regularity of pullback attractors for nonautonomous nonclassical diffusion equations abstract in this paper we consider the regularity of pullback attractors for a nonautonomous nonclassical diffusion equation with critical nonlinearity in particular under suitable assumptions we prove that there exists a pullback attractor a a t t r in h 0 1 for a nonautonomous nonclassical diffusion equation and for each t r a t is bounded in h 2 h 0 1

attractors for the nonclassical diffusion equations with fading memory abstract we discuss long time dynamical behavior of the nonclassical diffusion equation with fading memory when nonlinearity is critical the existence and regularity of global attractors in weak topological space and strong topological space are obtained while the forcing term only belongs to h 1 and l 2 respectively the results in this part are new and appear to be optimal corresponding to the forcing term

discontinuous galerkin finite element approximation of the cahn hilliard equation with convection the paper is concerned with the construction and convergence analysis of a discontinuous galerkin finite element method for the cahn hilliard equation with convection using discontinuous piecewise polynomials of degree p geq1 and backward euler discretization in time we show that the order parameter c is approximated in the broken rm l infty rm h 1 norm with optimal order cal o h p tau the associated chemical potential w phi c gamma 2 delta c is shown to be approximated with optimal order cal o h p tau in the broken rm l 2 rm h 1 norm here phi c frac 1 4 1 c 2 2 is a quartic free energy function and gamma 0 is an interface parameter numerical results are presented with polynomials of degree p 1 2 3

viscous cahn hilliard equation ii analysis the viscous cahn hilliard equation may be viewed as a singular limit of the phase field equations for phase transitions it contains both the allen cahn and cahn hilliard models of phase separation as particular cases by specific choices of parameters it may be formulated as a one parameter say homotopy connecting the cahn hilliard 0 and allen cahn 1 models the limit 0 is singular in the sense that the smoothing property of the analytic semigroup changes from being of the type associated with second order operators to the type associated with fourth order operators the properties of the gradient dynamical system generated by the viscous cahn hilliard equation are studied as varies in 0 1 continuity of the phase portraits near equilibria is established independently of 0 1 and using this a piecewise uniform in time perturbation result is proved for trajectories finally the continuity of the attractor is established and in one dimension the existence and continuity of inertial manifolds shown and the flow on the attractor detailed

an energetic variational approach for the cahn hilliard equation with dynamic boundary condition model derivation and mathematical analysis the cahn hilliard equation is a fundamental model that describes phase separation processes of binary mixtures in recent years several types of dynamic boundary conditions have been proposed in order to account for possible short range interactions of the material with the solid wall our first aim in this paper is to propose a new class of dynamic boundary conditions for the cahn hilliard equation in a rather general setting the derivation is based on an energetic variational approach that combines the least action principle and onsager s principle of maximum energy dissipation one feature of our model is that it naturally fulfills three important physical constraints conservation of mass dissipation of energy and force balance relations next we provide a comprehensive analysis of the resulting system of partial differential equations under suitable assumptions we prove the existence and uniqueness of global weak strong solutions to the initial boundary value problem with or without surface diffusion furthermore we establish the uniqueness of the asymptotic limit as t to infty and characterize the stability of local energy minimizers for the system

linearisation and potential symmetries of certain systems of diffusion equations abstract we consider systems of two pure one dimensional diffusion equations that have considerable interest in soil science and mathematical biology we construct non local symmetries for these systems these are determined by expressing the equations in a partially and wholly conserved form and then by performing a potential symmetry analysis on those systems that can be linearised we give several examples of such systems and in a specific case we show how linearising and hodograph type mappings can lead to new solutions of the diffusion system

potential symmetries of nonlinear diffusion convection equations in this paper potential symmetries are sought for the nonlinear diffusion convection equations the functional forms of f u and k u that admit such symmetries are completely classified

potential symmetries and direct reduction methods of order two for partial differential equations written in conservative form a remarkable link between potential symmetries and direct reduction methods of order two is enlightened

