long time behavior of solutions of gkdv equations abstract in this paper we consider the long time behavior of solutions of a version of gkdv equation on the real line the existence of the global attractor in h 2 r is proved when the force f belongs to h 2 r moreover we investigate the change of the attractor according to the change of f and the coefficient of damping term finally several numerical experiments confirm all the theoretical conclusions

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

attractors for the semilinear reaction diffusion equation with distribution derivatives in unbounded domains abstract in this paper we prove the existence of global attractors for a nonlinear reaction diffusion equation with a nonlinearity having a polynomial growth of arbitrary order p 1 p 2 and with distribution derivatives in the inhomogeneous term the global attractors are obtained in l 2 r n and l p r n respectively a new a priori estimate method has been used since the solutions of the equation have no higher regularity and the semigroup associated with the solutions is not continuous in l p r n the results are new and appear to be optimal

kernel sections and uniform attractors of multi valued semiprocesses abstract we present the existence of kernel sections which are all compact invariant and pullback attracting of an infinite dimensional general multi valued process constructed by the set valued backward extension of multi valued semiprocesses moreover the structure of the uniform attractors of a family of multi valued semiprocesses and the uniform forward attraction of kernel sections of a family of general multi valued processes are investigated finally we explain our abstract results by considering the mixed wave systems with supercritical exponent and ordinary differential equations

global attractor for degenerate damped hyperbolic equations abstract the aim of this paper is to consider a class of degenerate damped hyperbolic equations with the critical nonlinearity involving an operator l that is x elliptic with respect to a family of vector fields x we prove the global existence of solutions and characterize their long time behavior in particular we show the semigroup generated by the equation has a compact connected and invariant attractor

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

attractors for the semilinear reaction diffusion equation with distribution derivatives we investigate the asymptotic behavior of solutions to a nonlinear reaction diffusion equation with distribution derivatives in the inhomogeneous term because the solutions of this equation are not very regular i e u only belongs to lp rn h1 rn and ut is only in h 1 rn for the forcing term in h 1 rn the standard method does not directly work in our case we demonstrate the asymptotic regularity of the solution to obtain the l2 rn h1 rn asymptotic compactness of the semigroup and therefore the existence of a l2 rn h1 rn global attractor in particular cases our results enable us to improve on some previously known results

p sv wave propagation in heterogeneous media velocity stress finite difference method i present a finite difference method for modeling p sv wave propagation in heterogeneous media this is an extension of the method i previously proposed for modeling sh wave propagation by using velocity and stress in a discrete grid the two components of the velocity cannot be defined at the same node for a complete staggered grid the stability condition and the p wave phase velocity dispersion curve do not depend on the poisson s ratio while the s wave phase velocity dispersion curve behavior is rather insensitive to the poisson s ratio therefore the same code used for elastic media can be used for liquid media where s wave velocity goes to zero and no special treatment is needed for a liquid solid interface typical physical phenomena arising with p sv modeling such as surface waves are in agreement with analytical results the weathered layer and corner edge models show in seismograms the same converted phases obtained by previous authors this method gives stable results for step discontinuities as shown for a liquid layer above an elastic half space the head wave preserves the correct amplitude finally the corner edge model illustrates a more complex geometry for the liquid solid interface as the poisson s ratio v increases from 0 25 to 0 5 the shear converted phases are removed from seismograms and from the time section of the wave field

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

uniform compact attractors for the coupled suspension bridge equations in this paper we investigate the uniformly absorbing set for the coupled suspension bridge equations with a new class of external forces and then prove the existence of the uniform attractors for the family of process corresponding to the equations

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

upper semi continuity and regularity of random attractors on p times integrable spaces and applications abstract a sufficient condition for a family of random attractors to be upper semi continuous and regular is obtained when the state space is a p times integrable space with p 2 it is shown that both upper semi continuity and regularity under the topology of p norms do not rely on continuity convergence and absorption of the associated random dynamical systems in l p since the corresponding l 2 properties can offer all as an application of the abstract result it is shown that the family of random attractors for the stochastic reaction diffusion equations on a unbounded domain is upper semi continuous and regular at any point under the topology of p norms

