the riemann problem for reversible reactive flows with metastability a hyperbolic model for dynamic phase transitions is studied the model involves three phases liquid vapor and a mixture of them metastable regions are present both in the liquid and in the vapor phase results on the behavior of traveling wave profiles of the model involving viscosity species diffusion and relaxation are obtained these behaviors are consistent with physical intuitions admissibility criteria kinetic relations that mimic the behavior of traveling wave profiles are then proposed admissible basic waves of the model are liquefaction evaporation and isobaric waves in addition to lax shock and rarefaction waves based on these waves solutions of the riemann problem for the model are constructed for general riemann initial data most of the physical phenomena are embodied in the solver for some riemann initial data solutions are expected to be nonunique which solutions actually appear depends on whether nucleation already occurred or not the model admits both solutions as it sh

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

on a class of degenerate parabolic equations with dynamic boundary conditions we consider a quasi linear parabolic possibly degenerate equation with nonlinear dynamic boundary conditions the corresponding class of initial and boundary value problems has already been studied previously proving well posedness of weak solutions and the existence of the global attractor assuming that the nonlinearities are subcritical to a given exponent the goal of this article is to show that the previous analysis can be redone for supercritical nonlinearities by proving an additional l estimate on the solutions in particular we derive new conditions which reflect an exact balance between the internal and the boundary mechanisms involved even when both the nonlinear sources contribute in opposite directions then we show how to construct a trajectory attractor for the weak solutions of the associated parabolic system and prove that any solution belonging to the attractor is bounded which implies uniqueness finally we also prove for the semilinear reaction diffusion equation with nonlinear dynamic boundary conditions that the fractal dimension of the global attractor is of the order n 1 as diffusion 0 in any space dimension n 2 improving some recent results in gal 2012 23

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

elliptic 1 laplacian equations with dynamical boundary conditions abstract this paper is concerned with an evolution problem having an elliptic equation involving the 1 laplacian operator and a dynamical boundary condition we apply nonlinear semigroup theory to obtain existence and uniqueness results as well as a comparison principle our main theorem shows that the solution we found is actually a strong solution we also compare solutions with different data

transition to detonation in dynamic phase changes subsonically propagating phase boundaries kinks can be modelled by material discontinuities which satisfy integral conservation laws plus an additional jump condition governing the phase change kinetics the necessity of an additional jump condition distinguishes kinks from the conventional shocks which satisfy the lax criterion we study stability of kinks with respect to the breakup splitting into a sequence of waves we assume that all conventional shocks are admissible and that admissible kinks are selected by a prescribed kinetic relation as we show regardless of a particular choice of the kinetic relation sufficiently fast phase boundaries are unstable the mode of instability includes an emission of a centered riemann wave followed by a sonic shock chapman jouguet type phase boundary

perturbing lyapunov functions and stability criteria for initial time difference stability criteria for differential equations where the initial time for each solution is different is developed using the method of perturbing lyapunov functions

uniform attractors for the closed process and applications to the reaction diffusion equation with dynamical boundary condition abstract in this paper we first introduce the concept of a closed process in a banach space and we obtain the structure of a uniform attractor of the closed process by constructing a skew product flow on the extended phase space then the properties of the kernel section of closed process are investigated moreover we prove the existence and structure of the uniform attractor for the reaction diffusion equation with a dynamical boundary condition in l p without any restriction on the growth order of the nonlinear term

regular random attractors for non autonomous stochastic evolution equations with time varying delays on thin domains this paper deals with the limiting dynamical behavior of non autonomous stochastic reaction diffusion equations with time varying delays on thin domains first we prove the existence and uniqueness of the regular random attractor then we prove the upper semicontinuity of the regular random attractors for the equations on a family of n 1 dimensional thin domains that collapses onto an n dimensional domain

