decay of solutions of nonlinear wave equations in three space dimensions abstract given a solution of the cauchy problem for nonlinear wave equations of the type 2 u t 2 u f u 0 in three space dimensions the asymptotic behaviour in time is considered it is shown that for nonlinearities which behave like powers u 1 u uniform decay holds with a certain rate depending on if 5 1 2 1 2 13 and moreover scattering states exist if is not too small this improves former results of w a strauss j funct anal 2 1968 409 457

nonlinear scattering theory at low energy abstract we study the scattering theory of a conservative nonlinear one parameter group of operators on a hilbert space x relative to a group of linear unitary operators under certain hypotheses the scattering operator carries a neighborhood of 0 in x into x the theory is designed to apply to the semilinear schrodinger and klein gordon equations

a note on fuzzy relational equations with min implication composition the fuzzy relational equations with min implication composition are considered where the implication operation involved is the residuum with respect to the minimum operation the solution set of such a system of equations whenever nonempty can be characterized by a minimum solution and finitely many maximal solutions it is shown that a system of fuzzy relational equations with min implication composition can be reformulated in polynomial time into a system of 0 1 mixed integer linear inequalities and consequently the structure of its solution set can be revealed in a succinct manner besides it is shown that the determination of all maximal solutions is polynomially reducible to the transversal hypergraph generation problem of which some practically well performed algorithms are available

topological entropy for set valued maps abstract any continuous map t on a compact metric space x induces in a natural way a continuous map t on the space k x of all non empty compact subsets of x let t be a homeomorphism on the interval or on the circle it is proved that the topological entropy of the induced set valued map t is zero or infinity moreover the topological entropy of t c x is zero where c x denotes the space of all non empty compact and connected subsets of x for general continuous maps on compact metric spaces these results are not valid

strichartz estimates for the wave and schrodinger equations with the inverse square potential abstract we prove spacetime weighted l 2 estimates for the schrodinger and wave equation with an inverse square potential we then deduce strichartz estimates for these equations

quasi factors of zero entropy systems for minimal systems x t of zero topological entropy we demon strate the sharp difference between the behavior regarding entropy of the sys tems m x t and 2x t induced by t on the spaces m x of prob ability measures on x and 2x of closed subsets of x it is shown that the system m x t has itself zero topological entropy two proofs of this the orem are given the first uses ergodic theoretic ideas the second relies on the different behavior of the banach spaces ln and ln with respect to the existence of almost hilbertian central sections of the unit ball in contrast to this theorem we construct a minimal system x t of zero entropy with a minimal subsystem y t of 2 t whose entropy is positive school of mathematics tel aviv university ramat aviv israel e mail address glasner0math tau ac il institute of mathematics hebrew university jerusalem israel e mail address weissfmath huji ac il this content downloaded from 207 46 13 117 on sun 23 oct 2016 04 35 33 utc all use subject to http about jstor org terms

sup t norm and inf residuum are one type of relational product unifying framework and consequences we present a simple framework which enables us to consider the well known sup t norm and inf residuum products of relations as two particular cases of a single more general type of product we present basic properties of the framework and consequences for the theory of fuzzy relations informally the paper implies that in many cases of fuzzy relational modeling such as in solving fuzzy relational equations there is no need to develop the methods for sup t norm and inf residuum products separately because these methods are just two particular instances of a single method

global dynamics above the ground state energy for the focusing nonlinear klein gordon equation abstract the analysis of global dynamics of nonlinear dispersive equations has a long history starting from small solutions in this paper we study the focusing cubic nonlinear klein gordon equation in r 3 with large radial data in the energy space this equation admits a unique positive stationary solution q called the ground state in 1975 payne and sattinger showed that solutions u t with energy e u u strictly below that of the ground state are divided into two classes depending on a suitable functional k u if k u 0 then one has finite time blow up if k u 0 global existence moreover these sets are invariant under the flow recently ibrahim masmoudi and the first author 22 improved this result by establishing scattering to zero for k u 0 by means of a variant of the kenig merle method kenig and merle 2006 2008 25 26 in this paper we go slightly beyond the ground state energy and we give a complete description of the evolution in that case for example in a small neighborhood of q one encounters the following trichotomy on one side of a center stable manifold one has finite time blow up for t 0 on the other side scattering to zero and on the manifold itself one has scattering to q both as t in total the class of data with energy at most slightly above that of q is divided into nine disjoint non empty sets each displaying different asymptotic behavior as t which includes solutions blowing up in one time direction and scattering to zero on the other the analogue of the solutions found by duyckaerts and merle 2009 2008 13 14 for the energy critical wave and schrodinger equations appear here as the unique one dimensional stable unstable manifolds approaching q exponentially as t or t respectively the main technical ingredient in our proof is a one pass theorem which excludes the existence of almost homoclinic orbits between q as well as q and almost heteroclinic orbits connecting q with q in a companion paper nakanishi and schlag 2010 31 we establish analogous properties for the nls equation

