blow up of critical semilinear wave equations on schwarzschild spacetime abstract in this paper we established the blow up theorem for critical semilinear wave equations with focusing nonlinear term on schwarzschild spacetime concavity method is used to prove the main result which was introduced by levine payne in the papers levine and payne 1974 11 12 and levine 1973 7 in 1970s also a new auxiliary function with parameter is constructed following the idea from payne 1975 13 in order to guarantee that the result holds without any assumption on the initial data and initial energy

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

stable soliton resolution for exterior wave maps in all equivariance classes abstract in this paper we consider finite energy l equivariant wave maps from r t x 1 3 r b 0 1 s 3 with a dirichlet boundary condition at r 1 and for all l n each such l equivariant wave map has a fixed integer valued topological degree and in each degree class there is a unique harmonic map which minimizes the energy for maps of the same degree we prove that an arbitrary l equivariant exterior wave map with finite energy scatters to the unique harmonic map in its degree class i e soliton resolution this extends the recent results of the first second and fourth authors on the 1 equivariant equation to higher equivariance classes and thus completely resolves a conjecture of bizon chmaj and maliborski who observed this asymptotic behavior numerically the proof relies crucially on exterior energy estimates for the free radial wave equation in dimension d 2 l 3 which are established in the companion paper 13

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

lifespan of solutions to semilinear damping wave equations in de sitter spacetime abstract this paper studies blowup phenomena for two kind of initial value problem of semilinear damping wave equations in de sitter spacetime the first problem comes from the latter problem of conjecture of strauss in de sitter spacetime and the second problem arises from the latter problem of conjecture of glassey in de sitter spacetime we give blowup and lifespan of solutions on those two problems comparing with non damping case established in yan 2018 one can see that the damping term with a positive bounded time dependent coefficient does not effect blowup of solutions meanwhile we establish a new ode blowup result concerning damping term

global existence for energy critical waves in 3 d domains we prove that the defocusing quintic wave equation with dirichlet boundary conditions is globally well posed on for any smooth compact domain the main ingredient in the proof is an spectral projector estimate obtained recently by smith and sogge combined with a precise study of the boundary value problem

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

lifespan of semilinear wave equation with scale invariant dissipation and mass and sub strauss power nonlinearity abstract in this paper we study the blow up of solutions for semilinear wave equations with scale invariant dissipation and mass in the case in which the model is somehow wave like a strauss type critical exponent is determined as the upper bound for the exponent in the nonlinearity in the main theorems two blow up results are obtained for the subcritical case and for the critical case respectively in both cases an upper bound lifespan estimate is given

blow up for semilinear damped wave equations with subcritical exponent in the scattering case abstract it is well known that the critical exponent for semilinear damped wave equations is fujita exponent when the damping is effective lai takamura and wakasa in 2017 have obtained a blow up result not only for super fujita exponent but also for the one closely related to strauss exponent when the damping is scaling invariant and its constant is relatively small which has been recently extended by ikeda and sobajima introducing a multiplier for the time derivative of the spatial integral of unknown functions we succeed in employing the techniques on the analysis for semilinear wave equations and proving a blow up result for semilinear damped wave equations with sub strauss exponent when the damping is in the scattering range

improved kato s lemma on ordinary differential inequality and its application to semilinear wave equations abstract we are interested in the upper bound of the lifespan of solutions of semilinear wave equations for the sub critical case in high dimensions it has been believed that the basic tools of its analysis are kato s lemma on ordinary differential inequalities and the rescaling argument in the functional method but there is a small lack of delicate analysis and no published paper about this here we give a simple alternative proof by means of improved kato s lemma without any rescaling argument

the glassey conjecture on asymptotically flat manifolds we verify the 3 dimensional glassey conjecture on asymptotically flat manifolds r1 3 g where the metric g is a small space time perturbation or nontrapping time independent perturbation of the minkowski metric and approaches the minkowski metric like 1 x with 0 moreover for radial asymptotically flat manifolds r1 n g with n 3 we verify the glassey conjecture in the radial case high dimensional wave equations with higher regularity are also discussed the main idea is to exploit local energy and kss estimates with variable coefficients together with the weighted sobolev estimates including trace estimates

lifespan estimates of solutions to quasilinear wave equations with scattering damping abstract the formation of singularity of solutions to the quasilinear wave equations with scattering damping and divergence form nonlinearities is investigated the upper bound lifespan estimates of solutions to the problems with variable coefficient in the critical case and constant coefficient in the sub critical case are established by applying the rescaled test function method and iterative method

