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

graphical representations and cluster algorithms i discrete spin systems abstract graphical representations similar to the fk representation are developed for a variety of spin systems in several cases it is established that these representations have fkg monotonicity properties which enables characterization theorems for the uniqueness phase and the low temperature phase of the spin system certain systems with intermediate phases and or first order transitions are also described in terms of the percolation properties of the representations in all cases these representations lead in a natural fashion to swendsen wang type algorithms hence at least in the above mentioned instances these algorithms realize the program described by kandel and domany phys rev b 43 1991 8539 8548 all of the algorithms are shown to satisfy a li sokal bound which at least for systems with a divergent specific heat implies critical slowing down however the representations also give rise to invaded cluster algorithms which may allow for the rapid simulation of some of these systems at their transition points

cluster method for the ashkin teller model a cluster monte carlo algorithm for the ashkin teller at model is constructed according to the guidelines of a general scheme for such algorithms its dynamical behaviour is tested for the square lattice at model we perform simulations on the line of critical points along which the exponents vary continuously and find that critical slowing down is significantly reduced we find continuous variation of the dynamical exponent z along the line following the variation of the ratio alpha nu in a manner which satisfies the li sokal bound z cluster geq alpha nu that was so far proved only for potts models

instability of standing waves for klein gordon zakharov equations with different propagation speeds in three space dimensions abstract this paper is concerned with the standing wave for klein gordon zakharov equations with different propagation speeds in three space dimensions the existence of standing wave with the ground state is established by applying an intricate variational argument and the instability of the standing wave is shown by applying pagne and sattinger s potential well argument and levine s concavity method

convex envelopes generated from finitely many compact convex sets we consider the problem of constructing the convex envelope of a lower semi continuous function defined over a compact convex set we formulate the envelope representation problem as a convex optimization problem for functions whose generating sets consist of finitely many compact convex sets in particular we consider nonnegative functions that are products of convex and component wise concave functions and derive closed form expressions for the convex envelopes of a wide class of such functions several examples demonstrate that these envelopes reduce significantly the relaxation gaps of widely used factorable relaxation techniques

convex envelopes for edge concave functions deterministic global optimization algorithms frequently rely on the convex underestimation of nonconvex functions in this paper we describe the structure of the polyhedral convex envelopes of edge concave functions over polyhedral domains using geometric arguments an algorithm for computing the facets of the convex envelope over hyperrectangles in 3 is described sufficient conditions are described under which the convex envelope of a sum of edge concave functions may be shown to be equivalent to the sum of the convex envelopes of these functions

convex envelopes of some quadratic functions over the n dimensional unit simplex in this paper we discuss convex envelopes over the unit simplex for some quadratic functions the discussion exploits the combinatorics related to quadratic functions over the unit simplex after recalling the definition of the underlying convex graph of a quadratic function i e the graph whose nodes are the vertices of the unit simplex and whose edges correspond to the edges of the unit simplex along which the function is strictly convex it is shown that the analytical formula of the convex envelope for quadratic functions whose underlying graph is triangle free is the maximum of a finite number of functions each of which is related to a spanning forest of the graph the special case of an underlying star graph is discussed and computational experiments based on a decomposition of a general quadratic function into a sum of quadratic functions with an underlying star graph are presented

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

non perturbative criteria for gibbsian uniqueness for spin systems with an internal symmetry we provide sufficient conditions for unicity of the gibbs state and or complete analyticity by comparison to random cluster models

long time behavior of a class of thermoelastic plates with nonlinear strain abstract in recent years a class of vibrating plates with nonlinear strain of p laplacian type was studied by several authors the present paper contains a first thermoelastic model of that class of problems including both fourier and non fourier heat laws our main result establishes the existence of global and exponential attractors for the strongly damped problem through a stabilizability inequality in addition for the weakly damped problem we establish the exponential stability of its galerkin semiflows

agreement percolation and phase coexistence in some gibbs systems we extend some relations between percolation and the dependence of gibbs states on boundary conditions known for ising ferromagnets to other systems and investigate their general validity percolation is defined in terms of the agreement of a configuration with one of the ground states of the system this extension is studied via examples and counterexamples including the antiferromagnetic ising and hard core models on bipartite lattices potts models and many layered ising and continuum widom rowlinson models in particular our results on the hard square lattice model make rigorous observations made by hu and mak on the basis of computer simulations moreover we observe that the naturally defined clusters of the widom rowlinson model play for the wr model itself the same role that the clusters of the fortuin kasteleyn measure play for the ferromagnetic potts models the phase transition and percolation in this system can be mapped into the corresponding liquid vapor transition of a one component fluid

