stefan problems with nonlinear diffusion and convection abstract we consider a class of stefan type problems having a convection term and a pseudomonotone nonlinear diffusion operator assuming data in l 1 we prove existence uniqueness and stability in the framework of renormalized solutions existence is established from compactness and monotonicity arguments which yield stability of solutions with respect to l 1 convergence of the data uniqueness is proved through a classical l 1 contraction principle obtained by a refinement of the doubling variable technique which allows us to extend previous results to a more general class of nonlinear possibly degenerate operators

on the spaces lp x ω and wm p x ω in this paper we present some basic results on the generalized lebesgue spaces l p x omega and generalized lebesgue sobolev spaces w m w p x omega these results provide the necessary framework for the study of variational problems and elliptic equations with non standard p x growth conditions c 2001 academic press

variational properties and orbital stability of standing waves for nls equation on a star graph we study standing waves for a model of nonlinear schr odinger equation on a graph the graph is obtained joining n halflines at a vertex i e it is a star graph at the vertex an interaction occurs due to a boundary condition of delta type with strength alpha leqslant 0 which includes the free or kirchhoff case alpha 0 there imposed the nonlinearity is of focusing power type the dynamics is given by an equation of the form i frac d dt psi t h psi t psi t 2 mu psi t where h is the s a operator which generates the linear schr odinger dynamics on the graph in the hilbert space l 2 gg we show the existence of several families of standing waves solutions of the form phi t e i omega t psi omega where psi omega is a vector amplitude parametrized by a frequency omega they exist for every sign of the coupling at the vertex attractive alpha 0 and for every omega frac alpha 2 n 2 the number of the families depends on the number n of the edges of the graph we moreover determine the ground state which is the standing wave the amplitude of which minimizes the action on the natural nehari constraint and order the various families of standing waves according to their increasing action thus determining a nonlinear spectrum of the problem finally we show that the ground state is orbitally stable for every allowed omega if the nonlinearity is subcritical or critical for supercritical nonlinearities there exists an omega frac alpha 2 n 2 such that the ground state is orbitally stable for frac alpha 2 n 2 omega omega action minimization requires technical subtleties in particular an adaptation of decreasing rearrangements on a star graph is developed with a generalization of the p olya szeg h o inequality

the h text p version of the finite element method for domains with curved boundaries this paper analyzes the convergence of the h text p version of the finite element method for elliptic problems on curved domains it proves that when the boundary of the domain is piecewise analytic boundary conditions are piecewise analytic and the sides of the elements are analytic the h text p version has an exponential rate of convergence numerical computations are included the paper generalizes the results previously obtained by the authors the h text p version of finite element method part 1 the basic approximation results comp mech 1 1986 pp 21 41 the h text p version of finite element method part 2 general results and applications comp mech 1 1986 pp 203 220

calderon zygmund estimates in generalized orlicz spaces abstract we establish the w 2 solvability of the linear elliptic equations in non divergence form under a suitable essentially minimal condition of the generalized orlicz function x t by deriving calderon zygmund type estimates the class of generalized orlicz spaces we consider here contains as special cases classical lebesgue and orlicz spaces as well as non standard growth cases like variable exponent and double phase growth

renormalised solutions of nonlinear parabolic problems with l1 data existence and uniqueness in this paper we prove the existence and uniqueness of a renormalised solution of the nonlinear problem where the data f and u 0 belong to l 1 0 t and l 1 and where the function a 0 t n n is monotone but not necessarily strictly monotone and defines a bounded coercive continuous operator from the space into its dual space the renormalised solution is an element of c 0 0 t l 1 such that its truncates t k u belong to with this solution satisfies the equation formally obtained by using in the equation the test function s u where belongs to and where s belongs to c with

on the existence of weak solutions for quasilinear parabolic initial boundary value problems structure conditions on a strongly nonlinear operator a u are given under whichthe initial dirichlet boundary value problem for has weak solutions

a pocket guide to nonlinear differential equations in musielak orlicz spaces abstract the musielak orlicz setting unifies variable exponent orlicz weighted sobolev and double phase spaces they inherit technical difficulties resulting from general growth and inhomogeneity in this survey we present an overview of developments of the theory of existence of pdes in the setting including reflexive and non reflexive cases as well as isotropic and anisotropic ones particular attention is paid to problems with data below natural duality in absence of lavrentiev s phenomenon

renormalized solutions for nonlinear elliptic equations with variable exponents and l1 data abstract we prove existence and uniqueness of a renormalized solution to nonlinear elliptic equations with variable exponents and l 1 data the functional setting involves lebesgue sobolev space with variable exponents w 1 p