global existence for the one dimensional second order semilinear hyperbolic equations abstract the aim of the paper is to study necessary and sufficient conditions for the existence of the global solution of the one dimensional semilinear equation appearing in the boundary value problems of gas dynamics we investigate the cauchy problem for such equation in the domain where the operator is weakly hyperbolic we obtain the necessary condition for the existence of the self similar solutions for the semilinear gellerstedt type equation the approach used in the paper is based on the fundamental solution of the linear gellerstedt operator and the l p l q estimates

normal forms and quadratic nonlinear klein gordon equations on construit une transformation qui change l equation a non linearite cubique que l on sait resoudre a 3 1 dimensions on demontre que la solution du probleme original a le meme comportement asymptotique que l equation de klein gordon non lineaire

hs global well posedness for semilinear wave equations abstract we consider the cauchy problem for semilinear wave equations in r n with n 3 making use of bourgain s method in conjunction with the endpoint strichartz estimates of keel and tao we establish the hs global well posedness with s

nonrelativistic limit of scattering theory for nonlinear klein gordon equations abstract we study the nonrelativistic limit of the nonlinear klein gordon equation and prove that the wave operators the inverses of them and the scattering operator for the naturally modulated equation converge to those for the singular limit the nonlinear schrodinger equation

on devaney chaotic induced fuzzy and set valued dynamical systems it is well known that any given discrete dynamical system uniquely induces its fuzzified counterpart i e a discrete dynamical system on the space of fuzzy sets in this paper we study relations between dynamical properties of the original and fuzzified dynamical system especially we study conditions used in the definition of devaney chaotic maps i e periodic density and transitivity among other things we show that dynamical behavior of the set valued and fuzzy extensions of the original system mutually inherits some global characteristics and that the space of fuzzy sets admits a transitive fuzzification this paper contains the solution of the problem that was partially solved by roman flores and chalco cano some chaotic properties of zadeh s extension chaos solitons and fractals 35 3 2008 452 459

the global cauchy problem for the nls and nlkg with small rough data abstract by using the unit cube decomposition to the frequency spaces we study the cauchy problem for the nonlinear schrodinger equation and the nonlinear klein gordon equation some global well posedness results are obtained for the small cauchy data in some modulation spaces m p 1 s

nonlinear small data scattering for the wave and klein gordon equation on etudie l operateur diffusion qui appartient a la paire d equations u tt au f u 0 et u tt au 0 a represente l operateur j 1 n 2 x j 2 m 2 m r et f u u 1 u ou r 1

scattering and complete integrability in the massive ϑ4 theory abstract the wave and scattering operators for the equation m 2 3 0 with m 0 on four dimensional minkowski space are analytic on a neighborhood of the space of finite energy cauchy data l 2 1 r 3 l 2 r 3 this allows the construction of infinitely many independent conserved quantities with vanishing poisson brackets and implies that the massive 4 theory is completely integrable at low energy

ω chaos and topological entropy we present a new concept of chaos chaos and prove some properties of chaos then we prove that chaos is equivalent to positive entropy on the interval we also prove that chaos is equivalent to the definition of chaos given by devaney on the interval

sharp threshold of global existence and blow up of the combined nonlinear klein gordon equation abstract in this paper we show the global well posedness and blow up result of the solutions with the energy below the threshold for the combined nonlinear klein gordon equation u t t u u u p 1 u u q 1 u d 3 in the energy space h 1 r d l 2 r d where 1 4 d q p 1 4 d 2 we give a threshold of blow up and global well posedness using a modified variational approach in the spirit of ibrahim et al 2011 7

introduction to scattering for radial 3d nlkg below energy norm abstract we prove scattering for some radial 3d semilinear klein gordon equations with rough data first we prove strichartz type estimates in mixed norm spaces then by using these decays we establish some local bounds by combining these results to a morawetz type estimate and a radial sobolev inequality we control the variation of an almost conserved quantity on arbitrary large intervals once we have showed that this quantity is controlled we prove that some of these local bounds can be upgraded to global bounds this is enough to establish scattering all the estimates involved require a delicate analysis due to the nature of the nonlinearity and the lack of scaling

chaos on hyperspaces abstract let f be a continuous self map of a compact metric space x the transformation f induces in a natural way a self map f defined on the hyperspace k x of all nonempty closed subsets of x we study which of the most usual notions of chaos for dynamical systems induced by f are inherited by f and vice versa we consider distributional chaos li yorke chaos chaos devaney chaos topological chaos positive topological entropy specification property and their variants this answers questions stated independently by roman flores and banks

lp lp estimates for the schrodinger equation on the line and inverse scattering for the nonlinear schrodinger equation with a potential abstract in this paper i prove a l p l p estimate for the solutions to the one dimensional schrodinger equation with a potential in l 1 where in the generic case 3 2 and in the exceptional case i e when there is a half bound state of zero energy 5 2 i use this estimate to construct the scattering operator for the nonlinear schrodinger equation with a potential i prove moreover that the low energy limit of the scattering operator uniquely determines the potential and the coupling constant of the nonlinearity using a method that allows as well for the reconstruction of the potential and of the nonlinearity