rigorous analysis of discontinuous phase transitions via mean field bounds we consider a variety of nearest neighbor spin models defined on the d dimensional hypercubic lattice z d our essential assumption is that these models satisfy the condition of reflection positivity we prove that whenever the associated mean field theory predicts a discontinuous transition the actual model also undergoes a discontinuous transition which occurs near the mean field transition temperature provided the dimension is sufficiently large or the first order transition in the mean field model is sufficiently strong as an application of our general theory we show that for d sufficiently large the 3 state potts ferromagnet on z d undergoes a first order phase transition as the temperature varies similar results are established for all q state potts models with q 3 the r component cubic models with r 4 and the o n nematic liquid crystal models with n 3

dynamics around the ground state of a nonlinear evolution equation we analyze the qualitative behavior of solutions forwards and backwards we present characterizations of blowup boundedness and convergence to the ground state as t we define and use a potential well and several invariant and positive invariant sets 2005 elsevier ltd all rights reserved

critical and sup critical initial energy finite time blowup phenomena for the fourth order wave equations with nonlinear strain term abstract this paper is devoted to the blowup phenomena for a class of fourth order wave equation with nonlinear strain term first for the model equation with strain structure we prove the finite time blowup of solution for the critical initial energy i e e 0 d then the finite time blowup and the upper bound of the blowup time are established for the sup critical initial energy i e e 0 0

a weakly nonlinear analysis of elasto plastic microstructure models at certain critical values of the hardening modulus the governing equations of elasto plastic flow may lose their hyperbolicity and exhibit two modes of ill posedness shear band and flutter ill posedness these modes of ill posedness are characterized by the uncontrolled growth of modes at infinitely fine scales which ultimately violates the continuum assumption in previous work l an and a peirce siam j appl math 54 1994 pp 708 730 a continuum model accounting for microscale deformations was built linear analysis demonstrated the regularizing effect of the microstructure and provided a relationship between the width of the localized instabilities and the microlength scale in this paper a weakly nonlinear analysis is used to explore the immediate post critical behavior of the solutions for both one dimensional and anti plane shear models post critical deformations in the plastic regions are shown to be governed by the boussinesq equation one of the completely integrable pdes having so

a characterization of global and nonglobal solutions of nonlinear wave and kirchhoff equations we give necessary and sufficient conditions for existence of global and nonglobal solutions of a nonlinear wave equation in a bounded domain we consider nonlinear dissipation and a nonlinear source term we also analyze the qualitative behavior of solutions forwards and backwards for the wave equation without dissipation in this case we present characterizations of blow up and asymptotic behavior finally we extend some of our results to a nonlinear kirchhoff equation we use the concepts of stable and unstable sets introduced by payne and sattinger in 1975

on convex envelopes for bivariate functions over polytopes in this paper we discuss convex envelopes for bivariate functions satisfying suitable assumptions over polytopes we first propose a technique to compute the value and a supporting hyperplane of the convex envelope over a general two dimensional polytope through the solution of a three dimensional convex subproblem with continuously differentiable constraint functions then for quadratic functions as well as for some polynomial and rational ones again satisfying suitable assumptions we show how the same computations can be carried out through the solution of a single semidefinite problem

explicit convex and concave envelopes through polyhedral subdivisions in this paper we derive explicit characterizations of convex and concave envelopes of several nonlinear functions over various subsets of a hyper rectangle these envelopes are obtained by identifying polyhedral subdivisions of the hyper rectangle over which the envelopes can be constructed easily in particular we use these techniques to derive in closed form the concave envelopes of concave extendable supermodular functions and the convex envelopes of disjunctive convex functions

discontinuity of the percolation density in one dimensional 1 x y 2 percolation models we consider one dimensional percolation models for which the occupation probability of a bond kx y has a slow power decay as a function of the bond s length for independent models and with suitable reformulations also for more general classes of models it is shown that i no percolation is possible if for short bondskx y p 1 this dichotomy resembles one for the magnetization in 1 x y 2 ising models which was first proposed by thouless and further supported by the renormalization group flow equations of anderson yuval and hamann the proofs of the above percolation phenomena involve rigorous renormalization type arguments of a different sort