renormalized and entropy solutions for nonlinear parabolic equations with variable exponents and l1 data abstract in this paper we prove the existence and uniqueness of both renormalized solutions and entropy solutions for nonlinear parabolic equations with variable exponents and l 1 data and moreover we obtain the equivalence of renormalized solutions and entropy solutions

l1 contraction and uniqueness for quasilinear elliptic parabolic equations abstract we prove the l 1 contraction principle and uniqueness of solutions for quasilinear elliptic parabolic equations of the form formula where b is monotone nondecreasing and continuous we assume only that u is a weak solution of finite energy in particular we do not suppose that the distributional derivative t b u is a bounded borel measure or a locally integrable function

solitary wave dynamics in an external potential we study the behavior of solitary wave solutions of some generalized nonlinear schrodinger equations with an external potential the equations have the feature that in the absence of the external potential they have solutions describing inertial motions of stable solitary waves we consider solutions of the equations with a non vanishing external potential corresponding to initial conditions close to one of these solitary wave solutions and show that over a large interval of time they describe a solitary wave whose center of mass motion is a solution of newton s equations of motion for a point particle in the given external potential up to small corrections corresponding to radiation damping

regularity for double phase variational problems we prove sharp regularity theorems for minimisers of a class of variational integrals whose integrand switches between two different types of degenerate elliptic phases according to the zero set of a modulating coefficient a cdot the model case is given by the functional w mapsto int dw p a x dw q rm d x where q p and a cdot geqq 0

global existence of renormalized solutions to entropy dissipating reaction diffusion systems in the present work we introduce the notion of a renormalized solution for reaction diffusion systems with entropy dissipating reactions we establish the global existence of renormalized solutions in the case of integrable reaction terms our notion of a renormalized solution reduces to the usual notion of a weak solution our existence result in particular covers all reaction diffusion systems involving a single reversible reaction with mass action kinetics and possibly species dependent fick law diffusion more generally it covers the case of systems of reversible reactions with mass action kinetics which satisfy the detailed balance condition for such equations the existence of any kind of solution in general was an open problem thereby motivating the study of renormalized solutions

steady flow of non newtonian fluids monotonicity methods in generalized orlicz spaces abstract our purpose is to show the existence of weak solutions to steady flow of non newtonian incompressible fluids with nonstandard growth conditions of the cauchy stress tensor we are motivated by the fluids of strongly inhomogeneous behavior and characterized by rapid shear thickening since l p framework is not sufficient to capture the described model we describe the growth conditions with the help of a general x dependent convex function and formulate our problem in generalized orlicz spaces

nonlinear parabolic equations with measure data in this paper we give summability results for the gradients of solutions of nonlinear parabolic equations whose model isu div u p 2 u on 0 t p with homogeneous cauchy dirichlet boundary conditions wherep 1 and is a bounded measure on 0 t we also study how the summability of the gradient improves if the measure is a function inlm 0 t withm small moreover we give a new proof of the existence of a solution for problem p

equivalence groupoids and group classification of multidimensional nonlinear schrodinger equations abstract we study admissible and equivalence point transformations between generalized multidimensional nonlinear schrodinger equations and classify lie symmetries of such equations we begin with a wide superclass of schrodinger type equations which includes all the other classes considered in the paper showing that this superclass is not normalized we partition it into two disjoint normalized subclasses which are not related by point transformations further constraining the arbitrary elements of the superclass we construct a hierarchy of normalized classes of schrodinger type equations this gives us an appropriate normalized superclass for the non normalized class of multidimensional nonlinear schrodinger equations with potentials and modular nonlinearities and allows us to partition the latter class into three families of normalized subclasses after a preliminary study of lie symmetries of nonlinear schrodinger equations with potentials and modular nonlinearities for an arbitrary space dimension we exhaustively solve the group classification problem for such equations in space dimension two

well posedness of parabolic equations in the non reflexive and anisotropic musielak orlicz spaces in the class of renormalized solutions abstract we prove existence and uniqueness of renormalized solutions to general nonlinear parabolic equation in musielak orlicz space avoiding growth restrictions namely we consider t u div a x u f l 1 t on a lipschitz bounded domain in r n the growth of the weakly monotone vector field a is controlled by a generalized nonhomogeneous and anisotropic n function m the approach does not require any particular type of growth condition of m or its conjugate m neither 2 nor 2 the condition we impose on m is continuity of log holder type which results in good approximation properties of the space however the requirement of regularity can be skipped in the case of reflexive spaces the proof of the main results uses truncation ideas the young measures methods and monotonicity arguments uniqueness results from the comparison principle

