nontrivial periodic solutions for the asymptotically linear hamiltonian systems with resonance at infinity abstract in this paper we are concerned with the existence of nontrivial periodic solutions of an asymptotically linear hamiltonian systems when a kind of resonance condition is satisfied we prove several new existence results via the morse theory the garlerkin methods and the moslov index theory

global existence of classical solutions to the minimal surface equation in two space dimensions with slow decay initial value in this paper we prove the global existence of classical solutions to the cauchy problem for the minimal surface equation in the minkowski space r1 2 with slow decay initial value

global existence for the einstein vacuum equations in wave coordinates we prove global stability of minkowski space for the einstein vacuum equations in harmonic wave coordinate gauge for the set of restricted data coinciding with the schwarzschild solution in the neighborhood of space like infinity the result contradicts previous beliefs that wave coordinates are unstable in the large and provides an alternative approach to the stability problem originally solved for unrestricted data in a different gauge and with a precise description of the asymptotic behavior at null infinity by d christodoulou and s klainerman

homoclinic solutions for a class of the second order hamiltonian systems abstract we study the existence of homoclinic orbits for the second order hamiltonian system q v q t q f t where q r n and v c 1 r r n r v t q k t q w t q is t periodic in t a map k satisfies the pinching condition b 1 q 2 k t q b 2 q 2 w is superlinear at the infinity and f is sufficiently small in l 2 r r n a homoclinic orbit is obtained as a limit of 2 kt periodic solutions of a certain sequence of the second order differential equations

the equation for time like extremal surfaces in minkowski space r2 n in this paper we investigate the equation for time like extremal surfaces in the minkowski space r2 n and show that this kind of equation enjoys many interesting properties nonstrict hyperbolicity constant multiplicity of eigenvalues boundedness of characteristic propagation speeds linear degeneracy of all characteristic fields richness etc without any smallness assumption of initial data we give the necessary and sufficient condition on the global existence of classical solutions of the cauchy problem based on this we prove some global existence theorems on classical solutions of the dirichlet problem and neumann problem for this kind of equation finally we present an explicit exact representation involving two independent arbitrary functions of general solution

asymptotic properties of linear field equations in minkowski space the aim of this paper is to derive the uniform asymptotic behavior of solutions to linear field equations in minkowski space based on geometric consideration and generalized energy estimates our method relies on a systematic use of the invariance properties of the field equations with respect to the conformal group of the minkowski space and thus departs substantially from the classical method of analyzing the fundamental solution

global existence and asymptotic behavior for some multidimensional quasilinear hyperbolic systems abstract for the cauchy problem of multidimensional quasilinear hyperbolic systems of diagonal form without self interaction we show the global existence of classical solutions with small initial data we also prove that the global solution will converge to a solution of the linearized system as time tends to infinity and the convergence rate is also determined

scattering for the quasilinear wave equations with null conditions in two dimensions abstract we give a detailed description of the scattering theory of solutions to the quasilinear wave equations with null conditions and small initial data in two dimensions the scattering profile is shown to be the same as the one conjectured by alinhac 1 1995 our proof is based on a framework of fourier transform by deng and pusateri 6 2018 and an expansion of oscillatory integral together with the global well posedness of the solutions by alinhac 2 2001

remarks on energy approach for global existence of some one dimension quasilinear hyperbolic systems abstract we show that luli yang and yu s weighted energy estimate with positive weights 8 together with some variants of alinhac s ghost weight energy approach 1 can be used to prove the global existence of classical solutions to some one dimension quasilinear hyperbolic systems of diagonal form with small initial data

infinite dimensional cohomology groups and periodic solutions of asymptotically linear hamiltonian systems abstract in this paper we study the existence of nontrivial 2 periodic solutions of asymptotically linear hamiltonian systems we consider the case of resonance both at zero and at infinity and we permit time dependent asymptotic matrices our main tools are an infinite dimensional cohomology theory and a corresponding morse theory recently constructed by w kryszewski and the first author we develop a method to compute the new critical groups