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

the global attractor of g navier stokes equations with linear dampness on r2 in this paper the long time behaviors of g navier stokes equations with linear dampness on r 2 were investigated by using the energy equation method the existence of the global attractor for the equations was proved without the restriction of the forcing term belonging to some weighted sobolev space moreover the estimation of the hausdorff and fractal dimensions of such attractors were also obtained

the bisymmetric solutions of the matrix equation a1x1b1 a2x2b2 alxlbl c and its optimal approximation abstract a matrix a a ij r n n is said to be bisymmetric matrix if a ij a ji a n 1 j n 1 i for all 1 i j n in this paper an iterative method is constructed to find the bisymmetric solutions of matrix equation a 1 x 1 b 1 a 2 x 2 b 2 a l x l b l c where x 1 x 2 x l is real matrices group by this iterative method the solvability of the matrix equation can be judged automatically when the matrix equation is consistent for any initial bisymmetric matrix group x 1 0 x 2 0 x l 0 a bisymmetric solution group can be obtained within finite iteration steps in the absence of roundoff errors and the least norm bisymmetric solution group can be obtained by choosing a special kind of initial bisymmetric matrix group in addition the optimal approximation bisymmetric solution group to a given bisymmetric matrix group x 1 x 2 x l in frobenius norm can be obtained by finding the least norm bisymmetric solution group of new matrix equation a 1 x 1 b 1 a 2 x 2 b 2 a l x l b l c where c c a 1 x 1 b 1 a 2 x 2 b 2 a l x l b l

existence and upper semicontinuity of bi spatial pullback attractors for smoothing cocycles abstract in this work we establish several criteria for the existence as well as the upper semi continuity of bi spatial attractors under a closedness condition which dramatically weakens the usual requirement on the continuity of the cocycle it is also shown that though the continuity plays a less important role in the study of attractors it is impossible to establish an existence criteria for common attractors for systems without any continuity like properties however for such bad systems one can expect a mini attractor which is shown adequate well to depict the asymptotic behavior of non continuous systems finally we study the l 2 h 0 1 pullback attractor for a stochastic complex ginzburg landau equation a spectrum decomposition method is employed to overcome the lack of sobolev compactness embeddings in h 0 1

long time dynamics for a damped benjamin bona mahony equation in low regularity spaces abstract long time behavior of solutions for a damped bbm equation in low regularity space is investigated by the i method and high low frequency decomposition we prove the existence of a global attractor in l 2 if the force belongs to l 2 moreover we show that the global attractor has finite fractal dimension in the sharp regularity space h 2

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

singular value and generalized singular value decompositions and the solution of linear matrix equations abstract the solution of the linear matrix equations i axb cyd e and ii axb fxg e h are considered new necessary and sufficient conditions for the consistency of the equations are derived some using the generalized singular value decomposition special cases iii ax yd e and iv axb e are treated using the singular value decomposition numerical algorithms for the solutions are also suggested

existence of global attractors for two dimensional newton boussinesq equation abstract in this paper we study global attractors of two dimensional newton boussinesq equation we obtain the existence of h 0 1 h 0 1 global attractors and h 2 h 2 global attractors in two dimensional bounded domains meanwhile we get the existence of h 0 1 h 0 1 global attractors in two dimensional channels

