on the stochastic porous medium equation abstract in this paper we discuss an initial boundary value problem and a cauchy problem for the stochastic porous medium equation our basic estimates are based on the known results due to math sbornik 67 1965 609 642 arch rational mech anal 25 1967 64 80 by the procedure developed in trans amer math soc 354 2001 1117 1135 we obtain solutions over the given probability space rather than martingale solutions we will also establish the existence of invariant measures when the space domain is bounded

random attractors for quasi continuous random dynamical systems and applications to stochastic reaction diffusion equations abstract this paper first introduces the so called quasi continuous random dynamical system rds on a separable banach space the quasi continuity is weaker than all the usual continuities and thus is easier to check in practice we then establish a necessary and sufficient condition for the existence of random attractors for the quasi continuous rds we also give a general method to obtain the random attractors for the rds on the banach space l q d for q 2 as an application it is shown that the rds generated by the stochastic reaction diffusion equation possesses a finite dimensional random attractor in l q d for any q 2 a comparison result of fractal dimensions under the different l q norms is also obtained

minimum probability of error for asynchronous gaussian multiple access channels consider a gaussian multiple access channel shared by k users who transmit asynchronously independent data streams by modulating a set of assigned signal waveforms the uncoded probability of error achievable by optimum multiuser detectors is investigated it is shown that the k user maximum likelihood sequence detector consists of a bank of single user matched filters followed by a viterbi algorithm whose complexity per binary decision is o 2 k the upper bound analysis of this detector follows an approach based on the decomposition of error sequences the issues of convergence and tightness of the bounds are examined and it is shown that the minimum multiuser error probability is equivalent in the iow noise region to that of a single user system with reduced power these results show that the proposed multiuser detectors afford important performance gains over conventional single user systems in which the signal constellation carries the entire burden of complexity required to achieve a given performance level

higher order wong zakai approximations of stochastic reaction diffusion equations on rn abstract in this paper we consider the higher order wong zakai approximations of the non autonomous stochastic reaction diffusion equation driven by additive multiplicative white noises the solutions between the approximation equation and stochastic reaction diffusion equation are compared in higher order spaces in terms of the initial data based on these results and the known l 2 upper semi continuity we prove that the random attractor of the approximation random system converges to that of the non autonomous stochastic reaction diffusion equation with additive multiplicative white noises in l p r n h 1 r n when the size of the approximation shrinks to zero

z4 kerdock codes orthogonal spreads and extremal euclidean line sets when m is odd spreads in an orthogonal vector space of type omega 2m 2 2 are related to binary kerdock codes and extremal line sets in rr 2 m 1 with prescribed angles spreads in a 2m dimensional binary symplectic vector space are related to kerdock codes over zz 4 and extremal line sets in cc 2 m with prescribed angles these connections involve binary real and complex geometry associated with extraspecial 2 groups a geometric map from symplectic to orthogonal spreads is shown to induce the gray map from a corresponding zz 4 kerdock code to its binary image these geometric considerations lead to the construction for any odd composite m of large numbers of zz 4 kerdock codes they also produce new zz 4 linear kerdock and preparata codes 1991 mathematics subject classification primary 94b60 secondary 51m15 20c99

time delay estimation from low rate samples a union of subspaces approach time delay estimation arises in many applications in which a multipath medium has to be identified from pulses transmitted through the channel previous methods for time delay recovery either operate on the analog received signal or require sampling at the nyquist rate of the transmitted pulse in this paper we develop a unified approach to time delay estimation from low rate samples this problem can be formulated in the broader context of sampling over an infinite union of subspaces although sampling over unions of subspaces has been receiving growing interest previous results either focus on unions of finite dimensional subspaces or finite unions the framework we develop here leads to perfect recovery of the multipath delays from samples of the channel output at the lowest possible rate even in the presence of overlapping transmitted pulses and allows for a variety of different sampling methods the sampling rate depends only on the number of multipath components and the transmission rate but not on the bandwidth of the probing signal this result can be viewed as a sampling theorem over an infinite union of infinite dimensional subspaces by properly manipulating the low rate samples we show that the time delays can be recovered using the well known esprit algorithm combining results from sampling theory with those obtained in the context of direction of arrival estimation we develop sufficient conditions on the transmitted pulse and the sampling functions in order to ensure perfect recovery of the channel parameters at the minimal possible rate

asymptotic behavior of stochastic fitzhugh nagumo systems on unbounded thin domains abstract this article is concerned with the asymptotic behavior of solutions of a class of non autonomous stochastic fitzhugh nagumo systems driven by multiplicative white noise on unbounded thin domains the existence and uniqueness of tempered pullback random attractors are proved for the systems defined on n 1 dimensional unbounded narrow domains we also establish the upper semicontinuity of these attractors when a family of n 1 dimensional unbounded thin domains collapses onto an n dimensional unbounded domain a cut off method and a decomposition technique are employed to derive the pullback asymptotic compactness of solutions in order to overcome the difficulties caused by the non compactness of sobolev embeddings on unbounded domains as well as the lack of regularity of one component of the solutions

