comparison and stability of solutions to a class of quasilinear parabolic problems on presente des resultats de comparaison et d unicite pour les solutions de problemes aux valeurs limites quasilineaires paraboliques avec et sans obstacles un resultat de stabilite dans l 1 donne la stabilisation asymptotique dans cet espace quand t vers le probleme elliptique correspondant

an infinite dimensional morse theory with applications in this paper we construct an infinite dimensional extraordinary cohomology theory and a morse theory corresponding to it these theories have some special properties which make them useful in the study of critical points of strongly indefinite functionals by strongly indefinite we mean a functional unbounded from below and from above on any subspace of finite codimension several applications are given to hamiltonian systems the onedimensional wave equation of vibrating string type and systems of elliptic partial differential equations 0 introduction let e be a real hilbert space with an inner product and let be a twice continuously differentiable functional denote the frechet derivative and the gradient of at x by x and x respectively where as usual x y x y y e recall that a point x0 e is said to be critical if x0 0 or equivalently if x0 0 the level c r will be called regular if 1 c contains no critical points and critical if x0 0 for some x0 1 c let a b a b be two regular levels of denote m 1 a b and consider the restriction of to m in morse theory one is interested in the local topological structure of the level sets of m near a critical point and in the relation between this local structure and the topological structure of the set m to be more specific suppose that x0 m is an isolated critical point of then one defines a sequence of critical groups of at x0 by setting cq x0 hq c u u x0 q 0 1 2 0 1 where c x0 c x e x c hq is the q th singular homology group with coefficients in some field f and u is a neighbourhood of x0 define the morse index of x0 to be the maximal dimension of a subspace of e on which the quadratic form x0 y y is negative definite one shows that if x0 is a nondegenerate critical point i e if x0 e e is invertible then cq x0 f received by the editors march 20 1995 1991 mathematics subject classification primary 58e05 secondary 34c25 35j65 35l05 55n20 58f05

forced vibrations of superquadratic hamiltonian systems on etudie l existence des solutions t periodiques ter t 0 du systeme hamiltonien suivant z gh z f t ou g est la matrice asymetrique standard

free vibrations for a semilinear wave equation abstract existence and regularity of nontrivial time periodic solutions are proved for semilinear wave equations under mild smoothness monotonicity and superlinearity assumptions the forced vibration case is also treated

periodic solutions of large norm of hamiltonian systems abstract hamiltonian systems of ordinary differential equations are studied the only assumption made on the hamiltonian is appropriately rapid growth at infinity it is proved that for any given period there is an unbounded sequence of periodic solutions of the system having the given period

on some nonlinear elliptic equations involving derivatives of the nonlinearity the existence of weak solutions of the dirichlet problem for nonlinear elliptic equations of the type x j a ij x u x i b j u a x u f is proved for nonlinearities b j which are only continuous the proof is based on the observation that an l a priori estimate is valid which is independent of b j the second part contains several uniqueness results and counterexamples showing that uniqueness may fail in general

implicit spectral methods for wave propagation problems abstract the numerical solution of a non linear wave equation can be obtained by using spectral methods to resolve the unknown in space and the standard crank nicolson differencing scheme to advance the solution in time we have analyzed iterative techniques for solving the non linear equations that arise from such implicit time stepping schemes for the k dv and the k p equations we derived predictor corrector method that retain the full accuracy of the implicit method with minimal stability restrictions on the size of the time step some numerical examples show the propagation of interacting solitons

quasivariational solutions for first order quasilinear equations with gradient constraint we prove the existence of solutions for a quasi variational inequality of evolution with a first order quasilinear operator and a variable convex set which is characterized by a constraint on the absolute value of the gradient that depends on the solution itself the only required assumption on the nonlinearity of this constraint is its continuity and positivity the method relies on an appropriate parabolic regularization and suitable a priori estimates we also obtain the existence of stationary solutions by studying the asymptotic behaviour in time in the variational case corresponding to a constraint independent of the solution we also give uniqueness results

large amplitude time periodic solutions of a semilinear wave equation abstract this paper studies the existence of periodic solutions for a family of semilinear wave equations where the restoring force is independent of time monotone and grows at a more rapid rate than linear near infinity with appropriate technical assumptions it is shown that there is an unbounded sequence of such free vibrations i e there are solutions of arbitrarily large amplitude if the restoring force is independent of x the monotonicity assumption can be omitted

