nilpotent property of skew generalized power series rings let r be an ni ring with nil r nilpotent s a strictly totally ordered monoid satisfying the condition that 0 s for all s s a nd s end r be a compatible monoid homomorphism then the ring r s of the skew generalized power series with coefficients in r and exponents in s is a nilpotent p p ring if and only if r is a nilpotent p p ring furthermore if r satisfies the descending chain condition on weak annihilators then r s is a weak app ring if and only if r is a weak app ring consequently some properties of the base ring r can be profitably generalized to the skew generalized power series ring r s

weighted strichartz estimates for the schrodinger and wave equations on damek ricci spaces we prove strichartz estimates for radial solutions of the schrodinger and wave equations on damek ricci spaces and in particular on symmetric spaces of noncompact type and rank one using the perturbative theory with potentials the curvature of the noncompact manifold has an influence on the dispersive properties and indeed we obtain strichartz estimates with weights at spatial infinity which are stronger than the standard ones in the flat case

nonexistence of global solutions to critical semilinear wave equations in exterior domain in high dimensions abstract the aim of this paper is to study the long time behavior of solutions to the initial boundary value problem of critical semilinear wave equations with small data in exterior domain in high dimensions n 5 we prove that solutions cannot exist globally in time the novelty is that we first use a cutoff function to transform the exterior problem to a cauchy problem then we divide the time into two intervals in one time interval we establish an energy estimate to control the error terms while in the other interval the asymptotic behavior of two test functions is used furthermore we obtain the upper bound of the lifespan by following the idea of zhou and han 2014

blow up of solutions to semilinear wave equations with variable coefficients and boundary this paper is devoted to studying initial boundary value problems for semilinear wave equations and derivative semilinear wave equations with variable coefficients on exterior domain with subcritical exponents in n space dimensions we will establish blow up results for the initial boundary value problems it is proved that there can be no global solutions no matter how small the initial data are and also we give the life span estimate of solutions for the problems

blow up for a weakly coupled system of semilinear damped wave equations in the scattering case with power nonlinearities abstract in this work we study the blow up of solutions of a weakly coupled system of damped semilinear wave equations in the scattering case with power nonlinearities we apply an iteration method to study both the subcritical case and the critical case in the subcritical case our approach is based on lower bounds for the space averages of the components of local solutions in the critical case we use the slicing method and a couple of auxiliary functions recently introduced by wakasa yordanov to modify the definition of the functionals with the introduction of weight terms in particular we find as critical curve for the pair p q of the exponents in the nonlinear terms the same one as for the weakly coupled system of semilinear wave equations with power nonlinearities

small data blow up for semilinear klein gordon equations energy heuristics predict global existence from small data for semi linear klein gordon systems when the nonlinearity has order p 1 2 n we complement these heuristics and previous rigorous existence results by demonstrating blow up from small data for p 1 2 n

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

concerning l p resolvent estimates for simply connected manifolds of constant curvature abstract we prove families of uniform l r l s resolvent estimates for simply connected manifolds of constant curvature negative or positive that imply the earlier ones for euclidean space of kenig ruiz and the second author 7 in the case of the sphere we take advantage of the fact that the half wave group of the natural shifted laplacian is periodic in the case of hyperbolic space the key ingredient is a natural variant of the stein tomas restriction theorem

weighted strichartz estimates and global existence for semilinear wave equations in this paper we prove a sharp global existence theorem in all dimensions for nonlinear wave equations with power type nonlinearities the proof is based on a weighted strichartz estimate involving powers of the lorentz distance

an elementary proof of strauss conjecture abstract in this paper we consider the cauchy problem for semilinear wave equation in high dimensions first we use a morawetz energy estimate which is obtained by integrating on the light cone to get a weighted l 2 l 2 estimate of the solution and then give an elementary proof of the weighted strichartz estimate in georgiev et al 3 hence the strauss conjecture we also obtain a variant of the weighted strichartz estimates and give the sharp estimate of the lifespan for the semilinear wave equation with subcritical nonlinearity

