global existence and nonexistence of solution for cauchy problem of multidimensional double dispersion equations abstract in this paper we consider the cauchy problem of multidimensional generalized double dispersion equations u t t u u t t 2 u f u where f u a u p by potential well method we prove the existence and nonexistence of global weak solution without establishing the local existence theory and we derive some sharp conditions for global existence and lack of global existence solution

cauchy problem for the multi dimensional boussinesq type equation abstract the paper studies the existence and non existence of global weak solutions to the cauchy problem for the multi dimensional boussinesq type equation u t t u 2 u u it proves that the cauchy problem admits a global weak solution under the assumptions that c r s is of polynomial growth order say p 1 either 0 s s 0 s d s r where 0 is a constant or the initial data belong to a potential well and the weak solution is regularized and the strong solution is unique when the space dimension n 1 in contrast any weak solution of the cauchy problem blows up in finite time under certain conditions and two examples are shown

global existence and nonexistence of solution for cauchy problem of two dimensional generalized boussinesq equations abstract in this paper we consider the cauchy problem of two dimensional generalized boussinesq type equation u t t u u t t 2 u f u 0 under the assumption that f u is a function with exponential growth at infinity and under some assumptions on the initial data we prove the existence and nonexistence of global weak solution there are very few works on boussinesq equation with nonlinear exponential growth term by potential well theory

cauchy problems of semilinear pseudo parabolic equations abstract this paper concerns with the cauchy problems of semilinear pseudo parabolic equations after establishing the necessary existence uniqueness and comparison principle for mild solutions which are also classical ones provided that the initial data are appropriately smooth we investigate large time behavior of solutions it is shown that there still exist the critical global existence exponent and the critical fujita exponent for pseudo parabolic equations and that these two critical exponents are consistent with the corresponding semilinear heat equations

high energy blow up for two dimensional generalized exponential type boussinesq equation abstract a finite time blow up result for the cauchy problem of two dimensional generalized boussinesq type equation with exponential type source term is considered in this paper based on an adapted concave method we derive a sufficient condition on initial data leading to the arbitrarily positive initial energy finite time blow up

global existence and exponential growth of solution for the logarithmic boussinesq type equation abstract in this paper we consider the boussinesq type equation associated with logarithmic nonlinear term and initial boundary value conditions by using potential well method combined with the logarithmic sobolev inequality we obtain the existence of global solution at last we show that the l 2 norm of the solution will grow up as an exponential function as time goes to infinity under some suitable conditions for initial data the result shows that the polynomial nonlinearity is a critical condition of blow up in finite time for the solutions of boussinesq equations

initial boundary value problem of the generalized cubic double dispersion equation in this paper the existence and the uniqueness of the global generalized solution and the global classical solution for the initial boundary value problem of the generalized cubic double dispersion equation are proved the nonexistence of global solution for the initial boundary value problem of the generalized cubic double dispersion equation is discussed

computation of invariants for reductive groups abstract we will give an algorithm for computing generators of the invariant ring for a given representation of a linearly reductive group the algorithm basically consists of a single grobner basis computation we will also show a connection between some open conjectures in commutative algebra and finding good degree bounds for generating invariants

global existence and blow up of solutions for cauchy problem of generalized boussinesq equation abstract we study the cauchy problem of generalized boussinesq equation u t t u x x u x x f u x x 0 where f u u p or u p 1 u p 1 by introducing a family of potential wells we obtain invariant sets vacuum isolating and threshold result of global existence and nonexistence of solution

the evolution operator technique in solving the schrodinger equation and its application to disentangling exponential operators and solving the problem of a mass varying harmonic oscillator the authors present a method for finding the evolution operator for the schrodinger equation for the hamiltonian expressible as h t a1 t j4 a2 t j0 a3 t j where j j0 and j are the su 2 group generators such a method is applied to the disentangling technique for exponential operators which are not necessarily unitary as a demonstration of our general approach they solved the problem of a harmonic oscillator with a varying mass

local and global existence of solutions for the cauchy problem of a generalized boussinesq equation abstract we consider the local and global existence of solutions for a generalized boussinesq equation u t t u x x u x x x x u k 1 x x 0 k 4 with initial data in some homogenous besov type space