sub super solution method for a class of higher order evolution hemivariational inequalities abstract in this paper we extend the extremality results for the variational inequality to a higher order evolution hemivariational inequality more precisely we give an existence theorem of solution for the higher order evolution hemivariational inequality by using the sub super solution method we prove the compactness of the solution set within an order interval formed by the sub solution and super solution we also show an existence theorem of the extremal solution for the higher order evolution hemivariational inequality under consideration

on some sharp spectral inequalities for schrodinger operators on semiaxis in this paper we obtain sharp lieb thirring inequalities for a schrodinger operator on semiaxis with a matrix potential and show how they can be used to other related problems among them are spectral inequalities on star graphs and spectral inequalities for schrodinger operators on half spaces with robin boundary conditions

instabilities and oscillations of one and two dimensional kadomtsev petviashvili waves and solitons ii linear to nonlinear analysis we further investigate the dynamics of nonlinear structures that arise from the kadomtsev petviashvili equation when the analysis is linear assuming perturbations grow exponentially in time we find the growth rates of two important instabilities numerically these are the only purely growing modes the wavelength doubling instability is seen to dominate its rival that of benjamin and feir at least when the amplitude of the wave is not too large approximate formulae found to higher order than in part i referenced in 1 are checked against the numerically found values the models are seen to be better than expected for the dominant wavelength doubling instability our model extends beyond the assumed region of validity it is surprisingly close almost up to the soliton limit when we depart from linear stability analysis and include terms nonlinear in the perturbation a simple analysis shows that the linear instability eventually drives a doubly space periodic hyperbolic secans pulse in time after a long time initial conditions are reproduced a proof that the maximum amplitude achieved by the perturbation is approximately proportional to the linear growth rate is given within the limitations of the calculation this fact was suspected from numerics a second class of possible dynamic behaviour not arising from initially linear growth of a perturbation is found this class involves fully two dimensional stationary solutions and their possible oscillations

topological results on a certain class of functionals and application abstract we study in this paper problems of the type u u p 1 u x bounded r n u 0 i where x is given and where p 1 n 2 n 2 p 1 if n 2 our main result is that i has an infinite number of solutions for a residual set of in h 1 in particular for many n n there exists an open and dense subset of in h 1 such that i has n distinct solutions for such an this result is to be related to the conjecture developed in 1 of the existence of an infinite number of solutions to i the proof relies on a general characterization of level sets for a certain class of functionals when there are no critical value in a large enough interval in addition to the study of problem i we apply this characterization to give another proof using e g brouwer s fixed point theorem for some classical results about even functionals and saddle points

liouville theorems for stable solutions of semilinear elliptic equations with convex nonlinearities abstract we extend our recent results on the classification of stable solutions of the equation u f u in r n where f 0 is a general convex non decreasing function

periodic solutions of symmetric wave equations this paper concerns the existence of solutions to the following nonautonomous wave equation gu uttu f 2 f x t is nondecreasing in and f x t 0 if and only if 0 our first result deals with the case where f is superlinear at theorem 1 let f 1 and f 2 be satisfied assume moreover that f satisfies f 3 there are r 0 and 2 such that

cylindrical quasi solitons of the zakharov kuznetsov equation abstract evolutions and interactions of two dimensional solitary waves of the zakharov kuznetsov equation are investigated numerically formations of cylindrical bell shaped pulses are observed in the initial value problems a single bell shaped pulse propagates stably without any deformation like a soliton two similar pulses exchange their amplitudes without merging and two dissimilar ones undergo overtaking collision after a collision of two pulses the strong pulse becomes somewhat stronger and the weak one becomes weaker with radiation of ripples so that the collision process is slightly inelastic generated ripples are very small for center to center collision of similar pulses the cylindrically symmetric pulses of the zakharov kuznetov equation are thus found to behave approximately soliton like some properties of collision process are interpreted in terms of the conservation laws

petrov galerkin methods for nonlinear dispersive waves abstract petrov galerkin methods based on piecewise linear interpolants for the korteweg de vries and related equations are studied both accuracy and stability are analysed for the linearised case it is shown that in the nonlinear case the order of accuracy of the standard galerkin procedure is reduced and an alternative technique is therefore proposed which retains the fourth order accuracy in space this is found to perform well when compared with finite difference or other finite element schemes