a simplicial branch and bound algorithm for solving quadratically constrained quadratic programs we propose a branch and bound algorithm for solving nonconvex quadratically constrained quadratic programs the algorithm is novel in that branching is done by partitioning the feasible region into the cartesian product of two dimensional triangles and rectangles explicit formulae for the convex and concave envelopes of bilinear functions over triangles and rectangles are derived and shown to be second order cone representable the usefulness of these new relaxations is demonstrated both theoretically and computationally

on convex relaxations for quadratically constrained quadratic programming we consider convex relaxations for the problem of minimizing a possibly nonconvex quadratic objective subject to linear and possibly nonconvex quadratic constraints let mathcal f denote the feasible region for the linear constraints we first show that replacing the quadratic objective and constraint functions with their convex lower envelopes on mathcal f is dominated by an alternative methodology based on convexifying the range of the quadratic form genfrac 0 0pt 1 x genfrac 0 0pt 1 x t for x in mathcal f we next show that the use of alpha bb underestimators as computable estimates of convex lower envelopes is dominated by a relaxation of the convex hull of the quadratic form that imposes semidefiniteness and linear constraints on diagonal terms finally we show that the use of a large class of d c difference of convex underestimators is dominated by a relaxation that combines semidefiniteness with rlt constraints

reflection positivity of the random cluster measure invalidated for noninteger q we consider the random cluster potts measure on a lattice torus that weights each connected component by a positive number q we show by constructing a counterexample that this measure is not reflection positive unless q is integer

new and old bounds for standard quadratic optimization dominance equivalence and incomparability a standard quadratic optimization problem stqp consists in minimizing a quadratic form over a simplex among the problems which can be transformed into a stqp are the general quadratic problem over a polytope and the maximum clique problem in a graph in this paper we present several new polynomial time bounds for stqp ranging from very simple and cheap ones to more complex and tight constructions the main tools employed in the conception and analysis of most bounds are semidefinite programming and decomposition of the objective function into a sum of two quadratic functions each of which is easy to minimize we provide a complete diagram of the dominance incomparability or equivalence relations among the bounds proposed in this and in previous works in particular we show that one of our new bounds dominates all the others furthermore a specialization of such bound dominates schrijver s improvement of lovasz s function bound for the maximum size of a clique in a graph

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

the dynamics of a nonlinear wave equation we consider a wave equation in a bounded domain with linear dissipation and with a nonlinear source term we give characterizations of all the solutions with respect to their qualitative properties globality boundedness nonglobality blow up and convergence to equilibria

on the initial boundary problem for fourth order wave equations with damping strain and source terms abstract in this paper we study the global existence and blow up of solutions to the fourth order equations u t t u t 2 u u i 1 n x i i u x i f u x t 0 where 0 under appropriate assumptions on the initial data and parameters in the above equation we establish two results on blow up of solutions with arbitrary initial energy e 0 also by using a potential well we show the global existence of solutions for the fourth order wave equation with some i s and f s especially it is proved that the energy decays exponentially as t

invariance and existence analysis for semilinear hyperbolic equations with damping and conical singularity abstract in this paper we will discuss about the invariance of solution set and present the existence and non existence of the global solutions a class of initial boundary value problems with dissipative terms is considered for a class of semilinear degenerate hyperbolic equations on the cone sobolev spaces first we will discuss the invariance of some sets corresponding to the problem 1 1 and then by using a family of potential wells and concavity methods we obtain existence and non existence results of global solutions with exponential decay and show the blow up in finite time of solutions on a manifold with conical singularities

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

computable representations for convex hulls of low dimensional quadratic forms let mathcal c be the convex hull of points 1 choose x 1 choose x t x in mathcal f subset re n representing or approximating mathcal c is a fundamental problem for global optimization algorithms based on convex relaxations of products of variables we show that if n 4 and mathcal f is a simplex then mathcal c has a computable representation in terms of matrices x that are doubly nonnegative positive semidefinite and componentwise nonnegative we also prove that if n 2 and mathcal f is a box then mathcal c has a representation that combines semidefiniteness with constraints on product terms obtained from the reformulation linearization technique rlt the simplex result generalizes known representations for the convex hull of x 1 x 2 x 1x 2 x in mathcal f when mathcal f subset re 2 is a triangle while the result for box constraints generalizes the well known fact that in this case the rlt constraints generate the convex hull of x 1 x 2 x 1x 2 x in mathcal f when n 3 and mathcal f is a box we show that a representation for mathcal c can be obtained by utilizing the simplex result for n 4 in conjunction with a triangulation of the 3 cube