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

optimal control problem for viscous cahn hilliard equation abstract this paper is concerned with the viscous cahn hilliard equation which arises in the dynamics of viscous first order phase transitions in cooling binary solutions the optimal control under boundary condition is given and the existence of optimal solution to the equation is proved

fuzzy data envelopment analysis and its application to location problems in this paper fuzzy dea data envelopment analysis models are proposed for evaluating the efficiencies of objects with fuzzy input and output data the obtained efficiencies are also fuzzy numbers that reflect the inherent ambiguity in evaluation problems under uncertainty an aggregation model for integrating fuzzy attribute values is provided in order to rank objects objectively using the proposed method a case study involving a restaurant location problem is analyzed in detail rent of establishment traffic amount level of security consumer consumption level and competition level are identified as the primary factors in determining an ideal location for a japanese style rotisserie restaurant based on field investigation the uncertain information on primary factors is represented by fuzzy numbers using the fuzzy aggregation model the best location of restaurant is determined the case study shows that fuzzy dea models can be quite useful for solving business problems under uncertainty

on an application of tikhonov s fixed point theorem to a nonlocal cahn hilliard type system modeling phase separation abstract this paper investigates a nonlocal version of a model for phase separation on an atomic lattice that was introduced by p podio guidugli 2006 36 the model consists of an initial boundary value problem for a nonlinearly coupled system of two partial differential equations governing the evolution of an order parameter and the chemical potential singular contributions to the local free energy in the form of logarithmic or double obstacle potentials are admitted in contrast to the local model which was studied by p podio guidugli and the present authors in a series of recent publications in the nonlocal case the equation governing the evolution of the order parameter contains in place of the laplacian a nonlocal expression that originates from nonlocal contributions to the free energy and accounts for possible long range interactions between the atoms it is shown that just as in the local case the model equations are well posed where the technique of proving existence is entirely different it is based on an application of tikhonov s fixed point theorem in a rather unusual separable and reflexive banach space

non classical reductions of initial value problems for a class of nonlinear evolution equations we classify initial value problems for a class of one dimensional evolution equations which can be reduced to cauchy problems for some systems of first order ordinary differential equations the technique applied relies heavily on higher conditional symmetries of the equations under study which means in particular that the obtained reductions cannot be derived within the framework of the standard lie approach

spectral comparison principles for the cahn hilliard and phase field equations and time scales for coarsening we initiate a study of the instability properties of steady solutions of the cahn hilliard ch equation and the phase field pf system in terms of the coarseness of those solutions other spectral results are obtained as well steady state solutions for the ch equation with no flux or periodic boundary conditions are also stationary states for the bistable reaction diffusion equation and the pf system with constant temperature thus it is meaningful to linearize each of these equations about a common stationary solution and discuss the stability of this state in each dynamical process converting the linearized operators to self adjoint form and expressing eigenvalues in terms of modified rayleigh quotients enables us to relate the spectra of the three operators in particular we relate the indices of instability with respect to the three processes we also discuss the change in the spectrum of the linearized ch operator in one dimension when we change the stationary solution at which it is linearized we conjecture that the eigenvalues increase with the number of oscillations of the steady state for a given mass and prove that this is so in certain cases this lends credence to the idea that motion of these dynamical systems near coarse solutions takes place on longer time scales

potential symmetries of a porous medium equation potential symmetries which are not local symmetries are carried out for the porous medium equation where when it can be written in a conserved form these symmetries are realized as local symmetries of a related auxiliary system and lead to the construction of corresponding invariant solutions as well as to the linearization of the equation by non invertible mappings

pullback attractors for a nonautonomous nonclassical diffusion equation with variable delay we present the existence of pullback attractors in ch01 and ch2 h01 for the processes associated with the nonautonomous nonclassical diffusion equations with delays new methods for checking the asymptotical compactness of the solutions are used