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

on the lp norm of spectral clusters for compact manifolds with boundary we use microlocal and paradifferential techniques to obtain l8 norm bounds for spectral clusters associated with elliptic second order operators on two dimensional manifolds with boundary the result leads to optimal lq bounds in the range 2 q for l2 normalized spectral clusters on bounded domains in the plane and more generally for two dimensional compact manifolds with boundary we also establish new sharp lq estimates in higher dimensions for a range of exponents qn q

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

global existence for systems of quasilinear wave equations in 1 4 dimensions abstract hormander proved global existence of solutions for sufficiently small initial data for scalar wave equations in 1 4 dimensions of the form u q u u u where q vanishes to second order and u 2 q 0 0 0 0 without the latter condition only almost global existence may be guaranteed the first author and sogge considered the analog exterior to a star shaped obstacle both results relied on writing the lowest order terms u u 1 2 u 2 and as such do not immediately generalize to systems the current study remedies such and extends both results to the case of multiple speed systems

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

an elementary proof of the blow up for semilinear wave equation in high space dimensions abstract this paper concerns the blow up of solutions to u tt u u p in high dimensions for n 4 and 1 p p 0 n where p 0 n is a critical exponent we proved that the solutions blow up in finite time by estimating the solutions near the wave front using elementary inequalities

global strichartz estimates for solutions to the wave equation exterior to a convex obstacle hhin this paper we show that certain local strichartz estimates for solutions of the wave equation exterior to a convex obstacle can be extended to estimates that are global in both space and time this extends the work that was done previously by h smith and c sogge in odd spatial dimensions in order to prove the global estimates we explore weighted strichartz estimates for solutions of the wave equation when the cauchy data and forcing term are compactly supported

the role of critical exponents in blow up theorems the sequel abstract in 27 fujita showed that for positive solutions the initial value problem in r n for u t u u p with p 1 exhibited the following behavior if p p c 1 2 n then the initial value problem does not have any nontrivial non negative solution existing on r n 0 a global solution whereas if p p c there exist global small data positive solutions as well as solutions which are non global we call such a result a blow up theorem of fujita type in 50 levine discussed the various theorems of this type that had appeared in the literature prior to 1990 in this paper we revisit the literature since 1990

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

blow up for quasi linear wave equations in three space dimensions we consider equations of the form u phi x t u u u 1a for a function u u x 1 x n t u x t here denotes the d alembertian square frac partial 2 partial t 2 vartriangle 1b and u represents the vector of first derivatives u that of second derivatives of u with respect to the x k and t on linearizing 1a shall go over into the classical wave equation u 0 this means that and its first derivatives with respect to u u u shall vanish for u u u 0

global existence for energy critical waves in 3 d domains neumann boundary conditions we prove that the defocusing quintic wave equation with neumann boundary conditions is globally well posed on h 1 l 2 for any smooth compact domain r 3 the proof relies on one hand on l p estimates for the spectral projector and on the other hand on a precise analysis of the boundary value problem which turns out to be much more delicate than in the case of dirichlet boundary conditions

an efficient local search framework for the minimum weighted vertex cover problem the minimum weighted vertex cover mwvc problem an extension of the classical minimum vertex cover mvc problem is an important np complete combinatorial optimization problem with a wide range of applications the objective of this paper is to design an efficient local search algorithm to solve the mwvc problem first the weighted edge strategy is proposed to define the dynamic scoring strategy so that our algorithm can find different possible optimal solutions second the weighted configuration checking wcc strategy is proposed to overcome the cycling problem in local search by combining the wcc strategy with the scoring strategy we design the vertex selection strategy to determine the vertex to be selected as a candidate solution component based on these strategies a novel local search framework namely diversion local search based on weighted configuration checking dlswcc is presented dlswcc is evaluated against several state of the art algorithms on various benchmark instances experimental results show that dlswcc outperforms its competitors in terms of both solution quality and computational efficiency in most classical instances specifically dlswcc can obtain 22 new upper bounds of 71 moderate scale problem instances 5 new upper bounds of 15 large scale problem instances and 56 new upper bounds of 56 massive graph instances