random attractors for singular stochastic evolution equations abstract the existence of random attractors for singular stochastic evolution equations see perturbed by general additive noise is proven the drift is only assumed to satisfy the standard assumptions of the variational approach to see with compact embeddings in the gelfand triple and singular coercivity for ergodic monotone contractive random dynamical systems it is proven that the random attractor consists of a single random point in case of real linear multiplicative noise finite time extinction is obtained applications include stochastic generalized fast diffusion equations and stochastic generalized singular p laplace equations perturbed by levy noise with jump measure having finite first and second moments

global attractors for p laplacian equation abstract the existence of a l 2 w 0 1 p l q global attractor is proved for the p laplacian equation u t div u p 2 u f u g on a bounded domain r n n 3 with dirichlet boundary condition where p 2 the nonlinear term f is supposed to satisfy the polynomial growth condition of arbitrary order c 1 u q k f u u c 2 u q k and f u l where q 2 is arbitrary there is no other restriction on p and q the asymptotic compactness of the corresponding semigroup is proved by using a new a priori estimate method called asymptotic a priori estimate

the existence of global attractors for the norm to weak continuous semigroup and application to the nonlinear reaction diffusion equations abstract in this paper first we introduce a new concept called the norm to weak continuous semigroup in a banach space and give a technical theorem to verify this notion of continuity then we establish a general method which is necessary and sufficient to obtain the existence of the global attractor for this kind of semigroup as an application we obtain the existence of the global attractor for a nonlinear reaction diffusion equation with a polynomial growth nonlinearity of arbitrary order and with some weak derivatives in the inhomogeneous term the global attractors are obtained in l p h 0 1 and h 2 h 0 1 respectively a new a priori estimate method called asymptotic a priori estimate has been introduced since the solutions of the equation has no higher regularity and the semigroup associated the solutions is not continuous in l p h 0 1 and h 2 h 0 1 the results in this part are new and appear to be optimal

asymptotic behaviour of non linear parabolic equations with monotone principal part abstract we consider non linear parabolic equations with subdifferential principal part and give conditions under which they posses global attractors in spite of considering non lipschitz perturbations the case of globally lipschitz perturbations of a maximal monotone operator has been addressed in boll un mat ital b 8 2 2000 693 706 in the case of perturbations which are not globally lipschitz the main difficulty is the lack of uniqueness of solutions which at first does not even allow us to define attractors we overcome this difficulty for problems enjoying certain regularity and absorption properties that allow uniqueness of solutions after some time has been elapsed the results developed here are applied to the case when the subdifferential operator is the p laplacian to obtain existence of attractors and the existence of periodic solutions

pullback attractors for asymptotically compact non autonomous dynamical systems abstract first we introduce the concept of pullback asymptotically compact non autonomous dynamical system as an extension of the similar concept in the autonomous framework our definition is different from that of asymptotic compactness already used in the theory of random and non autonomous dynamical systems as developed by crauel flandoli kloeden schmalfuss amongst others which means the existence of a random or time dependent family of compact attracting sets next we prove a result ensuring the existence of a pullback attractor for a non autonomous dynamical system under the general assumptions of pullback asymptotic compactness and the existence of a pullback absorbing family of sets this attractor is minimal and in most practical applications it is unique finally we illustrate the theory with a 2d navier stokes model in an unbounded domain

non autonomous reaction diffusion model with dynamic boundary conditions abstract we investigate the dynamics of a non autonomous reaction diffusion model with dynamic boundary conditions we first show that under the same assumptions the known l 2 l 2 pullback d attractor indeed can attract in l 2 l 2 norm for any 0 then we prove the continuity of the solution in h 1 h 1 2 with respect to the initial data and finally show that such attractor can also attract in h 1 h 1 2 norm under a slightly stronger integrability condition on the time dependent external forcing term the proofs are based on a new nash moser alikakos type a priori estimate about the difference of solutions near the initial time

existence of a global attractor for the parabolic equation with nonlinear laplacian principal part in an unbounded domain abstract in this paper we study the long time behavior of solutions for the parabolic equation with non linear laplacian principal part in r n we prove the existence of a global l 2 r n l r n attractor when n p and the existence of a global l 2 r n l n p n p r n attractor when n p

existence of a global attractor for a p laplacian equation in rn abstract in this paper we prove the existence of the l 2 r n w 1 p r n l q r n global attractor for the p laplacian equation u t div u p 2 u u p 2 u f u g with a more general nonlinear term f f 1 a x f 2 in r n where a l 1 r n l r n and f 1 f 2 satisfy the arbitrary q order polynomial growth condition without any restriction on q p q 2 p 2

stochastic porous media equations and self organized criticality the existence and uniqueness of nonnegative strong solutions for stochastic porous media equations with noncoercive monotone diffusivity function and wiener forcing term is proven the finite time extinction of solutions with high probability is also proven in 1 d the results are relevant for self organized criticality behavior of stochastic nonlinear diffusion equations with critical states