the cauchy problem for a class of the multidimensional boussinesq type equation abstract in this paper we study the cauchy problem for a class of the multidimensional boussinesq type equation u t t u 2 u 2 u t t f u where f u a u p or a u p 1 u a 0 is a constant first we establish a local existence theorem for the solution then for m 1 n 1 m 2 3 4 n 3 we prove the existence of a global h m solution finally we prove the global nonexistence and finite time blow up of the solution

the initial boundary value problems for a class of nonlinear wave equations with damping term abstract in this paper we study the existence and uniqueness of the global generalized solution and the global classical solution the blowup of the solution and the energy decay of the solutions of the initial boundary value problems for a class of nonlinear wave equations

ground state solution for a fourth order elliptic equation with logarithmic nonlinearity modeling epitaxial growth abstract this paper deals with a fourth order elliptic equation with logarithmic nonlinearity modeling epitaxial growth and the existence of a ground state solution is obtained

global existence and blow up of solutions for a general class of doubly dispersive nonlocal nonlinear wave equations this study deals with the analysis of the cauchy problem of a general class of nonlocal nonlinear equations modeling the bi directional propagation of dispersive waves in various contexts the nonlocal nature of the problem is reflected by two different elliptic pseudodifferential operators acting on linear and nonlinear functions of the dependent variable respectively the well known doubly dispersive nonlinear wave equation that incorporates two types of dispersive effects originated from two different dispersion operators falls into the category studied here the class of nonlocal nonlinear wave equations also covers a variety of well known wave equations such as various forms of the boussinesq equation local existence of solutions of the cauchy problem with initial data in suitable sobolev spaces is proven and the conditions for global existence and finite time blow up of solutions are established

the initial boundary value problem for the good boussinesq equation on the bounded domain abstract the initial boundary value problem for the good boussinesq equation on the bounded domain is studied in this article the local and global well posedness of this initial boundary value problem is given

longtime dynamics of boussinesq type equations with fractional damping abstract the paper investigates the well posedness and longtime dynamics of boussinesq type equations with fractional damping u t t 2 u u t f u g x with 1 2 the main results focus on the relations among the dissipative exponent the growth exponent p of nonlinearity f u and the well posedness and the longtime dynamics of the equations we find a new critical exponent p n 2 2 1 n 2 2 1 rather than p n 2 n 2 n 3 as known before and show that when 1 p p i the equations are like parabolic that is not only the ibvp of the equations are well posedness but also their weak solutions are of higher global regularity as t 0 ii the related solution semigroup has a global attractor a in natural energy space and also has an exponential attractor a e x p in the sense of partially strong topology in particular when 1 p p n 2 n 2 p the partially strong topology becomes the strong one iii for any 0 1 2 the family of global attractors a is upper semicontinuous at the point 0

exactly solving a two dimensional time dependent coupled quantum oscillator based on the generalized linear quantum transformation theory we present a new method to exactly solve a two dimensional time dependent coupled quantum oscillator and obtain the exact formulae of the normal product form of the evolution operator evolution matrix element wavefunction and expectation value of an arbitrary observable

degree bounds for syzygies of invariants abstract suppose that g is a linearly reductive group good degree bounds for generators of invariant rings were given in proc amer math soc 129 4 2001 955 here we study minimal free resolutions of invariant rings for finite linearly reductive groups g it was recently shown in adv math 156 1 2000 23 electron res announc amer math soc 7 2001 5 adv math 172 2002 151 that rings of invariants are generated in degree at most the group order g in characteristic 0 this degree bound is a classical result by emmy noether see math ann 77 1916 89 given an invariant ring of a finite linearly reductive group g we prove that the ideal of relations of a minimal set of generators is generated in degree at most 2 g

