existence and upper semicontinuity of random attractors for stochastic degenerate parabolic equations with multiplicative noises the existence and the upper semicontinuity of random attractors in l 2 r n for stochastic semi linear degenerate parabolic equations with multiplicative noises are obtained though the spatial domain is the entire space r n a compact embedding is applied to get the compactness of random absorbing sets from which follows the compactness of the system and the precompactness of the union of all perturbed random attractors

random pullback attractor of a non autonomous local modified stochastic swift hohenberg equation with multiplicative noise in this paper we study the existence of the random d pullback attractor of a non autonomous local modified stochastic swift hohenberg equation with multiplicative noise in the stratonovich sense we show that a random d pullback attractor exists in h02 d when its external force has exponential growth due to the stochastic term a priori estimates are delicate we used the ornstein uhlenbeck transformation and its properties to overcome the difficulty that the stochastic term brings to us

global attractors for the wave equation with nonlinear damping based on a new a priori estimate method so called asymptotic a priori estimate the existence of a global attractor is proved for the wave equation utt kg ut u f u 0 on a bounded domain r3 with dirichlet boundary conditions the nonlinear damping term g is supposed to satisfy the growth condition c1 s c2 g s c3 1 s p where 1 p 0 is arbitrary the nonlinear term f is supposed to satisfy the growth condition f s c4 1 s q where q 2 it is remarkable that when 2 p 5 we positively answer an open problem in chueshov and lasiecka i chueshov i lasiecka long time behavior of second evolution equations with nonlinear damping math scuola norm sup 2004 and improve the corresponding results in feireisl e feireisl global attractors for damped wave equations with supercritical exponent j differential equations 116 1995 431 447

regularity of invariant sets in semilinear damped wave equations abstract under fairly general assumptions we prove that every compact invariant subset i of the semiflow generated by the semilinear damped wave equation u t t u t x u i j a i j x u x j x i f x u t x 0 u 0 t x 0 in h 0 1 l 2 is in fact bounded in d a h 0 1 here is an arbitrary possibly unbounded domain in r 3 a u x u i j a i j x u x j x i is a positive selfadjoint elliptic operator and f x u is a nonlinearity of critical growth the nonlinearity f x u needs not to satisfy any dissipativeness assumption and the invariant subset i needs not to be an attractor

self consistent projection operator method for describing the non markovian time evolution of subsystems it is pointed out that the usual projection operator method yields non positive definite reduced density operators for subsystems in order to obtain non negative probabilities a new self consistent projection operator method for treating non markovian behaviour is developed finally the applicability of the method is shown by an example

one dimensional random attractor and rotation number of the stochastic damped sine gordon equation abstract this paper is devoted to the study of the asymptotic dynamics of the stochastic damped sine gordon equation with homogeneous neumann boundary condition it is shown that for any positive damping and diffusion coefficients the equation possesses a random attractor and when the damping and diffusion coefficients are sufficiently large the random attractor is a one dimensional random horizontal curve regardless of the strength of noise hence its dynamics is not chaotic it is also shown that the equation has a rotation number provided that the damping and diffusion coefficients are sufficiently large which implies that the solutions tend to oscillate with the same frequency eventually and the so called frequency locking is successful

random attractors for the stochastic benjamin bona mahony equation on unbounded domains abstract we prove the existence of a compact random attractor for the stochastic benjamin bona mahony equation defined on an unbounded domain this random attractor is invariant and attracts every pulled back tempered random set under the forward flow the asymptotic compactness of the random dynamical system is established by a tail estimates method which shows that the solutions are uniformly asymptotically small when space and time variables approach infinity

finite fractal dimensions of random attractors for stochastic fitzhugh nagumo system with multiplicative white noise abstract in this paper we consider the asymptotic behavior of solutions for stochastic non autonomous fitzhugh nagumo system with multiplicative white noise first we prove the existence of random attractor of the random dynamical system generated by the solutions of considered system then we present some conditions for estimating an upper bound of the fractal dimension of a random invariant set of a random dynamical system on a separable banach space and apply these conditions to prove the finiteness of fractal dimension of random attractor