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

spde in hilbert space with locally monotone coefficients the aim of this paper is to extend the usual framework of spde with monotone coecients to include a large class of cases with merely locally monotone coecients this new framework is conceptually not more involved than the classical one but includes many more fundamental examples not included previously thus our main result can be applied to various types of spdes such as stochastic reaction diusion equations stochastic burgers type equation stochastic 2 d navier stokes equation stochastic p laplace equation and stochastic porous media equation with non monotone perturbations

comparison results for nonlinear parabolic equations with monotone principal part in this work we prove an abstract comparison principle for abstract quasi linear parabolic problems with monotone principal part this result is applied to parabolic problems having the p laplacian as principal part and with non lipschitz perturbations

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

asymptotic behavior of non autonomous fractional stochastic reaction diffusion equations abstract we first apply the galerkin method to prove the existence and uniqueness of solutions for a class of non autonomous fractional reaction diffusion equations driven by multiplicative noise we then establish the existence and uniqueness of tempered pullback random attractors for the equations in an appropriate hilbert space the upper semicontinuity of random attractors is also obtained when the intensity of noise approaches zero

random attractors for a class of stochastic partial differential equations driven by general additive noise abstract the existence of random attractors for a large class of stochastic partial differential equations spde driven by general additive noise is established the main results are applied to various types of spde as e g stochastic reaction diffusion equations the stochastic p laplace equation and stochastic porous media equations besides classical brownian motion we also include space time fractional brownian motion and space time levy noise as admissible random perturbations moreover cases where the attractor consists of a single point are also investigated and bounds for the speed of attraction are obtained

global attractors for degenerate semilinear parabolic equations abstract let be a smooth bounded domain in r n n 2 we consider the long time behavior of solutions to the following problem u t div x u f u g in r u 0 on r u x 0 u 0 x in where u 0 g l 1 the diffusion coefficient x is measurable nonnegative and is allowed to have a finite number of zeroes at some points we provide the existence and uniqueness results for the problem then we establish some asymptotic regularity results on the solution and consider its long time behavior the existence of a global attractor is obtained in the proper space

pullback attractors of non autonomous stochastic degenerate parabolic equations on unbounded domains abstract this paper is concerned with pullback attractors of the stochastic p laplace equation defined on the entire space r n we first establish the asymptotic compactness of the equation in l 2 r n and then prove the existence and uniqueness of non autonomous random attractors this attractor is pathwise periodic if the non autonomous deterministic forcing is time periodic the difficulty of non compactness of sobolev embeddings on r n is overcome by the uniform smallness of solutions outside a bounded domain

limiting dynamics for stochastic reaction diffusion equations on the sobolev space with thin domains abstract this paper is devoted to bi spatial random attractors of the stochastic reaction diffusion equation when the terminate space is the sobolev space on a thin domain where the nonlinearity can be decomposed into two functions with p q growth exponents by means of a computation method of induction it is shown that the difference of solutions near the initial time is integrable of k p 2 k 2 order this higher order integrability shows continuity of the solution operator from the square lebesgue space to k p 2 k 2 order lebesgue spaces in particular the 2 p 2 order integrability shows continuity of the solution operator from the square lebesgue space to the sobolev space which further shows existence of a random attractor in the sobolev space when the initial space is the square lebesgue space moreover the higher order integrability can be uniform with respect to all thin domains which provides uniformly asymptotic compactness of the random dynamical systems as a conclusion the upper semi continuity of attractors under the sobolev norm is established when the narrow domain degenerates onto a lower dimensional domain

stochastic generalized porous media and fast diffusion equations abstract we present a generalization of krylov rozovskii s result on the existence and uniqueness of solutions to monotone stochastic differential equations as an application the stochastic generalized porous media and fast diffusion equations are studied for finite reference measures where the drift term is given by a negative definite operator acting on a time dependent function which belongs to a large class of functions comparable with the so called n functions in the theory of orlicz spaces

random attractors for damped non autonomous wave equations with memory and white noise abstract we first prove the existence of random attractors for the continuous random dynamical systems generated by stochastic damped non autonomous wave equations with linear memory and additive white noise when the nonlinearity has a critically growing exponent then we study the upper semicontinuity of random attractors when the coefficient of random term approaches zero