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

pullback attractors of nonautonomous reaction diffusion equations abstract in this paper firstly we introduce the concept of norm to weak continuous cocycle in banach space and give a technical method to verify this kind of continuity then we obtain some abstract results for the existence of pullback attractors about this kind of cocycle using the measure of noncompactness as an application we prove the existence of pullback attractors in h 0 1 of the cocycle associated with the solutions for some nonlinear nonautonomous reaction diffusion equations the attractor pullback attracts all bounded subsets of h 0 1 in the norm of h 0 1

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

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

kinetic relations and the propagation of phase boundaries in solids this paper treats the dynamics of phase transformations in elastic bars the specific issue studied is the compatibility of the field equations and jump conditions of the one dimensional theory of such bars with two additional constitutive requirements a kinetic relation controlling the rate at which the phase transition takes place and a nucleation criterion for the initiation of the phase transition a special elastic material with a piecewise linear non monotonic stress strain relation is considered and the riemann problem for this material is analyzed for a large class of initial data it is found that the kinetic relation and the nucleation criterion together single out a unique solution to this problem from among the infinitely many solutions that satisfy the entropy jump condition at all strain discontinuities

dynamics for a class of non autonomous degenerate p laplacian equations abstract in this paper we investigate a class of non autonomous degenerate p laplacian equations t u div a x u p 2 u u f u g x t in where a x is allowed to vanish on a nonempty closed subset with lebesgue measure zero g x t l l o c p r d 1 p a and an unbounded domain in r n we first establish the well posedness of these equations by constructing a compact embedding then we show the existence of the minimal pullback d attractor and prove that it indeed attracts the d class in l 2 norm for any 0 our results extend some known ones in previously published papers

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

coupling of general lagrangian systems this work is devoted to the coupling of two fluid models such a s two euler systems in lagrangian coordinates at a fixed interface we define coupling conditi ons which can be expressed in terms of continuity of some well chosen variables and then solve the coupled riemann problem in the present setting where the interface is characteristic a particula r choice of dependent variables which are transmitted ensures the overall conservativity

asymptotic behavior for a semilinear second order evolution equation this paper is devoted to the qualitative analysis for a second order evolution equation u tt delta u delta u t epsilon delta u tt f u g x epsilon is an element of 0 1 with critical nonlinearity some uniformly w r t epsilon is an element of 0 1 asymptotic regularity about the solutions has been established for both g x is an element of l 2 omega and g x is an element of h 1 which shows that the solutions are exponentially approaching a more regular fixed subset uniformly w r t epsilon is an element of 0 1 as an application of this regularity result a family epsilon epsilon epsilon is an element of 0 1 of finite dimensional exponential attractors has been constructed moreover to characterize the relation with a strongly damped wave equation epsilon 0 the upper semicontinuity at epsilon 0 of the global attractors has been proved

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

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

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

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

existence of traveling waves in van der waals fluids with viscosity and capillarity effects abstract we establish the existence of traveling waves of non isentropic van der waals fluids with viscosity and capillarity effects the method developed the one for simpler models the nonconvex equation of state of the fluid causes much difficulty in evaluating the related quantities and so the argument and the analysis are much more involved than the convex equation of state the point is to estimate the pressure along the hugoniot curves such that a lyapunov function can be defined in an appropriate way

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

travelling wave solutions to the korteweg de vries burgers equation the existence and certain qualitative properties of travelling wave solutions to the korteweg de vries burgers equation are established the limiting behaviour of these waves when e tends to zero and when tends to zero is examined together with a singular limit wherein both e and tend to zero

existence of traveling waves in compressible euler equations with viscosity and capillarity abstract given any lax shock of the compressible euler dynamics equations we show that there exists the corresponding traveling wave of the system when viscosity and capillarity are suitably added for a traveling wave corresponding to a given lax shock the governing viscous capillary system is reduced to a system of two differential equations of first order which admits an asymptotically stable equilibrium point and a saddle point we then develop the method of estimating attraction domain of the asymptotically stable equilibrium point for the compressible euler equations and show that the saddle point in fact lies on the boundary of this set then we establish a saddle to stable connection by pointing out that there is a stable trajectory leaving the saddle point and entering the attraction domain of the asymptotically stable equilibrium point this gives us a traveling wave of the viscous capillary compressible euler equations