existence stability and bifurcation of random complete and periodic solutions of stochastic parabolic equations abstract in this paper we study the existence stability and bifurcation of random complete and periodic solutions for stochastic parabolic equations with multiplicative noise we first prove the existence and uniqueness of tempered random attractors for the stochastic equations and characterize the structures of the attractors by random complete solutions we then examine the existence and stability of random complete quasi solutions and establish the relations of these solutions and the structures of tempered attractors when the stochastic equations are incorporated with periodic forcing we obtain the existence and stability of random periodic solutions for the stochastic chafee infante equation we further establish the multiplicity and stochastic bifurcation of complete and periodic solutions

global attractors for wave equations with nonlinear interior damping and critical exponents abstract in this paper we study the global attractors for wave equations with nonlinear interior damping we prove the existence regularity and finite dimensionality of the global attractors without assuming a large value for the damping parameter when the growth of the nonlinear terms is critical

analytic continuation of generalized susceptibility and system reservoir correlation effects a fully non markovian calculation of the generalized susceptibility of a quantum system weakly coupled to a heat bath is presented in contrast to most of the treatments we provide a complete solution in the sense that it includes the analytic continuation to the lower half plane various applications are discussed including the consideration of boson and fermion heat baths it is shown that our result takes into account system reservoir correlations which are neglected in the markovian approximation

attractors for random dynamical systems a criterion for existence of global random attractors for rds is established existence of invariant markov measures supported by the random attractor is proved for spde this yields invariant measures for the associated markov semigroup the results are applied to reation diffusion equations with additive white noise and to navier stokes equations with multiplicative and with additive white noise

on miura transformations among nonlinear partial differential equationsa in this paper we study miura transformations u v from partial differential equations uxxx f u ux ut to nonlinear partial differential equations g v vx vt xlv t1v 0 defined using integrable systems on v we classify all such miura transformations under some restrictions and hence generalize the classical miura transformation to a large class of nonlinear partial differential equations for some examples by applying miura transformations found in this paper we derive exact solutions v from known solutions u in particular kink and soliton kink solutions of vt 32vxsin2v 12vx3 vxxx are obtained from constant solutions and soliton solutions of the mkdv equation as another application of miura transformations of this paper we deduce a new backlund transformation for each of vt 32vxsin2v 12vx3 vxxx and vt 32vxsinh2v 12vx3 vxxx from the known backlund transformations for the mkdv equations

dynamics for a stochastic reaction diffusion equation with additive noise abstract in this paper we present a new scheme to study the dynamics of a stochastic reaction diffusion equation with the nonlinearity satisfying a dissipative condition with polynomial growth of arbitrary order p 2 firstly we use this scheme to establish some new estimates higher order integrability of the difference of the solutions near the initial time instead of using the usual estimates about higher regularities and higher order integrability of solutions secondly we verify that the attraction in the usual l 2 q l 2 q d pullback random attractor indeed can be l 2 norm for any 0 and the solutions of the equation are continuous in h 0 1 q with respect to initial data thirdly we obtain the existence of pullback random attractor in h 0 1 q as an application of the continuity even for the corresponding deterministic cases the results and methods introduced in this paper are new

h1 random attractors for stochastic reaction diffusion equations with additive noise abstract in this paper a new theorem obtained by li and guo y li b guo random attractors for quasi continuous random dynamical systems and applications to stochastic reaction diffusion equations j differential equations 245 7 2008 1775 1800 is employed to some reaction diffusion equations with additive noise we prove the unique existence of d random attractors in h 0 1 for the corresponding random dynamical system we also obtain the existence of a unique random equilibrium for the generated random dynamical system under some restrictive assumption on the coefficients