polynomial growth estimates for multilinear singular integral operators on demontre un theoreme pour des mesures de carleson et on l applique a l operateur de kato en dimension 1 on demontre un critere de limitation general on donne une application aux commutateurs de coolderon on etudie les d commutateurs

long time random dynamics of stochastic parabolic p laplacian equations on rn abstract in this article we study the long time random dynamics of non autonomous stochastic parabolic p laplacian equations driven by a wiener type multiplicative noise by a new asymptotic a priori estimate method not involving the plus or minus sign of the nonlinearity the omega limit compact of solutions in l p r n l q r n is proved for any p q 2 and n 1 where q is the polynomial growth exponent of the nonlinearity if 2 p n by a boot strap technique we obtain the higher order integrability of difference of solutions near the initial time as an application we show that the obtained l 2 random attractor is attracting in the topology of l 2 r n for any 0

dynamics of the non autonomous stochastic p laplace equation driven by multiplicative noise we investigate the asymptotic behavior of solutions of the p laplace equation driven simultaneously by non autonomous deterministic forcing and multiplicative white noise on r n we show the tails of solutions of the equation are uniformly small outside a bounded domain which is used to derive asymptotic compactness of solution operators in l 2 r n by overcoming the non compactness of sobolev embeddings on unbounded domains we then prove existence and uniqueness of random attractors and further establish upper semicontinuity of attractors as the intensity of noise approaches zero

exploiting sparse user activity in multiuser detection the number of active users in code division multiple access cdma systems is often much lower than the spreading gain the present paper exploits fruitfully this a priori information to improve performance of multiuser detectors a low activity factor manifests itself in a sparse symbol vector with entries drawn from a finite alphabet that is augmented by the zero symbol to capture user inactivity the non equiprobable symbols of the augmented alphabet motivate a sparsity exploiting maximum a posteriori probability s map criterion which is shown to yield a cost comprising the l2 least squares error penalized by the p th norm of the wanted symbol vector p 0 1 2 related optimization problems appear in variable selection shrinkage schemes developed for linear regression as well as in the emerging field of compressive sampling cs the contribution of this work to such sparse cdma systems is a gamut of sparsity exploiting multiuser detectors trading off performance for complexity requirements from the vantage point of cs and the least absolute shrinkage selection operator lasso spectrum of applications the contribution amounts to sparsity exploiting algorithms when the entries of the wanted signal vector adhere to finite alphabet constraints

decision feedback multiuser detection a systematic approach a systematic approach to decision feedback multiuser detection is introduced for the joint detection of symbols of k simultaneously transmitting users of a synchronous correlated waveform multiple access cwma channel with gaussian noise a new performance criterion called symmetric energy is defined which is a low noise indicator of the joint error rate that at least one user is detected erroneously even the best linear detectors can perform poorly in terms of symmetric energy compared to the maximum likelihood detector a general class of decision feedback detectors dfds is defined with o k implementational complexity per user the symmetric energy of arbitrary dfd and bounds on their asymptotic effective energy aee performance are obtained along with an exact bit error rate and aee analysis for the decorrelating dfd the optimum dfd that maximizes symmetric energy is obtained each one of the k optimum decorrelating and conventional dfds that correspond to the k orders in which the users can be detected are shown to outperform the linear optimum decorrelating and conventional detectors respectively in terms of symmetric energy moreover algorithms are obtained for determining the choice of order of detection for the three dfds which guarantee that they uniformly user wise outperform their linear counterparts in addition to optimality in symmetric energy it is also shown that under certain conditions the optimum dfd achieves the aee performance of the exponentially complex maximum likelihood detector for all users simultaneously none of the results of this paper make the perfect feedback assumption the implications of our work on power control for multiuser detection are also discussed

global attractors for degenerate parabolic equations on unbounded domains we consider a nonlinear degenerate parabolic equation and show existence of the maximal compact attractor in weighted spacesl1 rn x dx with weight l1 rn going to zero at infinity

linear multiuser detectors for synchronous code division multiple access channels under the assumptions of symbol synchronous transmissions and white gaussian noise the authors analyze the detection mechanism at the receiver comparing different detectors by their bit error rates in the low background noise region and by their worst case behavior in a near far environment where the received energies of the users are not necessarily similar optimum multiuser detection achieves important performance gains over conventional single user detection at the expense of computational complexity that grows exponentially with the number of users it is shown that in the synchronous case the performance achieved by linear multiuser detectors is similar to that of optimum multiuser detection attention is focused on detectors whose linear memoryless transformation is a generalized inverse of the matrix of signature waveform crosscorrelations and on the optimum linear detector it is shown that the generalized inverse detectors exhibit the same degree of near far resistance as the optimum multiuser detectors the optimum linear detector is obtained