propagating phase boundaries formulation of the problem and existence via the glimm method this paper treats the hyperbolic elliptic system of two conservation laws which describes the dynamics of an elastic material having a non monotone strain stress function followingabeyaratne knowles we propose a notion of admissible weak solution for this system in the class of functions of bounded variation the formulation includes an entropy inequality a kinetic relation imposed along any subsonic phase boundary and an initiation criterion for the appearance of new phase boundaries we prove thel1 continuous dependence of the solution to the riemann problem our main result yields the existence and the stability of propagating phase boundaries the proofs are based onglimm s scheme and in particular on the techniques ofglimm andlax in order to deal with the kinetic relation we prove a result of pointwise convergence of the phase boundary

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

dynamics for a stochastic degenerate parabolic equation abstract in this paper we consider the long time behavior of a class of stochastic degenerate parabolic equations involving an operator which is x elliptic with respect to a family of locally lipschitz continuous vector fields x x 1 x 2 x m the nonlinearity satisfies a dissipative condition with polynomial growth of arbitrary order p 2 the existence of the random attractor in l 2 o higher order integrability and continuity in h are established

attractor and traveling waves of a fluid with nonlinear diffusion and dispersion abstract this work completes the description of the method of estimating attraction domain for traveling waves for several systems of conservation laws with viscosity and capillarity effects proposed in our earlier work thanh 2010 26 precisely we establish the global existence of traveling waves for an isentropic fluid with nonlinear diffusion and dispersion coefficients the shock wave can be classical or nonclassical interestingly we in particular show the existence of a traveling wave for a given lax shock but rather nonclassical when the straight line connecting the two left hand and right hand states crosses the graph of the pressure function two more times in the middle region furthermore we also discuss all the possibilities of saddle stable saddle saddle or stable stable connections

on a singularly perturbed wave equation with dynamic boundary conditions in this paper we analyse a singular perturbation problem for linear wave equations with interior and boundary damping we show how the solutions converge to the formal parabolic limit problem with dynamic boundary conditions conditions are given for uniform convergence in the energy space

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

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

pullback exponential attractors for the viscous cahn hilliard navier stokes system with dynamic boundary conditions abstract this paper is concerned with the existence of pullback exponential attractors for the viscous cahn hilliard navier stokes system with dynamic boundary conditions due to the lack of the holder continuity in time of the semigroup and the smoothing property of the difference of two weak solutions we construct a pullback exponential attractor in the original phase space by the l trajectories methods

stochastic optimal control and algorithm of the trajectory of horizontal wells this paper presents a nonlinear multi phase and stochastic dynamical system according to engineering background we show that the stochastic dynamical system exists a unique solution for every initial state a stochastic optimal control model is constructed and the sufficient and necessary conditions for optimality are proved via dynamic programming principle this model can be converted into a parametric nonlinear stochastic programming by integrating the state equation it is discussed here that the local optimal solution depends in a continuous way on the parameters a revised hooke jeeves algorithm based on this property has been developed computer simulation is used for this paper and the numerical results illustrate the validity and efficiency of the algorithm

delay integro differential inequalities with initial time difference and applications some comparison results for delay integro differential inequalities starting from different initial points are formulated the monotone iterative technique is applied to formulate sufficient conditions for existence of extremal solutions of delay integro differential equations with initial time difference

global solutions for quasilinear parabolic systems we present an approach for proving the global existence of classical solutions of certain quasilinear parabolic systems with homogeneous dirichlet boundary conditions in bounded domains with a smooth boundary c 2003 elsevier inc all rights reserved

