fractional i e 0 n and k in 0 k c where kc depends upon m and n it is further established that these wavefronts are of excitable rather than fisher kolmogorov type and have only semi infinite support that is the reaction is totally dormant ahead of the wavefront these features are in contrast to the case when m n geq 1 as studied by merkin needham and scott proc roy soc london ser a 424 1989 pp 187 209 merkin and needham j engrg math 23 1989 pp 343 356 and needham and merkin phil trans roy soc london ser a 337 1991 pp 261 274 it should be noted that the corresponding reaction diffusion equations are singular in the sense that the reaction and decay terms are not lipsch

the asymptotic critical wave speed in a family of scalar reaction diffusion equations abstract we study traveling wave solutions for the class of scalar reaction diffusion equations u t 2 u x 2 f m u where the family of potential functions f m is given by f m u 2 u m 1 u for each m 1 real there is a critical wave speed c crit m that separates waves of exponential structure from those which decay only algebraically we derive a rigorous asymptotic expansion for c crit m in the limit as m this expansion also seems to provide a useful approximation to c crit m over a wide range of m values moreover we prove that c crit m is c smooth as a function of m 1 our analysis relies on geometric singular perturbation theory as well as on the blow up technique and confirms the results obtained by means of asymptotic methods in d j needham a n barnes reaction diffusion and phase waves occurring in a class of scalar reaction diffusion equations nonlinearity 12 1 1999 41 58 t p witelski k ono t j kaper critical wave speeds for a family of scalar reaction diffusion equations appl math lett 14 1 2001 65 73

existence of pulsating waves of advection reaction diffusion equations of ignition type by a new method abstract the goal of this paper is to study the advection reaction diffusion equation of ignition type u t u q x y u f u xin j x xin existence of planar flame fronts in convective diffusive periodic media arch ration mech anal 121 1992 205 233 berestycki and hamel h berestycki f hamel front propagation in periodic excitable media comm pure appl math 55 2002 949 1032 proved the existence of pulsating waves using the theory of degenerate elliptic equations our goal is to give an alternative and rather natural proof of the same result first we transform the problem so that the existence of pulsating waves is equivalent to the existence of 1 periodic in time solutions of a nonlinear parabolic equation with 1 periodic in time coefficients next we prove the existence of such periodic solutions using a continuation method based on the implicit function theorem

two periodic solutions of n dimensional neutral functional difference systems in this paper we establish existence multiplicity and nonexistence of periodic solutions for a class of first order n dimensional neutral functional difference systems our approach requires the krasnoselskii fixed point theorem in cones

positive periodic solutions of nonlinear functional difference equations depending on a parameter in this paper we use the upper and lower solutions method to show that there existsa l such that the nonlinear functional difference equation of the form g n 1 a n g n lh n g n t n n e z has at least one positive t periodic solutions for l e 0 l and does not have any positive t periodic solutions for l l where a n h n and t n are t periodic functions

first order strong approximations of scalar sdes defined in a domain we are interested in strong approximations of one dimensional sdes which have non lipschitz coefficients and which take values in a domain under a set of general assumptions we derive an implicit scheme that preserves the domain of the sdes and is strongly convergent with rate one moreover we show that this general result can be applied to many sdes we encounter in mathematical finance and bio mathematics we will demonstrate flexibility of our approach by analyzing classical examples of sdes with sublinear coefficients cir cev models and wright fisher diffusion and also with superlinear coefficients 3 2 volatility ait sahalia model our goal is to justify an efficient multilevel monte carlo method for a rich family of sdes which relies on good strong convergence properties

the truncated milstein method for stochastic differential equations with commutative noise inspired by the truncated eulermaruyama method developed in mao 2015 we propose the truncated milstein method in this paper the strong convergence rate is proved to be close to 1 for a class of highly non linear stochastic differential equations with commutative noise numerical examples are given to illustrate the theoretical results

existence and nonexistence of curved front solution of a biological equation abstract this paper deals with the existence of curved front solution of a partial differential equation coming from a mathematical model of stroke the equation is of reaction diffusion type in a cylinder of radius r and of diffusion and absorption type outside of the cylinder we prove the nonexistence of a travelling front when r is small enough and the existence if r is large enough using a recent energy method we construct the travelling front as the limit in time of a solution with a well chosen initial condition in a travelling referential

entire solutions in monostable reaction advection diffusion equations in infinite cylinders abstract this paper is concerned with entire solutions of a monostable reaction advection diffusion equation in infinite cylinders without the condition f u f 0 by constructing a quasi invariant manifold we prove that there exist two classes of entire solutions furthermore we show that one class of such entire solutions is unique up to space and time translation