period recovery of strings over the hamming and edit distances a string t of length m is periodic in p of length p if p is a substring of t and t i t i p t i t i p for all 0 i m p 1 0 i m p 1 and m 2p m 2 p the shortest such prefix p is called the period of t i e p t 0 p 1 p t 0 p 1 in this paper we investigate the period recovery problem given a string s of length n find the primitive period s p such that the distance between s and a string t that is periodic in p is below a threshold we consider the period recovery problem over both the hamming distance and the edit distance for the hamming distance case we present an o nlog n o n log n time algorithm where is given as n 2 p for 0 0 for the edit distance case n 3 75 p and 0 0 we provide an o n 4 3 o n 4 3 time algorithm

finite time blow up for critical wave equations in high dimensions we prove that solutions to the critical wave equation 1 1 with dimension n 4 can not be global if the initial values are positive somewhere and nonnegative this completes the solution to the famous strauss conjecture about semilinear wave equations of the form u t2u u p 0 the rest of the cases the lower dimensional case n 3 and the sub or super critical cases were settled many years earlier by the work of several authors

on the existence and nonexistence of global solutions for certain semilinear exterior problems with nontrivial robin boundary conditions abstract we consider three types of semilinear equations elliptic parabolic and hyperbolic posed in the n dimensional exterior domain r n d where n 2 and d is the closed unit ball in r n a nontrivial robin boundary condition is imposed on the boundary of d using a test function approach with judicious choices of the test functions we show that the considered problems share a common critical behavior we discuss separately the cases n 2 and n 3 moreover in the case n 3 the dependence of the critical exponent on initial data is discussed to the best our knowledge the study of the critical behavior in an exterior domain with a nontrivial robin boundary condition has never been studied in the literature

global existence for the dissipative wave equations with space time dependent potential abstract this research paper aims at studying the long time behavior of solutions for wave equations with focusing nonlinearity we consider p c r n p n 2 n 2 and separable potential a t x x t with x 1 x and t 1 t satisfying conditions 3 4 which are given below the main result is that if the exponent p lies between p c r 1 4 1 2 n 1 2 p n 2 n 2 the solutions are global for sufficiently small data also the precise decay estimates for the energy l 2 and l p 1 norms of solutions in the region of global solutions have been found

adams inequalities for riesz subcritical potentials abstract we derive adams inequalities for potentials on general measure spaces extending and improving previous results in fontana and morpurgo 2011 the integral operators involved which we call riesz subcritical have kernels whose decreasing rearrangements are not worse than that of the riesz kernel on r n where the kernel is large but they behave better where the kernel is small the new element is a critical integrability condition on the kernel at infinity typical examples of such kernels are fundamental solutions of nonhomogeneous differential or pseudo differential operators another example is the riesz kernel itself restricted to suitable measurable sets which we name riesz subcritical domains such domains are characterized in terms of their growth at infinity as a consequence of the general results we obtain several new sharp adams and moser trudinger inequalities on r n on the hyperbolic space on riesz subcritical domains and on domains where the poincare inequality holds

a shift in the strauss exponent for semilinear wave equations with a not effective damping abstract in this note we study the global existence of small data solutions to the cauchy problem for the semilinear wave equation with a not effective scale invariant damping term namely v t t v 2 1 t v t v p v 0 x v 0 x v t 0 x v 1 x where p 1 n 2 we prove blow up in finite time in the subcritical range p 1 p 2 n and existence theorems for p p 2 n n 2 3 in this way we find the critical exponent for small data solutions to this problem our results lead to the conjecture p 2 n p 0 n 2 for n 2 where p 0 n is the strauss exponent for the classical semilinear wave equation with power nonlinearity

generalized strichartz inequalities for the wave equation strichartz inequalities for the wave equation u f are estimates of the solution u of the cauchy problem for that equation in the form of space time integral norms in terms of similar norms of the inhomogeneity f and of suitable norms of the initial data

on existence and scattering with minimal regularity for semilinear wave equations abstract we prove existence and scattering results for semilinear wave equations with low regularity data we also determine the minimal regularity that is needed to ensure local existence and well posedness and we give counterexamples to well posedness more specifically we show that equations of the type u u p with initial data u u t in r n 1 r n have a local solution if p n and we construct counterexamples if p n the existence results rely on mixed norm space time estimates of strichartz type