reduced dimension multiuser detection we present a reduced dimension multiuser detector rd mud structure for synchronous systems that significantly decreases the number of required correlation branches at the receiver front end while still achieving performance similar to that of the conventional matched filter mf bank rd mud exploits the fact that in some wireless systems the number of active users may be small relative to the total number of users in the system hence the ideas of analog compressed sensing may be used to reduce the number of correlators the correlating signals used by each correlator are chosen as an appropriate linear combination of the users spreading waveforms we derive the probability of symbol error when using two methods for recovery of active users and their transmitted symbols the reduced dimension decorrelating rdd detector which combines subspace projection and thresholding to determine active users and sign detection for data recovery and the reduced dimension decision feedback rddf detector which combines decision feedback matching pursuit for active user detection and sign detection for data recovery we derive probability of error bounds for both detectors and show that the number of correlators needed to achieve a small probability of symbol error is on the order of the logarithm of the number of users in the system the theoretical performance results are validated via numerical simulations

multiuser detection in a dynamic environment part i user identification and data detection in random access communication systems the number of active users varies with time and has considerable bearing on receiver s performance thus techniques aimed at identifying not only the information transmitted but also that number play a central role in those systems an example of application of these techniques can be found in multiuser detection mud in typical mud analyses receivers are based on the assumption that the number of active users is constant and known at the receiver and coincides with the maximum number of users entitled to access the system this assumption is often overly pessimistic since many users might be inactive at any given time and detection under the assumption of a number of users larger than the real one may impair performance the main goal of this paper is to introduce a general approach to the problem of identifying active users and estimating their parameters and data in a random access system where users are continuously entering and leaving the system the tool whose use we advocate is random set theory rst applying this we derive optimum receivers in an environment where the set of transmitters comprises an unknown number of elements in addition we can derive bayesian filter equations which describe the evolution with time of the a posteriori probability app density of the unknown user parameters and use this density to derive optimum detectors in this paper we restrict ourselves to interferer identification and data detection while in a companion paper we shall examine the more complex problem of estimating users parameters

strong solutions for stochastic partial differential equations of gradient type unique existence of analytically strong solutions to stochastic partial differential equations spde with drift given by the subdifferential of a quasi convex function and with general multiplicative noise is proven the proof applies a genuinely new method of weighted galerkin approximations based on the distance defined by the quasi convex function spatial regularization of the initial condition analogous to the deterministic case is obtained the results yield a unified framework which is applied to stochastic generalized porous media equations stochastic generalized reaction diffusion equations and stochastic generalized degenerated p laplace equations in particular higher regularity for solutions of such spde is obtained

spectral efficiency of cdma with random spreading the cdma channel with randomly and independently chosen spreading sequences accurately models the situation where pseudonoise sequences span many symbol periods furthermore its analysis provides a comparison baseline for cdma channels with deterministic signature waveforms spanning one symbol period we analyze the spectral efficiency total capacity per chip as a function of the number of users spreading gain and signal to noise ratio and we quantify the loss in efficiency relative to an optimally chosen set of signature sequences and relative to multiaccess with no spreading white gaussian background noise and equal power synchronous users are assumed the following receivers are analyzed a optimal joint processing b single user matched filtering c decorrelation and d mmse linear processing

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

random attractors for locally monotone stochastic partial differential equations abstract we prove the existence of random dynamical systems and random attractors for a large class of locally monotone stochastic partial differential equations perturbed by additive levy noise the main result is applicable to various types of spde such as stochastic burgers type equations stochastic 2d navier stokes equations the stochastic 3d leray model stochastic power law fluids the stochastic ladyzhenskaya model stochastic cahn hilliard type equations stochastic kuramoto sivashinsky type equations stochastic generalized porous media equations and stochastic generalized p laplace equations

sufficient and necessary criteria for existence of pullback attractors for non compact random dynamical systems abstract we study pullback attractors of non autonomous non compact dynamical systems generated by differential equations with non autonomous deterministic as well as stochastic forcing terms we first introduce the concepts of pullback attractors and asymptotic compactness for such systems we then prove a sufficient and necessary condition for existence of pullback attractors we also introduce the concept of complete orbits for this sort of systems and use these special solutions to characterize the structures of pullback attractors for random systems containing periodic deterministic forcing terms we show the pullback attractors are also periodic under certain conditions as an application of the abstract theory we prove the existence of a unique pullback attractor for reaction diffusion equations on r n with both deterministic and random external terms since sobolev embeddings are not compact on unbounded domains the uniform estimates on the tails of solutions are employed to establish the asymptotic compactness of solutions