almost global existence for 2 d incompressible isotropic elastodynamics author s lei z sideris tc zhou y abstract 2015 american mathematical society we consider the cauchy problem for 2 d incompressible isotropic elastodynamics standard energy methods yield local solutions on a time interval 0 t for initial data of the form u0 where t depends only on some sobolev norm of u0 we show that for such data there exists a unique solution on a time interval 0 exp t provided that is sufficiently small this is achieved by careful consideration of the structure of the nonlinearity the incompressible elasticity equation is inherently linearly degenerate in the isotropic case in other words the equation satisfies a null condition this is essential for time decay estimates the pressure which arises as a lagrange multiplier to enforce the incompressibility constraint is estimated in a novel way as a nonlocal nonlinear term with null structure the proof employs the generalized energy method of klainerman enhanced by weighted l2 estimates and the ghost weight introduced by alinhac

non autonomous second order hamiltonian systems abstract we study the existence of periodic solutions for a second order non autonomous dynamical system containing variable kinetic energy terms our assumptions balance the interaction between the kinetic energy and the potential energy with neither one dominating the other we study sublinear problems and the existence of non constant solutions

the global smooth symmetric solution to 2 d full compressible euler system of chaplygin gases abstract for one dimensional or multidimensional compressible euler system of polytropic gases it is well known that the smooth solution will generally develop singularities in finite time however for three dimensional chaplygin gases due to the crucial role of null condition in the potential equation which is derived by the irrotational and isentropic flow p godin in 9 has proved the global existence of a smooth 3 d spherically symmetric flow with variable entropy when the initial data are of small smooth perturbations with compact supports to a constant state it is noted that there are some clear differences for the global solution or blowup problems between 2 d and 3 d hyperbolic equations or systems in this paper we will focus on the global symmetric solution problem of 2 d full compressible euler system of chaplygin gases through carrying out involved analysis and finding an appropriate weight we can derive some uniform weighted energy estimates on the small symmetric solution to 2 d compressible euler system of chaplygin gases and further establish the global existence of the smooth solution by the continuous induction method

global existence of null form wave equations on small asymptotically euclidean manifolds abstract we prove the global existence of the small solutions to the cauchy problem for quasilinear wave equations satisfying the null condition on r 3 g where the metric g is a small perturbation of the flat metric and approaches the euclidean metric like 1 x 2 2 with 1 global and almost global existence for systems without the null condition are also discussed for certain small time dependent perturbations of the flat metric in appendix a

the life span of the solutions to cauchy problems for nonlinear wave equations on etudie le comportement des solutions des problemes de cauchy pour des equations d onde non lineaires u f u du ddx u t x er r n t 0 u ef x u t eg x xer n

multiple solutions of perturbed superquadratic second order hamiltonian systems abstract in this paper the author proves the existence of infinitely many distinct t periodic solutions of the perturbed second order hamiltonian systems q v q f t under the condition that v is c and superquadratic via minimax methods we also obtain similar results for general nonautonomous second order hamiltonian systems and perturbed lagrangian systems keywords invariance asymmetry a priori growth estimates multiple periodic solutions

uniform bound of sobolev norms of solutions to 3d nonlinear wave equations with null condition abstract this article concerns the time growth of sobolev norms of classical solutions to the 3d quasi linear wave equations with the null condition given initial data in h s h s 1 with compact supports the global well posedness theory has been established independently by klainerman 13 and christodoulou 3 respectively for a relatively large integer s however the highest order sobolev energy namely the h s energy of solutions may have a logarithmic growth in time in this paper we show that the h s energy of solutions is also uniformly bounded for s 5 the proof employs the generalized energy method of klainerman enhanced by weighted l 2 estimates and the ghost weight introduced by alinhac

global oscillatory waves for second order quasilinear wave equations in this paper we prove the global existence and describe the asymptotic behaviour of a family of oscillatory solutions of cauchy problems for a class of scalar second order quasilinear wave equations when the space dimension is odd and at least equal to 3 if time is bounded corresponding results for quasilinear first order systems were obtained by gues to prove our results we reduce our problems to bounded time problems with the help of a conformal inversion to obtain global results suitable geometric assumptions must be made on the set where the oscillations are concentrated at initial time

abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity critical point theorems and applications to some nonlinear problems with strong resonance at infinity p bartolo and v benci istituto di matematica applicata via re david x0 bari ital and istituto di analisi matematica palazzo ateneo l ia scolai 2 bari italy received in revised form 20 december 1952

infinitely many solutions for hamiltonian systems abstract we consider two classes of the second order hamiltonian systems with symmetry if the systems are asymptotically linear with resonance we obtain infinitely many small energy solutions by minimax technique if the systems possess sign changing potential we also establish an existence theorem of infinitely many solutions by morse theory

global solutions for the gravity water waves equation in dimension 3 we show existence of global solutions for the gravity water waves equation in dimension 3 in the case of small data the proof combines energy estimates which yield control of l 2 related norms with dispersive estimates which give decay in l 8 to obtain these dispersive estimates we use an analysis in fourier space the study of space and time resonances is then the crucial point

infinitely many periodic solutions for second order p q laplacian differential systems abstract some existence theorems are obtained for infinitely many periodic solutions for second order p q laplacian differential systems by using a general variational principle due to b ricceri

global existence of classical solutions to the cauchy problem for a kind of quasilinear hyperbolic system abstract this paper deals with the cauchy problem for a kind of quasilinear hyperbolic system with the linear elastodynamic system as its principal part and the nonlinear terms depending on quadratic terms of the first derivatives of u a system of new null conditions is given by which the author shows the global existence of the problem

codimension one stability of the catenoid under the vanishing mean curvature flow in minkowski space we study timelike hypersurfaces with vanishing mean curvature in the 3 1 dimensional minkowski space which are the hyperbolic counterparts to minimal embeddings of riemannian manifolds the catenoid is a stationary solution of the associated cauchy problem this solution is linearly unstable and we show that this instability is the only obstruction to the global nonlinear stability of the catenoid more precisely we prove in a certain symmetry class the existence in the neighborhood of the catenoid initial data of a codimension one lipschitz manifold transverse to the unstable mode consisting of initial data whose solutions exist globally in time and converge asymptotically to the catenoid

mixed problems for fully nonlinear hyperbolic equations 1 i n the summation convention is used here i j n f 2 c l r is a bounded domain with smooth boundary f v the outer normal f2t 0 t x f2 and ft 0 t x f 0 t oo f and g satisfy some natural conditions which assure hyperbolicity the main result is concerned with classical solutions for small t assume that uoe ws l f2 n and ul e ws f2 n where s is an integer bigger than 1 n 2 this implies uo e c2 0 n under suitable assumptions the following holds

explicit predictor multicorrector time discontinuous galerkin methods for non linear dynamics abstract explicit predictor multicorrector time discontinuous galerkin tdg methods developed for linear structural dynamics are formulated and implemented in a form suitable for arbitrary non linear analysis of structural dynamics problems the formulation is intended to inherit the accuracy properties of the exact parent implicit tdg methods to this end suitable predictors and correctors are designed to achieve third order accuracy large stability limits and controllable numerical dissipation by means of an algorithmic parameter as the study of a general non linear case is rather complex the analysis of the convergence properties of the resulting algorithms are restricted to conservative duffing oscillators for which closed form solutions are available it is shown that the main properties of the underlying parent scheme can be retained finally results of representative numerical simulations relevant to duffing oscillators and to a stiff spring pendulum discretized with finite elements illustrate the performance of the numerical schemes and confirm the analytical estimates