exponential attractors and inertial manifolds for singular perturbations of the cahn hilliard equations abstract we consider the singular perturbations of two boundary value problems concerning respectively the viscous and the nonviscous cahn hilliard equations in one dimension of space we show that the dynamical systems generated by these two problems admit exponential attractors and inertial manifolds in the phase space h01 0 h 1 0

existence of solutions and separation from singularities for a class of fourth order degenerate parabolic equations a nonlinear parabolic equation of the fourth order is analyzed the equation is characterized by a mobility coefficient that degenerates at 0 existence of at least one weak solution is proved by using a regularization procedure and deducing suitable a priori estimates if a viscosity term is added and additional conditions on the nonlinear terms are assumed then it is proved that any weak solution becomes instantaneously strictly positive this in particular implies uniqueness for strictly positive times and further time regularization properties the long time behavior of the problem is also investigated and the existence of trajectory attractors and under more restrictive conditions of strong global attractors is shown

global attractors for singular perturbations of the cahn hilliard equations abstract we consider the singular perturbations of two boundary value problems concerning respectively the viscous and the nonviscous cahn hilliard equations in one dimension of space we show that the dynamical systems generated by these two problems admit global attractors in the phase space h 0 1 0 h 1 0 and that these global attractors are at least upper semicontinuous with respect to the vanishing of the perturbation parameter

conditional lie backlund symmetry and reduction of evolution equations we suggest a generalization of the notion of invariance of a given partial differential equation with respect to a lie backlund vector field such a generalization proves to be effective and enables us to construct principally new ansatz reducing evolution type equations to several ordinary differential equations in the framework of the said generalization we obtain principally new reductions of a number of nonlinear heat conductivity equations ut uxx f u ux with poor lie symmetry and obtain their exact solutions it is shown that these solutions cannot be constructed by means of the symmetry reduction procedure

the behavior of finite element solutions of semilinear parabolic problems near stationary points the qualitative behavior of spatially semidiscrete finite element solutions of a semilinear parabolic problem near an unstable hyperbolic equilibrium overline u is studied it is shown that any continuous trajectory is approximated by an appropriate discrete trajectory and vice versa as long as they remain in a sufficiently small neighborhood of overline u error bounds of optimal order in the l 2 and h 1 norms hold uniformly over arbitrarily long time intervals in particular the local stable and unstable manifolds of the discrete problem converge to their continuous counterparts therefore the discretized dynamical system has the same qualitative behavior near overline u as the continuous system

pullback attractors for a class of non autonomous nonclassical diffusion equations abstract we consider the first initial boundary value problem for the non autonomous nonclassical diffusion equation u t e u t u f u g t e 0 1 in a bounded domain in r n under a sobolev growth rate of the nonlinearity f and a suitable exponential growth of the external force g using the asymptotic a priori estimate method we prove the existence of pullback d attractors a e in the space h 0 1 and the upper semicontinuity of a e at e 0

attractive invariant manifolds under approximation inertial manifolds a class of nonlinear dissipative partial differential equations that possess finite dimensional attractive invariant manifolds is considered an existence and perturbation theory is developed which unifies the cases of unstable manifolds and inertial manifolds into a single framework it is shown that certain approximations of these equations such as those arising from spectral or finite element methods in space one step time discretization or a combination of both also have attractive invariant manifolds convergence of the approximate manifolds to the true manifolds is established as the approximation is refined in this part of the paper applications to the behavior of inertial manifolds under approximation are considered from this analysis deductions about the structure of the attractor and the flow on the attractor under discretization can be made

the existence of contact transformations for evolution type equations we investigate the existence of a one parameter group of contact transformations for evolution type equations ut f t x u ux uxx u n subscripts denote differentiation unless otherwise indicated where u n is the nth derivative of u with respect to x we prove that contact transformations of evolution equations where f is expandable as a power series in terms of all derivatives of order higher than one are just extended lie point transformations this result is extended to the case with m independent space variables as a consequence we obtain an ansatz for determining lie point transformations for nth order evolution equations with m independent space variables examples are given to verify the results obtained as well as to show how lie point transformations of these evolution type partial differential equations can be calculated from this ansatz