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

global existence and lifespan for semilinear wave equations with mixed nonlinear terms abstract firstly we study the equation u u q c u p with small data where q c is the critical power of strauss conjecture and p q c we obtain the optimal estimate of the lifespan ln t e e q c q c 1 in n 3 and improve the lower bound of t e from exp c e q c 1 to exp c e q c 1 2 2 in n 2 then we study the cauchy problem with small initial data for a system of semilinear wave equations u v q v t u p in 3 dimensional space with q 2 we obtain that this system admits a global solution above a p q curve for spherically symmetric data on the contrary we get a new region where the solution will blow up

the equation utt δu u p for the critical value of p the equation u tt u u p is considered in two and three space dimensions smooth cauchy data of compact support are given at t 0 for the case of three space dimensions john has shown that solutions with sufficiently small data exist globally in time if but that small data solutions blow up in finite time if glassey has shown the two dimensional case is similar this paper shows that small data solutions blow up in finite time when p is the critical value in three dimensions and in two

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

optimal algorithms for well conditioned nonlinear systems of equations we propose solving nonlinear systems of equations by function optimization and we give an optimal algorithm which relies on a special canonical form of gradient descent the algorithm can be applied under certain assumptions on the function to be optimized that is an upper bound must exist for the norm of the hessian whereas the norm of the gradient must be lower bounded due to its intrinsic structure the algorithm looks particularly appealing for a parallel implementation as a particular case more specific results are given for linear systems we prove that reaching a solution with a degree of precision spl epsiv takes spl theta n sup 2 k sup 2 log sup k sub spl epsiv k being the condition number of a and n the problem dimension related results hold for systems of quadratic equations for which an estimation for the requested bounds can be devised finally we report numerical results in order to establish the actual computational burden of the proposed method and to assess its performances with respect to classical algorithms for solving linear and quadratic equations

life span of solutions to semilinear wave equation with time dependent critical damping for specially localized initial data this paper is concerned with the blowup phenomena for initial value problem of semilinear wave equation with critical time dependent damping term dw begin aligned left begin array ll partial t 2 u x t delta u x t dfrac mu 1 t partial t u x t u x t p x t in mathbb r n times 0 t u x 0 varepsilon f x x in mathbb r n partial t u x 0 varepsilon g x x in mathbb r n end array right end aligned where n in mathbb n mu in 0 frac n 2 n 2 n 2 and varepsilon 0 is a parameter describing the smallness of initial data given data f g are compactly supported in a small area inside of the unit ball the result is the upper bound of lifespan of solution u with respects to the small parameter varepsilon when p f n le p le p 0 n mu where p f n denotes the fujita exponent for the nonlinear heat equations and p 0 n denotes the strauss exponent for nonlinear wave equation dw in n dimension with mu 0 consequently by connecting the result of d abbicco lucente reissig j differ equ 259 5040 5073 2015 our result clarifies the threshold exponent p 0 n mu for dividing blowup phenomena and global existence of small solutions when n 3 the crucial idea is to construct suitable test functions satisfying the conjugate linear equation partial t 2 phi delta phi partial t frac mu 1 t phi 0 of dw including the gauss hypergeometric functions note that the construction of test functions is different from zhou han commun partial differ equ 39 439 451 2014

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

endpoint strichartz estimates we prove an abstract strichartz estimate which implies previously unknown endpoint strichartz estimates for the wave equation in dimension n 4 and the schrodinger equation in dimension n 3 three other applications are discussed local existence for a nonlinear wave equation and strichartz type estimates for more general dispersive equations and for the kinetic transport equation

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

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

life span of classical solutions to one dimensional initial boundary value problems for general quasilinear wave equations abstract this paper is devoted to studying the initial boundary value problem for one dimensional general quasilinear wave equations u t t u x x b u d u u x x 2 a 0 u d u u t x f u d u on exterior domain we obtain the sharp lower bound of the life span of classical solutions to the initial boundary value problem with small initial data and zero boundary data for one dimensional general quasilinear wave equations

on nonlinear generalized conjugate gradient methods the generalized conjugate gradient method see 1 is an iterative method for nonsymmetric linear systems we obtain generalizations of this method for nonlinear systems with nonsymmetric jacobians we prove global convergence results