global smooth axisymmetric solutions to 2d compressible euler equations of chaplygin gases with non zero vorticity abstract for 2d compressible isentropic euler equations of polytropic gases s alinhac establishes that the axisymmetric smooth solution blows up in the present paper for 2d compressible isentropic euler equations of chaplygin gases we show that the small perturbed smooth solution exists globally when the rotationally invariant initial data are a perturbation of size e 0 of a rest state near the light cone 2d euler equations of chaplygin gases can be transformed into a second order quasilinear wave equation which satisfies both the null conditions this will lead to that the corresponding second order quasilinear wave equation admits a global smooth solution near the light cone however away from the light cone the hydrodynamical waves have no decay in time and strongly affect the related acoustical waves by introducing a nonlinear ode we can distinguish the fast decay part and non decay part so that the global energy estimates are derived

homoclinic solutions for a class of second order hamiltonian systems abstract some new results for the existence of homoclinic orbits are obtained for the second order hamiltonian systems u t v t u t f t where t r u r n and v c 1 r r n r v t x k t x w t x is t periodic with respect to t t 0 and f r r n is a continuous and bounded function our assumptions on k and f are rather loose

on the global existence and stability of a multi dimensional supersonic conic shock wave we establish the global existence and stability of a three dimensional supersonic conic shock wave for a compactly perturbed steady supersonic flow past an infinitely long circular cone with a sharp angle the flow is described by a 3 d steady potential equation which is multi dimensional quasilinear and hyperbolic with respect to the supersonic direction making use of the geometric properties of the pointed shock surface together with the rankine hugoniot conditions on the conic shock surface and the boundary condition on the surface of the cone we obtain a global uniform weighted energy estimate for the nonlinear problem by finding an appropriate multiplier and establishing a new hardy type inequality on the shock surface based on this we prove that a multi dimensional conic shock attached to the vertex of the cone exists globally when the mach number of the incoming supersonic flow is sufficiently large moreover the asymptotic behavior of the 3 d supersonic conic shock solution which is shown to approach the corresponding background shock solution in the downstream domain for the uniform supersonic constant flow past the sharp cone is also explicitly given

uniform bound of the highest energy for the three dimensional incompressible elastodynamics this article concerns the time growth of sobolev norms of classical solutions to the three dimensional incompressible isotropic elastodynamics with small initial displacements given initial data in h k lambda for a fixed big integer k the global well posedness of this cauchy problem has been established by sideris and thomases in commun pure appl math 58 6 750 788 2005 and j hyperbolic differ equ 3 4 673 690 2006 commun pure appl math 60 12 1707 1730 2007 where the highest order generalized energy ek t may have a certain growth in time alinhac conjectured that such a growth in time may be a true phenomenon in geometric analysis of hyperbolic differential equations an introduction lecture note series 374 mathematical society london he proved that ek t is still uniformly bounded in time only for the three dimensional scalar quasilinear wave equation under a null condition in this paper we show that the highest order generalized energy ek t is still uniformly bounded for the three dimensional incompressible isotropic elastodynamics the equations of incompressible elastodynamics can be viewed as nonlocal systems of wave type and are inherently linearly degenerate in the isotropic case there are three ingredients in our proof the first is that we still have a decay rate of t frac 3 2 when we do the highest energy estimate away from the light cone even though in this case the lorentz invariance is not available the second one is that the l infty norm of the good unknowns in particular nabla v g omega is shown to have a decay rate of t frac 3 2 near the light cone the third one is that the pressure is estimated in a novel way as a nonlocal nonlinear term with null structure as has been recently observed in 16 the proof employs the generalized energy method of klainerman enhanced by weighted l2 estimates and the ghost weight introduced by alinhac

periodic solutions for second order systems with not uniformly coercive potential abstract the existence and multiplicity of periodic solutions are obtained for the nonautonomous second order systems with locally coercive potential that is f t x as x for a e t in some positive measure subset of 0 t by using an analogy of egorov s theorem the properties of subadditive functions the least action principle and a three critical point theorem proposed by brezis and nirenberg