symmetric singularity formation in lubrication type equations for interface motion fourth order degenerate diffusion equations arise in a lubrication approximation of a thin film or neck driven by surface tension numerical studies of the lubrication equation le h t h n h xxx x 0 with various boundary conditions indicate that singularity formation in which h x t t to 0 occurs for small enough n with anomalous or second type scaling inconsistent with usual dimensional analysis this paper considers locally symmetric or even singularities in the le and in the modified lubrication equation mle h t h n h xxxx 0 both equations have the property that entropy bounds forbid finite time singularities when n is sufficiently large power series expansions for local symmetric similarity solutions are proposed for equation le with n 1 and mle for all n in mathbb r in the latter case special boundary conditions that force singularity formation are required to produce singularities when n is large matching conditions at higher order terms in the

pullback attractors for nonclassical diffusion equations with delays in this paper we prove the existence of pullback attractors in ch01 for a nonclassical diffusion equation with delay term g t ut which contains some hereditary characteristics we consider two types of nonlinearity f one is the case of critical growth and the other one is the polynomial growth of arbitrary order p 1 p 2

modified piecewise linear dea model recently cook and zhu have proposed the piecewise linear data envelopment analysis pl dea model a situation in which a generalization of the dea methodology which incorporates piecewise linear functions of factors is considered standard dea models provide an efficiency score and targets for an inefficient unit but the pl dea model fails to produce acceptable targets thus this issue has been considered in the piecewise linear ccr model in which a non increasing set of multipliers describe the weight function also the piecewise linear ccr model has been enhanced by introducing two mip models for a two stage procedure in order to set targets precisely furthermore as it follows the above mentioned models are compared with each other and an example is provided for the sake of lucidity

potential symmetries and new solutions of a simplified model for reacting mixtures in this paper we investigate potential symmetries of a simplified model for reacting mixtures we find new similarity reductions and wider class of solutions through this approach further we explore an invertible mapping which linearizes the reacting mixture model

on the dynamics of a class of nonclassical parabolic equations abstract we consider the first initial and boundary value problem of nonclassical parabolic equations u t u t u g u f x on a bounded domain where 0 1 first we establish some uniform decay estimates for the solutions of the problem which are independent of the parameter then we prove the continuity of solutions as 0 finally we show that the problem has a unique global attractor a in v 2 h 2 h 0 1 in the topology of h 2 moreover a a 0 in the sense of hausdorff semidistance in h 0 1 as goes to 0

potential symmetries to systems of nonlinear diffusion equations in this paper the potential symmetry method is developed to study systems of nonlinear diffusion equations potential variables of the systems are introduced through conservation laws such conservation laws yield equivalent systems auxiliary systems of pdes with the given dependent and potential variables as new dependent variables lie point symmetries of the auxiliary systems which cannot be projected to the vector fields of the given dependent and independent variables yield potential symmetries of the systems classification for systems of nonlinear diffusion equations with two and three components is performed symmetry reductions associated with the potential symmetries are presented

upper semicontinuity of pullback attractors for nonclassical diffusion equations our aim in this paper is to study the long time behavior in terms of upper semicontinuous property of pullback attractors of nonclassical diffusion equations with nonautonomous perturbation specifically we prove that under some proper assumptions the pullback attractor ae t t r of eq 1 and the global attractor a of eq 1 with e 0 satisfy lime 0 distx ae t a 0 for any t r