modified scattering for the critical nonlinear schrodinger equation abstract we consider the nonlinear schrodinger equation i u t u u 2 n u in all dimensions n 1 where c and 0 we construct a class of initial values for which the corresponding solution is global and decays as t like t n 2 if 0 and like t log t n 2 if 0 moreover we give an asymptotic expansion of those solutions as t we construct solutions that do not vanish so as to avoid any issue related to the lack of regularity of the nonlinearity at u 0 to study the asymptotic behavior we apply the pseudo conformal transformation and estimate the solutions by allowing a certain growth of the sobolev norms which depends on the order of regularity through a cascade of exponents

analyticity of the scattering operator for the nonlinear klein gordon equation with cubic nonlinearity the wave and scattering operators for the equation left square m 2 right varphi lambda varphi 2 0 withm 0 and 0 on four dimensional minkowski space are analytic on the space of finite energy cauchy data i e l21 r3 l2 r3

a note on systems with max min and max product constraints we consider a system email protected x b where email protected r m x n is a non negative matrix and email protected r m is a non negative vector over the n dimensional variable l

scattering theory in the energy space for a class of non linear wave equations we study the asymptotic behaviour in time of the solutions and the theory of scattering in the energy space for the non linear wave equation square varphi f varphi 0 in n n 3 we prove the existence of the wave operators asymptotic completeness for small initial data and forn 4 asymptotic completeness for arbitrarily large data the assumptions onf cover the case wheref behaves slightly better than a single powerp 1 4 n 2 both near zero and at infinity see 1 5 1 6 and 1 8

a note on the 2d generalized zakharov kuznetsov equation local global and scattering results abstract we consider the generalized two dimensional zakharov kuznetsov equation u t x u x u k 1 0 where k 3 is an integer number for k 8 we prove local well posedness in the l 2 based sobolev spaces h s r 2 where s is greater than the critical scaling index s k 1 2 k for k 3 we also establish a sharp criteria to obtain global h 1 r 2 solutions a nonlinear scattering result in h 1 r 2 is also established assuming the initial data is small and belongs to a suitable lebesgue space

long time decay for 2d klein gordon equation abstract we obtain a dispersive long time decay in weighted energy norms for solutions of the 2d klein gordon equations the decay extends the results obtained by jensen kato and murata for the equations of schrodinger s type by the spectral approach for the proof we modify the approach to make it applicable to relativistic equations

nonlinear scattering with nonlocal interaction we consider the scattering problem for the hartree type equation in n withn 2 i frac partial u partial t frac 1 2 delta u v u 2 u where v x sum limits j 1 2 lambda j x gamma j lambda 1 lambda 2 ne 0 0 lambda j in mathbb r gamma j 0 and denotes the convolution in n we prove the existence of wave operators inh0 k l2 n x k l2 n for any positive integerk under the assumption 1 1 2 2 this is an optimal result in the sense that the existence of wave operators breaks down if min 1 2 1 the case where 1 1 2 2 is also treated according to the sign of 2

sensitivity and transitivity of fuzzified dynamical systems abstract in this paper it is proved that a set valued dynamical system is sensitively dependent on initial conditions resp f sensitive multi sensitive if and only if its g fuzzification is sensitively dependent on initial conditions resp f sensitive multi sensitive where f is a furstenberg family as an application it is shown that there exists a sensitive dynamical system whose g fuzzification does not have such sensitive dependence for any g in a certain domain moreover a sufficient condition is derived for ensuring that the g fuzzification of every nontrivial dynamical system is not transitive

energy scattering for nonlinear klein gordon and schrodinger equations in spatial dimensions 1 and 2 we show that when n 1 and 2 the scattering operators are well defined in the whole energy space for nonlinear klein gordon and schrodinger equations in r1 n with nonlinearity u p 1 u p 1 4 n such results have been known only for n 3

scattering and the geometry of the solution manifold of ƒ λƒ3 0 abstract the wave transforms for the equation 3 0 on r 4 can be represented as maps from the hilbert space h s of finite energy cauchy data for the equation 1 3 0 on r 4 s 3 to hilbert spaces h c of finite energy goursat data we show that the wave transforms and associated scattering transform are diffeomorphisms the action of the group g so 2 4 on solutions of the latter equation gives rise to actions of g as diffeomorphisms of h s and h c which are intertwined by the wave and scattering transforms moreover the spaces h s and h c have symplectic structures relative to which the group g and the wave and scattering transforms act as symplectomorphisms these symplectic structures extend to flat kahler structures that are invariant under the scaling extended poincare group p