nonresonance and global existence of prestressed nonlinear elastic waves this article considers the existence of global classical solutions to the cauchy problem in nonlinear elastodynamics the unbounded elastic medium is assumed to be homogeneous isotropic and hyperelastic as in the theory of 3d nonlinear wave equations in three space dimensions global existence hinges on two basic assumptions first the initial deformation must be a small displacement from equilibrium in this case a prestressed homogeneous dilation of the reference conflguration and equally important the nonlinear terms must obey a type of nonresonance or null condition the omission of either of these assumptions can lead to the breakdown of solutions in flnite time in particular nonresonance complements the genuine nonlinearity condition of f john under which arbitrarily small spherically symmetric displacements develop singularities although one expects this to carry over to the nonsymmetric case as well 4 john also showed that small solutions exist almost globally 5 see also 10 formation of singularities for large displacements was illustrated by tahvildar zadeh 16 the nonresonance condition introduced here represents a substantial improvement over our previous work on this topic 13 to explain the difierence roughly our earlier version of the null condition forced the cancellation of all nonlinear wave interactions to flrst order along the characteristic cones here only the cancellation of nonlinear wave interactions among individual wave families is required the di culty in realizing this weaker version is that the decomposition of elastic waves into their longitudinal and transverse components involves the nonlocal helmholtz projection which is ill suited to nonlinear analysis however our decay estimates make clear that only the leading contribution of the resonant interactions along the characteristic cones is potentially dangerous and this permits the usage of approximate local decompositions

on nonlinear elastic waves in 2 d abstract for the cauchy problem of 2 d nonlinear elastic waves of isotropic homogeneous and hyperelastic materials satisfying null conditions assuming the smallness of h 3 r 2 h 2 r 2 norm and the boundedness of h 4 r 2 h 3 r 2 norm for radially symmetric initial data we show the global existence of classical solutions similar results for exterior domain problems are also proved

on one dimension semi linear wave equations with null conditions abstract it is well known that in space dimensions at least three semilinear wave equations with null conditions admit global solutions for small initial data it is also known that in space dimension two such result still holds for a certain class of quasi linear wave equations with null conditions the proofs are based on the decay mechanism of linear waves however in one space dimension waves do not decay nevertheless we will prove that small data still lead to global solutions if the null condition is satisfied

the unit condition and global existence for a class of nonlinear klein gordon equations abstract global existence for a special class of nonlinear klein gordon equations in two space dimensions is proved in particular it is shown that for small initial data the equation tt u u u u 2 has a global solution

nonlinear fractional schrodinger equations in one dimension abstract we consider the question of global existence of small smooth and localized solutions of a certain fractional semilinear cubic nls in one dimension i t u u c 0 u 2 u c 1 u 3 c 2 u u 2 c 3 u 3 x x 1 2 where c 0 r and c 1 c 2 c 3 c this model is motivated by the two dimensional water wave equation which has a somewhat similar structure in the eulerian formulation in the case of irrotational flows we show that one cannot expect linear scattering even in this simplified model more precisely we identify a suitable nonlinear logarithmic correction and prove global existence and modified scattering of solutions

global existence for small initial data in the born infeld equations we prove global existence of a classical solution for small initial in the cauchy problem of the born infeld system describing nonlinear electromagnetism for the proof we crucially use the null form structure of the the nonlinear terms under the lorentz gauge condition

well posedness for a quadratic derivative nonlinear schrodinger system at the critical regularity abstract we consider the cauchy problem for a quadratic derivative nonlinear schrodinger system in sobolev space h s in this paper under the null condition and the mass resonance relation we prove large data local well posedness if the space dimension d 3 and s s c d 2 and s s c or d 1 and s 0 where s c is the scaling critical regularity moreover under the same assumptions we also prove small data global well posedness and scattering in h s if d 3 and s s c our proof is based on a contraction argument using the u p and v p type function spaces

global solutions of the euler maxwell two fluid system in 3d the fundamental two uid model for describing plasma dynamics is given by the euler maxwell system in which compressible ion and electron uids interact with their own self consistent electromagnetic eld we prove global stability of a constant neutral background in the sense that irrotational smooth and localized perturbations of a constant background with small amplitude lead to global smooth solutions in three space dimensions for the euler maxwell system our construction is robust in dimension 3 and applies equally well to other plasma models such as the euler poisson system for two uids and a relativistic euler maxwell system for two uids our solutions appear to be the rst nontrivial global smooth solutions in all of these models