asymptotic stability of ground states in some hamiltonian pdes with symmetry we consider a ground state soliton of a hamiltonian pde we prove that if the soliton is orbitally stable then it is also asymptotically stable the main assumptions are transversal nondegeneracy of the manifold of the ground states linear dispersion in the form of strichartz estimates and nonlinear fermi golden rule we allow the linearization of the equation at the soliton to have an arbitrary number of eigenvalues the theory is tailor made for the application to the translational invariant nls in space dimension 3 the proof is based on the extension of some tools of the theory of hamiltonian systems reduction theory darboux theorem normal form to the case of systems invariant under a symmetry group with unbounded generators

non linear elliptic and parabolic equations involving measure data in this paper we prove the existence of solutions for equations of the type div a du f in a bounded open set u 0 on where a is a possibly non linear function satisfying some coerciveness and monotonicity assumptions and f is a bounded measure we also consider the equation div a du g u f in u 0 on with f l1 or f m u u 0 and the parabolic equivalent of the first elliptic equation

existence and regularity of renormalized solutions for some elliptic problems involving derivatives of nonlinear terms the authors study the nonlinear elliptic equation div a x u du div u g x u f x in with the boundary condition u 0 on where is a bounded open subset of rn a u div a x u du is a nonlinear operator of leray lions type from w1 p0 into its dual c0 r n g x t t 0 and f w 1 p no growth hypothesis is assumed on the vector valued function u the term div u may be meaningless for usual weak solutions even as a distribution the authors deal with a weaker form of solutions of which in w1 p0 are called the renormalized solutions of the original problem this weaker form can be formally obtained from by means of pointwise multiplication by h u where h is a c1 function with compact support they prove the existence of renormalized solutions of and give sufficient conditions under which a renormalized solution is a usual weak solution furthermore the authors study ls and l regularity of renormalized solutions

generalized orlicz spaces and related pde abstract we prove the boundedness of the maximal operator in generalized orlicz spaces defined on subsets of r n the proof is based on an extension result for functions we study generalized sobolev orlicz spaces and establish density of smooth functions and the poincare inequality as applications we establish the existence of solutions of the laplace equation with zero and non zero right hand side further we systematize assumptions for functions and prove several basic tools needed for the study of differential equations of generalized orlicz growth

existence and uniqueness of renormalized solutions to nonlinear elliptic equations with variable exponents and l1 data abstract we prove the existence and uniqueness of a renormalized solution to nonlinear elliptic equations with variable exponents and l 1 data the functional setting involves sobolev spaces with variable exponents w 1 p

nonlinear parabolic problems in musielak orlicz spaces abstract our studies are directed to the existence of weak solutions to a parabolic problem containing a multi valued term the problem is formulated in the language of maximal monotone graphs we assume that the growth and coercivity conditions of a nonlinear term are prescribed by means of a time and space dependent n function this results in the formulation of the problem in generalized musielak orlicz spaces we are using density arguments hence an important step of the proof is a uniform boundedness of appropriate convolution operators in musielak orlicz spaces for this purpose we shall need to assume a kind of logarithmic holder regularity with respect to t and x

variable exponent linear growth functionals in image restoration we study a functional with variable exponent 1 leq p x leq 2 which provides a model for image denoising enhancement and restoration the diffusion resulting from the proposed model is a combination of total variation tv based regularization and gaussian smoothing the existence uniqueness and long time behavior of the proposed model are established experimental results illustrate the effectiveness of the model in image restoration

on the dynamics of solitons in the nonlinear schrodinger equation we study the behavior of the soliton solutions of the equation i frac partial psi partial t frac 1 2m delta psi frac 1 2 w varepsilon prime psi v x psi where w varepsilon prime is a suitable nonlinear term which is singular for varepsilon 0 we use the strong nonlinearity to obtain results on existence shape stability and dynamics of the soliton the main result of this paper theorem 1 shows that for varepsilon rightarrow 0 the orbit of our soliton approaches the orbit of a classical particle in a potential v x

renormalized solutions of nonlinear elliptic problems in generalized orlicz spaces abstract we study a general class of nonlinear elliptic problems associated with the differential inclusion x u div a x u f u f where f l 1 the vector field a is monotone in the second variable and satisfies a non standard growth condition described by an x dependent convex function that generalizes both l p x and classical orlicz settings using truncation techniques and a generalized minty method in the functional setting of non reflexive spaces we prove existence of renormalized solutions for general l 1 data under an additional strict monotonicity assumption uniqueness of the renormalized solution is established sufficient conditions are specified which guarantee that the renormalized solution is already a weak solution to the problem