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

pullback d attractors for non autonomous suspension bridge equations abstract in this paper we prove the existence of a pullback d attractor in v h for the non autonomous suspension bridge equations

blow up of the nonclassical diffusion equation in this paper we consider the blow up phenomenon of the nonclassical diffusion equation under suitable conditions on the initial data we prove a finite time blow up result

initial value problems for evolutionary partial differential equations and higher order conditional symmetries we suggest a new approach to the problem of dimensional reduction of initial boundary value problems for evolution equations in one spatial variable the approach is based on higher order generalized conditional symmetries of the equations involved it is shown that reducibility of an initial value problem for an evolution equation to a cauchy problem for a system of ordinary differential equations can be fully characterized in terms of conditional symmetries which leave invariant the equation in question we also give some examples of the solution of initial value problems for second and third order nonlinear differential equations by reduction by their conditional symmetries we give a systematic classification of general second order partial differential equations admitting second order conditional symmetries based on lie s classification of invariant second order ordinary differential equations this yields five classes of principally new initial value problems for nonlinear evolution equations which admit no lie symmetries and are reducible via second order conditional symmetries

pullback d attractors for nonautonomous sine gordon equations abstract in this paper we consider the pullback d attractor for the nonautonomous sine gordon wave equation we will show the existence of the pullback d attractor in v h and in d a v to this end some abstract results are established we use the methods provided by q f ma s h wang c k zhong necessary and sufficient conditions for the existence of global attractors for semigroups and applications j indiana univ math 51 6 2002 to establish some sufficient conditions for the existence of the pullback d attractor of an infinite dimensional dynamical system

on single stage dea models with weight restrictions the literature on data envelopment analysis dea often employs multiplier models that incorporate very small theoretically infinitesimal lower bounds on the input and output weights computational problems arising from the solution of such programs are well known in this paper we identify an additional theoretical problem that may arise if such bounds are used in a multiplier model with weight restrictions namely we show that the use of small lower bounds may lead to the identification of an efficient target with negative inputs we suggest a corrected model that overcomes this problem

computation of efficient targets in dea models with production trade offs and weight restrictions recently new models of data envelopment analysis dea were introduced that incorporate production trade offs between inputs and outputs or based on them weight restrictions in this paper we develop a computational procedure suitable for the practical application of such models we show that the standard two stage optimisation procedure used in dea to test the full efficiency of units and identify their efficient targets may work incorrectly in the new models the modified procedure consists of three stages the first evaluates the radial efficiency of the unit the second identifies its efficient target and the third its reference set of efficient peers each stage requires solving one linear program for each unit

potential symmetries and invariant solutions for the inhomogeneous nonlinear diffusion equation abstract the work reported here is a sequel to our paper physica a 304 2002 395 wherein we have obtained the invariant solutions of the generalized one dimensional fokker planck equation via the so called potential symmetry method herein the application of the same method has been carried over to the inhomogeneous nonlinear diffusion inld equation to obtain the determining equations in most general form which determine the one parameter transformation group to get the invariant solutions in explicit form we confine our attention to the physically interesting situations this study has led us to some new exact solutions that does not seem to have been reported in the literature

preliminary group classification of a class of fourth order evolution equations we perform preliminary group classification of a class of fourth order evolution equations in one spatial variable following the approach developed by basarab horwath et al acta appl math 69 43 2001 we construct all inequivalent partial differential equations belonging to the class in question which admit semisimple lie groups in addition we describe all fourth order evolution equations from the class under consideration which are invariant under solvable lie groups of dimension n 4 we have constructed all galilei invariant equations belonging to the class of evolution differential equations under study the list of so obtained invariant equations contains both the well known fourth order evolution equations and a variety of new ones possessing rich symmetry and as such may be used to model nonlinear processes in physics chemistry and biology

on the cahn hilliard equation in h 1 r n in this paper we exhibit the dissipative mechanism of the cahn hilliard equation in h 1 r n we show a weak form of dissipativity by showing that each individual solution is attracted in some sense by the set of equilibria we also indicate that strong dissipativity that is asymptotic compactness in h 1 r n cannot be in general expected then we consider two types of perturbations a nonlinear perturbation and a small linear perturbation in both cases we show that for the resulting equations the dissipative mechanism becomes strong enough to obtain the existence of a compact global attractor