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

global attractors for nonlinear parabolic equations with nonstandard growth and irregular data abstract we investigate the large time behavior of solutions to the following nonlinear parabolic equations u t div u p x 2 u f x u g in r u 0 on r u x 0 u 0 x in where u 0 g l 1 we first provide the existence and uniqueness of an entropy solution for the problem then through some delicate analysis we establish some regularity results on the solution by which we prove the existence of a global attractor for the solution semigroup

lp uniform attractor for nonautonomous reaction diffusion equations in unbounded domains the goal of this paper is to consider the asymptotic behavior of solutions of nonautonomous classical reaction diffusion equations in unbounded domains with nonlinearity having a polynomial growth of arbitrary order the existence and structure of a uniform attractor are obtained in the spaces l2 rn and lp rn respectively

a gronwall type lemma with parameter and dissipative estimates for pdes abstract we discuss the problem of establishing dissipative estimates for certain differential equations for which the usual methods apparently do not work the main tool is a new gronwall type lemma with parameter as an application we consider a semilinear equation of viscoelasticity with low dissipation

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

global well posedness of solutions for fourth order dispersive wave equation with nonlinear weak damping linear strong damping and logarithmic nonlinearity abstract to understand the characteristics of dynamical behavior especially the kinetic evolution for logarithmic nonlinearity we aim to study the global well posedness of nonlinear fourth order wave equations with logarithmic source term where the dispersive the nonlinear weak damping and linear strong damping are taken into account based on the potential well method the main ingredient of this paper is to construct several conditions for initial data leading to the solution global existence or infinite time blow up with subcritical initial energy moreover we extend the results to the critical initial energy which is realized by introducing scaling technique finally surrounding the blow up at arbitrarily high initial energy we compare and discuss the research program stemming from two different strategies in the first one it is assumed that the blow up result is bound to the original logarithmic source by weakening the dispersive dissipative structure while the second one is based on the nonlinear wave equation with complete dispersive dissipative structure but the logarithmic source is replaced by an enhanced version through these two strategies we explain the mechanism of these two structures respectively and establish two kinds of sufficient conditions for initial data leading to high energy blow up

uniform attractor for non autonomous hyperbolic equation with critical exponent abstract in this paper we study the uniform attractor for semilinear wave equation with mixed differential quotient terms and critical nonlinearity we prove the existence of the uniform attractor in h 0 1 l 2

existence of global attractors for the benjamin bona mahony equation in unbounded domains in this paper we investigate the asymptotic behavior of the benjamin bona mahony equation in unbounded domains we prove the existence of a global attractor when the equation is defined in a three dimensional channel the asymptotic compactness of the solution operator is obtained by the uniform estimates on the tails of solutions

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

existence of strong solutions and global attractors for the coupled suspension bridge equations abstract in this paper we show the existence of the strong solutions for the coupled suspension bridge equations furthermore existence of the strong global attractors is investigated using a new semigroup scheme since the solutions of the coupled equation have no higher regularity and the semigroup associated with the solutions is not continuous in the strong hilbert space the results are new and appear to be optimal

uniform attractor for non autonomous plate equation with a localized damping and a critical nonlinearity abstract in this paper we consider dynamical behavior of non autonomous plate type evolutionary equations with critical nonlinearity we prove the existence of a uniform attractor in the space h 0 2 l 2

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

on the symmetric solutions of linear matrix equations abstract the symmetric solutions of linear matrix equations ax c and axb c are considered the necessary and sufficient conditions for the consistency of the equations with a symmetric condition on solutions are derived using the singular value decomposition and the generalized singular value decomposition numerical algorithms for solving the equations are suggested

pullback attractors for the non autonomous reaction diffusion equations in rn in this paper we prove the existence of the pullback attractors in l2 rn lp rn and h1 rn for a nonlinear reaction diffusion equation in unbounded domains in which the nonlinearity f satisfies the polynomial growth of arbitrary order p 1 p 2 and the external forcing g lloc2 r l2 rn only subjects to the weaker integrability condition te s g s 22ds in this paper we prove the existence of the pullback attractors in l2 rn lp rn and h1 rn for a nonlinear reaction diffusion equation in unbounded domains in which the nonlinearity f satisfies the polynomial growth of arbitrary order p 1 p 2 and the external forcing g lloc2 r l2 rn only subjects to the weaker integrability condition te s g s 22ds