stochastic porous media equations and self organized criticality convergence to the critical state in all dimensions if x x t is the solution to the stochastic porous media equation in mathcal o subset mathbf r d 1 le d le 3 modelling the self organized criticality barbu et al in commun math phys 285 901 923 2009 and x c is the critical state then it is proved that int infty 0m mathcal o setminus mathcal o t 0 dt infty mathbb p hbox a s and lim t to infty int mathcal o x t x c d xi ell infty mathbb p hbox a s here m is the lebesgue measure and mathcal o t c is the critical region xi in mathcal o x t xi x c xi and x c x 0 a e xi in mathcal o if the stochastic gaussian perturbation has only finitely many modes but is still function valued lim t to infty int k x t x c d xi 0 exponentially fast for all compact k subset mathcal o with probability one if the noise is sufficiently strong we also recover that in the deterministic case l 0

existence of strong solutions for stochastic porous media equation under general monotonicity conditions this paper addresses existence and uniqueness of strong solutions to stochastic porous media equations dx x dt b x dw t in bounded domains of r d with dirichlet boundary conditions here is a maximal monotone graph in r r possibly multivalued with

asymptotic behavior of stochastic discrete wave equations with nonlinear noise and damping in this article we study the asymptotic behavior for a class of discrete wave equations with nonlinear noise and damping defined on a k dimensional integer set the well posedness of the system is established when the nonlinear drift function and the nonlinear diffusion term are only locally lipschitz continuous the mean random dynamical system associated with the non autonomous system is shown to possess a unique tempered weak pullback mean random attractor in l2 f l2 l2 the existence of invariant measures for the autonomous system is also derived by using the krylov bogolyubov method the difficulty in proving the tightness of a family of distribution laws of the solutions is overcome by using the idea of uniform estimates on the tails of solutions

synchronization by noise we provide sufficient conditions for synchronization by noise i e under these conditions we prove that weak random attractors for random dynamical systems consist of single random points in the case of sde with additive noise these conditions are also essentially necessary in addition we provide sufficient conditions for the existence of a minimal weak point random attractor consisting of a single random point as a result synchronization by noise is proven for a large class of sde with additive noise in particular we prove that the random attractor for an sde with drift given by a multidimensional double well potential and additive noise consists of a single random point all examples treated in tearne probab theory relat fields 141 1 2 1 18 2008 are also included

continuity and pullback attractors for a non autonomous reaction diffusion equation in r n in this paper we study the dynamics of a non autonomous reaction diffusion equation in r n with the nonlinearity f satisfying the polynomial growth of arbitrary order p 1 p 2 firstly we prove the existence of the pullback d attractor in l 2 r n secondly we use a 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 thirdly we verify that the usual l 2 r n l 2 r n pullback d attractor indeed can pullback attract the d class in l 2 norm for any 0 and h 1 norm and the solutions of the equation in h 1 r n are continuous with respect to initial data finally we obtain the pullback d attractor in l p r n and h 1 r n as an application of the higher order integrability and the continuity respectively

asymptotic behavior of non autonomous fractional stochastic p laplacian equations abstract this article is concerned with the asymptotic behavior of solutions of a wide class of non autonomous non local fractional stochastic p laplacian equations driven by multiplicative white noise the time dependent nonlinear drift term of the equation has a polynomial growth of arbitrary order in its third component which is allowed to be greater than the exponent p we first employ the faedo galerkin method to prove the well posedness of the equation in an appropriate hilbert space we then establish the existence uniqueness and periodicity of tempered pullback random attractors for the equations the upper semi continuity of these attractors is also derived as the density of noise tends to zero the results of this paper are new even when the stochastic equation reduces to the deterministic one

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

random dynamics of stochastic p laplacian equations on rn with an unbounded additive noise abstract this article is concerned with the random dynamics of solutions of the non autonomous parabolic p laplacian equations on r n driven by an unbounded additive noise where the nonlinearity has a p q growth exponents including the general polynomial type functions firstly by using an inductive method we establish the higher order integrability of difference of solutions near the initial time for arbitrary space dimension secondly based on this very estimate the asymptotical compactness of solutions in l p r n l q r n for any p q 2 is proved by a new technique instead of the usual truncation estimate method and then we obtain the existence of pullback attractor in l p r n l q r n for p laplacian equations with an unbounded additive noise thirdly we prove that the obtained l 2 pullback attractor is attracting and translation bounded in l r n topology for any 2 and arbitrary space dimension n n 1

global attractor for the m semiflow generated by a quasilinear degenerate parabolic equation using theory of global attractors for multi valued semiflows we prove the existence of a global attractor for the m semiflow generated by a parabolic equation involving the nonlinear degenerate operator in a bounded domain

non monotone stochastic generalized porous media equations abstract by using the nash inequality and a monotonicity approximation argument existence and uniqueness of strong solutions are proved for a class of non monotone stochastic generalized porous media equations moreover we prove for a large class of stochastic pde that the solutions stay in the smaller l 2 space provided the initial value does so that some recent results in the literature are considerably strengthened