scattering for the two dimensional energy critical wave equation we investigate existence and asymptotic completeness of the wave operators for the nonlinear klein gordon equation with a defocusing exponential nonlinearity in two space dimensions a certain threshold is defined based on the value of the conserved hamiltonian below which the exponential potential energy is dominated by the kinetic energy via a trudinger moser type inequality we prove that if the energy is below or equal to the critical value then the solution approaches a free klein gordon solution at the time infinity an interesting feature in the critical case is that the strichartz estimate together with sobolev type inequalities cannot control the nonlinear term uniformly on each time interval it crucially depends on how much the energy is concentrated thus we have to trace concentration of the energy along time in order to set up favorable nonlinear estimates and only after that we can apply bourgain s induction argument or any other similar one

system of fuzzy relation equations with inf composition complete set of solutions the problem of solvability of a system of equations with inf composition is considered on finite universes equations are expressed using operations of a bl algebra we study complete set of solutions of the respective system in the particular case one equation and in the general case in both cases various conditions of solvability are found and proved we characterize all maximal solutions and prove that under certain conditions each solution of the system an equation is less than or equal to a respective maximal one as a result we are able to characterize the complete set of solutions examples of single equations and their systems are considered and complete sets of solutions are found for each

global nonexistence results for a class of hyperbolic systems abstract our concern in this paper is to prove blow up results to the non autonomous nonlinear system of wave equations u t t u a t x v p v t t v b t x u q t 0 x r n in any space dimension we show that a curve f p q 0 depending on the space dimension on the exponents p q and on the behavior of the functions a t x and b t x exists such that all nontrivial solutions to the above system blow up in a finite time whenever f p q 0 our method of proof uses some estimates developed by galaktionov and pohozaev in 11 for a single non autonomous wave equation enabling us to obtain a system of ordinary differential inequalities from which the desired result is derived our result generalizes some important results such as the ones in del santo et al 1996 12 and galaktionov and pohozaev 2003 11 the advantage here is that our result applies to a wide variety of problems

on space time means and everywhere defined scattering operators for nonlinear klein gordon equations on demontre l existence d operateurs de diffusion definis partout pour certaines equations de klein gordon non lineaires

finite time blow up to critical semilinear wave equation outside the ball in 3 d abstract in this paper we study the initial boundary value problem of critical semilinear wave equation with small initial data in 3 d blow up result will be established no matter how small the data are we find a radial and positive solution of the corresponding linear problem then our blow up result follows from the idea of zhou and han 2014 what is more we get the upper bound of the lifespan estimate

finite time blowup of solutions to semilinear wave equation in an exterior domain abstract we consider the initial boundary value problem of semilinear wave equation with nonlinearity u p in exterior domain in r n n 3 especially the lifespan of blowup solutions with small initial data are studied the result gives upper bounds of lifespan which is essentially the same as the cauchy problem in r n at least in the case n 4 their estimates are sharp in view of the work by zha zhou 21 the idea of the proof is to use special solutions to linear wave equation with dirichlet boundary condition which are constructed via an argument based on wakasa yordanov 15

scattering operator for nonlinear klein gordon equations in higher space dimensions abstract we prove the existence of the scattering operator in the neighborhood of the origin in the weighted sobolev space h 1 with max 3 2 1 2 n for the nonlinear klein gordon equation with a power nonlinearity u t t u u u 1 u t x r r n where 1 4 n 2 1 4 n for n 3 c

quantum interaction φ44 the construction of quantum field defined as a bilinear form we construct the solution t x of the quantum wave equation m2 3 0 as a bilinear form which can be expanded over wick polynomials of the free in field and where 3 t x is defined as the normal ordered product with respect to the free in field the constructed solution is correctly defined as a bilinear form on d d where d is a dense linear subspace in the fock space of the free in field on d d the diagonal of the wick symbol of this bilinear form satisfies the nonlinear classical wave equation

on fuzzifications of discrete dynamical systems let x denote a locally compact metric space and f x x be a continuous map in the 1970s zadeh presented an extension principle helping us to fuzzify the dynamical system x f i e to obtain a map f for the space of fuzzy sets on x we extend an idea mentioned in p diamond a pokrovskii chaos entropy and a generalized extension principle fuzzy sets syst 61 1994 277 283 to generalize zadeh s original extension principle in this paper we study basic properties of so called g fuzzifications such as their continuity properties we also show that for any g fuzzification i a uniformly convergent sequence of uniformly continuous maps on x induces a uniformly convergent sequence of fuzzifications on the space of fuzzy sets and ii a conjugacy resp a semi conjugacy between two discrete dynamical systems can be extended to a conjugacy resp a semi conjugacy between fuzzified dynamical systems throughout this paper we consider different topological structures in the space of fuzzy sets namely the sendograph endograph and levelwise topologies

conserved quantities for the yang mills equations abstract there are infinitely many gauge invariant conserved quantities for sufficiently regular solutions of the yang mills equations on 4 dimensional minkowski space for example those that extend to c2 classical solutions on the universal cover of the conformal compactification which may be identified with the einstein universe r s3 these conserved quantities are expressed in terms of the in or out fields constructed via the formulation of scattering as a global goursat problem on the einstein universe

