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

iterative method for the study of the thermodynamic behavior of hard sphere yukawa fluids abstract semi analytic expressions for the calculation of the thermodynamic properties of the hard sphere yukawa fluid are derived from the solution of the ornstein zernike equation according to the mean spherical approximation these expressions are written in terms of a cumulative parameter that becomes the physically meaningful root of a fourth or larger order polynomial which can be determined directly by a proposed iterative method the method is illustrated by comparison against well known results including predictions from perturbation theories and its limitations are discussed

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

comparison results for quasilinear elliptic equations via a picone type identity part i quasilinear cases abstract using some identities of picone s types comparison results and uniqueness theorems are established for some general quasilinear elliptic equations

optimal wloc2 2 regularity pohozhaevʼs identity and nonexistence of weak solutions to some quasilinear elliptic equations abstract we begin by establishing a sharp optimal w loc 2 2 regularity result for bounded weak solutions to a nonlinear elliptic equation with the p laplacian p u def div u p 2 u 1 p we develop very precise optimal regularity estimates on the ellipticity of this degenerate for 2 p or singular for 1 p 2 problem we apply this regularity result to prove pohozhaev s identity for a weak solution u w 1 p of the elliptic neumann problem p p u w u f x in u 0 on here is a bounded domain in r n whose boundary is a c 2 manifold x 0 denotes the outer unit normal to at x 0 x x 1 x n is a generic point in and f l w 1 1 the potential w r r is assumed to be of class c 1 and of the typical double well shape of type w s 1 s for s r where 1 are some constants finally we take an advantage of the pohozhaev identity to show that problem p with f 0 in has no phase transition solution u w 1 p 1 p n such that 1 u 1 in with u 1 in 1 and u 1 in 1 where both 1 and 1 are some nonempty subdomains of such a scenario for u is possible only if n 1 and 1 1 are finite unions of suitable subintervals of the open interval r 1

global strong solution of fourth order nonlinear wave equation abstract in this paper we consider a class of fourth order nonlinear wave equation in a bounded domain with smooth boundary and discuss the existence and uniqueness of the global strong solutions by gradually smoothing method

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

global existence nonexistence and asymptotic behavior of solutions for the cauchy problem of semilinear heat equations abstract in this paper we study the cauchy problem of semilinear heat equations by introducing a family of potential wells we first prove the invariance of some sets and isolating solutions then we obtain a threshold result for the global existence and nonexistence of solutions finally we discuss the asymptotic behavior of the solution

uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation for independent translation invariant irreducible percolation models it is proved that the infinite cluster when it exists must be unique the proof is based on the convexity or almost convexity and differentiability of the mean number of clusters per site which is the percolation analogue of the free energy the analysis applies to both site and bond models in arbitrary dimension including long range bond percolation in particular uniqueness is valid at the critical point of one dimensional 1 x y 2 models in spite of the discontinuity of the percolation density there corollaries of uniqueness and its proof are continuity of the connectivity functions and except possibly at the critical point of the percolation density related to differentiability of the free energy are inequalities which bound the specific heat critical exponent in terms of the mean cluster size exponent and the critical cluster size distribution exponent e g 1 2 1 1

global well posedness for a nonlocal semilinear pseudo parabolic equation with conical degeneration abstract this paper deals with a class of nonlocal semilinear pseudo parabolic equation with conical degeneration u t b u t b u u p 1 u 1 b b u p 1 u d x 1 x 1 d x on a manifold with conical singularity where b is fuchsian type laplace operator with totally characteristic degeneracy on the boundary x 1 0 by using the modified method of potential well with galerkin approximation and concavity the global existence uniqueness finite time blow up and asymptotic behavior of the solutions will be discussed at the low initial energy j u 0 d and critical initial energy j u 0 d respectively furthermore we investigate the global existence and finite time blow up of the solutions with the high initial energy j u 0 d by the variational method especially we also derive the threshold results of global existence and nonexistence for the solutions at two different initial energy levels i e low initial level and critical initial level

an intermediate phase with slow decay of correlations in one dimensional 1 x y 2 percolation ising and potts models we rigorously establish the existence of an intermediate ordered phase in one dimensional 1 x y 2 percolation ising and potts models the ising model truncated two point function has a power law decay exponent which ranges from its low and high temperature value of two down to zero as the inverse temperature and nearest neighbor coupling vary similar results are obtained for percolation and potts models

existence and nonexistence of nontrivial solutions of some nonlinear degenerate elliptic equations in connection with the maximizing problem for the functional r u u lq u lp in w01 p 0 we consider the equation div u p 2 u x u q 2 u x x 1 p resp q p e has no nontrivial resp nonnegative nontrivial solution belonging to the class p u w01 p lq xi u q 2u lp p 1 i 1 2 n w01 p provided that is star shaped the crucial point of the proof of our result is to obtain an l estimate of weak solutions and to verify a certain pohozaev type inequality for weak solutions belonging to p

