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

unified approach to h optimization hankel approximation and balanced realization problems a unified approach is proposed here to solve three problems simultaneously namely h optimization hankel approximation and minimal balanced realization for scalar systems it is shown that all these problems can be reduced to the same standard form wherein merely the singular value decomposition svd of an easily constructed square matrix is required to establish the final solutions the formulation is solely in terms of the coefficients of a transfer function and the resulting algorithm can be performed in a very computationally efficient way

vacuum isolating blow up threshold and asymptotic behavior of solutions for a nonlocal parabolic equation in this paper we consider a nonlocal parabolic equation associated with initial and dirichlet boundary conditions first we discuss the vacuum isolating behavior of solutions with the help of a family of potential wells then we obtain a threshold of global existence and blow up for solutions with critical initial energy furthermore for those solutions that satisfy j u0 d and i u0 0 we show that global solutions decay to zero exponentially as time tends to infinity and the norm of blow up solutions increases exponentially

on weighted integrability of double sine series abstract we introduce a new class of double sequences d g m r called double general monotone and some new class of weight functions to study the weighted integrability of double sine series some results of d yu p zhou and s zhou are also generalized

total blow up versus single point blow up traditional thermal explosion theory is used to describe reaction initiation in condensed explosives and is limited formally to nondeformable materials kassoy and poland 4 significantly extended this theory to develop an ignition model for a reactive gas in a bounded container in order to describe the induction period during this induction period there is a spatially homogeneous pressure rise in the system which causes a com pressive heating effect in the constant volume container mathematically this compressibility of the gas is expressed by means of an integral term in the induction model for the temperature perturbation 0 x t this model is given by e ak 1 j e x t dx y x 2 0 x 0 d x 2 0

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 asymptotical behavior and some new blow up conditions of solutions to a thin film equation abstract we consider a thin film equation by exploiting the boundary condition and the variational structure of the equation we consider the global existence and blow up of the solutions for the global solutions we study the asymptotic behavior and for the blow up solutions we study the upper and lower bounds of the blow up time especially we have derived the necessary and sufficient conditions for the solution blowing up in finite time when the initial energy j u 0 d where d is the mountain pass level for j u 0 d we also study the conditions for the solution exists globally and blows up in finite time and we also obtain some necessary and sufficient conditions for the solution blowing up in finite time when the initial energy at arbitrary level furthermore we study the decay behavior of both the global solutions and its corresponding energy functional and the decay ratios are given specially the results generalize the former studies on this equation such as the papers li et al 2016 16 dong and j zhou 2017 4 and sun et al 2018 23

on weighted integrability of double cosine series abstract in this paper we study weighted integrability of the sum of double cosine series the results obtained extend a theorem of 5 onlp integrability of double cosine series and two theorems of 1 on weightedlp norms

trigonometric series with general monotone coefficients we study trigonometric series with general monotone coefficients convergence results in the different metrics are obtained also we prove a hardy type result on the behavior of the series near the origin

internally balanced minimal realization of discrete siso systems given the transfer function or the state equations in phase canonical form of a discrete siso system a method is proposed to obtain an internally balanced state space representation of it

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

a new blow up condition for a parabolic equation with singular potential abstract this paper is aimed at giving a new blow up condition of the following parabolic problem with singular potential x s u t u u p 2 u which was studied extensively in recent years by constructing a new functional we obtain some blow up conditions which include the possibility of the initial energy j u 0 d where d is a positive constant which will be given in the main part of this paper the results of this paper extend previous results where only the case j u 0 d was considered

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

blow up in a system of partial differential equations with conserved first integral part ii problems with convection a reaction diffusion convection equation with a nonlocal term is studied the nonlocal operator acts to conserve the spatial integral of the unknown function as time evolves the equations are parameterised by and for 1 the equation arises as a similarity solution of the navier stokes equations and the nonlocal term plays the role of pressure for 0 the equation is a nonlocal reaction diffusion problem the aim of the paper is to determine for which values of the parameter blow up occurs and to study its form in particular interest is focused on the three cases 1 2 and 1 it is observed that for any 0 1 2 nonuniform global blow up occurs if 1 2 1 then the blow up is global and uniform while for 1 the navier stokes equations there are exact solutions with initial data of arbitrarily large l l 2 and h 1 norms that decay to zero furthermore one of these exact solutions is proved to be nonlinearly stable in l 2 for arbitrarily large supremum norm an understanding of this transition from blow up behaviour to decay behaviour is achieved by a combination of analysis asymptotics and numerical techniques