the generalized harmonic oscillator and the infinite square well with a moving boundary for the one dimensional generalized harmonic oscillator we obtain in this paper its wavefunctions in closed form by means of two independent methods which are based respectively in i the algebraic properties of the dynamical symmetry of the system and ii the construction of an invariant operator the total equivalence of these two formulations is shown and quantal properties are described in terms of a classical solution of the equations of motion two possible reductions for the system exist the static harmonic oscillator and the free particle in the latter case the quantum system becomes a fermi oscillator or equivalently it can describe a free particle in a well with one moving boundary which in turn follows certain classical rules the time dependent boundary conditions in the well play the role of an effective interaction acting on the particle the formalism is shown to be compatible with the gauge principle of minimal coupling and several different gauges are constructed and analysed

the basic invariants and high n squeezed states of the time dependent harmonic oscillator a new concept of the basic invariants is introduced with the help of the basic invariants new high n squeezed states of the time dependent harmonic oscillator are constructed from the basic invariants the author also finds a general form of invariants which has not been seen thus far in the literature

large time behavior of solutions to the nonlinear pseudo parabolic equation abstract in this paper we investigate the initial value problem for the nonlinear pseudo parabolic equation global existence and optimal decay estimate of solution are established provided that the initial value is suitably small moreover when n 2 and the nonlinear term f u disappears we prove that the global solutions can be approximated by the linear solution as time tends to infinity when n 1 and the nonlinear term f u disappears we show that as time tends to infinity the global solution approaches the nonlinear diffusion wave described by the self similar solution of the viscous burgers equation

global existence and blow up of solution for semilinear hyperbolic equation with logarithmic nonlinearity abstract in this paper we consider the semilinear wave equation with logarithmic nonlinearity by modifying and using potential well combined with logarithmic sobolev inequality we derive the global existence and infinite time blow up of the solution at low energy level e 0 d then these results are extended in parallel to the critical case e 0 d besides with additional assumptions on initial data the infinite time blow up result is given with arbitrary positive initial energy e 0 0

global solution and blow up of a semilinear heat equation with logarithmic nonlinearity abstract we study the initial boundary value problem of a semilinear heat equation with logarithmic nonlinearity by using the logarithmic sobolev inequality and a family of potential wells we obtain the existence of global solution and blow up at under some suitable conditions on the other hand the results for decay estimates of the global solutions are also given our result in this paper means that the polynomial nonlinearity is a critical condition of blow up in finite time for the solutions of semilinear heat equations

on the cauchy problem for a generalized boussinesq equation in this paper we consider the cauchy problem for a generalized boussinesq equation we show that under suitable conditions a global solution for the initial value problem exists in addition we derive the sufficient conditions for the blow up of the solution to the problem

existence and decay rate estimates for the wave equation with nonlinear boundary damping and source term the wave equation with a source term is considered utt u u uin 0 we prove the existence and uniform decay rates of the energy by assuming a nonlinear feedback ut acting on the boundary provided that has necessarily not a polynomial growth near the origin to obtain the existence of global solutions we make use of the potential well method combined with the faedo galerkin procedure and constructing a special basis furthermore we prove that the energy of the system decays uniformly to zero and we obtain an explicit decay rate estimate adapting the ideas of lasiecka and tataru differential integral equations 6 3 1993 507 and patrick martinez esain control optimisation calc var 4 1999 419 the resulting problem generalizes martinez results and complements the works of lasiecka and tataru 1993 and vitillaro glasgow math j 44 2002 375

on potential wells and vacuum isolating of solutions for semilinear wave equations abstract in this paper we study the initial boundary value problem of semilinear wave equations u tt u u p 1 u x t 0 u x 0 u 0 x u t x 0 u 1 x x u x t 0 x t 0 where r n is a bounded domain 1 1 n 2 n 2 for n 3 first by using a new method we introduce a family of potential wells which include the known potential well as a special case then by using it we obtain some new existence theorems of global solutions and prove that for any e 0 d d is the depth of the known potential well all solutions with initial energy e 0 satisfying 0 h 0 1

null controllability of a pseudo parabolic equation with moving control abstract in this paper we study the internal controllability of the pseudo parabolic equation on the one dimensional torus our control function is acting on a moving small interval with a constant velocity with this moving distributed control we obtain the system is null controllable for given data with certain regularity

on quasi periodic solutions for a generalized boussinesq equation abstract in this paper a one dimensional generalized boussinesq equation u t t u x x u 3 u x x x x 0 with hinged boundary conditions is considered it is proved that the above equation admits small amplitude quasi periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional hamiltonian system the proof is based on an infinite dimensional kam theorem and the birkhoff normal form