upper semicontinuity of attractors of stochastic delay reaction diffusion equations in the delay a system of stochastic delayed reaction diffusion equations with multiplicative noise and deterministic non autonomous forcing is considered we first prove the existence and uniqueness of a bi spatial pullback attractor for these equations when the initial space is c 0 l2o and the terminate space is c 0 h01o the asymptotic compactness of solutions in c 0 h01o is established by combining positive and negative truncations technique and some new estimates on solutions then the periodicity of the random attractors is proved when the stochastic delay equations are forced by periodic functions finally upper semicontinuity of the global random attractors in the delay is obtained as the length of time delay approaches zero a system of stochastic delayed reaction diffusion equations with multiplicative noise and deterministic non autonomous forcing is considered we first prove the existence and uniqueness of a bi spatial pullback attractor for these equations when the initial space is c 0 l2o and the terminate space is c 0 h01o the asymptotic compactness of solutions in c 0 h01o is established by combining positive and negative truncations technique and some new estimates on solutions then the periodicity of the random attractors is proved when the stochastic delay equations are forced by periodic functions finally upper semicontinuity of the global random attractors in the delay is obtained as the length of time delay approaches zero

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

non classical riemann solvers with nucleation we introduce a new nonclassical riemann solver for scalar conservation laws with concave convex flux function this solver is based on both a kinetic relation which determines the propagation speed of undercompressive nonclassical shock waves and a nucleation crite rion which makes a choice between a classical riemann solution and a nonclassical one we establish the existence of nonclassical entropy solutions of the cauchy problem and discuss several examples of wave interactions we also show the existence of a class of solutions called splitting merging solutions which are made of two large shocks and small bv bounded variation perturbations the nucleation solvers as we call them are applied to and actually motivated by the theory of thin film flows they help explain numerical results observed for such flows

impulsive optimal control model for the trajectory of horizontal wells this paper presents an impulsive optimal control model for solving the optimal designing problem of the trajectory of horizontal wells we take fully into account the effect of unknown disturbances in drilling the optimal control problem can be converted into a nonlinear parametric optimization by integrating the state equation we discuss here that the locally optimal solution depends in a continuous way on the parameters disturbances and utilize this property to propose a revised hooke jeeves algorithm the uniform design technique is incorporated into the revised hooke jeeves algorithm to handle the multimodal objective function the numerical simulation is in accordance with theoretical results the numerical results illustrate the validity of the model and efficiency of the algorithm

on pullback attractors in lp for nonautonomous reaction diffusion equations abstract using the method introduced in zhong et al 2006 17 together with a new way of dealing with well known estimates of solutions introduced in ukaszewicz in press 9 we prove the existence of a unique minimal pullback attractor for the evolutionary process associated with a nonautonomous nonlinear reaction diffusion system in l p p 2 in which the right hand side satisfies only a certain integrability condition in particular we generalize a result obtained recently in li et al 2009 13 where an at most exponential growth of the right hand side has been assumed for times going to both plus and minus infinity

why many theories of shock waves are necessary kinetic functions equivalent equations and fourth order models we consider several systems of nonlinear hyperbolic conservation laws describing the dynamics of nonlinear waves in presence of phase transition phenomena these models admit under compressive shock waves which are not uniquely determined by a standard entropy criterion but must be characterized by a kinetic relation building on earlier work by lefloch and collaborators we investigate the numerical approximation of these models by high order finite difference schemes and uncover several new features of the kinetic function associated with physically motivated second and third order regularization terms especially viscosity and capillarity terms on one hand the role of the equivalent equation associated with a finite difference scheme is discussed we conjecture here and demonstrate numerically that the numerical kinetic function associated with a scheme approaches the analytic kinetic function associated with the given model especially since its equivalent equation approaches the regularized model at a higher order on the other hand we demonstrate numerically that a kinetic function can be associated with the thin liquid film model and the generalized camassa holm model finally we investigate to what extent a kinetic function can be associated with the equations of van der waals fluids whose flux function admits two inflection points

random attractor for the ladyzhenskaya model with additive noise this paper studies the dynamical behavior of the ladyzhenskaya model with additive noise with some conditions we prove that the generated random dynamical system has a compact random attractor which is a random compact set absorbing any bounded nonrandom subset of the phase space