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

discontinuity of the magnetization in one dimensional 1 x y 2 ising and potts models results from percolation theory are used to study phase transitions in one dimensional ising and q state potts models with couplings of the asymptotic form j sub x y approx const absolute value x y sup 2 for translation invariant systems with well defined lim sub x infinity x sup 2 j sub x j sup possibly o or infinity we establish 1 there is no long range order at inverse temperatures beta with beta j sup less than or equal to1 2 if beta j sup q then by sufficiently increasing j sub 1 the spontaneous magnetization m is made positive 3 in models with 0 j sup infinity the magnetization is discontinuous at the transition point as originally predicted by thouless and obeys m beta sub c less than or equal to 1 beta sub c j sup sup 1 2 4 for ising q 2 models with j sup infinity it is noted that the correlation function decays as beta approx c beta absolute value x y sup 2 whenever beta beta sub c points 1 3 are deduced from previous percolation results by utilizing the fortuin kasteleyn representation which also yields other results of independent interest relating potts models with different values of q

ising models on hyperbolic graphs we consider ising models with ferromagnetic interactions and zero external magnetic field on the hyperbolic graph v f where v is the number of neighbors of each vertex and f is the number of sides of each face let tc be the critical temperature and t c sup t tc f 2 where f is the free boundary condition b c gibbs state is the plus b c gibbs state and is the minus b c gibbs state we prove that if the hyperbolic graph is self dual i e v f or if v is sufficiently large how large depends on f e g v 35 suffices for any f 3 and v 17 suffices for any f 17 then 0 t c tc in contrast with that t c tc for ising models on the hypercubic lattice zd with d 2 a result due to lebowitz 22 while whenever t t c f 2 the last result is an improvement in comparison with the analogous statement in refs 28 and 33 in which it was only proved that f 2 when t t c and it remains to show in both papers that f 2 whenever t t c therefore t c and tc divide 0 into three intervals 0 t c t c tc and tc in which but f 2 and f 2 and respectively

global well posedness asymptotic behavior and blow up of solutions for a class of degenerate parabolic equations abstract the initial boundary value problem for a class of degenerate parabolic equations is studied some new results on global existence of solutions are established by introducing a family of potential wells in addition asymptotic behavior and finite time blow up of solutions are obtained in the case of subcritical initial energy and critical initial energy respectively

addendum to global existence and finite time blow up for a class of semilinear pseudo parabolic equations j func anal 264 12 2013 2732 2763 abstract in this remark we correct the proof of the asymptotic behavior of solution for the initial boundary value problem of semilinear pseudo parabolic equations with critical initial energy obtained in xu and su 2013 1

sharp threshold of global existence for nonlinear schrodinger equation with partial confinement abstract this paper deals with the nonlinear schrodinger equation with partial confinement which may model the attractive bose einstein condensate under a partial trap potential by using the variational characteristic of the classic nonlinear scalar field equation and the hamilton conservation we get the threshold for global existence of the cauchy problem on mass moreover by a novel scaling argument we prove that this threshold is sharp in addition by a numerical computation we obtain the numerical result of the threshold

on periodic solutions of superlinear parabolic problems on etudie l existence des solutions periodiques non triviales positives des problemes paraboliques semilineaires on demontre des resultats de bifurcation

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

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

threshold results for the existence of global and blow up solutions to kirchhoff equations with arbitrary initial energy abstract in this paper we will apply the modified potential well method and variational method to the study of the long time behaviors of solutions to a class of parabolic equation of kirchhoff type global existence and blow up in finite time of solutions will be obtained for arbitrary initial energy to be a little more precise we will give a threshold result for the solutions to exist globally or to blow up in finite time when the initial energy is subcritical and critical respectively the decay rate of the l 2 norm is also obtained for global solutions in these cases moreover some sufficient conditions for the existence of global and blow up solutions are also derived when the initial energy is supercritical

the contact process on trees the contact process on an infinite homogeneous tree is shown to exhibit at least two phase transitions as the infection parameter is varied for small values of a single infection eventually dies out for larger the infection lives forever with positive probability but eventually leaves any finite set the survival probability is a continuous function of and the proof of this is much easier than it is for the contact process on d dimensional integer lattices for still larger the infection converges in distribution to a nontrivial invariant measure for an n ary tree with n large the first of these transitions occurs when 1 n and the second occurs when 1 2 n e n nonhomogeneous trees whose vertices have degrees varying between 1 and n behave essentially as homogeneous n ary trees provided that vertices of degree n are not too rare in particular letting n go to galton watson trees whose vertices have degree n with probability that does not decrease exponentially with n may have both phase transitions occur together at 0 the nature of the second phase transition is not yet clear and several problems are mentioned in this regard