on potential wells and applications to semilinear hyperbolic equations and parabolic equations abstract in this paper we generalize the family of potential wells to the initial boundary value problem of semilinear hyperbolic equations and parabolic equations u tt u f u x t 0 u x 0 u 0 x u t x 0 u 1 x x u x t 0 x t 0 and u t u f u x t 0 u x 0 u 0 x x u x t 0 x t 0 not only give a threshold result of global existence and nonexistence of solutions but also obtain the vacuum isolating of solutions finally we prove the global existence of solutions for above problem with critical initial conditions i u 0 0 e 0 d or i u 0 0 j u 0 d so payne and sattinger s results are generalized and improved in essential

thresholds for global existence and blow up in a general class of doubly dispersive nonlocal wave equations abstract in this paper we study the global existence and blow up of solutions for a general class of nonlocal nonlinear wave equations with power type nonlinearities u t t l u x x b u p 1 u x x p 1 where the nonlocality enters through two pseudo differential operators l and b we establish thresholds for global existence versus blow up using the potential well method which relies essentially on the ideas suggested by payne and sattinger our results improve the global existence and blow up results given in the literature for the present class of nonlocal nonlinear wave equations and cover those given for many well known nonlinear dispersive wave equations such as the so called double dispersion equation and the traditional boussinesq type equations as special cases

the noether bound in invariant theory of finite groups abstract let r be a commutative ring v a finitely generated free r module and g glr v a finite group acting naturally on the graded symmetric algebra a sym v let ag denote the minimal number m such that the ring ag of invariants can be generated by finitely many elements of degree at most m furthermore let h g be a normal subgroup such that the index g h is invertible in r in this paper we prove the inequality a g a h g h for h 1 and g invertible in r we obtain noether s bound ag g which so far had been shown for arbitrary groups only under the assumption that the factorial of the group order g is invertible in r

global existence of cauchy problem for boussinesq paradigm equation in this paper we study the cauchy problem for the boussinesq paradigm equation u t email protected b email protected t email protected email protected email protected u where f u email protected u p new functionals are introduced and their sign preserving properties under the flow of the equation are studied the existence of a global weak solution with supercritical initial energy is proved the explicit value of the critical energy constant is found

bures fidelity for diagonalizable quadratic hamiltonians in multi mode systems fidelity as a measure of the distinguishability of states is an important concept in quantum mechanics quantum optics and quantum information theory recently the explicit expressions of fidelity for single mode squeezed states have been given however in experimental studies especially in non degenerate parametric down conversion two photons are generated and one studies two or more mode systems in this paper we study the bures fidelity for thermal states of a diagonalizable quadratic hamiltonian in multi mode fock space to the best of our knowledge no one has yet attempted to give an explicit general formula of fidelity of mixed states in multi mode systems

a sharp bound for the castelnuovo mumford regularity of subspace arrangements abstract we show that the ideal of an arrangement of d linear subspaces of projective space is d regular in the sense of castelnuovo and mumford answering a question of b sturmfels in particular this implies that the ideal of an arrangement of d subspaces is generated in degrees less than or equal to d

exact solution for a time dependent multi mode coupled quadratic bose system by utilizing generalized linear quantum transformation theory the evolution operator normal and anti normal wigner characteristic functions p and q representations of a multi mode coupled quadratic boson system are presented the squeezing properties of a time dependent double mode coupled quadratic bose system are investigated as a specific example

strong instability of solitary wave solutions of a generalized boussinesq equation abstract in this paper we prove the strong instability of solitary wave solutions with small speed of propagation for a generalized boussinesq equation