periodic and subharmonic solutions of second order hamiltonian systems abstract some solvability conditions of periodic solutions and subharmonic solutions are obtained for a class of new superquadratic non autonomous second order hamiltonian systems by the minimax methods in critical point theory

existence and multiplicity of periodic solutions for a class of second order hamiltonian systems we study the existence and multiplicity of periodic solutions of the following second order hamiltonian system email protected t email protected k t u t w t u t 0 the existence of a nontrivial periodic solution is obtained when w is asymptotically linear at infinity and the existence of infinitely many periodic solutions is also obtained when w is superlinear

global existence of radial solutions for general semilinear hyperbolic systems in 3d abstract we study the well posedness of radial solutions for general nonlinear hyperbolic systems in three dimensions we give a proof of the global existence of radial solutions for general semilinear hyperbolic systems in 3d under null condition with small scaling invariant w 2 1 r 3 data we obtain a bilinear estimate that is effective for hyperbolic systems which do not have any time decay it allows us to achieve the boundedness of the weighted bv norm of the radial solutions

remarks on global solutions for nonlinear wave equations under the standard null conditions abstract a combination of some weighted energy estimates is applied for the cauchy problem of quasilinear wave equations with the standard null conditions in three spatial dimensions alternative proofs for global solutions are shown including the exterior domain problems

the small data solutions of general 3 d quasilinear wave equations ii abstract this paper is a continuation of the work in 8 where the authors established the global existence of smooth small data solutions to the general 3 d quasilinear wave equation i j 0 3 g i j u u i j 2 u 0 when the weak null condition holds in the present paper we show that the smooth small data solutions of equation i j 0 3 g i j u u i j 2 u 0 will blow up in finite time when the weak null condition does not hold and a generic nondegenerate condition on the initial data is satisfied moreover a precise blowup time is completely determined therefore collecting the main results in this paper and 8 we have given a basically complete study on the blowup or global existence of small data solutions to the 3 d quasilinear wave equation i j 0 3 g i j u u i j 2 u 0

nontrivial periodic solutions for asymptotically linear hamiltonian systems with resonance abstract in this paper we establish some existence results of nontrivial 1 periodic solutions to the first order asymptotically linear hamiltonian systems under the assumptions that the linear operators have different maslove indices at the origin and at infinity the results obtained in this paper are also valued for the case that both of the asymptotic matrices may be degenerate resonant type and time dependent

global solutions of nonlinear wave equations in time dependent inhomogeneous media we consider the problem of small data global existence for a class of semilinear wave equations with null condition on a lorentzian background mathbb r 3 1 g with a time dependent metric g coinciding with the minkowski metric outside the cylinder t x x leq r we show that the small data global existence result can be reduced to two integrated local energy estimates and demonstrate that these estimates work in the particular case when g is merely c1 close to the minkowski metric one of the novel aspects of this work is that it applies to equations on backgrounds which do not settle to any particular stationary metric

the cauchy problem for the pavlov equation with large data abstract we prove a local solvability of the cauchy problem for the pavlov equation with large initial data by the inverse scattering method the pavlov equation arises in studies einstein weyl geometries and dispersionless integrable models our theory yields a local solvability of cauchy problems for a quasi linear wave equation with a characteristic initial hypersurface

global existence for systems of wave equations with nonresonant nonlinearities and null forms abstract we consider the cauchy problem for systems of nonlinear wave equations with different propagation speeds in three space dimensions we prove global existence of small amplitude solutions for systems with some nonresonant nonlinearities which may depend on both of the unknowns and their derivatives our method here can be also adopted to treat the null forms