pullback attractors for reaction diffusion delay equations on unbounded domains with non autonomous deterministic and stochastic forcing terms the existence of a pullback attractor in a space of higher regularity is established for the non autonomous non compact dynamical system associated with the reaction diffusion delay equation with both non autonomous deterministic and stochastic forcing terms defined on all n dimensional space the asymptotical compactness of the solutions is demonstrated by using the uniform estimates for far field values of solutions and a new method

attractors for reaction diffusion equations in unbounded domains abstract in this paper we study the asymptotic behaviour of solutions for parabolic non linear evolution equations in r n we prove the asymptotic compactness of the solutions and then establish the existence of the global attractor in l 2 r n

fractal dimension of random attractors for stochastic non autonomous reaction diffusion equations in this paper we first give some conditions for bounding the fractal dimension of a random invariant set for a non autonomous random dynamical system on a separable banach space then we apply these conditions to prove the finiteness of fractal dimension of the random attractors for stochastic reaction diffusion equations with multiplicative white noise and additive white noise

random attractors for the stochastic coupled fractional ginzburg landau equation with additive noise this paper is concerned with the stochastic coupled fractional ginzburg landau equation with additive noise we first transform the stochastic coupled fractional ginzburg landau equation into random equations whose solutions generate a random dynamical system then we prove the existence of random attractor for random dynamical system

random exponential attractor for stochastic reaction diffusion equation with multiplicative noise in r3 abstract in this paper we first improve the existing conditions for the existence of a random exponential attractor for a continuous cocycle on a separable banach space then we consider the existence of a random exponential attractor for stochastic non autonomous reaction diffusion equation with multiplicative noise defined in r 3 which implies the existence of a random attractor with finite fractal dimension the essential difficulty here is the continuity of the spectrum of the linear part of the equation which can be overcome by the tail estimation of solutions of equation and carefully decomposing the solution into a sum of three parts of whose one part is finite dimensional and other two parts are quickly decay in mean sense

flattening squeezing and the existence of random attractors the study of qualitative properties of random and stochastic differential equations is now one of the most active fields in the modern theory of dynamical systems in the deterministic case the properties of flattening and squeezing in infinite dimensional autonomous dynamical systems require the existence of a bounded absorbing set and imply the existence of a global attractor the flattening property involves the behaviour of individual trajectories while the squeezing property involves the difference of trajectories it is shown here that the flattening property is implied by the squeezing property and is in fact weaker since the attractor in a system with the flattening property can be infinite dimensional whereas it is always finite dimensional in a system with the squeezing property the flattening property is then generalized to random dynamical systems for which it is called the pullback flattening property it is shown to be weaker than the random squeezing property but equivalent to pullback asymptotic compactness and pullback limit set compactness and thus implies the existence of a random attractor the results are also valid for deterministic non autonomous dynamical systems formulated as skew product flows

invariant manifolds for stochastic partial differential equations invariant manifolds provide the geometric structures for describing and understanding dynamics of nonlinear systems the theory of invariant manifolds for both finite and infinite dimensional autonomous deterministic systems and for stochastic ordinary differential equations is relatively mature in this paper we present a unified theory of invariant manifolds for infinite dimensional random dynamical systems generated by stochastic partial differential equations we first introduce a random graph transform and a fixed point theorem for nonautonomous systems then we show the existence of generalized fixed points which give the desired invariant manifolds

pullback attractors for multi valued non compact random dynamical systems generated by reaction diffusion equations on an unbounded domain in this paper we study pullback attractors of reaction diffusion equations on an unbounded domain with non autonomous deterministic as well as stochastic forcing terms for which the uniqueness of solutions need not hold we first present the existence and structure of pullback attractors of multi valued non compact random dynamical systems then we prove the existence of pullback attractors in l2 rn lp rn and h1 rn for multi valued non compact random dynamical systems associated with the reaction diffusion equations on rn and the identical relation of pullback attractors in different spaces is also provided in particular the measurability of pullback attractors is established for reaction diffusion equations on rn