on potential wells to a semilinear hyperbolic equation with damping and conical singularity abstract this article deals with a semilinear hyperbolic equation with damping and conical singularity that was built by alimohammady et al 2017 1 where the weak solution with low initial energy i 0 d d is the potential well depth was considered we extend the previous results in the following three aspects firstly we consider the vacuum isolating behavior of the solution for initial energy i 0 0 and 0 i 0 d respectively we find that there are two explicit vacuum regions an annulus and a ball moreover we obtain the asymptotic behavior of the energy functional as t tends to the maximal existence time and then two necessary and sufficient conditions for the weak solution existing globally and blowing up in finite time are obtained secondly we discuss the weak solution with critical initial energy and establish the global existence and nonexistence results finally the weak solution with arbitrary positive initial energy is studied in this case the initial conditions such that the weak solution exists globally and blows up in finite time are given

der kanonische modul eines invariantenrings abstract we calculate the canonical module of a ring of invariants r of a reductive group acting on an affine variety from that we derive a criterion for the gorenstein property to hold for r in case r is the ring of invariants of a graded ring we obtain lower and upper bounds for the degree of the generating function of r

longtime dynamics of the damped boussinesq equation abstract the paper studies the longtime dynamics of the damped boussinesq equation u t t 2 u u t g u f x first the existence of global solutions to the initial boundary value problem of the equation is obtained provided that the growth exponent of g u say p is either non supercritical subcritical and critical or supercritical especially the stability of solutions is established when p is non supercritical second the existence of a global attractor and an exponential attractor for the related solution semigroup s t are respectively established in the non supercritical case

the cauchy problem for the rosenau equation abstract this paper considers the existence and the uniqueness of the solution for the cauchy problem of the rosenau equation u t t u x 4 u x 4 t t u x x f u x x we prove the existence and the uniqueness of the global solution under some conditions of f u and give the condition for nonexistence of a global solution for f u u p u with 0 and p 0 the global existence and finite time blow up for the problem are proved with the aid of the potential well method

a presentation of the trace algebra of three 3 3 matrices abstract the trace algebra c n d is generated by all traces of products of d generic n n matrices minimal generating sets of c n d and their defining relations are known for n 3 and n 3 d 2 this paper states a minimal generating set and their defining relations for n d 3 furthermore the computations yield a description of c 33 as a free module over the ring generated by a homogeneous system of parameters

global existence and finite time blow up for a class of semilinear pseudo parabolic equations abstract in this paper we study a class of semilinear pseudo parabolic equations by introducing a family of potential wells we prove the invariance of some sets global existence nonexistence and asymptotic behavior of solutions with initial energy j u 0 d moreover we obtain finite time blow up with high initial energy j u 0 d by comparison principle

global existence of solution of cauchy problem for nonlinear pseudo parabolic equation abstract in this paper we prove that the cauchy problem for the nonlinear pseudo parabolic equation v t v x x t v x x v x f v x v x x g v g v x x admits a unique global generalized solution in c 1 0 h s r a unique global classical solution and asymptotic behavior of the solution we also prove that the cauchy problem for the equation v t v x x t v x x g v g v x x has a unique global generalized solution in c 1 0 w m p r l r a unique global classical solution and asymptotic behavior of the solution

blow up at infinity of solutions to a semilinear heat equation with logarithmic nonlinearity abstract in this short note the author establishes a blow up result for a semilinear heat equation with logarithmic nonlinearity by using the logarithmic sobolev inequality this improves a recent blow up result obtained in chen et al 2015 1

global existence to generalized boussinesq equation with combined power type nonlinearities abstract the cauchy problem to the generalized boussinesq equation with combined power type nonlinearities is studied global solvability or finite time blow up of the solutions with subcritical initial energy is proved by means of the sign preserving property of the nehari functional for generalized lienard or generalized bernoulli nonlinear terms the critical energy constant is explicitly evaluated a new method that can be considered as a modification of the potential well method is developed the performed numerical experiments support the theoretical results

the existence of many periodic non travelling solutions to the boussinesq equation abstract this paper uses variational methods in particular a generalization of the mountain pass lemma of rabinowitz together with an invariance argument to demonstrate the existence of weak sobolev periodic non travelling solutions to the boussinesq equation formula in certain cases the author demonstrates the existence of infinitely many such solutions the assumptions on the non linear terms are that they are odd and roughly polynomial and that they satisfy g g1 g2 for someg c2 r2 r