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

an attractor for a nonlinear dissipative wave equation of kirchhoff type abstract in this paper we prove the existence and some absorbing properties of an attractor in a local sense for the initial boundary value problem of a quasilinear wave equation of kirchhoff type with a standard dissipation u t

global attractor for plate equation with nonlinear damping abstract in this paper we study the long time behavior of plate equation with nonlinear damping and critical nonlinearity we prove the existence of a global attractor in the space h 0 2 l 2

upper semicontinuity of attractors for nonclassical diffusion equations in h1 r3 abstract in this paper we mainly investigate upper semicontinuity of global attractors in h 1 r 3 for a perturbed nonclassical diffusion equations for this purpose some necessary abstract results are established

gromov hausdorff stability of global attractors of reaction diffusion equations under perturbations of the domain abstract in this paper we use the gromov hausdorff distances between two global attractors which belong to disjoint phase spaces and two dynamical systems to consider the continuous dependence of the global attractors and the stability of the dynamical systems on global attractors induced by the reaction diffusion equation under perturbations of the domain the novelty of the paper is to compare any two systems in different phase spaces without the process of pull backing the perturbed systems to the original domain

long time behavior of reaction diffusion equations with dynamical boundary condition abstract in this paper we study the long time behavior of the reaction diffusion equation with dynamical boundary condition where the nonlinear terms f and g satisfy the polynomial growth condition of arbitrary order some asymptotic regularity of the solution has been proved as an application of the asymptotic regularity results we can not only obtain the existence of a global attractor a in h 1 l p l q immediately but also can show further that a attracts every l 2 l 2 bounded subset with h 1 l p l q norm for any 0

long time behavior for a nonlinear parabolic problem with variable exponents abstract this paper addresses the question of the asymptotic behavior of solutions to the p x laplacian problem u t div u p x 2 u f x u g with general assumptions on f x u and the exponent p x we prove the existence of global attractors in proper spaces then we consider the fractal dimension of global attractors for the problem under suitable conditions we show that the problem admits an infinite dimensional global attractor

random attractors and averaging for non autonomous stochastic wave equations with nonlinear damping abstract this paper is devoted to the asymptotic behavior of solutions to a non autonomous stochastic wave equation with nonlinear damping and multiplicative white noise defined on an unbounded domain by showing the pullback asymptotic compactness of the cocycle in a certain parameter region we prove the existence of a random attractor when the intensity of noise is sufficiently small for the stochastic wave equation with rapidly oscillating external force we prove that the hausdorff distance between the random attractor a of the original equation and the random attractor a 0 of the averaged equation is in the order of o

attractors in lp rn and h1 rn for a class of reaction diffusion equations abstract this paper is concerned with a class of nonlinear reaction diffusion equations we prove that the semigroup of the solutions for this class of equations has global attractors in l p r n and h 1 r n

necessary and sufficient conditions for the existence of time dependent global attractor and application in this paper we are concerned with infinite dimensional dynamical systems in time dependent space first we characterize some necessary and sufficient conditions for the existence of the time dependent global attractor by using a measure of noncompactness then we give a new method to verify the sufficient condition as a simple application we prove the existence of the time dependent global attractor for the damped equation in strong topological space

on the existence of a global attractor for the wave equation with nonlinear strong damping perturbed by nonmonotone term abstract the long time behavior of the wave equation with nonmonotone interior damping is considered it is shown that the semigroup generated by this equation possesses a global attractor in h 0 1 l 2

asymptotic behavior and attractors for reaction diffusion equations in unbounded domains abstract in this paper we give general and flexible conditions for a reaction diffusion equation to be dissipative in an unbounded domain the functional setting is based on standard lebesgue and sobolev lebesgue spaces we show how the reaction and diffusion mechanisms have to work together to obtain the asymptotic compactness of solutions and therefore the existence of the compact attractor in particular cases our results allow us to improve some previous known results

random attractors of reaction diffusion equations with multiplicative noise in lp in this paper we study the random dynamical system rds generated by the reaction diffusion equation with multiplicative noise and prove the existence of a random attractor for such rds in l p d for any p 2

asymptotic behavior of the newton boussinesq equation in a two dimensional channel abstract we prove the existence of a global attractor for the newton boussinesq equation defined in a two dimensional channel the asymptotic compactness of the equation is derived by the uniform estimates on the tails of solutions we also establish the regularity of the global attractor