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

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

thin films with high surface tension this paper is a review of work on thin fluid films where surface tension is a driving mechanism its aim is to highlight the substantial amount of literature dealing with relevant physical models and also analytic work on the resultant equations in general the introduction of surface tension into standard lubrication theory leads to a fourth order nonlinear parabolic equation pad h t pad x left c frac h 3 3 frac partial 3 h partial x 3 f h h x h xx right 0 label abeq1 where h h x t is the fluid film height for steady situations this equation may be integrated once and a third order ordinary differential equation is obtained appropriate forms of this equation have been used to model fluid flows in physical situations such as coating draining of foams and the movement of contact lenses in the introduction a form of the above equation is derived for flow driven by surface tension surface tension gradients gravity and long range molecular forces modifications to the equation due to slip the effect of two free surfaces two phase fluids and higher dimensional forms are also discussed the second section of this paper describes physical situations where surface tension driven lubrication models apply and the governing equations are given the third section reviews analytical work on the model equations as well as the generalized lubrication equation pad h t pad x left h n h xxx right 0 in particular the discussion focusses on asymptotic results travelling waves stability and similarity solutions numerical work is also discussed while for analytical results the reader is directed to existing literature

framework for potential systems and nonlocal symmetries algorithmic approach an algorithmic framework is presented to find an extended tree of nonlocally related systems for a given system of differential equations des each system in an extended tree is equivalent in the sense that the solution set for any system in a tree can be found from the solution set for any other system in the tree useful conservation laws play an essential role in the construction of an extended tree a useful conservation law yields potential variables and equivalent nonlocally related potential systems and subsystems for any given system nonlocal symmetries for a given system of des can arise from any system in its extended tree we construct extended trees for the systems of planar gas dynamics and nonlinear telegraph equations and in both cases obtain new nonlocal symmetries more importantly due to the equivalence of solution sets any coordinate independent method of analysis qualitative numerical perturbation etc can be applied to any system within the tree and may yield simpler computations and or results that cannot be obtained when the method is directly applied to the given system

group theoretical framework for potential symmetries of evolution equations we develop algebraic approach to the problem of classification of potential symmetries of nonlinear evolution equations it is essentially based on the recently discovered fact r zhdanov j math phys 50 053522 2009 that any such symmetry is mapped into a contact symmetry the approach enables using the classical results on classification of contact symmetries of nonlinear evolution equations by sokolov and magadeev to classify evolution equations admitting potential symmetries we construct several examples of new nonlinear fourth order evolution equations admitting potential symmetries since the symmetries obtained depend on nonlocal variables they cannot be derived by the infinitesimal lie approach

classification of potential symmetries of generalised inhomogeneous nonlinear diffusion equations we consider the class of generalised nonlinear diffusion equations f x ut g x unux x which are of considerable interest in mathematical physics we classify the nonlocal symmetries which are known as potential symmetries for these equations it turns out that potential symmetries exist only if the parameter n takes the values 2 or 23 also certain relations must be satisfied by the functions f x and g x for the cases where we obtain infinite parameter potential symmetries linearising mappings are constructed furthermore we employ the potential symmetries to derive similarity solutions

new classes of symmetries for partial differential equations new classes of symmetries for partial differential equations are introduced by writing a given partial differential equation s in a conserved form a related system t with potentials as additional dependent variables is obtained the lie group of point transformations admitted by t induces a symmetry group of s new symmetries may be obtained for s that are neither point nor lie backlund symmetries they are determined by a completely algorithmic procedure significant new symmetries are found for the wave equation with a variable wave speed and the nonlinear diffusion equation

symmetry groups and separation of variables of a class of nonlinear diffusion convection equations in this paper a class of nonlinear diffusion convection equations ut d u uxn x p u ux which has quite a large number of physical applications is analysed by using symmetry group methods which include the classical method the potential symmetry method and the generalized conditional symmetry method a complete classification of the functional forms of the diffusion and convection coefficients is presented when the equation admits lie s point symmetry groups and potential symmetry groups the separation of variables for the equation is investigated using the generalized conditional symmetry approach for some interesting cases exact solutions using the method of separation of variables are discussed in detail