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

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

a note on blow up of solution for a class of semilinear pseudo parabolic equations abstract in this short note we revisit the blow up of solution for the initial boundary value problem of semilinear pseudo parabolic equations with low critical initial energy stated in xu and su 2013 4 and amend the proofs of the original paper

a note on finite time blowup for dissipative klein gordon equation abstract this is a continuation of the study on an arbitrarily positive initial energy finite time blowup result of the solution to the initial value problem of strongly damped klein gordon equation based on an adapted concavity method both the conclusions and arguments of the present paper are not bounded by the coefficients of dissipative damping terms

the ising model on diluted graphs and strong amenability say that a graph has persistent transition if the ising model on the graph can exhibit a phase transition nonuniqueness of gibbs measures in the presence of a nonzero external field we show that for nonamenable graphs for bernoulli percolation with p close to 1 all the infinite clusters have persistent transition on the other hand we show that for transitive amenable graphs the infinite clusters for any stationary percolation do not have persistent transition this extends a result of georgii for the cubic lattice a geometric consequence of this latter fact is that the infinite clusters are strongly amenable i e their anchored cheeger constant is 0 finally we show that the critical temperature for the ising model with no external field on the infinite clusters of bernoulli percolation with parameter p on an arbitrary bounded degree graph is a continuous function of p

periodic solutions of the evolutionary p laplacian with nonlinear sources abstract this paper is concerned with the evolutionary p laplacian with nonlinear and periodic sources we will give a rather complete characterization in terms of the parameter p and the exponent q of the source of whether or not the positive periodic solutions exist

global solutions and blow up solutions to a class of pseudo parabolic equations with nonlocal term abstract in this paper we investigate an initial boundary value problem to a class of pseudo parabolic partial differential equations with newtonian nonlocal term first the local existence and uniqueness of a weak solution is established in virtue of the energy functional and the related nehari manifold we also describe the exponent decay behavior and the blow up phenomenon of weak solutions with different kinds of initial data our second conclusion states that some solutions starting in a potential well exist globally whereas solutions with suitable initial data outside the potential well must blow up furthermore the instability of a ground state equilibrium solution is studied

eigenvalue estimates for the p laplace operator on manifolds abstract in this paper some new geometric estimates for the first eigenvalue and the fundamental tone of the p laplacian on manifolds in terms of admissible vector fields are obtained also a new spectral invariant and its relation with the geometry of the manifold is defined

a note on mean field behavior for self avoiding walk on branching planes we consider the critical behavior of the susceptibility of the self avoiding walk on the grapht z wheret is a bethe lattice with degreek andz is the one dimensional lattice by directly estimating the two point function using a method of grimmett and newman we show that the bubble condition is satisfied whenk 2 and therefore the critical exponent associated with the susceptibility equals 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

well posedness and ill posedness of a multidimensional chemotaxis system in the critical besov spaces abstract we study the cauchy problem for a multidimensional chemotaxis system in critical besov spaces b p 1 d p 2 r d b p 1 d p 1 r d d for 1 p 2 d we prove locally well posedness for large initial data and globally well posedness for small initial data of this system and more importantly we show the ill posedness in the sense that a norm inflation phenomenon occurs for p 2 d more precisely we construct a specific initial data which can be arbitrarily small in the besov spaces meanwhile the corresponding solution u can be arbitrarily large after an arbitrarily short time

blow up phenomena for a pseudo parabolic equation with p laplacian and logarithmic nonlinearity terms abstract this paper deals with a pseudo parabolic equation with p laplacian and logarithmic nonlinearity terms under homogeneous dirichlet boundary condition in a smooth bounded domain which was studied in 12 and the global existence and finite time blow up of the weak solution were studied under different energy levels we generalize and extend those results by discussing the asymptotic behavior of the weak solution and proving that the weak solution converges to the corresponding stationary solution as time tends to infinity moreover a lower and an upper bound estimation for blow up time and rate are obtained for the blow up weak solution in different initial energy cases furthermore we establish the weak solution with high initial energy that is global bounded or blow up under some sufficient conditions as well as a nonblow up criterion and algebraic decay result in other assumptions