special properties of generalized power series abstract this is a sequel to my previous papers on generalized power series for the convenience of the reader i gather in the first section the definitions and results which shall be required any missing proof is either very easy or is already in one of the above quoted papers after the preliminaries i characterize under suitable conditions the generalized pow er series which are powers the essential idea is to extend the validity of the usual binomial series a short section gives conditions for a ring of generalized pou er series to be a real ring as known the ring of usual power series with coefficients in a field in any number of indeterminates is a unique factorization domain i show that the result holds for generalized power series with exponents in a free ordered monoid which is noetherian and narrow this leads to interesting examples of unique factorization domains completely integrally closed domains of generalized power series are also characterized in terms of their ring of coefficients and monoid of exponents the final section is devoted to seminormal domains the main results about usual power series are extended to generalized power series

krull domains of generalized power series satisfies the acc on cyclic submonoids on the otherhand there are at least three distinct power series rings in infinitely manyindeterminates in the literature in 5 gilmer proved that each of thesethree types of power series rings in infinitely many indeterminates is also akrull domain in case the coefficient ring is a krull domain in 10 ribenboim investigated the following question if a ring

existence and blow up of small amplitude nonlinear waves with a sign changing potential abstract we study the nonlinear wave equation with a sign changing potential in any space dimension if the potential is small and rapidly decaying then the existence of small amplitude solutions is driven by the nonlinear term if the potential induces growth in the linearized problem however solutions that start out small may blow up in finite time

combined effects of two nonlinearities in lifespan of small solutions to semi linear wave equations this paper investigates the combined effects of two distinctive power type nonlinear terms with parameters p q 1 in the lifespan of small solutions to semi linear wave equations we determine the full region of p q to admit global existence of small solutions at least for spatial dimensions n 2 3 moreover for many p q when there is no global existence we obtain sharp lower bound of the lifespan which is of the same order as the upper bound of the lifespan

nonexistence of global solutions to semilinear wave equations in high dimensions on considere le probleme de cauchy pour l equation d onde semi lineaire 2 t 2 u x t u x t p p 1 u x o f x u t x o g x avec f g c o r n

semilinear schrodinger flows on hyperbolic spaces scattering in h 1 we prove global well posedness and scattering in h1 for the defocusing nonlinear schrodinger equations left begin array ll i partial t delta g u u u 2 sigma u 0 phi end array right on the hyperbolic spaces mathbb h d d 2 for exponents sigma in 0 2 d 2 the main unexpected conclusion is scattering to linear solutions in the case of small exponents for comparison on euclidean spaces scattering in h1 is not known for any exponent sigma in 1 d 2 d and is known to fail for sigma in 0 1 d our main ingredients are certain noneuclidean global in time strichartz estimates and noneuclidean morawetz inequalities

on abstract strichartz estimates and the strauss conjecture for nontrapping obstacles we establish the strauss conjecture concerning small data global existence for nonlinear wave equations in the setting of exterior domains to compact obstacles for space dimensions n 3 and 4 the obstacle is assumed to be nontrapping and the solution is assumed to satisfy either dirichlet or neumann conditions along the boundary of the obstacle the key step in the proof is establishing certain abstract strichartz estimates for the linear wave equation on exterior domains

long time existence for small amplitude semilinear wave equations we prove that a certain class of semilinear wave equations has global solutions if the initial data is small these existence results rely on mixed norm angular radial space time estimates more specifically we consider power nonlinearities u f p u where f p u u p and 2 t denotes the d alembertian on inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i in 1979 john investigated this equation for small initial data in dimension n 3 he proved that if p 1 2 then there is a global solution whereas for p n dimensions and the critical power p c above which global solutions exist should be the positive root of n 1 p 2 c n 1 p c 2 0 sideris proved that for all n the solutions can blow up if p p c recently zhou proved that for n 4 one has global solutions if p p c 2 here we prove that for n 8 one has global solutions if p p c furthermore we proved that for all n one has global solutions if p p c and the initial data are spherically symmetric in the radial case we use l p estimates derived from the explicit form of the fundamental solution using inequalities from classical analysis in the nonradial case we write the fundamental solution in polar coordinates and use fourier integral and special function estimates to handle the angular part