global solutions for incompressible viscoelastic fluids we prove the existence of both local and global smooth solutions to the cauchy problem in the whole space and the periodic problem in the n dimensional torus for the incompressible viscoelastic system of oldroyd b type in the case of near equilibrium initial data the results hold in both two and three dimensional spaces the results and methods presented in this paper are also valid for a wide range of elastic complex fluids such as magnetohydrodynamics liquid crystals and mixture problems

almost global existence for quasilinear wave equations in three space dimensions this article studies almost global existence for solutions of quadratically quasi linear systems of wave equations in three space dimensions the approach here uses only the classical invariance of the wave operator under translations spatial rotations and scaling using these techniques we can handle wave equations in minkowski space or dirichlet wave equations in the exterior of a smooth star shaped obstacle we can also apply our methods to systems of quasilinear wave equations having different wave speeds this extends our work 11 for the semilinear case previous almost global ex istence theorems for quasilinear equations in three space dimensions were for the non obstacle case in 9 john and klainerman proved almost global existence on minkowski space for quadratic quasilinear equations using the lorentz invariance of the wave operator in addition to the symmetries listed above subsequently in 14 klainerman and sideris obtained the same result for a class of quadratic divergence form nonlinearities without relying on lorentz invariance this line of thought was refined and applied to prove global in time results for null form equa tions related to the theory of elasticity in sideris 22 23 and for multiple speed systems of null form quasilinear equations in sideris and tu 24 and yokoyama 29 the main difference between our approach and the earlier ones is that we ex ploit the 0 x 1 decay of solutions of wave equations with sufficiently decaying initial data as much as we involve the stronger 0 t l decay here of course x x x2 x is the spatial component and t the time component of a space time vector t x g m x e3 establishing 0 x 1 decay is considerably easier and can be achieved using only the invariance with respect to translations and spatial rotation a weighted l2 space time estimate for inhomogeneous wave equations proposition 3 1 below from 11 is important in making the spatial decay useful for the long time existence argument for semilinear systems one can show almost global existence from small data using only this spatial decay 11 for quasilinear systems however we also have to show that both first and second derivatives of u decay like 1 t fortunately we can do this using a variant of some l1 l estimates of john h rmander

multiplicity of periodic solutions for a class of second order perturbed hamiltonian systems abstract consider the following second order perturbed hamiltonian systems u t f u u g t u t r u 0 u t u 0 u t 0 t 0 where f u k u w u k w c 2 r n r g c 2 r r n r with g t t u g t u k and w is even but g t u maybe has no parity in u assume that w is superquadratic and g is subquadratic at the infinity and k satisfies pinched condition the existence of infinitely many periodic solutions of the above perturbed systems is proved via the bolle s perturbation method which is a generalization of 21 theorem 1 2 and 3 theorem 2 in some sense

space time resonances and the null condition for first order systems of wave equations in this manuscript we prove global existence and linear asymptotic behavior of small solutions to nonlinear wave equations we assume that the quadratic part of the nonlinearity satisfies a nonresonance condition that is a generalization of the null condition given by klainerman 2013 wiley periodicals inc

semilinear hyperbolic systems violating the null condition we consider systems of semilinear wave equations in three space dimensions with quadratic nonlinear terms not satisfying the null condition we prove small data global existence of the classical solution under a new structural condition related to the weak null condition for two component systems satisfying this condition we also observe a new kind of asymptotic behavior only one component is dissipated and the other one behaves like a non trivial free solution in the large time

global existence of small solutions to the initial value problem for nonlinear thermoelasticity abstract we prove the existence of global small solutions to the initial value problem in three dimensional nonlinear thermoelasticity with quadratic completely coupled nonlinear terms the asymptotic behaviour of the solution as t is also described

a periodic solution for a second order asymptotically linear hamiltonian system abstract we study the existence of a periodic solution of the following second order hamiltonian system u t f t u t 0 t 0 t where f t x k t x w t x assuming that k satisfies the pinching condition and w is asymptotically linear at infinity the existence of a nontrivial periodic solution is obtained via the mountain pass theorem