sensitivity of set valued discrete systems abstract consider the surjective continuous map f x x and the continuous map f of k x into itself induced by f where x is a compact metric space and k x is the space of all non empty compact subsets of x endowed with a hausdorff metric in this paper we give examples showing that sensitivity of f does not imply sensitivity of f furthermore we prove that if f is a surjective continuous interval map then f is sensitive if and only if f is sensitive

inviscid limit for the damped boussinesq equation abstract in this paper we consider the limit behavior of solutions to the cauchy problem for damped boussinesq equation in the regime of small viscosity in r n it is shown that the strong solutions of damped boussinesq equation converge to a strong solution of boussinesq equation in any time interval 0 t the key idea is that we only take into account the effect of dispersion and overlook the effect of dissipation in the damped boussinesq equation which is a kind of new view we analyze the dispersion by the techniques of high low frequency decomposition and littlewood paley dyadic decomposition stationary phase estimate and some properties of bessel function

final state problem for korteweg de vries type equations we study the final state problem for the korteweg de vries type equations ut 1 x 1ux u2ux t x r r u t fs t l2 0 as t where r the function fs t we call a final state defined by the final data u we show that there does not exist a nontrivial solution of this equation in the case of fs t u t u where u t is the free evolution group of this equation we construct the modified wave operator for the korteweg de vries type equations under the conditions that the final data u arc real valued functions and the fourier transform u vanishes at the origin

time asymptotics for solutions of vlasov poisson equation in a circle we prove that there exists a class of solutions of the nonlinear vlasov poisson equation vpe on a circle that converges weakly as t to a stationary homogeneous solution of vpe this behavior is called in the linear case landau damping the result is obtained by constructing a suitable scattering problem for the solutions of the vlasov poisson problem a consequence of this result is that a class of stationary solutions of the vlasov poisson equation is unstable in a weak topology

decay estimates for a class of wave equations abstract in this paper we use a unified way studying the decay estimate for a class of dispersive semigroup given by e i t where r r is smooth away from the origin especially the decay estimates for the solutions of the klein gordon equation and the beam equation are simplified and slightly improved

nonlinear scattering theory for a class of wave equations in hs abstract for the semi linear higher order wave equation and the nonlinear higher order schrodinger equation we show that the scattering operators map a band in h s into h s if the nonlinearities have sub critical powers in h s the smoothness of the scattering operators and the uniform boundedness of strong solutions for the defocusing nls equation are also shown provided that the nonlinearities have subcritical growth in h 1 moreover the spatial decaying behavior of solutions in energy space for the defocusing nls equation are obtained

latticized linear optimization on the unit interval this paper considers the latticized linear optimization llo problem and its variants which are a special class of optimization problems constrained by fuzzy relational equations or inequalities we show that an optimal solution to such a problem can be obtained in polynomial time as long as the objective function is a max separable function with continuous monotone components we further show that the set of all optimal solutions is fully determined by one maximum optimal solution and a finite number of minimal optimal solutions the maximum optimal solution can be constructed in polynomial time once the optimal objective value is known while the detection of all minimal optimal solutions in an efficient manner remains as a challenging problem the relation between llo and max separable optimization and related issues are also investigated

smoothing properties and retarded estimates for some dispersive evolution equations smoothing properties in the form of space time integrability properties play an important role in the study of dispersive evolution equations a number of them follow from a combination of general arguments and specific estimates we present a general formulation which makes the separation between the two types of ingredients as clear as possible and we illustrate it with the examples of the schrodinger equation of the wave equation and of a class of 1 1 dimensional equations related to the benjamin ono equation of special interest for the cauchy problem are retarded estimates expressed in terms of those properties we derive a number of such estimates associated with the last example and we mention briefly an application of those estimates to the cauchy problem for the generalized benjamin ono equation

some chaotic and mixing properties of fuzzified dynamical systems abstract let x be a compact metric space let be a continuous self map on x and let f x denote the space of fuzzy sets on x equipped with the levelwise topology in this paper we study relations between a given crisp dynamical system x and its g fuzzification f x g among other things we study various weak strong mild etc mixing properties and also several kinds of chaotic behaviors li yorke chaos chaos distributional chaos topological chaos etc we specified subspaces of fuzzy sets which makes sense to study further additionally we proved that numerous local characteristics of the original dynamical system can be found in its fuzzy extension while many global characteristics are preserved in the opposite direction thus among other things we obtained easy to check criteria for checking absence of complex behavior in induced fuzzy dynamical systems

final state problem for the cubic nonlinear klein gordon equation we study the final state problem for the nonlinear klein gordon equation utt u uxx u3 t r x r where r we prove the existence of solutions in the neighborhood of the approximate solutions 2 re u t w t where u t is the free evolution group defined by u t f 1e it f x 1 x2 f and f 1 are the direct and inverse fourier transformations respectively and w t x f 1 u e 3 2 i 2 u 2log t with a given final data u is a real valued function and 3 i u l is small