asymptotic behavior of a class of non autonomous degenerate parabolic equations abstract we investigate the asymptotic behavior of solutions of a class of non autonomous degenerate parabolic equations on r n with unbounded external forcing terms the existence of a pullback global attractor is proved in l 2 r n l p r n w 1 r r n for any p r 2

extremal equilibria for monotone semigroups in ordered spaces with application to evolutionary equations abstract we consider monotone semigroups in ordered spaces and give general results concerning the existence of extremal equilibria and global attractors we then show some applications of the abstract scheme to various evolutionary problems from odes and retarded functional differential equations to parabolic and hyperbolic pdes in particular we exhibit the dynamical properties of semigroups defined by semilinear parabolic equations in r n with nonlinearities depending on the gradient of the solution we consider as well systems of reaction diffusion equations in r n and provide some results concerning extremal equilibria of the semigroups corresponding to damped wave problems in bounded domains or in r n we further discuss some nonlocal and quasilinear problems as well as the fourth order cahn hilliard equation

invariant forward attractors of non autonomous random dynamical systems abstract forward attractors especially invariant ones of non autonomous and random dynamical systems have not been as well studied as pullback attractors this is mainly due to the different role that time plays in each case pullback attractors involve convergence at each instant of current time as the initial data is pulled back to distant past whereas forward attractors are concerned with what happens in the distant future moreover if they exist they need not be unique in this paper ideas introduced in kloeden 20 for asymptotically invariant forward attracting sets of non autonomous odes are generalized and extended to study strictly invariant forward attractors of non autonomous random dynamical systems on metric spaces these are formulated as processes that include the effects of both noise and deterministic time variation it is shown that the random forward attractors when they exist can be constructed by a pullback argument inside a positively invariant set however this is only a necessary condition so additional sufficiency conditions are also given since forward attractors need not exist in dissipative systems an alternative object specifically the closed union of the forward omega limit sets for each initial time of a forward absorbing set is investigated it is shown to be asymptotically invariant under certain conditions so looks more and more like an invariant attractor as conventionally understood it resembles the vishik uniform attractor but need not attract uniformly in the initial times and does not even require the system to be defined in the distant past

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

many access for small packets based on precoding and sparsity aware recovery modern mobile terminals produce massive small data packets for these short length packets it is inefficient to follow the current multiple access schemes to allocate transmission resources due to heavy signaling overhead we propose a non orthogonal many access scheme that is well suited for the future communication systems equipped with many receive antennas the system is modeled as having a block sparsity pattern with unknown sparsity level i e unknown number of transmitted messages block precoding is employed at each single antenna transmitter to enable the simultaneous transmissions of many users the number of simultaneously served active users is allowed to be even more than the number of receive antennas sparsity aware recovery is designed at the receiver for joint user detection and symbol demodulation to reduce the effects of channel fading on signal recovery normalized block orthogonal matching pursuit bomp algorithm is introduced and based on its approximate performance analysis we develop interference cancellation based bomp icbomp algorithm the icbomp performs error correction and detection in each iteration of the normalized bomp simulation results demonstrate the effectiveness of the proposed scheme in small packet services as well as the advantages of icbomp in improving signal recovery accuracy and reducing computational cost

torus breakdown and noise induced dynamics in the randomly driven morse oscillator strictly the kam torus no longer exists for a randomly driven hamiltonian system we firstly investigate the process of torus breakdown in the randomly driven morse oscillator by transforming the original system into another one in action angle variables and introducing the specific poincare map by makarov et al though the strength of random perturbations is large some thick closed like curves can still appear to characterize these noisy dynamics the pseudo periodic surrogate approach along with the algorithm on the mean divergence is applied to evaluate the correlation dimensions of the original data and their corresponding surrogates two kinds of noisy dynamics are picked out from the randomly driven morse oscillator

a note onbmoand bloch space of generalized α β harmonic functions abstract in this paper we study functions in the unit ball in c n annihilated by the differential operators c given by formula we obtain two equivalent carleson measure characterizations of those of such functions having boundary value in bmo we also give analogous carleson bergman measure characterizations of harmonic bloch functions

higher order asymptotic attraction of pullback attractors for a reaction diffusion equation in non cylindrical domains abstract in this paper we consider the higher order asymptotic attraction of pullback attractors for a reaction diffusion equation with domains expanding in time firstly to make the test functions meaningful a maximum principle is proved for the corresponding variational solution then we establish some higher order integrability results about the difference of variational solutions and finally we prove the main result that the known l 2 l 2 pullback d attractor can attract indeed the d class in l q norm for any q 2