linear second order and unconditionally energy stable schemes for the viscous cahn hilliard equation with hyperbolic relaxation using the invariant energy quadratization method abstract in this paper we consider numerical approximations for the viscous cahn hilliard equation with hyperbolic relaxation this type of equations processes energy dissipative structure the main challenge in solving such a diffusive system numerically is how to develop high order temporal discretization for the hyperbolic and nonlinear terms allowing large time marching step while preserving the energy stability i e the energy dissipative structure at the time discrete level we resolve this issue by developing two second order time marching schemes using the recently developed invariant energy quadratization approach where all nonlinear terms are discretized semi explicitly in each time step one only needs to solve a symmetric positive definite spd linear system all the proposed schemes are rigorously proven to be unconditionally energy stable and the second order convergence in time has been verified by time step refinement tests numerically various 2d and 3d numerical simulations are presented to demonstrate the stability accuracy and efficiency of the proposed schemes

passage to the limit over small parameters in the viscous cahn hilliard equation abstract we study singular passage to the limit over different small parameters for the viscous cahn hilliard equation under weak growth assumptions on the nonlinearity a rigorous proof of convergence to solutions of either the cahn hilliard equation or of the allen cahn equation or of the sobolev equation depending on the choice of the parameter is provided we also study the singular limit of the cahn hilliard equation as the parameter in the fourth order term goes to zero in particular we show that a radon measure valued solution of the limiting ill posed problem can arise depending on the behavior of the nonlinearity at infinity

analysis of the viscous cahn hilliard equation in r n abstract solvability of cauchy s problem in r n for an extended viscous cahn hilliard equation is studied the problem is considered first in a standard sobolev space h 1 r n next a notion of the h solution is introduced that is well adapted to the structure of the viscous cahn hilliard equation several properties of the solutions are reported in particular those connected with their asymptotic behavior we collect here also known properties of the unbounded operator 1 in r n that are needed in our considerations

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

on the cahn hilliard equation with dynamic boundary conditions and a dominating boundary potential the cahn hilliard and viscous cahn hilliard equations with singular and possibly nonsmooth potentials and dynamic boundary condition are considered and some well posedness and regularity results are proved

dynamics of the viscous cahn hilliard equation abstract we study generalized viscous cahn hilliard problems with nonlinearities satisfying critical growth conditions in w 0 1 p where is a bounded smooth domain in r n n 3 in the critical growth case we prove that the problems are locally well posed and obtain a bootstrapping procedure showing that the solutions are classical for p 2 and almost critical dissipative nonlinearities we prove global well posedness existence of global attractors in h 0 1 and uniformly with respect to the viscosity parameter l bounds for the attractors finally we obtain a result on continuity of regular attractors which shows that if n 3 4 the attractor of the cahn hilliard problem coincides in a sense to be specified with the attractor for the corresponding semilinear heat equation

local and nonlocal symmetries for nonlinear telegraph equation in this paper local and nonlocal symmetry classifications are considered for four equivalent nonlinear telegraph equations a complete potential symmetry classification of a scalar nonlinear telegraph equation is given through the point symmetry classification of a related potential system six new classes of equations are shown to admit potential symmetries the relationships between local including contact and nonlocal potential symmetries of these equations are explored a physical example is considered for possible applications of the obtained potential symmetries

attractors for the non autonomous nonclassical diffusion equations with fading memory abstract the long time dynamical behavior of the non autonomous nonclassical diffusion equation with fading memory when nonlinearity is critical is discussed for in the weak topological space h 0 1 l 2 r h 0 1 first the asymptotic regularity of solutions is proven and then the existence of a compact uniform attractor together with its structure and regularity is obtained while the time dependent forcing term is only translation bounded instead of translation compact the result extends and improves some results given in y xiao attractors for a nonclassical diffusion equation acta math appl sin engl ser 18 2002 273 276 c sun m yang dynamics of the nonclassical diffusion equations asympt anal 59 2008 51 81