on backlund transformations for nonlinear partial differential equations in this paper backlund transformations for nonlinear partial differential equations are obtained without using any structure associated with the equations we classify all the nonlinear partial differential equations of the form uxxx f u ux ut that have backlund transformations whose definition involves only u ux uxx and a function determined by u ux and uxx via an integrable system of two first order partial differential equations and we obtain all such backlund transformations in particular a new nonlinear partial differential equation with a backlund transformation i e uxxx 32 sin 2uux 12u3x ut is found and the known backlund transformations for the kdv equation the mkdv equation the potential kdv equation and the potential mkdv equation are recovered without using any knowledge of these equations our method is applicable to many other classes of nonlinear partial differential equations

asymptotic compactness and absorbing sets for 2d stochastic navier stokes equations on some unbounded domains we introduce a notion of an asymptotically compact ac random dynamical system rds we prove that for an ac rds the limit set b of any bounded set b is nonempty compact strictly invariant and attracts the set b we establish that the 2d navier stokes equations nses in a domain satisfying the poincare inequality perturbed by an additive irregular noise generate an ac rds in the energy space h as a consequence we deduce existence of an invariant measure for such nses our study generalizes on the one hand the earlier results by flandoli crauel 1994 and schmalfuss 1992 obtained in the case of bounded domains and regular noise and on the other hand the results by rosa 1998 for the deterministic nses

strongly driven dissipative systems an approach by robertson s theory abstract robertson s theory is used for quantumstatistical treatment of dissipative systems driven at high power assuming a weak internal interaction and a markoff like approximation tractable nonlinear equations of motion result for expectation values the systematic part of the driving fields is exactly treated two examples of thermal open systems demonstrate the significance of the presented theory

box counting dimensions and upper semicontinuities of bi spatial attractors for stochastic degenerate parabolic equations on an unbounded domain abstract the paper contributes box counting dimensions and upper semicontinuities of random bi spatial attractors for stochastic degenerate parabolic equations on the whole euclid space under some weak assumptions for the force and the nonlinearity we first prove the existence of a unique l 2 d 0 1 2 l q random attractor for any q 2 p 2 i 2 where p 1 is the order of the nonlinearity and i is a given integer such that the force is i 1 times integrable by using truncation and splitting techniques and also induction methods we then prove that a priori estimate is uniform with respect to the density of noise which leads to the upper semicontinuity result of the obtained attractors as the density tends to a constant including zero under the topology of the terminative space furthermore we give a new framework to discuss the bound of l q box counting dimensions of random attractors for spde on an unbounded domain

wong zakai approximations and attractors for stochastic reaction diffusion equations on unbounded domains abstract in this paper we study the wong zakai approximations given by a stationary process via the wiener shift and their associated long term behavior of the stochastic reaction diffusion equation driven by a white noise we first prove the existence and uniqueness of tempered pullback attractors for the wong zakai approximations of stochastic reaction diffusion equation then we show that the attractors of wong zakai approximations converges to the attractor of the stochastic reaction diffusion equation for both additive and multiplicative noise

the seke self consistent projection operator approach for the calculation of quantum mechanical eigenvalues and eigenstates abstract originally the seke self consistent projection operator method has been developed for treating non markovian time evolution of probability amplitudes of a relevant set of state vectors in the so called born approximation the method leads automatically to an hamiltonian restricted to a subspace and thus enables the construction of effective hamiltonians in the present paper in order to explain the efficiency of seke s method its algebraic operator structure is analyzed and a new successive approximation technique for the calculation of eigenstates and eigenvalues of an arbitrary quantum mechanical system is developed since unlike most perturbative techniques in the present approach a well defined effective restricted hamiltonian in each order of the applied approximation exists the self consistency of all obtained results is self evident finally to illustrate the efficiency of the developed formalism the optical polaron model is investigated

asymptotic behavior for non autonomous fractional stochastic ginzburg landau equations on unbounded domains in this paper we first prove the existence and uniqueness of tempered pullback random attractors for a non autonomous stochastic fractional ginzburg landau equation driven by multiplicative noise with 0 1 in l2r3 then we obtain the upper semicontinuity of random attractors when the intensity of noise approaches zero due to the lack of the compactness of sobolev embeddings on unbounded domains we establish the pullback asymptotic compactness of solutions in l2 r3 by the tail estimates of solutions