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

push a generalized operator for the maximum vertex weight clique problem the maximum vertex weight clique problem mvwcp is an important generalization of the well known np hard maximum clique problem in this paper we introduce a generalized move operator called push which generalizes the conventional add and swap operators commonly used in the literature and can be integrated in a local search algorithm for mvwcp the push operator also offers opportunities to define new search operators by considering dedicated candidate push sets to demonstrate the usefulness of the proposed operator we implement two simple tabu search algorithms which use push to explore different candidate push sets the computational results on 142 benchmark instances from different sources dimacs bhoslib and winner determination problem indicate that these algorithms compete favorably with the leading mvwcp algorithms

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

the sharp upper bound of the lifespan of solutions to critical semilinear wave equations in high dimensions the final open part of strauss conjecture on semilinear wave equations was the blow up theorem for the critical case in high dimensions this problem was solved by yordanov and zhang 2006 18 or zhou 2007 21 independently but the estimate for the lifespan the maximal existence time of solutions was not clarified in both papers in this paper we refine their theorems and introduce a new iteration argument to get the sharp upper bound of the lifespan as a result with the sharp lower bound by li and zhou 1995 10 the lifespan t e of solutions of utt u u2 in r4 0 with the initial data u x 0 ef x ut x 0 eg x of a small parameter e 0 compactly supported smooth functions f and g has an estimate exp ce 2 t e exp ce 2 where c and c are positive constants depending only on f and g this upper bound has been known to be the last open optimality of the general theory for fully nonlinear wave equations

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

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

almost global existence to nonlinear wave equations in three space dimensions on considere l existence a t grand de solutions de petite amplitude pour des equations d onde d ordre 2 non lineaires a 4 dimensions d espace temps

new blow up phenomena for hyperbolic inequalities with combined nonlinearities abstract we study for the first time the large time behavior of solutions to certain hyperbolic inequalities with nonlinearities involving the function and its gradient namely we consider the problem u t t u u p u q f t x in 0 r n where p q 1 and f 0 f 0 we obtain general criteria for the nonexistence of global solutions to this problem next we discuss some special cases of the inhomogeneous term f in particular when n 3 and f depends only on the variable space we obtain a discontinuous fujita critical exponent p n q namely p n q 1 2 n 2 if q 1 1 n 1 and p n q if q 1 1 n 1 next we extend our study to the exterior problem u t t u u p u q in 0 d c under the boundary condition u n f t x in 0 d where d is the unit open ball in r n d c r n d and n is the outward relative to d c unit normal of d

blow up theorem for semilinear wave equations with non zero initial position abstract one of the features of solutions of semilinear wave equations can be found in blow up results for non compactly supported data in spite of finite propagation speed of the linear wave we have no global in time solution for any power nonlinearity if the spatial decay of the initial data is weak this was first observed by asakura 1986 2 finding out a critical decay to ensure the global existence of the solution but the blow up result is available only for zero initial position having positive speed in this paper the blow up theorem for non zero initial position by uesaka 2009 22 is extended to higher dimensional case and the assumption on the nonlinear term is relaxed to include an example u p 1 u moreover the critical decay of the initial position is clarified by example

a competition between fujita and strauss type exponents for blow up of semi linear wave equations with scale invariant damping and mass abstract we obtain a blow up result for solutions to a semi linear wave equation with scale invariant dissipation and mass and power non linearity in the case in which the model has a wave like behavior we perform a change of variables that transforms our starting equation in a strictly hyperbolic semi linear wave equation with time dependent speed of propagation applying kato s lemma we prove a blow up result for solutions to the transformed equation under some assumptions on the initial data the limit case that is when the exponent p is exactly equal to the upper bound of the range of admissible values of p yielding blow up needs special considerations in this critical case an explicit integral representation formula for solutions of the corresponding linear cauchy problem in 1d is derived finally carrying out the inverse change of variables we get a non existence result for global in time solutions to the original model