classification and geometrical properties of the xθ valued function spaces abstract this paper is devoted to investigate the x valued function spaces based on the notions of bounded topological lattice banach space net continuous modular net and the dual space net we divide x valued function spaces into two classes norm modular spaces and modular modular spaces for the first class we study the separability of l p i x give a representation of the dual space l p i x find sufficient conditions of the equality l p i x l p i x and prove the reflexivity of l p i x under some reasonable conditions and for the second class we prove the completeness and uniform convexity of l i x in suitable situations to show the naturality and rationality of these results some concrete function spaces such as l p t i l p x t l p t i w 1 p x t and l p t i w 0 1 p x t together with l p p t i l p x t are taken into account

perturbation expansion and nth order fermi golden rule of the nonlinear schrodinger equations in this paper we consider generalized nonlinear schrodinger equations with external potentials we find the expressions for the fourth and the sixth order fermi golden rules fgrs conjectured in gang and sigal rev math phys 17 1143 1207 2005 geom funct anal 16 no 7 1377 1390 2006 the fgr is a key condition in a study of the asymptotic dynamics of trapped solitons

multichannel nonlinear scattering for nonintegrable equations ii the case of anisotropic potentials and data abstract the nonlinear scattering and stability results of soffer and weinstein comm math phys 1990 are extended to the case of anisotropic potentials and data the range of nonlinearities for which the theory is shown to be valid is also extended considerably

relaxation of solitons in nonlinear schrodinger equations with potential abstract in this paper we study dynamics of solitons in the generalized nonlinear schrodinger equation nls with an external potential in all dimensions except for 2 for a certain class of nonlinearities such an equation has solutions which are periodic in time and exponentially decaying in space centered near different critical points of the potential we call those solutions which are centered near the minima of the potential and which minimize energy restricted to l 2 unit sphere trapped solitons or just solitons in this paper we prove under certain conditions on the potentials and initial conditions that trapped solitons are asymptotically stable moreover if an initial condition is close to a trapped soliton then the solution looks like a moving soliton relaxing to its equilibrium position the dynamical law of motion of the soliton i e effective equations of motion for the soliton s center and momentum is close to newton s equation but with a dissipative term due to radiation of the energy to infinity

existence of renormalized solutions to elliptic equation in musielak orlicz space abstract we prove existence of renormalized solutions to general nonlinear elliptic equation in musielak orlicz space avoiding growth restrictions namely we consider div a x u f l 1 on a lipschitz bounded domain in r n the growth of the monotone vector field a is controlled by a generalized nonhomogeneous and anisotropic n function m the approach does not require any particular type of growth condition of m or its conjugate m neither 2 nor 2 the condition we impose is log holder continuity of m which results in good approximation properties of the space the proof of the main results uses truncation ideas the young measures methods and monotonicity arguments

spectral element methods for parabolic problems a spectral element method for solving parabolic initial boundary value problems on smooth domains using parallel computers is presented in this paper the space domain is divided into a number of shape regular quadrilaterals of size h and the time step k is proportional to h 2 at each time step we minimize a functional which is the sum of the squares of the residuals in the partial differential equation initial condition and boundary condition in different sobolev norms and a term which measures the jump in the function and its derivatives across inter element boundaries in certain sobolev norms the sobolev spaces used are of different orders in space and time we can define a preconditioner for the minimization problem which allows the problem to decouple error estimates are obtained for both the h and p versions of this method

sobolev embedding theorems for spaces wk p x ω this paper gives a sobolev type embedding theorem for the generalized lebesgue sobolev space wk p x where is an open domain in rn n 2 with cone property and p x is a lipschitz continuous function defined on satisfying 1 p p nk the main result can be stated as follows for any measurable function q x x formula with p x q x p x np x n kp x there exists a continuous embedding from wk p x to lq x

existence of renormalized solutions of degenerate elliptic parabolic problems we consider a general class of degenerate elliptic parabolic problems associated with the equation b t div a d f existence of renormalized solutions is established for general l 1 data uniqueness of renormalized solutions has already been shown in a previous work

stability of the solitary manifold of the perturbed sine gordon equation abstract we study the perturbed sine gordon equation t t x x sin f e x where f is of differentiability class c n in e and the first k derivatives vanish at e 0 i e e l f 0 0 for 0 l k we construct implicitly a virtual solitary manifold by deformation of the classical solitary manifold in n iteration steps our main result establishes that the initial value problem with an appropriate initial state e n close to the virtual solitary manifold has a unique solution which follows up to time 1 c e k 1 2 and errors of order e n a trajectory on the virtual solitary manifold the trajectory on the virtual solitary manifold is described by two parameters which satisfy a system of odes in contrast to previous works our stability result yields arbitrarily high accuracy as long as the perturbation f is sufficiently often differentiable