traveling waves for a boundary reaction diffusion equation abstract we prove the existence of a traveling wave solution for a boundary reaction diffusion equation when the reaction term is the combustion nonlinearity with ignition temperature a key role in the proof is plaid by an explicit formula for traveling wave solutions of a free boundary problem obtained as singular limit for the reaction diffusion equation the so called high energy activation energy limit this explicit formula which is interesting in itself also allows us to get an estimate on the decay at infinity of the traveling wave which turns out to be faster than the usual exponential decay

existence and asymptotics of travelling waves in a thermo diffusive model in half cylinders part ii oblique boundary conditions abstract in this paper we investigate a system of reaction diffusion equations set in half cylinders and involving oblique boundary conditions this type of system arises in the modelization of burner flames and approach in a suitable sense conical shaped fronts in the whole space we prove the existence of solutions and some properties of the speed of propagation of the travelling fronts in some asymptotic limit

large time behavior of solutions for degenerate p degree fisher equation with algebraic decaying initial data abstract this paper is concerned with the large time behavior of solutions to the one dimensional degenerate p degree fisher equation where the initial data are assumed to be asymptotically front like and to decay to zero non exponentially at one end by applying sub super solution method we first prove the lyapunov stability of all the wave fronts with noncritical speeds in some polynomially weighted space further by using semigroup estimates sub super solution method and the nonlinear stability of the traveling waves we obtain the asymptotic estimates of the solutions which indicate that the solution still moves like a wave front and the asymptotic spreading speed of the level set of the solution can be finite or infinite which is solely determined by the detailed decaying rate of the initial data

singular limit and homogenization for flame propagation in periodic excitable media this paper is concerned with a class of singular equations modelling the combustion of premixed gases in periodic media the model involves two parameters the period of the medium l and a singular parameter related to the activation energy the existence of pulsating travelling fronts for fixed and l was proved by berestycki hamel in bh in the present paper we investigate the behaviour of such solutions when more precisely we establish that pulsating travelling fronts behave like travelling waves when the period l is small and we also study the convergence of the solution as goes to zero and l is fixed toward a solution of a free boundary problem

preserving stability implicit euler method for nonlinear volterra and neutral functional differential equations in banach space this paper is concerned with the contractivity and asymptotic stability properties of the implicit euler method iem for nonlinear functional differential equations fdes these properties are first analyzed for volterra fdes and then the analysis is extended to the case of neutral fdes nfdes such an extension is particularly important since nfdes are more general and have received little attention in the literature the main result we establish is that the iem with linear interpolation can completely preserve these stability properties of the analytical solution to such fdes

stability of travelling fronts in a model for flame propagation part ii nonlinear stability this paper is concerned with the nonlinear stability of the travelling wave solutions of the multidimensional thermodiffusive model for flame propagation with unit lewis number the model consists in a semilinear parabolic equation in an infinite cylinder with neumann boundary conditions we prove that any solution which is initially close to a travelling wave will converge to a translate of that wave

exponential stability in p th mean of solutions and of convergent euler type solutions of stochastic delay differential equations one concept of the stability of a solution of an evolutionary equation relates to the sensitivity of the solution to perturbations in the initial data there are other stability concepts notably those concerned with persistent perturbations results are presented on the stability in p th mean of solutions of stochastic delay differential equations with multiplicative noise and of stochastic delay difference equations the difference equations are of a type found in numerical analysis and we employ our results to obtain mean square stability criteria for the solution of the euler maruyama discretization of stochastic delay differential equations the analysis proceeds as follows we show that an inequality of halanay type derivable via comparison theory can be employed to derive conditions for p th mean stability of a solution we then produce a discrete analogue of the halanay type theory that permits us to develop a p th mean stability analysis of analogous stochastic difference equations the application of the theoretical results is illustrated by deriving mean square stability conditions for solutions and numerical solutions of a constant coefficient linear test equation

fast propagation for kpp equations with slowly decaying initial conditions abstract this paper is devoted to the analysis of the large time behavior of solutions of one dimensional fisher kpp reaction diffusion equations the initial conditions are assumed to be globally front like and to decay at infinity towards the unstable steady state more slowly than any exponentially decaying function we prove that all level sets of the solutions move infinitely fast as time goes to infinity the locations of the level sets are expressed in terms of the decay of the initial condition furthermore the spatial profiles of the solutions become asymptotically uniformly flat at large time this paper contains the first systematic study of the large time behavior of solutions of kpp equations with slowly decaying initial conditions our results are in sharp contrast with the well studied case of exponentially bounded initial conditions