the multi symplectic fourier pseudospectral method for solving two dimensional hamiltonian pdes in this paper the multi symplectic fourier pseudospectral msfp method is generalized to solve two dimensional hamiltonian pdes with periodic boundary conditions using the fourier pseudospectral method in the space of the two dimensional hamiltonian pde 2d hpde the semi discrete system obtained is proved to have semi discrete multi symplectic conservation laws and a global symplecticity conservation law then the implicit midpoint rule is employed for time integration to obtain the msfp method for the 2d hpde the fully discrete multi symplectic conservation laws are also obtained in addition the proposed method is applied to solve the zakharov kuznetsov zk equation and the kadomtsev petviashvili kp equation numerical experiments on soliton solutions of the zk equation and the kp equation show the high accuracy and effectiveness of the proposed method

periodic solutions of a dirac equation with concave and convex nonlinearities abstract this paper is concerned with multiplicity of periodic solutions of a dirac equation with concave and convex nonlinearities we establish a variational setting and obtain two sequences of periodic solutions with large and small energies respectively for the problem by variational method

free and forced vibrations of nonlinear wave equations in ball first we shall deal with the free vibrations of a nonlinear radially symmetric wave equation t2 u f r u in n dimensional ball ba with center at the origin and radius a where f is smooth monotone decreasing in u and satisfies f r 0 0 f r u has asymptotic properties f r u o u3 u 0andu for n 1 3 we shall show the existence of infinitely many radially symmetric time periodic solutions with different periods of irrational multiple of a second we shall deal with bvp for a forced nonlinear wave equation t2 u eg r t u where g is t periodic in t and e is a small parameter under some diophantine condition on a t we shall show the existence of time periodic solutions of the bvp for 1 n 5 we shall construct infinitely many t satisfying the above diophantine inequality using asymptotic expansions of the zero points of the bessel functions

borsuk ulam theorems for arbitrary s1 actions and applications abstract an s to the first power version of the borsuk ulam theorem is proved for a situation where fix s to the first power may be nontrivial the proof is accomplished with the aid of a new relative index theory applications are given to intersection theorems and the existence of multiple critica points is established for a class of functionals invariant under an s to the first power symmetry minimax arguments from the calculus of variations serve as an important tool in establishing the existence of nonlinear vibrations of discrete mechanical systems as modelled by hamilton s equations in these arguments one obtains the solutions of the differential equations as critical points of an associated lagrangian by minimaxing the lagrangian over appropriate classes of sets intersection theorems such as are proved in this paper play a crucial role in this process in addition to obtaining some intersection theorems this report illustrates their use by proving an existence theorem for multiple critical points of the functional invarianet under an s to the first power symmetry group

on the schrodinger equation and the eigenvalue problem if k is thekth eigenvalue for the dirichlet boundary problem on a bounded domain in n h weyl s asymptotic formula asserts that lambda k sim c n left frac k v d right 2 n hence sum limits i 1 k lambda i sim frac nc n n 2 k frac n 2 n v d 2 n we prove that for any domain and for all sum limits i 1 k lambda i geqq frac nc n n 2 k frac n 2 n v d 2 n a simple proof for the upper bound of the number of eigenvalues less than or equal to for the operator v x defined on n n 3 in terms of int limits mathbb r n v alpha n 2 is also provided

existence of forced oscillations for some nonlinear differential equations abstract this article studies the existence of t periodic solutions for systems of nonlinear second order ordinary differential equations assuming function v to be superquadratic it is shown that this system possesses infinitely many t periodic solutions the proof of this result rests on showing that certain homotopy groups of level sets of the functional associated with the system are not trivial some more general results concerning systems of another type are also presented

sub supersolution method for quasilinear parabolic variational inequalities this paper is about a systematic attempt to apply the sub supersolution method to parabolic variational inequalities we define appropriate concepts of sub supersolutions and derive existence comparison and extremity results for such inequalities

a conservative spectral method for several two dimensional nonlinear wave equations a conservative spectral method is proposed to solve several two dimensional nonlinear wave equations the conventional fast fourier transform is used to approximate the spatial derivatives and a three level difference scheme with a free parameter is to advance the solution in time our time discretization is semi implicit in the sense that the linear terms are treated implicitly while the nonlinear terms are evaluated only by previous time levels and thus treated explicitly however the cost of the algorithm is no greater than that of a fully explicit method because the linear boundary value problem that must be solved at each time step is almost trivial in a spectral spatial discretization a linear stability analysis shows that the method leads to a less restrictive stability condition than the corresponding explicit one the method is conservative and the ratio of the numerical dispersion to the physical dispersion is of the order o t2 applications of our method to the kadomtsev petviashvili and the zakharov kuznetsov equations exhibit excellent results