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

weighted strichartz estimates for the wave equation in even space dimensions we prove the weighted strichartz estimates for the wave equation in even space dimensions with radial symmetry in space although the odd space dimensional cases have been treated in our previous paper 5 the lack of the huygens principle prevents us from a similar treatment in even space dimensions the proof is based on the two explicit representations of solutions due to rammaha 11 and takamura 14 and to kubo kubota 6 as in the odd space dimensional cases 5 we are also able to construct self similar solutions to semilinear wave equations on the basis of the weighted strichartz estimates

concerning the strauss conjecture on asymptotically euclidean manifolds abstract in this paper we verify the strauss conjecture for semilinear wave equations on asymptotically euclidean manifolds when n 3 4 we also give an almost sharp lifespan for the subcritical case 2 p p c when n 3 the main ingredients include a keel smith sogge type estimate with 0 1 2 and weighted strichartz estimates of order two

nonlinear waves on 3d hyperbolic space in this article global in time dispersive estimates and strichartz estimates are explored for the wave equation on three dimensional hyperbolic space due to the negative curvature extra dispersion is noted as compared to the euclidean case and a wider range of strichartz estimates is proved using these small data global existence to semilinear wave equations is shown for a range of powers that is broader than that known for euclidean space

wave equation on riemannian symmetric spaces we consider the wave equation on riemannian symmetric spaces of the noncompact type we prove that the solutions of the homogeneous equation have dispersion properties and we deduce strichartz type estimates for these solutions we use these estimates to treat nonlinear equations with power type as a byproduct we prove the global well posedness for small energy

regular articlegeneralized strichartz inequalities for the wave equation we make a synthetic exposition of the generalized strichartz inequalities for the wave equation obtained in 6 together with the limiting cases recently obtained in 13 with as simple proofs as possible the proofs combine stationary phase estimates dyadic decompositions the hardy littlewood inequality or the holder inequality in time and abstract duality and interpolation arguments

profile decompositions for wave equations on hyperbolic space with applications the goal for this paper is twofold our first main objective is to develop bahouri gerard type profile decompositions for waves on hyperbolic space recently such profile decompositions have proved to be a versatile tool in the study of the asymptotic dynamics of solutions to nonlinear wave equations with large energy with an eye towards further applications we develop this theory in a fairly general framework which includes the case of waves on hyperbolic space perturbed by a time independent potential our second objective is to use the profile decomposition to address a specific nonlinear problem namely the question of global well posedness and scattering for the defocusing energy critical semi linear wave equation on three dimensional hyperbolic space possibly perturbed by a repulsive time independent potential using the concentration compactness rigidity method introduced by kenig and merle we prove that all finite energy initial data lead to a global evolution that scatters to linear waves as t rightarrow pm infty this proof serves as a blueprint for the arguments in the work 45 by the authors where we study the asymptotic behavior of large energy equivariant wave maps on the hyperbolic plane

bounded solutions to an energy subcritical non linear wave equation on r3 abstract in this work we consider an energy subcritical semi linear wave equation 3 p 5 t 2 u u x u p 1 u x t r 3 r u t 0 u 0 h s p r 3 t u t 0 u 1 h s p 1 r 3 where s p 3 2 2 p 1 and the function r 3 1 1 is a radial continuous function with a limit at infinity we prove that unless the elliptic equation w x w p 1 w has a nonzero radial solution w c 2 r 3 h s p r 3 any radial solution u with a finite uniform upper bound on the critical sobolev norm u t t u t h s p h s p r 3 for all t in the maximal lifespan must be a global solution in time and scatter

fourier integral operators on noncompact symmetric spaces of real rank one let x g k be a noncompact symmetric space of real rank one the purpose of this paper is to investigate lp boundedness properties of a certain class of radial fourier integral operators on the space x we will prove that if u is the solution at some fixed time of the natural wave equation on x with initial data f and g and 1 p then u lp x cp f lpbp x 1 g lpbp 1 x we will obtain both the precise behavior of the norm cp and the sharp regularity assumptions on the functions f and g i e the exponent bp that make this inequality possible in the second part of the paper we deal with the analog of e m stein s maximal spherical averages and prove exponential decay estimates of a highly non euclidean nature on the lp norm of supt t 1 f d z where d is a normalized spherical measure

the strauss conjecture on kerr black hole backgrounds we examine solutions to semilinear wave equations on black hole backgrounds and give a proof of an analog of the strauss conjecture on the schwarzschild and kerr with small angular momentum black hole backgrounds the key estimates are a class of weighted strichartz estimates which are used near infinity where the metrics can be viewed as small perturbations of the minkowski metric and a localized energy estimate on the black hole background which handles the behavior in the remaining compact set