regularity of random attractors for fractional stochastic reaction diffusion equations on rn abstract we investigate the regularity of random attractors for the non autonomous non local fractional stochastic reaction diffusion equations in h s r n with s 0 1 we prove the existence and uniqueness of the tempered random attractor that is compact in h s r n and attracts all tempered random subsets of l 2 r n with respect to the norm of h s r n the main difficulty is to show the pullback asymptotic compactness of solutions in h s r n due to the noncompactness of sobolev embeddings on unbounded domains and the almost sure nondifferentiability of the sample paths of the wiener process we establish such compactness by the ideas of uniform tail estimates and the spectral decomposition of solutions in bounded domains

finite time extinction for stochastic sign fast diffusion and self organized criticality we prove finite time extinction for stochastic sign fast diffusion equations driven by linear multiplicative space time noise corresponding to the bak tang wiesenfeld model for self organized criticality this solves a problem posed and left open in several works barbu methods appl sci 36 1726 1733 2013 rockner and wang j lond math soc 2 87 545 560 2013 barbu et al j math anal appl 389 147 164 2012 barbu and rockner comm math phys 311 539 555 2012 barbu et al comm math phys 285 901 923 2009 c r math acad sci paris 347 1 2 81 84 2009 the highly singular degenerate nature of the drift in interplay with the stochastic perturbation causes the need for new methods in the analysis of mass diffusion and several new estimates and techniques are introduced

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

regularity of random attractors for a degenerate parabolic equations driven by additive noises abstract we study the regularity of random attractors for a class of stochastic degenerate parabolic equations with the leading term involving a diffusion variable which many be non smooth or unbounded without any restrictions on the upper growth order p of the nonlinearity except that p 2 we show that the associated random dynamical system admits a unique compact random attractor in the space d 0 1 2 d n l d n for any 2 2 p 2 where d n is an arbitrary bounded or unbounded domain in r n n 2

asymptotic regularity for p laplacian equation this paper is devoted to proving some asymptotic regularity of the solutions of the p laplacian equation ut div u p 2 u f u g x p 2 n considered on a bounded domain rn n 3 the nonlinear term f satisfies the polynomial growth condition of arbitrary order c1 u q k f u u c2 u q k where q 2 is arbitrary as an application of the asymptotic regularity results we not only can obtain the existence of a l2 w01 p lq global attractor a immediately but also can show further that a attracts every bounded subsets of l2 under the w01 p lq norm for any 0 furthermore the fractal dimension of a is finite in lq for any 0

some new function spaces and their applications to harmonic analysis abstract in this paper a family of spaces is introduced which seems well adapted for the study of a variety of questions related to harmonic analysis and its applications these spaces are the tent spaces they provide the natural setting for the study of such things as maximal functions the relevant space here is t p and also square functions where the space t2p is relevant as such these spaces lead to unifications and simplifications of some basic techniques in harmonic analysis thus they are closely related to lp and hardy spaces important parts of whose theory become corollaries of the description of tent spaces also as proc conf harmonic analysis cortona lect notes in math vol 992 springer verlag berlin new york 1983 already indicated where these spaces first appeared explicitly the tent spaces can be used to simplify some of the results related to the cauchy integral on lipschitz curves and multilinear analysis in retrospect one can recognize that various ideas important for tent spaces had been used if only implicitly for quite some time here one should mention carleson s inequality its simplifications and extensions the theory of hardy spaces and atomic decompositions

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

pathwise upper semi continuity of random pullback attractors along the time axis abstract the pullback attractor of a non autonomous random dynamical system is a time indexed family of random sets typically having the form a t t r with each a t a random set this paper is concerned with the nature of such time dependence it is shown that the upper semi continuity of the mapping t a t for each fixed has an equivalence relationship with the uniform compactness of the local union s i a s where i r is compact applied to a semi linear degenerate parabolic equation with additive noise and a wave equation with multiplicative noise we show that in order to prove the above locally uniform compactness and upper semi continuity no additional conditions are required in which sense the two properties appear to be general properties satisfied by a large number of real models

dynamics of fractional stochastic reaction diffusion equations on unbounded domains driven by nonlinear noise abstract this paper is concerned with the asymptotic behavior of the solutions of the fractional reaction diffusion equations with polynomial drift terms of arbitrary order driven by locally lipschitz nonlinear diffusion terms defined on r n we first prove the well posedness of the equation based on pathwise uniform estimates as well as uniform estimates on average we then define a mean random dynamical system via the solution operators and prove the existence and uniqueness of weak pullback mean random attractors we finally establish the existence of invariant measures when the diffusion terms are globally lipschitz continuous the uniform estimates on the tails of solutions are employed to prove the tightness of a family of probability distributions of solutions in order to overcome the non compactness of usual sobolev embeddings on unbounded domains