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

nonexistence theorems for singular solutions of quasilinear partial differential equations on etudie les etats fondamentaux singuliers de l equation de poisson semilineaire u f u 0 et de son analogue quasi lineaire div a du du f u 0 ou est le laplacien a n dimensions et a p et f u sont des fonctions donnees

phase transitions on nonamenable graphs we survey known results about phase transitions in various models of statistical physics when the underlying space is a nonamenable graph most attention is devoted to transitive graphs and trees

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

c1 α local regularity of weak solutions of degenerate elliptic equations abstract it is demonstrated the local c 1 alpha nature of weak solutions of elliptic equations of the type 1 1 in the introduction under the degeneracy or singularity assumptions a sub 1 a sub 3

on the analytical solution of the ornstein zernike equation with yukawa closure we discuss the solution of the ornstein zernike equation for most general closure consisting of a sum ofm yukawa type exponentials a formal solution for the factored case is bound for an arbitrary mixture of hard spheres introducing a general scaling matrix of dimensions m m a sufficient number of equations for this matrix is obtained from symmetry considerations and the boundary condition we discuss also restricted and semirestricted case for which explicit solutions in terms of the scaling parameters and input parameters are found

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

existence results for some nonlinear elliptic system with lack of compactness abstract we are interested in existence of solutions for some typical elliptic systems governed by two pseudo laplacian operators with critical sobolev exponents we show this result by a local minimization for an adapted variational problem a concentration compactness principle allows us to obtain some convergence results

global solution and blow up for a class of pseudo p laplacian evolution equations with logarithmic nonlinearity the main goal of this work is to study an initialboundary value problem for a nonlinear pseudoparabolic equation with logarithmic nonlinearity by using the potential well method and a logarithmic sobolev inequality we obtain results of existence or nonexistence of global weak solutions in addition we also provide sufficient conditions for the large time decay of global weak solutions and the finite time blow up of weak solutions

nonexistence of solutions to hele shaw equations abstract we study nonexistence of the solutions to quasilinear elliptic differential equations arising from nonisothermal non newtonian hele shaw flows the proof is based on the trial function method developed by pohozaev without recourse to comparison theorems and to the maximum principle

effect of aggregation on population recovery modeled by a forward backward pseudoparabolic equation in this paper we study the equation u t u f u u t f u in a bounded domain of r d d 1 with homogeneous boundary conditions of the neumann type as a model of aggregating population with a migration rate determined by and total birth and mortality rates characterized by f we will show that the aggregating mechanism induced by u allows the survival of a species in danger of extinction numerical simulations suggest that the solutions stabilize asymptotically in time to a not necessarily homogeneous stationary solution this is shown to be the case for a particular version of the function u

hyperbolic and parabolic equations with several opposite sign source at critical initial energy level in heat and vibrating systems abstract we study semilinear hyperbolic equations and parabolic equations with several opposite sign source terms under the dirichlet boundary condition in heat and vibrating systems for the above problems we prove the global nonexistence of solution and give the sharp conditions at critical energy level furthermore we prove that the existence of global solution of parabolic equation vanishes to zero as time goes to infinity which is different from the behavior of solution of the hyperbolic equation

on family of potential wells and applications to semilinear hyperbolic equations and parabolic equations abstract we study the initial boundary value problem for semilinear hyperbolic equations and parabolic equations with convex function nonlinear terms by introducing a family of potential wells we not only obtain the invariant sets and vacuum isolation of solutions but also give a threshold result of global existence and nonexistence of solutions moreover we discuss the global existence of solution for problem with critical initial condition e 0 d or j u 0 d which fills some important gaps regarding this problem finally we prove the global nonexistence of solution and asymptotic behavior of solution for semilinear parabolic equations

multiplicity of phase transitions and mean field criticality on highly non amenable graphs we consider independent percolation ising and potts models and the contact process on infinite locally finite connected graphs

second critical exponent and life span for pseudo parabolic equation this paper studies the second critical exponent and life span of solutions for the pseudo parabolic equation ut k ut u up in rn 0 t with p 1 k 0 it is proved that the second critical exponent i e the decay order of the initial data required by global solutions in the coexistence region of global and non global solutions is independent of the pseudo parabolic parameter k nevertheless it is revealed that the viscous term k ut relaxes restrictions on the amplitude of the initial data required by the global solutions moreover it is observed that the life span of the non global solutions will be delayed by the third order viscous term finally some numerical examples are given to illustrate all these results