3d quadratic nls equation with electromagnetic perturbations abstract in this paper we study the asymptotic behavior of a quadratic schrodinger equation with electromagnetic potentials we prove that small solutions scatter the proof builds on earlier work of the author for quadratic nls with a non magnetic potential the main novelty is the use of various smoothing estimates for the linear schrodinger flow in place of boundedness of wave operators to deal with the loss of derivative as a byproduct of the proof we obtain boundedness of the wave operator of the linear electromagnetic schrodinger equation on an l 2 weighted space for small potentials as well as a dispersive estimate for the corresponding flow

blow up for small amplitude semilinear wave equations with mixed nonlinearities on asymptotically euclidean manifolds abstract in this work we investigate the problem of finite time blow up as well as the upper bound estimates of lifespan for solutions to small amplitude semilinear wave equations with mixed nonlinearities c 1 u t p c 2 u q posed on asymptotically euclidean manifolds which is related to both the strauss conjecture and the glassey conjecture in some cases we obtain existence results where the lower bound of the lifespan agrees with the upper bound in order in addition our results apply for semilinear damped wave equations when the coefficient of the dissipation term is integrable without sign condition and space independent

critical exponent for nonlinear damped wave equations with non negative potential in 3d abstract we are studying possible interaction of damping coefficients in the subprincipal part of the linear 3d wave equation and their impact on the critical exponent of the corresponding nonlinear cauchy problem with small initial data the main new phenomena is that certain relation between these coefficients may cause very strong jump of the critical strauss exponent in 3d to the critical 5d strauss exponent for the wave equation without damping coefficients

the lifespan of solutions of semilinear wave equations with the scale invariant damping in two space dimensions abstract in this paper we study the initial value problem for semilinear wave equations with the time dependent and scale invariant damping in two dimensions similarly to the one dimensional case by kato takamura and wakasa in 2019 we obtain the lifespan estimates of the solution for a special constant in the damping term which are classified by total integral of the sum of the initial position and speed the key fact is that only in two space dimensions such a special constant in the damping term is a threshold between wave like domain and heat like domain as a result we obtain a new type of estimate especially for the critical exponent

finding community structure in very large networks the discovery and analysis of community structure in networks is a topic of considerable recent interest within the physics community but most methods proposed so far are unsuitable for very large networks because of their computational cost here we present a hierarchical agglomeration algorithm for detecting community structure which is faster than many competing algorithms its running time on a network with n vertices and m edges is o md phantom rule 0 2em 0ex mathrm log phantom rule 0 2em 0ex n where d is the depth of the dendrogram describing the community structure many real world networks are sparse and hierarchical with m ensuremath sim n and d ensuremath sim mathrm log phantom rule 0 2em 0ex n in which case our algorithm runs in essentially linear time o n phantom rule 0 2em 0ex mathrm log 2 phantom rule 0 2em 0ex n as an example of the application of this algorithm we use it to analyze a network of items for sale on the web site of a large on line retailer items in the network being linked if they are frequently purchased by the same buyer the network has more than 400 000 vertices and 2 ifmmode times else texttimes fi 10 6 edges we show that our algorithm can extract meaningful communities from this network revealing large scale patterns present in the purchasing habits of customers

life span of classical solutions to one dimensional initial boundary value problems for general quasilinear wave equations with robin boundary conditions abstract this paper is devoted to studying the initial boundary value problem for one dimensional general quasilinear wave equations u t t u x x b u d u u x x 2 a 0 u d u u t x f u d u with robin boundary conditions on an exterior domain we obtain the sharp lower bound of the life span of classical solutions to the initial boundary value problem with small initial data and zero boundary data for one dimensional general quasilinear wave equations our result in the general case and the special case is shorter than that of the initial boundary value problem for one dimensional general quasilinear wave equations with dirichlet boundary conditions the results in this paper are not the trivial generalization of that in the case of dirichlet boundary conditions the lower bound estimates of life span of classical solutions to initial boundary value problems are consistent with the actual physical meaning the physical phenomenon also explains our results

maximal functions associated to filtrations abstract let t be a bounded linear or sublinear operator from lp y to lq x a maximal operator t f x supj t f yj x is associated to any sequence of subsets yj of y under the hypotheses that q p and the sets yj are nested we prove that t is also bounded classical theorems of menshov and zygmund are obtained as corollaries multilinear generalizations of this theorem are also established these results are motivated by applications to the spectral analysis of schrodinger operators