blowup in a partial differential equation with conserved first integral a reaction diffusion equation with a nonlocal term is studied the nonlocal term acts to conserve the spatial integral of the unknown function as time evolves such equations give insight into biological and chemical problems where conservation properties predominate the aim of the paper is to understand how the conservation property affects the nature of blowup the equation studied has a trivial steady solution that is proved to be stable existence of nontrivial steady solutions is proved and their instability established numerically blowup is proved for sufficiently large initial data by using a comparison principle in fourier space the nature of the blowup is investigated by a combination of asymptotic and numerical calculations

blow up for a thin film equation with positive initial energy abstract in this paper we consider a thin film equation with nonlocal source which was studied by qu and zhou 2016 11 where the authors derived the conditions for global existence blow up and extinction we consider the case that the initial energy is positive and establish a blow up result for this case furthermore the upper bound of the blow up time is derived

finite time blow up for a thin film equation with initial data at arbitrary energy level abstract in this paper we consider the initial boundary value problem for a class of thin film equations in r n with a p laplace term and a nonlocal source term u q 2 u 1 u q 2 u dx we prove that there exist weak solutions for the problem with arbitrarily initial energy that blow up in finite time we also obtain the upper bounds for the blow up time

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

a reaction diffusion model for a single species with age structure i travelling wavefronts on unbounded domains in this paper we derive the equation for a single species population with two age classes and a fixed maturation period living in a spatially unbounded environment we show that if the mature deat

high initial energy finite time blowup with upper bound of blowup time of solution to semilinear parabolic equations abstract a sufficient condition on initial profile ensuring high initial energy finite time blowup along with the upper bound of the corresponding blowup time for the classical semilinear parabolic equations involving initial boundary value problem is established in this paper

persistence of wavefronts in delayed nonlocal reaction diffusion equations we develop a perturbation argument based on existing results on asymptotic autonomous systems and the fredholm alternative theory that yields the persistence of traveling wavefronts for reaction diffusion equations with nonlocal and delayed nonlinearities when the time lag is relatively small this persistence result holds when the nonlinearity of the corresponding ordinary reaction diffusion system is either monostable or bistable we then illustrate this general result using five different models from population biology epidemiology and bio reactors

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

blow up versus extinction in a nonlocal p laplace equation with neumann boundary conditions abstract this paper studies a fast diffusive p laplace equation with the nonlocal source u q u q d x in a bounded domain subject to homogeneous neumann boundary value condition a critical criterion is determined that the changing sign solutions blow up in finite time with q 1 and non positive initial energy associated and must be global for any initial energy if q 1 in particular the conditions are obtained under which the changing sign solutions vanish in finite time

generalized convexity and the existence of finite time blow up solutions for an evolutionary problem abstract in this paper we study a class of nonlinearities for which a nonlocal parabolic equation with neumann robin boundary conditions for p laplacian has finite time blow up solutions

global existence and finite time blow up of the solution for a thin film equation with high initial energy abstract this paper is devoted to the study of a thin film equation which was considered extensively in recent years we obtain the conditions for global existence and finite time blow up of solution for the high initial energy case i e j u 0 d while the previous works only considered the case j u 0 d here d is the mountain pass level moreover for the blow up solution with j u 0 d we obtain the lower bound estimate of the blow up time and blow up rate

balanced minimal realization via singular value decomposition of sarason operator abstract in this paper a method is proposed to obtain a balanced minimal realization of a stable rational transfer function matrix the allison young algorithm for the interpolation problems is directly extended to matrix value cases and is applied in the realization procedure this method uses the finite matrix of a sarason operator instead of the infinite block hankel matrix and the coefficients of the transfer function matrix instead of any initial realization the advantage of this approach in connection with the solution of l optimization problems and the balanced realization is also discussed

blow up and extinction for a thin film equation with initial boundary value conditions abstract in this paper we consider an initial boundary value problem for a thin film equation with the nonlocal source u q u q d x in a bounded domain due to the comparison principle being invalid we use the potential well method to give a threshold result of global existence and non existence for the sign changing weak solutions further we obtain the conditions under which the global solutions extinct in finite time and study the asymptotic behavior of the non extinct solutions

arbitrarily positive initial energy blowup and blowup time for some fourth order viscous wave equation abstract this paper is devoted to the finite time blowup phenomena for a class of fourth order wave equation with viscous damping term it is shown that the high initial energy finite time blowup associated with the upper bound of the blowup time is established by introducing a new auxiliary function and employing an adapted concavity method

singular perturbational approximations for discrete time balanced systems a new zero order singular perturbational type approximation is developed for model reduction of discrete time linear systems this approximation is also suitable for model reduction of systems which have fast subsystems and which are represented in the internally balanced format subsystem elimination as suggested by moore for continuous time systems does not generalize for the discrete time case and the singular perturbational approximation gives a particular solution to the discrete time internally balanced model reduction problem