renormalized solution for a nonlinear parabolic problems with noncoercivity in divergence form in orlicz spaces we prove the existence result in the setting of orlicz sobolev spaces the n function m does not satisfy the 2 condition existence of a renormalized solution for a class of nonlinear parabolic equations the data belongs to l 1 and no continuous assumption is made on divergence form in this paper we prove the existence results for renormalized solution of the following nonlinear parabolic problems in orlicz sobolev spaces 1 p b x u t div a x t u u div x t u f in q t 0 t b u t 0 b u 0 in u 0 on 0 t where b x u is unbounded function of u the term div a x t u u is a leray lions operator defined on a subset of w 0 1 x l m q t where m is a n function without assuming a 2 condition the second term f l 1 q t and is a noncoercive function which satisfies only the growth condition

existence and stability of ground states for fully discrete approximations of the nonlinear schrodinger equation in this paper we study the long time behavior of a discrete approximation in time and space of the cubic nonlinear schrodinger equation on the real line more precisely we consider a symplectic time splitting integrator applied to a discrete nonlinear schrodinger equation with additional dirichlet boundary conditions on a large interval we give conditions ensuring the existence of a numerical ground state which is close in energy norm to the continuous ground state such result is valid under a cfl condition of the form tau h 2 le c where tau and h denote the time and space step size respectively furthermore we prove that if the initial datum is symmetric and close to the continuous ground state eta then the associated numerical solution remains close to the orbit of eta gamma cup alpha e i alpha eta for very long times

local higher integrability of the gradient of a quasiminimizer under generalized orlicz growth conditions abstract we study local quasiminimizers of the dirichlet energy under generalized growth conditions special cases include standard variable exponent and double phase growths we show that the gradient of a local quasiminimizer has local higher integrability

stability theory of solitary waves in the presence of symmetry ii abstract consider an abstract hamiltonian system which is invariant under a group of operators we continue to study the effect of the group invariance on the stability of solitary waves applications are given to bound states and traveling wave solutions of nonlinear wave equations

long time dynamics of variable coefficient modified korteweg de vries solitary waves we study the long time behavior of solutions to the korteweg de vries type equation partial derivative t u partial derivative x partial derivative 2 x u f u b t x u with initial conditions cl

existence and uniqueness for a degenerate parabolic equation with l1 data in this paper we study existence and uniqueness of solutions for the boundary value problem with initial datum in l1 ut div a x du in 0 u a u on 0 u x 0 u0 x in where a is a caratheodory function satisfying the classical leray lions hypothesis a is the neumann boundary operator associated to a du the gradient of u and is a maximal monotone graph in r r with 0 0

spectral methods for hyperbolic initial boundary value problems on parallel computers abstract in this paper a parallel spectral algorithm is developed for hyperbolic initial boundary value problems in one space dimension the galerkin collocation method which is spectrally accurate in both space and time is parallelized by using domain decomposition this procedure leads to a minimization problem in which there is coupling at inter domain boundaries we construct a decoupled preconditioner which can be used to iteratively solve the minimization problem symmetric formulation of the problem which is needed to compute the residual for the normal equations is discussed the methodology outlined for computing the normal equations applies equally well to computation of the residual for the p and h p versions of the finite element method there is therefore no need to compute the mass and stiffness matrices to obtain the residual as is normally done this leads to a great saving in time and memory particularly for solving nonlinear problems using the p and h p versions of the finite element method the method we discuss in this paper generalizes to hyperbolic initial boundary value problems in multidimensions too provided the computational boundaries we have introduced are noncharacteristic and the system is symmetrizable finally we show that for the case of analytic coefficients and data satisfying all the required compatibility conditions so that the solution is analytic the numerical solution is exponentially accurate in n where n is proportional to the number of subdomains and the number of degrees of freedom in each element

long time dynamics of highly concentrated solitary waves for the nonlinear schrodinger equation abstract in this paper we study the behavior of solutions of a nonlinear schrodinger equation in presence of an external potential which is allowed to be singular at one point we show that the solution behaves like a solitary wave for long time even if we start from an unstable solitary wave and its dynamics coincides with that of a classical particle evolving according to a natural effective hamiltonian