the sharp sobolev type inequalities in the lorentz sobolev spaces in the hyperbolic spaces abstract let w 1 l p q h n 1 q p denote the lorentz sobolev spaces of order one in the hyperbolic spaces h n our aim in this paper is three fold first of all we establish a sharp poincare inequality in w 1 l p q h n with 1 q p which generalizes the result in 41 to the setting of lorentz sobolev spaces second we prove several sharp poincare sobolev type inequalities in w 1 l p q h n with 1 q p n which generalize the results in 45 to the setting of lorentz sobolev spaces finally we provide the improved moser trudinger type inequalities in w 1 l n q h n in the critical case p n with 1 q n which generalize the results in 43 and improve the results in 57 in the proof of the main results we shall prove a polya szego type principle in w 1 l p q h n with 1 q p which maybe is of independent interest

the wave equation on hyperbolic spaces abstract we study the dispersive properties of the wave equation associated with the shifted laplace beltrami operator on real hyperbolic spaces and deduce new strichartz estimates for a large family of admissible pairs as an application we obtain local well posedness results for the nonlinear wave equation

a generalization of the weighted strichartz estimates for wave equations and an application to self similar solutions weighted strichartz estimates with homogeneous weights with critical exponents are proved for the wave equation without a support restriction on the forcing term the method of proof is based on expansion by spherical harmonics and on the sobolev space over the unit sphere by which the required estimates are reduced to the radial case as an application of the weighted strichartz estimates the existence and uniqueness of self similar solutions to nonlinear wave equations are proved on up to five space dimensions 2006 wiley periodicals inc

strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation the aim of this article is twofold first we consider the wave equation in the hyperbolic space hi and obtain the counterparts of the strichartz type estimates in this context next we examine the relationship between semilinear hyperbolic equations in the minkowski space and in the hyperbolic space this leads to a simple proof of the recent result of georgiev lindblad and sogge on global existence for solutions to semilinear hyperbolic problems with small data shifting the space time strichartz estimates from the hyperbolic space to the minkowski space yields weighted strichartz estimates in rn x 1r which extend the ones of georgiev lindblad and sogge

weighted decay estimates for the wave equation abstract in this work we study weighted sobolev spaces in r n generated by the lie algebra of vector fields 1 x 2 1 2 x j j 1 n interpolation properties and sobolev embeddings are obtained on the basis of a suitable localization in r n as an application we derive weighted l q estimates for the solution of the homogeneous wave equation for the inhomogeneous wave equation we generalize the weighted strichartz estimate established by v georgiev 1997 amer j math 119 1291 1319 and establish global existence results for the supercritical semilinear wave equation with non compact small initial data in these weighted sobolev spaces

blow up of solutions to critical semilinear wave equations with variable coefficients abstract we verify the critical case p p 0 n of strauss conjecture 30 concerning the blow up of solutions to semilinear wave equations with variable coefficients in r n where n 2 the perturbations of laplace operator are assumed to be smooth and decay exponentially fast at infinity we also obtain a sharp lifespan upper bound for solutions with compactly supported data when p p 0 n the unified approach to blow up problems in all dimensions combines several classical ideas in order to generalize and simplify the method of zhou 43 and zhou han 45 exponential eigenfunctions of the laplacian 37 are used to construct the test function q for linear wave equation with variable coefficients and john s method of iterations 13 is augmented with the slicing method of agemi kurokawa and takamura 1 for lower bounds in the critical case

ads cft correspondence in the euclidean context we study two possible prescriptions for the ads cft correspondence by means of functional integrals the considerations are non perturbative and reveal certain divergencies which turn out to be harmless in the sense that reflection positivity and conformal invariance are not destroyed

small data global existence for the semilinear wave equation with space time dependent damping abstract in this paper we consider the critical exponent problem for the semilinear wave equation with space time dependent damping when the damping is effective it is expected that the critical exponent agrees with that of only the space dependent coefficient case we shall prove that there exists a unique global solution for small data if the power of nonlinearity is larger than the expected exponent moreover we do not assume that the data are compactly supported however it is still open whether there exists a blow up solution if the power of nonlinearity is smaller than the expected exponent