global existence and blow up of solutions for some hyperbolic systems with damping and source terms abstract this paper deals with the global existence and blow up of solutions to some nonlinear hyperbolic systems with damping and source terms in a bounded domain by using the potential well method we obtain the global existence moreover for the problem with linear damping terms blow up of solutions is considered and some estimates for the lifespan of solutions are given

global optimization of mixed integer quadratically constrained quadratic programs miqcqp through piecewise linear and edge concave relaxations we propose a deterministic global optimization approach whose novel contributions are rooted in the edge concave and piecewise linear underestimators to address nonconvex mixed integer quadratically constrained quadratic programs miqcqp to epsilon global optimality the facets of low dimensional n 3 edge concave aggregations dominating the termwise relaxation of miqcqp are introduced at every node of a branch and bound tree concave multivariable terms and sparsely distributed bilinear terms that do not participate in connected edge concave aggregations are addressed through piecewise linear relaxations extensive computational studies are presented for point packing problems standard and generalized pooling problems and examples from globallib meeraus globallib http www gamsworld org global globallib htm

well posedness and optimal decay rates for the wave equation with nonlinear boundary damping source interaction we establish subject to some natural additional assumptions imposed on the relation between the source and the damping both well posedness and effective optimal decay rates for the solutions of a semilinear model of the wave equation the theory presented allows to consider both superlinear and sublinear behaviours of the dissipation in the presence of unstructured sources

nonlinear parabolic equations with critical initial conditions j u0 d or i u0 0 abstract in this paper we study the initial boundary value problem of nonlinear parabolic equation u t i 1 n x i u x i p 2 u x i u 1 x t 0 u x 0 u 0 x x u x t 0 x t 0 where r n is a bounded domain p p n p np n p if p n by using potential well method we prove that if 0 u 0 x w 0 1 p j u 0 d i u 0 0 or i u 0 0 0 j u 0 d then the problem admits a global solution u t l 0 w 0 1 p with u t t l 2 0 l 2 and u t w for 0 t

jointly constrained biconvex programming this paper presents a branch and bound algorithm for minimizing the sum of a convex function in x a convex function in y and a bilinear term in x and y over a closed set such an objective function is called biconvex with biconcave functions similarly defined the feasible region of this model permits joint constraints in x and y to be expressed the bilinear programming problem becomes a special case of the problem addressed in this paper we prove that the minimum of a biconcave function over a nonempty compact set occurs at a boundary point of the set and not necessarily an extreme point the algorithm is proven to converge to a global solution of the nonconvex program we discuss extensions of the general model and computational experience in solving jointly constrained bilinear programs for which the algorithm has been implemented

multidimensional viscoelasticity equations with nonlinear damping and source terms abstract in this paper we study the initial boundary value problem of multidimensional viscoelasticity equations u tt u t i 1 n x i i u x i f u t g 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 i s1 i s2 s1 s2 0 f s s 0 g s s 0 and satisfy some growth conditions first we introduce a family of potential wells by using a new method then using it we obtain some new existence theorems of global solutions and behaviour of vacuum isolation of solutions

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

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

the kirkwood salsburg equations for random continuum percolation we develop two different hierarchies of kirkwood salsburg equations for the connectedness functions of random continuum percolation these equations are derived by writing the kirkwood salsburg equations for the distribution functions of thes state continuum potts model cpm taking thes 1 limit and forming appropriate linear combinations the first hierarchy is satisfied by a subset of the connectedness functions used in previous studies it gives rigorous low order bounds for the mean number of clusters n c and the two point connectedness function the second hierarchy is a closed set of equations satisfied by the generalized blocking functions each of which is defined by the probability that a given set of connections between particles is absent these auxiliary functions are shown to be a natural basis for calculating the properties of continuum percolation models they are the objects naturally occurring in integral equations for percolation theory also the standard connectedness functions can be written as linear combinations of them using our second kirkwood salsburg hierarchy we show the existence of an infinite sequence of rigorous upper and lower bounds for all the quantities describing random percolation including the mean cluster size and mean number of clusters these equations also provide a rigorous lower bound for the radius of convergence of the virial series for the mean number of clusters most of the results obtained here can be readily extended to percolation models on lattices and to models with positive repulsive pair potentials