reduction technique for matrix nonlinear evolution equations solvable by the spectral transform the main purpose of this paper is to describe a technique of reduction whereby from the class of evolution equations for matrices of order n solvable via the spectral transform associated to the matrix linear schrodinger eigenvalue problem one derives subclasses of nonlinear evolution equations involving less than n2 fields to illustrate the method from the equations for matrices of order 2 two subclasses of equations for 2 fields rather than 4 are obtained the first class coincides or rather includes that solvable via the spectral transform associated to the generalized zakharov shabat spectral problem further reduction to nonlinear evolution equations for a single field reproduces a number of well known equations but also yields a novel one highly nonlinear the second class also yields highly nonlinear equations some examples are given including another novel evolution equation for a single field

a note on the strong convergence towards attractors of damped forced kdv equations abstract this note is a continuation of a previous article which appeared in this journal 1988 74 369 390 in the previous work we showed that a weak dissipation is suffiient to impose a finite dimensional behavior for korteweg de vries equations with space periodic boundary conditions more precisely it was shown that all the trajectories are captured by a finite dimensional attractor however we were only able to show that the convergence of the trajectories holds good in the sense of weak topology the goal of this note is to show that in fact this convergence holds true in the sense of strong topology

random attractor for stochastic reaction diffusion equation with multiplicative noise on unbounded domains abstract in this paper we study the asymptotic dynamics for stochastic reaction diffusion equation with multiplicative noise defined on unbounded domains we investigate the existence of a random attractor for the random dynamical system associated with the equation the asymptotic compactness of the random dynamical system is established by using uniform a priori estimates for far field values of solutions and a cut off technique

asymptotic behavior of stochastic wave equations with critical exponents on mathbb r 3 the existence of a random attractor in h 1 3 x l 2 3 is proved for the damped semilinear stochastic wave equation defined on the entire space 3 the nonlinearity is allowed to have a cubic growth rate which is referred to as the critical exponent the uniform pullback estimates on the tails of solutions for large space variables are established the pullback asymptotic compactness of the random dynamical system is proved by using these tail estimates and the energy equation method

existence and continuity of bi spatial random attractors and application to stochastic semilinear laplacian equations abstract a concept of a bi spatial random attractor for a random dynamical system is introduced a unified result about existence and upper semi continuity for a family of bi spatial random attractors is obtained if a family of random systems is convergent uniformly absorbing in an initial space and uniformly omega compact in both initial and terminate spaces the upper semi continuity result improves all existing results even for single spatial attractors as an application of the abstract result it is shown that every semilinear laplacian equation on the entire space perturbed by a multiplicative and stochastic noise possesses an l 2 l q random attractor with q 2 moreover it is proved that the family of obtained attractors is upper semi continuous at any density of noises and the family of attractors for the corresponding compact systems is both upper and lower semi continuous at infinity under the topology of both spaces

a stochastic pitchfork bifurcation in a reaction diffusion equation we study in some detail the structure of the random attractor for the chafee infante reaction diffusion equation perturbed by a multiplicative white noise d u u u u 3 d t u o d w t x d r m first we prove for m 5 a lower bound on the dimension of the random attractor which is of the same order in as the upper bound we derived in an earlier paper and is the same as that obtained in the deterministic case then we show for m 1 that as passes through 1 the first eigenvalue of the negative laplacian from below the system undergoes a stochastic bifurcation of pitchfork type we believe that this is the first example of such a stochastic bifurcation in an infinite dimensional setting central to our approach is the existence of a random unstable manifold

random attractors for stochastic reaction diffusion equations on unbounded domains the existence of a pullback attractor is established for a stochastic reaction diffusion equation on all n dimensional space the nonlinearity is dissipative for large values of the state and the stochastic nature of the equation appears as spatially distributed temporal white noise the reaction diffusion equation is recast as a random dynamical system and asymptotic compactness for this is demonstrated by using uniform a priori estimates for far field values of solutions