spectral element method for three dimensional elliptic problems with smooth interfaces abstract in this paper we propose a least squares spectral element method for three dimensional elliptic interface problems the differentiability estimates and the main stability theorem using non conforming spectral element functions are proven the proposed method is free from any kind of first order reformulation a suitable preconditioner is constructed with help of the regularity estimate and proposed stability estimates which is used to control the condition number we show that these preconditioners are spectrally equivalent to the quadratic forms by which we approximate them we obtain the error estimates which show the exponential accuracy of the method numerical results are obtained for both straight and curved interfaces to show the efficiency of the proposed method

spectral methods for initial boundary value problems with nonsmooth data in this paper we consider hyperbolic initial boundary value problems with nonsmooth data we show that if we extend the time domain to minus infinity replace the initial condition by a growth condition at minus infinity and then solve the problem using a filtered version of the data by the galerkin collocation method using laguerre polynomials in time and legendre polynomials in space then we can recover pointwise values with spectral accuracy provided that the actual solution is piecewise smooth for this we have to perform a local smoothing of the computed solution

a model porous medium equation with variable exponent of nonlinearity existence uniqueness and localization properties of solutions abstract we study the localization properties of weak solutions to the dirichlet problem for the degenerate parabolic equation u t div u x t u f with variable exponent of nonlinearity we prove the existence and uniqueness of weak solutions and establish conditions on the problem data and the exponent x t sufficient for the existence of such properties as finite speed of propagation of disturbances the waiting time effect finite time vanishing of the solution it is shown that the solution may instinct in a finite time even if x 0 in the problem domain but max 0

renormalized solutions to nonlinear parabolic problems in generalized musielak orlicz spaces abstract we will present the proof of existence of renormalized solutions to a nonlinear parabolic problem t u div a d u f with right hand side f and initial data u 0 in l 1 the growth and coercivity conditions on the monotone vector field a are prescribed by a generalized n function m which is anisotropic and inhomogeneous with respect to the space variable in particular m does not have to satisfy an upper growth bound described by a 2 condition therefore we work with generalized musielak orlicz spaces which are not necessarily reflexive moreover we provide a weak sequential stability result for a more general problem t u div a d u f u f where is a monotone function with respect to the second variable and f is locally lipschitz continuous within the proof we use truncation methods young measure techniques the integration by parts formula and monotonicity arguments which have been adapted to nonreflexive musielak orlicz spaces

nonconforming least squares spectral element method for european options several methods have been proposed in the literature for solving the black scholes equation for european options the method proposed in the current study achieves spectral accuracy in both space and time the method is based on minimization of a functional given in terms of the sum of squares of the residuals in the partial differential equation and initial condition in different sobolev norms and a term which measures the jump in the function and its derivatives across inter element boundaries in appropriate fractional sobolev norms to obtain values of the solution and its derivatives the initial condition is mollified and the computed solution is post processed error estimates are obtained for this method specific numerical examples are given to show the efficiency of this method

renormalized solutions for a nonlinear parabolic equation with variable exponents and l 1 data abstract we prove the well posedness existence and uniqueness of a renormalized solution to nonlinear parabolic equations with variable exponents and l 1 data the functional setting involves lebesgue sobolev spaces with variable exponents

stability of traveling waves of nonlinear schrodinger equation with nonzero condition at infinity we study the stability of traveling waves of the nonlinear schrodinger equation with nonzero condition at infinity obtained via a constrained variational approach two important physical models for this are the gross pitaevskii gp equation and the cubic quintic equation first under a non degeneracy condition we prove a sharp instability criterion for 3d traveling waves of gp which had been conjectured in the physical literature this result is also extended for general nonlinearity and higher dimensions including 4d gp and 3d cubic quintic equations second for cubic quintic type nonlinearity we construct slow traveling waves and prove their nonlinear instability in any dimension for dimension two the non degeneracy condition is also proved for these slow traveling waves for general traveling waves without vortices that is nonvanishing and with general nonlinearity in any dimension we find a sharp condition for linear instability third we prove that any 2d traveling wave of gp is transversally unstable and we find the sharp interval of unstable transversal wave numbers near unstable traveling waves of all of the above cases we construct unstable and stable invariant manifolds

on weak interaction between a ground state and a non trapping potential abstract we show that ground states of the nls moving at nonzero speed are asymptotically stable if they either stay far from the potential or the potential is small or the ground state has large speed we search an effective hamiltonian using the birkhoff normal forms argument in 11 treating the potential as a perturbation the so called fermi golden rule which is used to describe the decay to 0 of the internal discrete modes of the ground state is similar to that in 12 the continuous modes dispersion requires the theory in 40 on charge transfer models

on some class of nonlinear parabolic inequalities in orlicz spaces abstract the paper deals with the existence of solutions of some unilateral problems in the framework of orlicz spaces