global existence and nonexistence of the initial boundary value problem for the dissipative boussinesq equation abstract a class of multidimensional dissipative boussinesq equations is considered under three different cases of initial energy we prove the existence and uniqueness of global solutions in the energy space and provide sufficient conditions of finite time blow up solutions the sharp result is given as e 0 d particular attention is paid to initial energy at supercritical initial energy e 0 d level which is novel contribution of this paper and new methods will be introduced and some analysis techniques are needed in addition the asymptotic behavior of the global solutions is studied

kirchhoff type system with linear weak damping and logarithmic nonlinearities abstract for the nonlinear kirchhoff type wave system with logarithmic nonlinearities and weak dissipation the global well posedness of initial boundary value problem is analyzed focusing on the interplay between kirchhoff terms and logarithmic sources we investigate the kirchhoff system controlled by logarithmic forces thus amplifying the difficulties in blow up analysis which is the primary scenario of interest by employing potential well method and concavity method we obtain several results related to the sufficient conditions posed on subcritical initial energy and critical initial energy which is used to classify initial data for global existence and finite time blow up finally via careful analysis involving the unstable invariant set under supercritical initial energy we are able to show an affirmative result that the solution blows up in finite time when initial data satisfy some suitable assumptions

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

on the initial and terminal value problem for a class of semilinear strongly material damped plate equations abstract we study the semilinear strongly damped plate equation by considering its two different problems for initial value problem we prove the local well posedness and blow up results of solution for the problem with polynomial nonlinear source terms for terminal value problem given the ill posedness in the sense of hadamard we propose a regularization method for the problem with logarithmic nonlinearities under the a priori assumption on the exact solution belonging to a gevrey space we propose the fourier truncation method to stabilize the ill posed problem also by establishing some stability estimates of logarithmic type in l q space

on certain second order ordinary differential equations associated with sobolev poincare type inequalities on considere l equation e i 1 n u x i p 2 u x i x i u q 2 u 0 dans u x 0 sur on donne les proprietes des solutions de e pour le cas n 1 et a b mais pour p q 1 avec p q regularite existence et unicite de solutions positives

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

multiple positive solutions for a class of p superlinear elliptic problems abstract we study nonlinear elliptic dirichlet problems driven by the p laplacian and with a nonlinearity which consists of a p superlinear term perturbed by a p linear one using critical point theory coupled with suitable truncation techniques we show that the problem has two positive smooth solutions when the conditions are symmetric in the two semiaxes we have four nontrivial smooth solutions of constant sign finally for the semilinear problem using morse theory we produce a fifth nontrivial smooth solution

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

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

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

static structure factor and thermodynamic properties of a binary yukawa mixture abstract we use the solution of the ornstein zernike equation in the mean spherical approximation to find the static structure factor for a hard spheres yukawa fluid the thermodynamic and the structure properties of this fluid are given in terms of an accumulative parameter which satisfies a polynomial equation of degree n 4 this parameter is obtained mumerically by an iterative method we study binary mixtures with a factored interaction for which the classical lorentz berthelot rules are satisfied our result for the static structure factor and thermodynamics properties are in good agreement with the computer simulations and former numerical solutions

multiple solutions for the semilinear elliptic equations with the sign changing logarithmic nonlinearity abstract in this paper we study the semilinear elliptic equations with the sign changing logarithmic nonlinearity under some conditions we obtain at least two nontrivial solutions by nehari manifold and logarithmic sobolev inequality here the results are quite different from these in the polynomial case

markov fields on branching planes we investigate ising models indexed by the sites of a branching plane mathbb t which is the product of a regular tree mathbb t and the line there are three parameter regimes corresponding to 1 a unique gibbs distribution 2 nonunique gibbs distributions with treelike structure the free boundary condition field is not a mixture of the plus and minus b c fields 3 nonunique gibbs distributions with planelike structure the free b c field is a mixture of the plus and minus b c fields

instability of positive periodic solutions for semilinear pseudo parabolic equations with logarithmic nonlinearity abstract in this paper we consider the periodic problem for semilinear heat equation and pseudo parabolic equation with logarithmic source after establishing the existence of positive periodic solutions we discuss the instability of such solutions

blow up and decay for a class of pseudo parabolic p laplacian equation with logarithmic nonlinearity abstract we consider the pseudo parabolic p laplacian equation u t u t p u u q 2 u log u in a bounded domain with homogeneous dirichlet boundary conditions when 2 p q p 1 2 n we obtain results of the decay and the finite time blow up for weak solutions

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