measures of chaos and a spectral decomposition of dynamical systems on the interval let f 0 1 0 1 be continuous for x y e 0 1 the upper and lower distance distribution functions fx y and fxy are defined for any t 0 as the lim sup and lim inf as n oc of the average number of times that the distance ifi x fi y i between the trajectories of x and y is less than t during the first n iterations the spectrum of f is the system f of lower distribution functions which is characterized by the following properties 1 the elements of f are mutually incomparable 2 for any f e f there is a perfect set pf 8 0 such that fuv f and fuv 1 for any distinct u v e pf 3 if s is a scrambled set for f then there are f g in f and a decomposition s sf u sg sg may be empty such that fuv f if u v e sf and fuv g if u v e sg our principal results are 1 if f has positive topological entropy then f is nonempty and finite and any f e f is zero on an interval 0 e where e 0 and hence any pf is a scrambled set in the sense of li and yorke 2 if f has zero topological entropy then f f where f 1 it follows that the spectrum of f provides a measure of the degree of chaos of f in addition a useful numerical measure is the largest of the numbers fo 1 f t dt where f e f

on scattering and everywhere defined scattering operators for nonlinear klein gordon equations asymptotic properties of solutions of the nonlinear klein gordon equation t2u u m2u f u 0 nlkg u 0 tu 0 are investigated which are inherited from the corresponding solutions v of the linear klein gordon equation t2v v m2v 0v 0 tv 0 kg in particular the finiteness of time integrals in lq over r of certain sobolevnorms in space of the solution is proved to be such a hereditary property together with a device by w a strauss and a weak decay result for the kg due to r s strichartz this is used to prove that under suitable restrictions on the nonlinearity the scattering operator for the nlkg is defined on all of l21 l2 for n 3

new blow up results for nonlinear boundary value problems in exterior domains abstract we consider exterior boundary value problems with nontrivial boundary conditions including three types of semilinear equations heat equation wave equation and damped wave equation using a test function approach with a choice of a suitable family of test functions we obtain the critical exponents in the sense of fujita for the considered problems our obtained results close some open questions from the literature

analyticity of scattering for the ϕ 4 theory we consider scattering for the equation m2 3 0 on four dimensional minkowski space form 0 one to one and onto wave operatorsw h h are known to exist for all 0 whereh denotes the hilbert space of finite energy cauchy data we prove that the maps u w u and u w 1 u are continuous from 0 h toh and extend to real analytic functions from an open neighborhood of 0 h 0 h to the hilbert spaceh 1 of cauchy data with poincare invariant norm form 0 wave operatorsw are known to exist as diffeomorphisms ofh for all 0 where hereh denotes the hilbert space of finite einstein energy cauchy data in this case we prove that the maps u w u and u w 1 u extend to real analytic functions from a neighborhood of 0 h h toh

analysis in space time bundles i general considerations and the scalar bundle abstract mathematical features involved in the systematic development of elementary particle theory on an alternative cosmos space time are presented bundles representative of physical fields are studied in the unique variant of minkowski space m 0 enjoying similar properties of causality and symmetry the universal cosmos m is the universal cover of the causal compactification of m 0 the bundles studied are induced from representations of the scale extended poincare group which forms the isotropy group in m of the universal cover g s u 2 2 of the connected component of the group of all causal transformations on m discrete symmetries and higher dimensional cases are also discussed the primary focus is on the temporal evolution especially stability involving positivity of the energy wave equations implicative of finite propagation velocity and the unitarity and or composition series of associated actions of g general spin bundles on m are treated parallelized and correlated with bundles on m 0 associated covariant wave equations and the spectral resolution of fundamental quantum numbers are studied in detail in the scalar case

critical criteria of fujita type for a system of inhomogeneous wave inequalities in exterior domains abstract we consider blow up results for a system of inhomogeneous wave inequalities in exterior domains we will handle three type boundary conditions dirichlet type neumann type and mixed boundary conditions we use a unified approach to show the optimal criteria of fujita type for each case our study yields naturally optimal nonexistence results for the corresponding stationary wave system and equation we provide many new results and close some open questions

necessary and sufficient conditions for the existence of solution of generalized fuzzy relation equations a x b abstract in 2013 li and jin studied a particular type of fuzzy relational equations on finite sets where the introduced min bi implication composition is based on ukasiewicz equivalence in this paper such fuzzy relation equations are studied on a more general level namely complete residuated lattice valued fuzzy relation equations of type y y a x y x y b x are analyzed and the existence of solutions s is studied first a necessary condition for the existence of solution is established then conditions for lower and upper limits of solutions are given and finally sufficient conditions for the existence of the smallest and largest solutions respectively are characterized if such general or global solutions do not exist there might still be partial or point wise solutions this is a novel way to study fuzzy relation equations such point wise solutions are studied on ukasiewicz product and godel t norm based residuated lattices on the real unit interval