nonclassical shocks and kinetic relations finite difference schemes we consider hyperbolic systems of conservation laws that are not genuinely nonlinear the solutions generated by diffusive dispersive regularizations may include nonclassical n c shock waves that do not satisfy the classical liu entropy criterion we investigate the numerical approximation of n c shocks via conservative difference schemes constrained only by a single entropy inequality the schemes are designed by comparing their equivalent equations with the continuous model and include discretizations of the diffusive and dispersive terms limits of these schemes are characterized via the kinetic relation introduced earlier by the authors we determine the kinetic function numerically for several examples of systems and schemes this study demonstrates that the kinetic relation is a suitable tool for the selection of unique n c solutions and for the study of their sensitive dependence on the critical parameters the ratios of diffusion dispersion and diffusion mesh size the shock strength and the order of discretization of the flux

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

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

finite energy method for compressible fluids the navier stokes korteweg model this is the first of a series of papers devoted to the initial value problem for the one dimensional euler system of compressible fluids and augmented versions containing higher order terms in the present paper we are interested in dispersive shock waves and analyze the zero viscosity capillarity limit associated with the navier stokes korteweg system specifically we establish the existence of finite energy solutions as well as their convergence toward entropy solutions to the euler system our method of proof combines energy and effective energy estimates a nonlinear sobolev inequality high integrability properties for the mass density and for the velocity and compactness properties based on entropies 2015 wiley periodicals inc

diffusive dispersive traveling waves and kinetic relations part i nonconvex hyperbolic conservation laws abstract motivated by the theory of phase transition dynamics we consider one dimensional nonlinear hyperbolic conservation laws with nonconvex flux function containing vanishing nonlinear diffusive dispersive terms searching for traveling wave solutions we establish general results of existence uniqueness monotonicity and asymptotic behavior in particular we investigate the properties of the traveling waves in the limits of dominant diffusion dominant dispersion and asymptotically small or large shock strength as the diffusion and dispersion parameters tend to 0 the traveling waves converge to shock wave solutions of the conservation law which either satisfy the classical oleinik entropy criterion or are nonclassical undercompressive shocks violating it

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

attractors for nonclassical diffusion equations with dynamic boundary conditions abstract in this paper we study the asymptotic behavior of solutions for a class of nonclassical diffusion equation t u e u u u f u g in a smooth bounded domain in r n n 3 with dynamic boundary condition t u e u n e t u u u f u g on under a sobolev type growth condition of the nonlinear functions we prove the existence and uniqueness of weak solution for the problem by using the galerkin approximation method and then establish the existence and upper semicontinuity of global attractors a e at e 0 0 1 by using the asymptotic a priori estimate method

pullback attractors for non autonomous reaction diffusion equations with dynamical boundary conditions abstract in this paper we prove the existence and uniqueness of a weak solution for a non autonomous reaction diffusion model with dynamical boundary conditions after that a continuous dependence result is established via an energy method including in particular some compactness properties finally the precedent results are used in order to ensure the existence of minimal pullback attractors in the frameworks of universes of fixed bounded sets and that given by a tempered growth condition the relation among these families is also discussed

pullback attractors for a semilinear heat equation in a non cylindrical domain abstract the existence and uniqueness of a variational solution satisfying energy equality is proved for a semilinear heat equation in a non cylindrical domain with homogeneous dirichlet boundary condition under the assumption that the spatial domains are bounded and increase with time in addition the non autonomous dynamical system generated by this class of solutions is shown to have a global pullback attractor

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

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

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

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

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

continuity and random dynamics of the non autonomous stochastic fitzhugh nagumo system on rn abstract in this article we use the so called difference estimate method to investigate the continuity and random dynamics of the non autonomous stochastic fitzhugh nagumo system with a general nonlinearity firstly under weak assumptions on the noise coefficient we prove the existence of a pullback attractor in l 2 r n l 2 r n by using the tail estimate method and a certain compact embedding on bounded domains secondly although the difference of the first component of solutions possesses at most p times integrability where p is the growth exponent of the nonlinearity we overcome the absence of higher order integrability and establish the continuity of solutions in l p r n h 1 r n l 2 r n with respect to the initial values belonging to l 2 r n l 2 r n as an application of the result on the continuity the existence of a pullback attractor in l p r n h 1 r n l 2 r n is proved for arbitrary n 1 and p 2