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

reaction diffusion equations with non autonomous force in h 1 and delays under measurability conditions on the driving delay term abstract in this paper we analyze the existence of solutions for a reaction diffusion problem with hereditary effects and a time dependent force term with values in h 1 the main novelty is that the delay term may be driven by a function under very minimal assumptions namely just measurability this is due to the fact that we only deal with a phase space of functions continuous in time allowing this general setting which might be more useful when less regularity is known in the hereditary mechanism after that we obtain uniform estimates and asymptotic compactness properties via an energy method that allow us to ensure the existence of pullback attractors for the associated process to the problem actually we obtain two different families of minimal pullback attractors namely those of fixed bounded sets but also for a class of time dependent families universe given by a tempered condition finally from comparison results we establish relations among them and under suitable additional assumptions we conclude that these families of attractors are in fact the same object

the existence of uniform attractors for non autonomous reaction diffusion equations on the whole space in this paper we introduce a new class of functions satisfying spacial absolutely continuous see definition 3 1 denoted by lsac2 r rn which are translation bounded but not normal see s s lu h q wu and c k zhong attractors for non autonomous 2d navier stokes equations with normal external forces discrete contin dyn syst a 13 3 701 719 2005 10 3934 dcds 2005 13 701 and definition 3 1 in lloc2 r rn then the asymptotic a priori estimate is applied to some nonlinear reaction diffusion equations with external forces g x s lsac2 r rn we obtain the existence of uniform attractor together with its structure in the bi spaces l2 rn l2 rn and l2 rn lp rn p 2 without any restriction on the growing order of the nonlinear term

global attractors for the p laplacian equations with nonregular data abstract let be a smooth bounded domain in r n n 3 we consider the long time behavior of solutions to the p laplacian equation u t p u f u g where 2 p n p div u p 2 u is the p laplace operator assume that g and the initial condition u 0 lie in l 1 f r r is of class c 1 and satisfies proper growth conditions firstly we prove the uniqueness of the entropy solution and establish some regularity results then we show the existence of a global attractor a in l r 1 w 0 1 s with s max n p 1 n 1 p r 1 r to obtain the results a decomposition method and a bootstrap technique are used

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

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

time dependent asymptotic behavior of the solution for wave equations with linear memory abstract in this article we consider the long time behavior of solutions for the wave equation with linear memory within the theory of process on time dependent spaces we investigate the existence of the time dependent attractor by using the operator decomposition technique and compactness of translation theorem and more detailed estimates furthermore the asymptotic structure of time dependent attractor which converges to the attractor of parabolic equation with memory is proved besides we obtain a further regular result about u t

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

attractors for partly dissipative reaction diffusion systems in rn in this paper we study the asymptotic behavior of solutions for the partly dissipative reaction diffusion equations in n we prove the asymptotic compactness of the solutions and then establish the existence of the global attractor in 2 n 2 n l l 2000 academic press

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

weighted least squares solutions to general coupled sylvester matrix equations this paper is concerned with weighted least squares solutions to general coupled sylvester matrix equations gradient based iterative algorithms are proposed to solve this problem this type of iterative algorithm includes a wide class of iterative algorithms and two special cases of them are studied in detail in this paper necessary and sufficient conditions guaranteeing the convergence of the proposed algorithms are presented sufficient conditions that are easy to compute are also given the optimal step sizes such that the convergence rates of the algorithms which are properly defined in this paper are maximized and established several special cases of the weighted least squares problem such as a least squares solution to the coupled sylvester matrix equations problem solutions to the general coupled sylvester matrix equations problem and a weighted least squares solution to the linear matrix equation problem are simultaneously solved several numerical examples are given to illustrate the effectiveness of the proposed algorithms