decomposition and resolution of min implication fuzzy relation equations based on s implications the problem of solving min implication fuzzy relation equations based on s implications is studied in this paper decomposition of min implication fuzzy relation equations is given in a finite case the solution existence problem of min implication fuzzy relation equations is discussed and for ns implications a new solvability criterion based upon solution matrices introduced in this paper is studied it is shown that for each solution r of a min implication fuzzy relation equation there exists a maximal solution r of this equation such that r is greater than or equal to r whenever the solution set of this equation is nonempty the complete solution set of min implication fuzzy relation equation can be determined by the maximal solution set of this equation which is finite an effective method to solve min implication fuzzy relation equations based on s implication is proposed

cauchy like functional equation based on continuous t conorms and representable uninorms commuting is an important property in any two step information merging procedure where the results should not depend on the order in which the single steps are performed in the case of bisymmetric aggregation operators with the neutral elements saminger mesiar and dubois have already reduced characterization of commuting n ary operators to resolving the unary distributive functional equations but only some sufficient conditions of unary functions distributive over two particular classes of uninorms are given out indeed it is very difficult to get the full characterizations of these equations because they are bound up with the famous cauchy functional equation which has so far not been completely solved along this way of thinking in this paper we will respectively investigate and fully characterize the following two functional equations f u x y s f x f y and f s x y u f x f y where f 0 1 0 1 is an unknown function u is a representable uninorm and s is a continuous t conorm these results are an important step toward obtaining complete characterization of the other unary distributive functional equations previously mentioned our results show that commuting is suitable for only the second equation but not for the first one this is because solutions of the first equation are all piecewise constant functions while those of the second equation are formulas with parameters in addition the two equations both have nonmonotone solutions completely different from those ones obtained

weighted energy decay for 3d klein gordon equation abstract we obtain a dispersive long time decay in weighted energy norms for solutions of the 3d klein gordon equation with generic potential the decay extends the results obtained by jensen and kato for the 3d schrodinger equation for the proof we modify the spectral approach of jensen and kato to make it applicable to relativistic equations

dgm a deep learning algorithm for solving partial differential equations abstract high dimensional pdes have been a longstanding computational challenge we propose to solve high dimensional pdes by approximating the solution with a deep neural network which is trained to satisfy the differential operator initial condition and boundary conditions our algorithm is meshfree which is key since meshes become infeasible in higher dimensions instead of forming a mesh the neural network is trained on batches of randomly sampled time and space points the algorithm is tested on a class of high dimensional free boundary pdes which we are able to accurately solve in up to 200 dimensions the algorithm is also tested on a high dimensional hamilton jacobi bellman pde and burgers equation the deep learning algorithm approximates the general solution to the burgers equation for a continuum of different boundary conditions and physical conditions which can be viewed as a high dimensional space we call the algorithm a deep galerkin method dgm since it is similar in spirit to galerkin methods with the solution approximated by a neural network instead of a linear combination of basis functions in addition we prove a theorem regarding the approximation power of neural networks for a class of quasilinear parabolic pdes

f sensitivity and multi sensitivity of hyperspatial dynamical systems abstract let f be a continuous self map defined on a compact metric space x and f be a continuous self map naturally induced by f on the hyperspace k x of all nonempty compact subsets of x endowed with a hausdorff metric firstly a few sufficient and necessary conditions to ensure a dynamical system be f sensitive or multi sensitive are obtained then the following results are proved 1 if x f is a non minimal m system then k x f has f s sensitive pairs almost everywhere 2 if k x f or k y g is f sensitive then k x y f g is f sensitive 3 k x y f g is multi sensitive if and only if k x f or k y g is multi sensitive if and only if x f or y g is multi sensitive moreover it is proved that f g is multi sensitive if and only if f or g is multi sensitive this is a positive answer to a question posed in r li and x zhou 2013 18

stable directions for small nonlinear dirac standing waves we prove that for a dirac operator with no resonance at thresholds nor eigenvalue at thresholds the propagator satisfies propagation and dispersive estimates when this linear operator has only two simple eigenvalues sufficiently close to each other we study an associated class of nonlinear dirac equations which have stationary solutions as an application of our decay estimates we show that these solutions have stable directions which are tangent to the subspaces associated with the continuous spectrum of the dirac operator this result is the analogue in the dirac case of a theorem by tsai and yau about the schrodinger equation to our knowledge the present work is the first mathematical study of the stability problem for a nonlinear dirac equation

inverse scattering on the line for the nonlinear klein gordon equation with a potential abstract in this paper we obtain a lp lp estimate for the klein gordon equation with a potential on the line this result follows from the previously known estimates in the case without potential and from the continuity on the sobolev spaces of the schrodinger wave operators that we proved in a previous paper using this estimate we construct the scattering operator for the nonlinear klein gordon equation with a potential moreover we prove that the small amplitude scattering operator uniquely determines both the potential and the nonlinearity our proof also gives a method for the reconstruction of the potential and of the nonlinearity