conservation laws and hierarchies of potential symmetries for certain diffusion equations abstract we show that the so called hidden potential symmetries considered in a recent paper m l gandarias new potential symmetries for some evolution equations physica a 387 2008 2234 2242 are ordinary potential symmetries that can be obtained using the method introduced by bluman and collaborators g w bluman s kumei symmetries and differential equations springer new york 1989 g w bluman g j reid s kumei new classes of symmetries for partial differential equations j math phys 29 1988 806 811 in fact these are simplest potential symmetries associated with potential systems which are constructed with single conservation laws having no constant characteristics furthermore we classify the conservation laws for classes of porous medium equations and then using the corresponding conserved potential systems we search for potential symmetries this is the approach one needs to adopt in order to determine the complete list of potential symmetries the provenance of potential symmetries is explained for the porous medium equations by using potential equivalence transformations point and potential equivalence transformations are also applied to deriving new results on potential symmetries and corresponding invariant solutions from known ones in particular in this way the potential systems potential conservation laws and potential symmetries of linearizable equations from the classes of differential equations under consideration are exhaustively described infinite series of infinite dimensional algebras of potential symmetries are constructed for such equations

on relation between potential and contact symmetries of evolution equations we prove that any evolution equation that can be written in the form of a conservation law reduces to an evolution equation admitting the group of contact symmetries leaving the temporal variable invariant and vice versa one of the consequences of this fact is that any evolution equation admitting the potential symmetries can be transformed into another evolution equation so that the potential symmetries map into contact symmetries of the latter based on this fact is out group approach to classification of evolution equations possessing nonlocal symmetries we present several examples of classifications of second order evolution equations admitting potential symmetries

on some methods for performance ranking and correspondence analysis in the dea context two novel methods named performance baseline and performance correspondence matrices are proposed to evaluate the performance of decision making units dmus based on the techniques of singular value decomposition svd the performance baseline matrix can be used to rank all the dmus because it provides a common basis for performance comparison the performance correspondence matrix can be used to conduct performance cluster analysis with which to explore the structure of input output variables that are associated with dmus the analysis can reveal the performance difference of the dmus and the key input output variables determining the efficiency of a certain dmu and provides valuable quantitative information for adjusting variables to improve efficiency of the dmu three case studies are presented to demonstrate that the proposed methods in this work are effective and easy to use and can provide insights into proper selection of input output variables for performance comparison to avoid over manipulating dea models in practice

contact symmetries and solutions by reduction of partial differential equations the correspondence between a contact symmetry of a second order pde e and a point symmetry of the equivalent first order system s is used to determine a class of solutions for e which may not always be invariant under contact transformation we name these solutions pseudo invariant because they are determined from a point symmetry of s via a reduction method invariant solutions under the related contact transformation are included in the family of pseudo invariant solutions

on the convective cahn hilliard equation with degenerate mobility abstract in this paper we study the existence of weak solutions for the convective cahn hilliard equation with degenerate mobility based on the schauder type estimates we establish the global existence of classical solutions for regularized problems after establishing some necessary uniform estimates on the approximate solutions we prove the existence of weak solutions the nonnegativity and the finite speed of propagation of perturbations of solutions are also discussed

potential symmetries and invariant solutions for the generalized one dimensional fokker planck equation abstract pucci and saccomandi j phys a 26 1993 681 have obtained invariant solutions for the fokker planck fp equation via potential symmetries herein we analyze the generalized one dimensional fp equation for various cases corresponding to the physically interesting situations through the application of the same technique some interesting outcomes of this study are the deductions of the new exact solutions that do not seem to have been reported in the literature

potential symmetry generators and associated conservation laws of perturbed nonlinear equations some recent results on approximate lie group methods and previously developed concepts on potential symmetries are extended and applied to mainly nonlinear systems perturbed by a small parameter the potential or auxilliary form of the perturbed system necessarily requires a knowledge of an approximate conservation law of the system we then also use knowledge of approximate potential symmetries to calculate new approximate potential conservation laws