asymptotic smoothing and the global attractor of a weakly damped kdv equation on the real line abstract the existence of the global attractor of a weakly damped forced korteweg de vries equation in the phase space l 2 r is proved an optimal asymptotic smoothing effect of the equation is also shown namely that for forces in l 2 r the global attractor in the phase space l 2 r is actually a compact set in h 3 r the energy equation method is used in conjunction with a suitable splitting of the solutions the dispersive regularization properties of the equation in the context of bourgain spaces are extensively exploited

an energy equation for the weakly damped driven nonlinear schro dinger equations and its application to their attractors abstract in this note we present an energy equation concerning the h 2 norm of a forced one dimensional nonlinear schrodinger equation as an application we are able to corroborate recent results on attractors of weakly damped forced nonlinear schrodinger equation by proving that the attractor is an attractor in the strong topology of h 2 and its hausdorff and fractal dimensions are finite in h 2 the energy equation for the h 1 norm can be derived in the same fashion and it implies corresponding results for the attractor in h 1

random attractors for the stochastic fitzhugh nagumo system on unbounded domains abstract the existence of a random attractor for the stochastic fitzhugh nagumo system defined on an unbounded domain is established the pullback asymptotic compactness of the stochastic system is proved by uniform estimates on solutions for large space and time variables these estimates are obtained by a cut off technique

l2 lp random attractors for stochastic reaction diffusion equation on unbounded domains abstract in this paper the existence of l 2 l p random attractor is established for a stochastic reaction diffusion equation on the whole space r n this random attractor is a compact and invariant tempered set which attracts every tempered random subset of l 2 in the topology of l p the nonlinearity f is supposed to satisfy some growth of arbitrary order p 1 where p 2 the l 2 l p asymptotic compactness of the random dynamical system is proved by an asymptotic a priori estimate of the unbounded part of solutions

asymptotically autonomous dynamics for parabolic equations abstract given an asymptotically autonomous system it is shown that the components of the pullback attractor converge to the global attractor if and only if the pullback attractor is forward compact this result reduces two uniformness conditions in the theoretical result given by kloeden and simsen j math anal appl 2015 2017 some constructions from the forward limit set of a pullback attractor are established some applications are given for two classes of non autonomous quasi linear parabolic equations one has a parabolic operator with spatially variable exponents and another has a weakly dissipative nonlinearity for two equations it is shown that the pullback attractor is forward compact and upper semi continuous to the global attractor moreover there is an inclusion between the global attractor and the forward limit set of pullback attractors

the global attractor for the 2d navier stokes flow on some unbounded domains we extend previous results on the existence of the global attractor for the 2d navier stokes equations on some unbounded domains in the sense that the forcing term need not lie in any weighted space nor is the boundary of the domain required to be smooth the existence of the global attractor is obtained on arbitrary open sets such that the poincare inequality holds and for forces in the natural dual space v the proof is based on the energy equation and the concept of asymptotic compactness an estimate of the hausdorff and fractal dimensions of the attractor is also given

on classification of backlund transformations abstract in this paper we study backlund transformations u v from nonlinear partial differential equations uxx f u ux ut to g v v x v t x l v t l v 0 defined via integrable systems on v we classify all such backlund transformations under some restrictions our results show that there exist many such backlund transformations

analysis of the projection operator method inconsistencies and their removal it is pointed out that the application of the projection operator method leads to the violation of a consistency condition if the so called born approximation is made in the nonmarkovian master equation since only the validity of this consistency condition guarantees obtaining non negative probabilities its violation makes the applicability of the whole method in this approximation meaningless therefore it is demonstrated that an additional markov approximation restores the validity of the consistency condition in the master equation and thus re establishes the applicability of the projection operator method furthermore the possibility of obtaining a self consistent projection operator method is also examined