decay and scattering of small solutions of a generalized boussinesq equation abstract we study the long time behavior of small solutions of the initial value problem for a generalized boussinesq equation we obtain a lower bound for the degrees of nonlinearity which allows us to establish a nonlinear scattering result for small perturbations that is the small solutions of the nonlinear problem behave asymptotically like the solution of the associated linear problem under certain hypotheses we can construct a scattering operator for the boussinesq equation which carries a neighborhood of 0 in the energy space x into x

analyticity of the scattering operator for semilinear dispersive equations we present a general algorithm to show that a scattering operator associated to a semilinear dispersive equation is real analytic and to compute the coefficients of its taylor series at any point we illustrate this method in the case of the schrodinger equation with power like nonlinearity or with hartree type nonlinearity and in the case of the wave and klein gordon equations with power nonlinearity finally we discuss the link of this approach with inverse scattering and with complete integrability

scattering and complete integrability in conformally invariant nonlinear theories conformally invariant nonlinear wave equations in four dimensions corresponding to multicomponent massless scalar fields with a quartic interaction are studied it is proved that the scattering operator s on the space h of finite einstein energy cauchy data has infinitely many fixed points as well as periodic points of all orders there are also h such that sn is almost periodic but not periodic and h such that sn is not almost periodic it is also proved that h admits no conformally invariant kahler metrics but infinitely many distinct kahler metrics invariant under the poincare group and scale transformations moreover it is proved that time evolution for these nonlinear wave equations is completely integrable on the space h

the scattering problem for the three dimensional cubic nonlinear klein gordon equation with rapidly decreasing input data abstract the scattering problem for the three dimensional cubic nonlinear klein gordon equation is studied it has been shown that the scattering operator s is well defined on a neighborhood in the critical space h 1 2 r 3 h 1 2 r 3 of 0 we prove that if functions f and g are in the schwartz space s r 3 and small in the sense of the critical space then the corresponding output data f g s f g also belongs to s r 3 s r 3 furthermore we give sufficient conditions for f g such that all order partial derivatives of f and g decay more rapidly than a same exponential function

approximately solving fuzzy relation equations some mathematical results and some heuristic proposals abstract fuzzy relation equations are intimately connected with the approach toward fuzzy control unfortunately systems of such equations are solvable only under quite restrictive conditions therefore a notion of approximate solution is discussed which applied also in cases where such systems do not have solutions for some types of fuzzy equations best possible approximate solutions are known for other types some strategies are considered to suitably guess approximate solutions furthermore that approach is applied to some types of cascaded fuzzy relation equations which are connected with ideas to join neural network structures with fuzzy control

low energy scattering for nonlinear klein gordon equations the scattering operator which belongs to a pair of pdes consisting of the klein gordon equation and a perturbation of it by a power like nonlinearity z hfl u is studied it is shown that this operator can be defined on a whole neighbourhood of the origin in energy space if z hfl u u p 1u or u p where 1 4 n 1 p 1 4 n 2 and the space dimension n 2 is arbitrary

necessary and sufficient conditions for existence of maximal solutions for inf α composite fuzzy relational equations on complete brouwerian lattices an inf a composite fuzzy relational equation and its equation system are investigated in finite domains a necessary and sufficient solvability condition is proposed for the equation then all its maximal solutions and the whole solution set are determined subsequently the whole solution set for the equation system is determined in infinite domains sufficient conditions for existence of a maximal solution for the equation and the equation system are shown respectively afterwards a necessary and sufficient condition that there exists a maximal solution which is more than or equal to any solution is presented for the equation

the global goursat problem and scattering for nonlinear wave equations abstract the goursat problem for nonlinear scalar equations on the einstein universe m with finite energy datum has a unique global solution in the positive energy sobolev controllable case such equations include those of the form h 0 where h denotes a hamiltonian that is a fourth order polynomial bounded below in components of the multicomponent scalar section in particular the conformally invariant equation 1 3 0 0 is included in the higher dimensional analog r s n to the einstein universe the same result holds under the stronger conditions on h required for sobolev controllability irrespective of energy positivity there is a unique local in time solution for arbitrary finite energy goursat datum for all n 3 establishing evolution from the given lightcone to any sufficiently close lightcone these results show the existence of wave operators in the sense of scattering theory and their continuity in the einstein energy metric for positive energy equations of the indicated type they also permit the comprehensive reduction of scattering theory for conformally invariant wave equations in minkowski space m 0 to the goursat problem in m in particular any solution of the equation arising from a nonnegative conformally invariant biquadratic interaction lagrangian on multicomponent scalar sections having finite einstein energy at any one time is asymptotic to solutions of the corresponding multicomponent free wave equation as the minkowski time x 0 thus given a finite einstein energy solution of the equation 3 0 on m 0 0 there exist unique solutions of the free wave equation which approach in the minkowski energy norm as x 0 and every finite einstein energy solution of the free wave equation is of the form or for a unique solution of the nonlinear equation this generalizes in part in maximality sharp form earlier results of strauss for this equation