local discontinuous galerkin methods for two classes of two dimensional nonlinear wave equations in this paper we develop analyze and test local discontinuous galerkin dg methods to solve two classes of two dimensional nonlinear wave equations formulated by the kadomtsev petviashvili kp equation and the zakharov kuznetsov zk equation our proposed scheme for the kadomtsev petviashvili equation satisfies the constraint from the pde which contains a non local operator and at the same time has the local property of the discontinuous galerkin methods the scheme for the zakharov kuznetsov equation extends the previous work on local discontinuous galerkin method solving one dimensional nonlinear wave equations to the two dimensional case l2 stability of the schemes is proved for both of these two nonlinear equations numerical examples are shown to illustrate the accuracy and capability of the methods

cylindrical solitary pulses in a two dimensional stabilized kuramoto sivashinsky system abstract by linearly coupling a generalized zakharov kuznetsov equation alias the two dimensional 2d benney equation to an extra linear dissipative equation a 2d extension of a recently proposed stabilized kuramoto sivashinsky system is developed the model applies to the description of surface waves on 2d liquid layers in various physical settings the extra equation provides for the stability of the zero state in the system thus paving a way to the existence of stable 2d localized solitary pulses sps a perturbation theory based on a family of cylindrical solitons existing in the conservative counterpart of the system is developed by treating dissipation and gain in the model as small perturbations it is shown that the system may select two steady state solitons from the continuous family provided by the conservative counterpart of which the one with larger amplitude is expected to be stable numerical simulations support the analytical predictions quite well additionally it is found that a shallow quasi one dimensional 1d trough is attached to the stable sp if the integration domain is not very large and an explanation to this feature is proposed stable double humped bound states of two pulses are found too

extremal solutions of quasilinear parabolic inclusions with generalized clarke s gradient abstract in this paper we consider an initial boundary value problem for a parabolic inclusion whose multivalued nonlinearity is characterized by clarke s generalized gradient of some locally lipschitz function and whose elliptic operator may be a general quasilinear operator of leray lions type recently extremality results have been obtained in case that the governing multivalued term is of special structure such as multifunctions given by the usual subdifferential of convex functions or subgradients of so called dc functions the main goal of this paper is to prove the existence of extremal solutions within a sector of appropriately defined upper and lower solutions for quasilinear parabolic inclusions with general clarke s gradient the main tools used in the proof are abstract results on nonlinear evolution equations regularization comparison truncation and special test function techniques as well as tools from nonsmooth analysis

a finite difference method for the korteweg de vries and the kadomtsev petviashvili equations a linearized implicit finite difference method for the korteweg de vries equation is proposed and straightforwardly extended to the kadomtsev petviashvili equation we investigate the order of accuracy of the method and prove the method to be unconditionally linearly stable the numerical experiments for the korteweg de vries and the kadomtsev petviashvili equations are carried out with various conditions numerical results for the collision of two lump type solitary wave solutions to the kadomtsev petviashvili equation are also reported

sums of laplace eigenvalues rotationally symmetric maximizers in the plane abstract the sum of the first n 1 eigenvalues of the laplacian is shown to be maximal among triangles for the equilateral triangle maximal among parallelograms for the square and maximal among ellipses for the disk provided the ratio area 3 moment of inertia for the domain is fixed this result holds for both dirichlet and neumann eigenvalues and similar conclusions are derived for robin boundary conditions and schrodinger eigenvalues of potentials that grow at infinity a key ingredient in the method is the tight frame property of the roots of unity for general convex plane domains the disk is conjectured to maximize sums of neumann eigenvalues

analytical and numerical aspects of certain nonlinear evolution equations 1v numerical modified korteweg de vries equation various numerical methods are used in order to approximate the korteweg de vries equation namely i zabusky kruskal scheme ii hopscotch method iii a scheme due to goda iv a proposed local scheme v a proposed global scheme vi a scheme suggested by kruskal vii split step fourier method by tappert viii an improved split step fourier method and ix pseudospectral method by fornberg and whitham comparisons between our proposed scheme which is developed using notions of the inverse scattering transform and the other utilized schemes are obtained

the rabinowitz floer homology for a class of semilinear problems and applications abstract in this paper we construct a rabinowitz floer type homology for a class of non linear problems having a starshaped potential we consider some equivariant cases as well we give an explicit computation of the homology and we apply it to obtain results of existence and multiplicity of solutions for several model equations