finite time blow up for a class of parabolic or pseudo parabolic equations abstract in this paper we study the initial boundary value problem for a class of parabolic or pseudo parabolic equations u t a u t u b u k t u p 2 u x t 0 t where a 0 b l 1 with l 1 being the principal eigenvalue for on h 0 1 and k t 0 by using the potential well method levine s concavity method and some differential inequality techniques we obtain the finite time blow up results provided that the initial energy satisfies three conditions i j u 0 0 0 ii j u 0 0 d where d is a nonnegative constant iii 0 j u 0 0 c 0 where 0 involves the l 2 norm or h 0 1 norm of the initial data we also establish the lower and upper bounds for the blow up time in particular we obtain the existence of certain solutions blowing up in finite time with initial data at the nehari manifold or at arbitrary energy level

global existence of solution for cauchy problem of multidimensional generalized double dispersion equations abstract we study the cauchy problem of a class of multidimensional generalized double dispersion equations u t t u u t t 2 u k u t f u we prove the global existence of weak solution under some assumptions on the nonlinear term and the initial data

initial boundary value problem for a class of semilinear pseudo parabolic equations with logarithmic nonlinearity abstract in this paper we study the initial boundary value problem for a class of semilinear pseudo parabolic equations with logarithmic nonlinearity by using the logarithmic sobolev inequality and a family of potential wells we obtain the existence of global solution blow up at and behavior of vacuum isolation of solutions on the other hand the asymptotic behavior of solutions is also discussed our result implies that the polynomial nonlinearity is important for the solutions of such kinds of semilinear pseudo parabolic equations to be blow up in finite time

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

decay estimates for a dissipative dispersive linear semigroup and application to the viscous boussinesq equation abstract at the core of this article is the new estimate for a class of dissipative dispersive linear semigroup e t i p t arising in the study of the viscous boussinesq equation we combine the decay estimate with introducing a set of time weighted sobolev spaces where the time weights and the regularity of the sobolev spaces are determined by our decay estimate to show the global existence and asymptotic behavior of solutions to the viscous boussinesq equation in r n

the squeezed state approach of the semiclassical limit of the time dependent schrodinger equation it is by now well recognized how the quantum time evolution for time dependent hamiltonians that are quadratic both in position and momentum variables generates squeezing on the other hand squeezed states and their high n generalizations are known to provide a simple classical mechanical description of general quantum time evolution in this paper these ideas are combined to show the relevance of generalized squeezed states in the semiclassical limit of the evolution of time dependent wave packets for general time dependent hamiltonians including the multidimensional case

multisymmetric polynomials in dimension three abstract the polarizations of one relation of degree five and two relations of degree six minimally generate the ideal of relations among a minimal generating system of the algebra of multisymmetric polynomials in an arbitrary number of three dimensional vector variables in the general case of n dimensional vector variables a relation of degree 2 n among the polarized power sums is presented such that it is not contained in the ideal generated by lower degree relations

global existence of smooth solutions and stability of solitary waves for a generalized boussinesq equation certain generalizations of one of the classical boussinesq type equations u tt u xx u 2 u xx xx are considered it is shown that the initial value problem for this type of equation is always locally well posed it is also determined that the special solitary wave solutions of these equations are nonlinearly stable for a range of their phase speeds these two facts lead to the conclusion that initial data lying relatively close to a stable solitary wave evolves into a global solution of these equations this contrasts with the results of blow up obtained recently by kalantarov and ladyzhenskaya for the same type of equation and casts additional light upon the results for the original version of this class of equations obtained via inverse scattering theory by deift tomei and trubowitz

squeezing by tuning the oscillator frequency the author examines the possibility of generating squeezed states out of coherent states by external changes of the oscillator frequency using the evolution operator method developed by cheng and fung he investigates the evolution of a coherent state of a time dependent harmonic oscillator as well as its squeezing and coherence property two specific models of frequency variation are explored namely one whose frequency is exponentially decreasing in time and the other with a periodic frequency his analyses indicates that the wavefunction of the time dependent oscillator starts as a coherent state at t o and evolves as a squeezed state at a later time it is also shown that squeezing cannot be generated by an adiabatic change of the oscillator frequency because the variances of q and p turn out to be adiabatic invariants whereas a sudden change can produce squeezing