a clique based exact method for optimal winner determination in combinatorial auctions winner determination wdp is a key problem in combinatorial auctions with many applications we develop a clique based branch and bound algorithm for the wdp we show extensive computational results on three test suites of popular wdp benchmark instances we compare the performance with the general cplex 12 4 solver we demonstrate both approaches complement each other given a set of items to sell and a set of combinatorial bids the winner determination problem wdp in combinatorial auctions is to determine an allocation of items to bidders such that the auctioneer s revenue is maximized while each item is allocated to at most one bidder wdp is at the core of numerous relevant applications in multi agent systems e commerce and many others we develop a clique based branch and bound approach for wdp which relies on a transformation of wdp into the maximum weight clique problem to ensure the efficiency of the proposed search algorithm we introduce specific bounding and branching strategies using a dedicated vertex coloring procedure and a specific vertex sorting technique we assess the performance of the proposed algorithm on a large collection of benchmark instances in comparison with the cplex 12 4 solver and other approaches computational results show that this clique based method constitutes a valuable and complementary approach for wdp relative to the existing methods

on the empirical scaling of run time for finding optimal solutions to the travelling salesman problem abstract the travelling salesman problem tsp is one of the most prominent np hard combinatorial optimisation problems after over fifty years of intense study the tsp continues to be of broad theoretical and practical interest using a novel approach to empirical scaling analysis which in principle is applicable to solvers for many other problems we demonstrate that some of the most widely studied types of tsp instances tend to be much easier than expected from previous theoretical and empirical results in particular we show that the empirical median run time required for finding optimal solutions to so called random uniform euclidean rue instances one of the most widely studied classes of tsp instances scales substantially better than 2 n with the number n of cities to be visited the concorde solver for which we achieved this result is the best performing exact tsp solver we are aware of and has been applied to a broad range of real world problems furthermore we show that even when applied to a broad range of instances from the prominent tsplib benchmark collection for the tsp concorde exhibits run times that are surprisingly consistent with our empirical model of concorde s scaling behaviour on rue instances this result suggests that the behaviour observed for the simple random structure underlying rue is very similar to that obtained on the structured instances arising in various applications

blow up for semilinear wave equations with the scale invariant damping and super fujita exponent abstract the blow up for semilinear wave equations with the scale invariant damping has been well studied for sub fujita exponent however for super fujita exponent there is only one blow up result which is obtained in 2014 by wakasugi in the case of non effective damping in this paper we extend his result in two aspects by showing that i the blow up will happen for bigger exponent which is closely related to the strauss exponent the critical number for non damped semilinear wave equations ii such a blow up result is established for a wider range of the constant than the known non effective one in the damping term

the fundamental analysis method for the blow up of wave equations in first quadrant abstract this paper is devoted to prove the sharpness on the lower bound of the life span of classical solutions to the initial boundary value problem for one dimensional general quasilinear wave equations in the general case obtained in 10 what s more the originality of the paper is the fundamental analysis method of the blow up and lifespan property that is to say instead of differential inequality and the pure mathematical analysis this is the first attempt to this direction the highlight of the paper is that we will use the most fundamental d alembert s formula of the solutions of wave equation and the corresponding actual physical meaning by means of the scaling argument dilation translation translation transformation rotation transformation we connect our problem with a goursat problem and then we can intuitively obtain our blow up and lifespan result for the problem this is a new idea in the paper

the glassey conjecture for nontrapping obstacles abstract we verify the 3 dimensional glassey conjecture for exterior domain m g where the metric g is asymptotically euclidean provided that certain local energy assumption is satisfied the radial glassey conjecture exterior to a ball is also verified for dimension three or higher the local energy assumption is satisfied for many important cases including exterior domain with nontrapping obstacles and flat metric exterior domain with star shaped obstacle and small asymptotically euclidean metric as well as the nontrapping asymptotically euclidean manifolds r n g

a review on algorithms for maximum clique problems the maximum clique problem mcp is to determine in a graph a clique i e a complete subgraph of maximum cardinality the mcp is notable for its capability of modeling other combinatorial problems and real world applications as one of the most studied np hard problems many algorithms are available in the literature and new methods are continually being proposed given that the two existing surveys on the mcp date back to 1994 and 1999 respectively one primary goal of this paper is to provide an updated and comprehensive review on both exact and heuristic mcp algorithms with a special focus on recent developments to be informative we identify the general framework followed by these algorithms and pinpoint the key ingredients that make them successful by classifying the main search strategies and putting forward the critical elements of the most relevant clique methods this review intends to encourage future development of more powerful methods and motivate new applications of the clique approaches