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

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

anisotropic wave propagation through finite difference grids an algorithm is presented to solve the elastic wave equation by replacing the partial differentials with finite differences it enables wave propagation to be simulated in three dimensions through generally anisotropic and heterogeneous models the space derivatives are calculated using discrete convolution sums while the time derivatives are replaced by a truncated taylor expansion a centered finite difference scheme in cartesian coordinates is used for the space derivatives leading to staggered grids the use of finite difference approximations to the partial derivatives results in a frequency dependent error in the group and phase velocities of waves for anisotropic media the use of staggered grids implies that some of the elements of the stress and strain tensors must be interpolated to calculate the hook sum this interpolation induces an additional error in the wave properties the overall error depends on the precision of the derivative and interpolation operators the anisotropic symmetry system its orientation and the degree of anisotropy the dispersion relation for the homogeneous case was derived for the proposed scheme since we use a general description of convolution sums to describe the finite difference operators the numerical wave properties can be calculated for any space operator and an arbitrary homogeneous elastic model in particular phase and group velocities of the three wave types can be determined in any direction we demonstrate that waves can be modeled accurately even through models with strong anisotropy when the operators are properly designed

inverse eigenvalue problem in structural design a method is presented for the design of a structure with specified low order natural frequencies the method can also be used for design problems in which certain bands devoid of any frequency constraints are specified this can further be used to generate initial feasible designs for optimum design problems with frequency constraints

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

the common bisymmetric nonnegative definite solutions with extreme ranks and inertias to a pair of matrix equations abstract we in this paper consider the bisymmetric nonnegative definite solution with extremal ranks and inertias to a system of quaternion matrix equations ax c xb d we derive the extremal ranks and inertias of the common bisymmetric nonnegative definite solution to the system the general expressions of the bisymmetric nonnegative definite solution with extremal ranks and inertias to the system mentioned above are also presented in addition we give a numerical example to illustrate the results of this paper

finite differences on minimal grids conventional approximations to space derivatives by finite differences use orthogonal grids to compute second order space derivatives in a given direction two points are used thus 2n points are required in a space of dimension n however a centered finite difference approximation to a second order derivative may be obtained using only three points in 2 d the vertices of a triangle four points in 3 d the vertices of a tetrahedron and in general n 1 points in a space of dimension n a grid using n 1 points to compute derivatives is called minimal the use of minimal grids does not introduce any complication in programming and suppresses some artifacts of the nonminimal grids for instance the well known decoupling between different subgrids for isotropic elastic media does not happen when using minimal grids because all the components of a given tensor e g displacement or stress are known at the same points some numerical tests in 2 d show that the propagation of waves is as accurate as when performed with conventional grids although this method may have less intrinsic anisotropies than the conventional method no attempt has yet been made to obtain a quantitative estimation

a new family of finite difference schemes to solve the heterogeneous acoustic wave equation equivalent staggered grid scheme esg is a new family of schemes based on the finite difference method fdm the method is applied to acoustic wave propagation in variable density media and the results are compared with those from some classic fdm approaches the main feature of this new family is that it is designed to generate results numerically equivalent to those using the standard staggered grid formulations ssg but with the same memory requirements of simple grid schemes hence it results in a reduction of memory usage by 33 in 2d and 50 in 3d problems compared to the memory usage of ssg the first order system of equations in terms of pressure and velocity is not used here instead the formulation is based on applying new central difference operators to the secondorder acoustic wave equation in terms of pressure obtaining the same level of accuracy and stability as the ssg schemes the equivalence between the esg and ssg is mathematically demonstrated and issues concerning the application of seismic sources and the boundary conditions are addressed