the global stability of minkowski space time in harmonic gauge we give a new proof of the global stability of minkowski space originally established in the vacuum case by christodoulou and klainerman the new approach which relies on the classical harmonic gauge shows that the einstein vacuum and the einstein scalar field equations with asymptotically flat initial data satisfying a global smallness condition produce global causally geodesically complete solutions asymptotically convergent to the minkowski space time

the null condition for quasilinear wave equations in two space dimensions ii we consider quasilinear wave equations with small data of size e in two space dimensions assuming the null condition is satisfied we proved in a previous paper the conjectured large time existence result in this paper using the geometric blowup techniques introduced in our previous work we prove that under generic conditions on the data blowup does occur at the specified time more precisely the blowup can be described as a geometric blowup of cusp type uniformly in e in appropriate variables taking advantage of the techniques introduced in the paper we also prove the blowup result for the technically similar case of quasilinear wave equations without null condition in three space dimensions

global solutions of nonlinear hyperbolic equations for small initial data on considere des systemes quasilineaires d equations hyperboliques d ordre 2 qui sont des deformations non lineaires de l equation d onde on etudie l existence globale et le comportement asymptotique des solutions du probleme de cauchy quand la donnee initiale est suffisamment petite

global solutions of quasilinear wave equations we show global small data existence for a class of quasilinear wave equations related to einstein s equations in harmonic coordinates these equations do not satisfy the classical null condition and the asymptotic behavior of solutions is not free but the light cones bend at infinity

global existence of null form wave equations in exterior domains we provide a proof of global existence of solutions to quasilinear wave equations satisfying the null condition in certain exterior domains in particular our proof does not require estimation of the fundamental solution for the free wave equation we instead rely upon a class of keel smith sogge estimates for the perturbed wave equation using this a notable simplification is made as compared to previous works concerning wave equations in exterior domains one no longer needs to distinguish the scaling vector field from the other admissible vector fields

periodic and subharmonic solutions of a class of subquadratic second order hamiltonian systems some solvability conditions of periodic and subharmonic solutions are obtained for a class of subquadratic nonautonomous second order hamiltonian systems by the minimax methods in critical point theory

harmonic maps in three dimensional minkowski space with image near geodesics shatah i sh has constructed global weak solutions of the nonlinear sigma model in h1 2 for arbitrary dimension moreover for the case of maps from four dimensional minkowski space r 3 1 to sphere s 3 he has found an example of smooth symmetric initial data for which the corresponding solution develops singularities on the second order derivatives in finite time showing that the large initial data may contribute the formation of singularities to avoid this situation sideris i si proved the existence of global smooth solution from r 3 1 into an n dimensional complete riemannian manifold n as a perturbation of a special class of harmonic maps geodesic geometrically the solution lies in a small tubular region of a given geodesic on n instead of a small neighborhood of a point on n therefore the initial data is partly small the purpose of this paper is to extend sideris s result to the case where domain is the minkowski space r 2 1 in this case the existence of smooth solution of small amplitude was proved by kovalyov ko but ours is more general since the parameter along the geodesic may vary freely formally a harmonic map from the minkowski space r 2 1 into an n dimensional complete riemannian manifold n h is described as a critical point r 2 1 n of the energy integral

the dynamics of relativistic strings moving in the minkowski space mathbb r 1 n in this paper we investigate the dynamics of relativistic in particular closed strings moving in the minkowski space mathbb r 1 n n ge 2 we first derive a system with n nonlinear wave equations of born infeld type which governs the motion of the string this system can also be used to describe the extremal surfaces in mathbb r 1 n we then show that this system enjoys some interesting geometric properties based on this we give a sufficient and necessary condition for the global existence of extremal surfaces without space like point in mathbb r 1 n with given initial data this result corresponds to the global propagation of nonlinear waves for the system describing the motion of the string in mathbb r 1 n we also present an explicit exact representation of the general solution for such a system moreover a great deal of numerical analyses are investigated and the numerical results show that in phase space various topological singularities develop in finite time in the motion of the string finally some important discussions related to the theory of extremal surfaces of mixed type in mathbb r 1 n are given