global existence asymptotic behavior and blowup of solutions for a class of nonlinear wave equations with dissipative term abstract the paper studies the global existence asymptotic behavior and blowup of solutions to the initial boundary value problem for a class of nonlinear wave equations with dissipative term it proves that under rather mild conditions on nonlinear terms and initial data the above mentioned problem admits a global weak solution and the solution decays exponentially to zero as t respectively in the states of large initial data and small initial energy in particular in the case of space dimension n 1 the weak solution is regularized to be a unique generalized solution and if the conditions guaranteeing the global existence of weak solutions are not valid then under the opposite conditions the solutions of above mentioned problem blow up in finite time and an example is given

coexistence of partially disordered ordered phases in an extended potts model we consider a generalization of the standard potts model in which there are q r s states with an interaction that distinguishes the two subspecies we develop a graphical representation of the fk type for the system and show that this representation may be incorporated directly into reflection positivity arguments using combinations of these techniques we establish detailed properties of the phase diagram including the existence of sharp triple points whenever relevant the phases are characterized by the percolation properties of the underlying representation

global solutions and finite time blow up for fourth order nonlinear damped wave equation in this paper we study the initial boundary value problem and global well posedness for a class of fourth order wave equations with a nonlinear damping term and a nonlinear source term which was introduced to describe the dynamics of a suspension bridge the global existence decay estimate and blow up of solution at both subcritical e 0 0 in this paper we study the initial boundary value problem and global well posedness for a class of fourth order wave equations with a nonlinear damping term and a nonlinear source term which was introduced to describe the dynamics of a suspension bridge the global existence decay estimate and blow up of solution at both subcritical e 0 0

discontinuity of the spin wave stiffness in the two dimensional xy model using a graphical representation based on the wolff algorithm the classical d dimensional xy model and some related spin systems are studied it is proved that in d 2 the predicted discontinuity in the spin wave stiffness indeed occurs further the critical properties of the spin system are related to percolation properties of the graphical representation in particular a suitably defined notion of percolation in the graphical representation is proved to be the necessary and sufficient condition for positivity of the spontaneous magnetization

the convex envelope of n 1 convex functions the question of determining strong convex underestimators for nonlinear functions is theoretically and practically of major interest unfortunately results along these lines are quite limited as very few general procedures are at hand that can be applied to general classes of functions in this paper we show how to reduce the question of determining a convex envelope to lower dimensional optimization problems when the underlying function is indefinite and n 1 convex our structural result about this reduction technique enables us to give descriptions for the convex envelope of a variety of two dimensional functions

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

graphical representations and cluster algorithms ii we continue the study initiated in part i of graphical representations and cluster algorithms for various models in or related to statistical mechanics for certain models e g the blume emery griffths model and various generalizations we develop fortuin kasteleyn type representations which lead immediately to swendsen wang type algorithms for other models e g the random cluster model that are defined by a graphical representation we develop cluster algorithms without reference to an underlying spin system in all cases phase transitions are related to percolation or incipient percolation in the graphical representation which via the ic algorithm allows for the rapid simulation of these systems at the transition point pertinent examples include the continuum widom rowlinson model the restricted 1 step solid on solid model and the xy model

a class of fourth order wave equations with dissipative and nonlinear strain terms we study the initial boundary value problem for a class of fourth order wave equations with dissipative and nonlinear strain terms by introducing a family of potential wells we not only obtain the invariant sets and vacuum isolating of solutions but also give some threshold results of global existence and nonexistence of solutions

finite dimensional attractors for the kirchhoff models with critical exponents the paper studies the existence of the finite dimensional global attractor and exponential attractor for the dynamical system associated with the kirchhoff models utt u m 1 u ut 2u h ut g u f x it proves that for the subcritical and critical cases 1 m n 2 n 2 with n 6 the above mentioned dynamical system possesses in phase spaces x1 a finite dimensional global attractor and a weak exponential attractor respectively

existence and asymptotic stability for evolution problems on manifolds with damping and source terms abstract one considers the nonlinear evolution equation with source and damping terms u tt au g u t u u on 0 where is a compact manifold we prove the global existence of solutions making use of the potential well method furthermore we study the asymptotic behaviour of solutions adapting the ideas introduced by patrick martinez in esaim control optim calc var 4 1999 419 444 when the frictional damping mechanism g on the manifold does not necessarily have a polynomial growth near the origin