stable manifolds for all monic supercritical focusing nonlinear schrodinger equations in one dimension keywords critical schrodinger equation stable manifolds modulation theory spectral theory klein gordon equations non linear schrodinger blow up rate standing waves nonintegrable equations invariant manifolds stability theory solitary waves ground states instability reference epfl article 155183doi 10 1090 s0894 0347 06 00524 8 record created on 2010 11 19 modified on 2017 05 12

soliton dynamics for a general class of schrodinger equations the soliton dynamics for a general class of nonlinear focusing schrodinger problems in presence of non constant external local and nonlocal potentials is studied by taking as initial datum the ground state solution of an associated autonomous elliptic equation

lyapunov stability of ground states of nonlinear dispersive evolution equations on presente une nouvelle demonstration de la stabilite orbitale des solitons d etat fondamental de l equation de schrodinger non lineaire pour une grande classe de non linearites on demontre la stabilite de l onde solitaire pour l equation de korteweg de vries generalisee

structural stability for variable exponent elliptic problems i the p x laplacian kind problems abstract we study the structural stability i e the continuous dependence on coefficients of solutions of the elliptic problems under the form b u n div a n x u n f n the equation is set in a bounded domain of r n and supplied with the homogeneous dirichlet boundary condition on here b is a non decreasing function on r and a n x n is a family of applications which verifies the classical leray lions hypotheses but with a variable summability exponent p n x 1 p p n p the need for varying p x arises for instance in the numerical analysis of the p x laplacian problem uniqueness and existence for these problems are well understood by now we apply the stability properties to further generalize the existence results the continuous dependence result we prove is valid for weak and for renormalized solutions notice that besides being interesting on its own the renormalized solutions framework also permits us to deduce optimal convergence results for the weak solutions our technique avoids the use of a fixed duality framework like the w 0 1 p x w 1 p x duality and thus it is suitable for the study of problems where the summability exponent p also depends on the unknown solution itself in a local or in a non local way the sequel of this paper will be concerned with well posedness of some p u laplacian kind problems and with existence of solutions to elliptic systems with variable solution dependent growth exponent

on the cauchy problem for boltzmann equations global existence and weak stability we study the large data cauchy problem for boltzmann equations with general collision kernels we prove that sequences of solutions which satisfy only the physically natural a priori bounds converge weakly in l to a solution from this stability result we deduce global existence of a solution to the cauchy problem our method relies upon recent compactness results for velocity averages a new formulation of the boltzmann equation which involves nonlinear normalization and an analysis of subsolutions and supersolutions it allows us to overcome the lack of strong a priori estimates and define a meaningful collision operator for general configurations

long time dynamics of kdv solitary waves over a variable bottom we study the variable bottom generalized korteweg de vries bkdv equation tu x u f u b t x u where f is a nonlinearity and b is a small bounded and slowly varying function related to the varying depth of a channel of water many variable coefficient kdv type equations including the variable coefficient variable bottom kdv equation can be rescaled into the bkdv we study the long time behavior of solutions with initial conditions close to a stable b 0 solitary wave we prove that for long time intervals such solutions have the form of the solitary wave whose center and scale evolve according to a certain dynamical law involving the function b t x plus an h1 small fluctuation 2005 wiley periodicals inc

least squares spectral element preconditioners for fourth order elliptic problems abstract in this paper we propose preconditioners for the system of linear equations that arise from a discretization of fourth order elliptic problems in two and three dimensions d 2 3 using spectral element methods these preconditioners are constructed using separation of variables and can be diagonalized and hence easy to invert for second order elliptic problems this technique has proven to be successful and performs better than other preconditioners in the framework of least squares methods we show that these preconditioners are spectrally equivalent to the quadratic forms by which we approximate them numerical results for the condition number reflects the effectiveness of the preconditioners

uniqueness of renormalized solutions of degenerate elliptic parabolic problems we consider a general class of degenerate elliptic paraboic problems associated with the equation b v t div a v dv f using kruzhkov s method of doubling variables both in space and time we prove uniqueness and a comparison principle in l 1 for renormalized solutions

effective dynamics of double solitons for perturbed mkdv we consider the perturbed mkdv equation partial t u partial x partial x 2u 2u 3 b x t u where the potential b x t b 0 hx ht 0 h ll 1 is slowly varying with a double soliton initial data on a dynamically interesting time scale the solution is mathcal o h 2 close in h2 to a double soliton whose position and scale parameters follow an effective dynamics a simple system of ordinary differential equations these equations are formally obtained as hamilton s equations for the restriction of the mkdv hamiltonian to the submanifold of solitons the interplay between algebraic aspects of complete integrability of the unperturbed equation and the analytic ideas related to soliton stability is central in the proof