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

blow up phenomena for a nonlocal semilinear parabolic equation with positive initial energy abstract this paper is concerned with the blow up of solutions to the following semilinear parabolic equation u t u u p 1 u 1 u p 1 u d x x t 0 under homogeneous neumann boundary condition in a bounded domain r n n 1 with smooth boundary for all p 1 we prove that the classical solutions to the above equation blow up in finite time when the initial energy is positive and initial data is suitably large this result improves a recent result by gao and han 2011 which asserts the blow up of classical solutions for n 3 provided that 1 p n 2 n 2

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

blowup in diffusion equations a survey this paper deals with quasilinear reaction diffusion equations for which a solution local in time exists if the solution ceases to exist for some finite time we say that it blows up in contrast to linear equations blowup can occur even if the data are smooth and well defined for all times depending on the equation either the solution or some of its derivatives become singular we shall concentrate on those cases where the solution becomes unbounded in finite time this can occur in quasilinear equations if the heat source is strong enough there exist many theoretical studies on the question on the occurrence of blowup in this paper we shall recount some of the most interesting criteria and most important methods for analyzing blowup the asymptotic behavior of solutions near their singularities is only completely understood in the special case where the source is a power a better knowledge would be useful also for their numerical treatment thus not surprisingly the numerical analysis of this type of problems is still at a rather early stage the goal of this paper is to collect some of the known results and algorithms and to direct the attention to some open problems

global existence blow up and extinction for a class of thin film equation abstract in this paper we use the modified method of potential wells to study the properties of solutions for a class of higher order nonlinear parabolic equations with p laplace term u x p 2 u x x and nonlocal source u q 1 u 1 u q 1 u d x global existence uniqueness blow up in finite time and asymptotic behavior of solutions will be proved under different initial conditions furthermore a numerical example is given to illustrate the blow up of solutions in finite time

on l convergence of trigonometric series abstract in the present paper we consider the trigonometric series with r general monotone and r rest bounded variation coefficients necessary and sufficient conditions of l convergence for such series are obtained in terms of the coefficients moreover we generalize and extend the tikhonov results j math anal appl 347 2008 416 427 to the class gm r or the class rbvs r

global nonexistence theorems for a class of evolution equations with dissipation we study abstract evolution equations with nonlinear damping terms and source terms including as a particular case a nonlinear wave equation of the type ba cl u tt delta u b u t m 2 u t c u p 2 u t x in 0 t times omega 6pt u t x 0 t x in 0 t times partial omega 6pt u 0 cdot u 0 in h 0 1 omega quad u t 0 cdot v 0 in l 2 omega es ea where 0 p 2 m 1 we prove a global nonexistence theorem for positive initial value of the energy when 1 1

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

blowup of solutions of semilinear parabolic equations on etudie le comportement d eclatement des solutions de u t uu x u xx f u a 0 u a t 0 t 0 u x 0 x a 2

integrability theorem of multiple trigonometric series x is not in general lebesgue integrable in 0 n 2 p 7881 he also proved that for 0 y 1 a function like xp near 0 has a cosine series whose coefficients behave at infinity like n7 l and conversely 4 p 31 boas 3 proved the following theorem

global existence and blow up of solutions to a nonlocal parabolic equation with singular potential abstract in this paper we study a nonlocal parabolic equation with singular potential on a bounded smooth domain with homogeneous neumann boundary condition firstly we find a threshold of global existence and blow up to the solutions of the problem when the initial data is at the low energy level i e j u 0 d where j u 0 is the initial energy and d is the mountain pass level moreover when j u 0 d the vacuum isolating behavior of the solutions is also discussed secondly we prove that there exist solutions of the problem with arbitrary initial energy that blow up in finite time we also obtain the upper bounds of the blow up time for blow up solutions