concerning the strauss conjecture for the subcritical and critical cases on the exterior domain in two space dimensions abstract the aim of this paper is to verify the strauss conjecture for semilinear wave equations with variable coefficients and for the subcritical and critical cases on the exterior domain for when the number of space dimensions n 2 the main ingredients include generalized sideris ode blow up results and the existence and a priori estimates of the test functions see section 2 for this purpose we study the following initial boundary value problem for two dimensional semilinear wave equations with variable coefficients on the exterior domain with subcritical and critical exponents 0 1 u t t i j 1 2 i a i j x j u u p x t c 0 n 2 where u u x t is a real valued scalar unknown function in c 0 where is a smooth compact obstacle in r 2 and c is its complement two dimensional initial boundary value problems for semilinear wave equations with variable coefficients on the exterior domain are difficult to deal with because the fundamental solution of the harmonic equation is different from that for the higher dimensional case and thus there will be a growth factor in the logarithmic term in the key test function 0 see section 2 one of the key points of our paper is the improvement of a well known ode blow up result lemma 2 1 of 12 p 386 for when the differential inequality involves a logarithmic term and we obtain an estimate of the upper bound for the blow up time on the basis of the key results lemma 2 3 lemma 2 4 obtained we will establish a blow up result for the above initial boundary value problem it is proved that there can be no global solutions no matter how small the initial data are and a lifespan estimate for the solutions for this problem will also be given

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

a discrete newton algorithm for minimizing a function of many variables a newton like method is presented for minimizing a function ofn variables it uses only function and gradient values and is a variant of the discrete newton algorithm this variant requires fewer operations than the standard method whenn 39 and storage is proportional ton rather thann2

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

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

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

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

strichartz estimates for dirichlet wave equations in two dimensions with applications we establish the strauss conjecture for nontrapping obstacles when the spatial dimension n is two as pointed out in 7 this case is more subtle than n 3 or 4 due to the fact that the arguments of the first two authors 11 burq 1 and metcalfe 9 showing that local strichartz estimates for obstactles imply global ones require that the sobolev index equal 1 2 when n 2 we overcome this difficulty by interpolating between energy estimates 0 and ones for 1 2 that are generalizations of minkowski space estimates of fang and the third author 4 5 the second author 12 and sterbenz 14

cabob a fast optimal algorithm for winner determination in combinatorial auctions combinatorial auctions where bidders can bid on bundles of items can lead to more economically efficient allocations but determining the winners is complete and inapproximable we present cabob a sophisticated optimal search algorithm for the problem it uses decomposition techniques upper and lower bounding also across components elaborate and dynamically chosen bid ordering heuristics and a host of structural observations cabob attempts to capture structure in any instance without making assumptions about the instance distribution experiments against the fastest prior algorithm cplex 8 0 show that cabob is often faster seldom drastically slower and in many cases drastically faster especially in cases with structure cabob s search runs in linear space and has significantly better anytime performance than cplex we also uncover interesting aspects of the problem itself first problems with short bids which were hard for the first generation of specialized algorithms are easy second almost all of the cats distributions are easy and the run time is virtually unaffected by the number of goods third we test several random restart strategies showing that they do not help on this problem the run time distribution does not have a heavy tail

life span of small solutions to a system of wave equations abstract we study the cauchy problem with small initial data for a system of semilinear wave equations u v q v t u p in n dimensional space when n 2 we prove that blow up can occur for arbitrarily small data if p q lies below a curve in the p q plane on the other hand we show a global existence result for n 3 which asserts that a portion of the curve is in fact the borderline between global in time existence and finite time blow up we also estimate the maximal existence time and get its upper bound which is sharp at least for n p q 2 2 2 and 3 2 2

the stability of gradient like methods this paper deals with the stability properties of gradient like methods for nonlinear system of equations the fridman variant and the conjugate gradient method the main part of the article is to analyse the total exponential stability of these methods with respect to strongly consistent disturbances of the jacobian some numerical experiments are also given

almost global existence for nonlinear wave equations in an exterior domain in two space dimensions in this paper we deal with the exterior problem for a system of nonlinear wave equations in two space dimensions assuming that the initial data is small and smooth we establish the same type of lower bound of the lifespan for the problem as that for the cauchy problem despite of the weak decay property of the solution in two space dimensions