convergence of a full discretization of quasi linear parabolic equations in isotropic and anisotropic orlicz spaces convergence of a subsequence of approximate solutions arising from a full discretization is shown for a general class of quasi linear parabolic problems the numerical method combines the backward euler method for the time discretization with a generalized internal approximation scheme for the spatial discretization the governing monotone elliptic differential operator is described by a nonlinearity that may have anisotropic and nonpolynomial growth but fulfills a coercivity condition in terms of a generalized mathscr n function if the problem admits a unique solution which is shown in the case of a strictly monotone nonlinearity and in a class of sufficiently smooth solutions then the whole sequence of approximate solutions converges moreover an a priori error estimate for the temporal semidiscretization is provided and a numerical illustration is given

renormalized solutions to parabolic equations in time and space dependent anisotropic musielak orlicz spaces in absence of lavrentiev s phenomenon abstract the paper concerns existence and uniqueness of solutions to a nonlinear parabolic equation with merely integrable data on a lipschitz bounded domain in r n our focal point is to involve the leading nonlinear part of the operator whose growth is described by anisotropic n function m inhomogeneous in the space and the time variables the main goals are proven in absence of lavrentiev s phenomenon to ensure which we impose a certain type of balance of interplay between the behavior of m for large and small changes of time and space variables its instances are log holder continuity of variable exponent inhomogeneous in time and space or optimal closeness condition for powers in double phase spaces possibly changing in time new delicate approximation in time result is proven and applied in the construction of renormalized solutions

long time oscillation of solutions of nonlinear schrodinger equations near minimal mass ground state abstract in this paper we consider the long time dynamics of radially symmetric solutions of nonlinear schrodinger equations nls having a minimal mass ground state in particular we show that there exist solutions with initial data near the minimal mass ground state that oscillate for long time more precisely we introduce a coordinate defined near the minimal mass ground state which consists of finite and infinite dimensional part associated to the discrete and continuous part of the linearized operator then we show that the finite dimensional 2 d part approximately obeys newton s equation of motion for a particle in an anharmonic potential well showing that the infinite dimensional part is well separated from the finite dimensional part we will have long time oscillation

stability of position control of traveling waves in reaction diffusion systems we consider the stability of position control of traveling waves in reaction diffusion systems as proposed in l ober and engel phys rev lett 112 148305 2014 instead of analyzing the controlled reaction diffusion system stability is studied on the reduced level of the equation of motion for the position over time of perturbed traveling waves we find an interval of perturbations of initial conditions for which position control is stable this interval can be interpreted as a localized region where traveling waves are susceptible to perturbations for stationary solutions of reaction diffusion systems with reflection symmetry this region does not exist analytical results are in qualitative agreement with numerical simulations of the controlled schl ogl model

asymptotic dynamics of nonlinear schrodinger equations resonance dominated and dispersion dominated solutions we consider a linear schrodinger equation with a nonlinear perturbation in r 3 assume that the linear hamiltonian has exactly two bound states and its eigenvalues satisfy some resonance condition we prove that if the initial data is sufficiently small and is near a nonlinear ground state then the solution approaches to certain nonlinear ground state as the time tends to infinity furthermore the difference between the wave function solving the nonlinear schrodinger equation and its asymptotic profile can have two different types of decay the resonancedominated solutions decay as t 1 2 or the dispersion dominated solutions decay at least like t 3 2 c 2002 john wiley sons inc

preconditioners for spectral element methods for elliptic and parabolic problems in this paper we propose preconditioners for spectral element methods for elliptic and parabolic problems these preconditioners are constructed using separation of variables and are easy to invert moreover they are spectrally equivalent to the quadratic forms which they are used to approximate

on the solitary wave dynamics under slowly varying medium for nonlinear schrodinger equations we consider the problem of a solitary wave propagation in a slowly varying medium for a variable coefficients nonlinear schrodinger equation we prove global existence and uniqueness of solitary wave solutions for a large class of slowly varying media moreover we describe for all time the behavior of these solutions which include refracted and reflected solitary waves depending on the initial energy

inelastic interaction of nearly equal solitons for the quartic gkdv equation this paper describes the interaction of two solitons with nearly equal speeds for the quartic gkdv equation partial tu partial x partial x 2u u 4 0 quad t x in mathbb r 0 1 we call soliton a solution of 0 1 of the form u t x qc x ct y0 where c 0 y0 and q c q c 4 cq c since 0 1 is not an integrable model the general question of the collision of two given solitons q c 1 x c 1 t q c 2 x c 2 t with c1 c2 is an open problem