practical stability of nonlinear differential equation with initial time difference this paper develops the concept of practical stability to nonlinear differential equations with solutions starting with different initial times several practical stability criteria of nonlinear dynamical systems relative to initial time difference are presented by employing two lyapunov like functions the method of two lyapunov like functions enables us to obtain results under weaker assumptions

pullback attractors for a semilinear heat equation on time varying domains abstract the existence of a pullback attractor is established for the nonautonomous dynamical system generated by the weak solutions of a semilinear heat equation on time varying domains with homogeneous dirichlet boundary conditions it is assumed that the spatial domains o t in r n are obtained from a bounded base domain o by a c 2 diffeomorphism which is continuously differentiable in the time variable and are contained in the past in a common bounded domain

a note on quenching for parabolic equations with dynamic boundary conditions we present a quenching result for semilinear parabolic equations with dynamic boundary conditions in bounded domains with a smooth boundary

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

attractors for parabolic equations with dynamic boundary conditions abstract in this paper we prove existence of global solutions and l 2 l 2 h 1 l p l p global attractors for semilinear parabolic equations with dynamic boundary conditions in bounded domains with a smooth boundary where there is no other restriction on p 2

asymptotic behaviour of nonautonomous difference inclusions abstract nonautonomous difference inclusions are formulated as difference inclusion cocycles which generalize set valued semigroups corresponding to autonomous difference inclusions and the existence of a difference inclusion cocycle attractor is established under the assumption of upper semi continuity of the cocycle mappings if the difference inclusion cocycle has an absorbing set several illustrative examples are provided

non classical shocks and kinetic relations scalar conservation laws this paper analyzes the non classical shock waves which arise as limits of certain diffusive dispersive approximations to hyperbolic conservation laws such shocks occur for non convex fluxes and connect regions of different convexity they have negative entropy dissipation for a single convex entropy function but not all convex entropies and do not obey the classical oleinik entropy criterion we derive necessary conditions for the existence of non classical shock waves and construct them as limits of traveling wave solutions for several diffusive dispersive approximations

on the pullback attractor for the non autonomous sir equations with diffusion abstract in this paper we study the dynamics of a non autonomous sir model with diffusion we first prove that the l 2 l 2 pullback attractor attracts any bounded subset of l 2 in the topology of l 2 for any 0 then we show that the l 2 l 2 pullback attractor is indeed a l 2 l 2 pullback attractor

diffusive dispersive travelling waves and kinetic relations v singular diffusion and nonlinear dispersion we consider scalar hyperbolic conservation laws with non convex flux and vanishing nonlinear and possibly singular diffusion and dispersion terms the diffusion has the form r u u x x and we cover for instance the singular diffusion u x p u x x where p 0 is arbitrary we investigate the existence uniqueness and various properties of classical and non classical travelling waves and of the kinetic function the latter serves to characterize non classical shock waves via an additional algebraic constrain called a kinetic relation we discover that p is a somewhat unexpected critical value for p we obtain properties that are qualitatively similar to those we established earlier for regular and linear diffusion however for p the behaviour of the kinetic function is very different as for instance non classical shocks can have arbitrary small strength the behaviour of the kinetic function near the origin is carefully investigated and depends on whether p p or p in particular in the special case of the cubic flux function and for the regularization u x p u x x with p 0 or 1 the kinetic function can be computed explicitly when p the kinetic function is simply a linear function of its argument

exponential and global stability of nonlinear dynamical systems relative to initial time difference specialized research fund for the doctoral program of higher education 20100121120018 natural science foundation of fujian province of china 2010j05013 fundamental research funds for the central universities 2010121004

continuity of strong solutions of the reaction diffusion equation in initial data abstract this paper is concerned with the continuity of strong solutions of the reaction diffusion equation with respect to initial conditions it is proved that the solutions of the equation are continuous in h 1 with respect to initial data provided the nonlinearity satisfies a dissipative condition with polynomial growth order p 2