blow up and global existence for a nonlocal degenerate parabolic system abstract this paper investigates the blow up and global existence of nonnegative solutions of the system u t u m a v p v t v n b u q x t 0 t with homogeneous dirichlet boundary data where r n is a bounded domain with smooth boundary m n 1 1 p q a b 0 and dx it is proved that if pq mn there exist both global and blow up nonnegative solutions when pq mn we show that if the domain is sufficiently small every nonnegative solution is global while if the domain is large enough that is if it contains a sufficiently large ball there exists no global solution in particular when p n q m we show that every positive solution exists globally iff x dx 1 ab where x is the unique positive solution of the linear elliptic problem x 1 x x 0 x

spectral theory for the periodic sine gordon equation a concrete viewpoint a summary of the spectral theory for quasiperiodic sine and sinh gordon equations is given analogies with whole line solitons and scattering theory motivates the discussion the relation between the ingredients in the inverse spectral solution of the periodic sine gordon equation and physical characteristics of sine gordon waves is emphasized the explicit topics covered are summarized in the table of contents in the introduction

on nonlocal problems for systems of parabolic equations abstract the initial boundary value problems for systems of two parabolic equations are studied when the conditions with respect to the time variable are given only for one of the unknown functions the first problem is considered in the case where along with the initial data for one of the functions the value of the same function is given at the final moment of time the other one is the problem where two nonlocal conditions with respect to time are known for the same function the global existence and uniqueness of the solution to these problems are established

a boundary blow up problem with a nonlocal reaction abstract in this paper we deal with several kinds of boundary blow up problems whose main feature is exhibiting a nonlocal reaction term which depends on the integral of some function of the concentration u as a representative of such problems we consider u f u 1 1 g u in u on where is a bounded smooth domain of r n and the boundary condition is understood as u x when dist x 0 we find necessary and sufficient conditions for the existence of positive solutions in some cases we also obtain uniqueness or multiplicity of solutions other classes of nonlocal dependence of the reaction term are also discussed

global existence and blowup of a localized problem with free boundary abstract this paper is concerned with a double fronts free boundary problem for the heat equation with a localized nonlinear reaction term the local existence and uniqueness of the solution are given by applying the contraction mapping theorem then we present some conditions so that the solution blows up in finite time finally the long time behavior of the global solution is discussed we show that the solution is global and fast if the initial data is small and that a global slow solution is possible when the initial data is suitably large

uniform blow up profiles and boundary layer for a parabolic system with localized sources abstract this paper deals with the dirichlet problem for a parabolic system with localized sources we first obtain some sufficient conditions for blow up in finite time and then deal with the possibilities of simultaneous blow up under suitable assumptions moreover when simultaneous blow up occurs we also establish the uniform blow up profiles in the interior and estimate the boundary layer

linear integral equations and nonlinear difference difference equations abstract in this paper we present a systematic method to obtain various integrable nonlinear difference difference equations and the associated linear integral equations from which their solutions can be inferred it is argued that these difference difference equations can be regarded as arising from bianchi identities expressing the commutativity of backlund transformations applying an appropriate continuum limit we first obtain integrable nonlinear differential difference equations together with the associated linear integral equations and after a second continuum limit we can obtain the corresponding integrable nonlinear partial differential equations and their linear integral equations as special cases we treat the difference difference versions and the differential difference versions of the korteweg de vries equation the modified korteweg de vries equation the nonlinear schrodinger equation the isotropic classical heisenberg spin chain and the complex and real sine gordon equation

a numerical study of the 3 periodic wave solutions to kdv type equations abstract in this paper by using the direct method of calculating periodic wave solutions proposed by akira nakamura we present a numerical process to calculate the 3 periodic wave solutions to several kdv type equations the korteweg de vries equation the sawada koterra equation the boussinesq equation the ito equation the hietarinta equation and the 2 1 dimensional kadomtsev petviashvili equation some detailed numerical examples are given to show the existence of the three periodic wave solutions numerically

single point blow up for a semilinear initial value problem abstract the initial value problem on r r is considered u t t x u xx t x u t x u t r 0 u 0 x x where 0 and is a fixed large number it is known that for some initial values the solution u t x exists only up to some finite time t and that u t as t t for the specific initial value k where 0 xx 0 r 0 k is sufficiently large it is shown that if x 0 then lim t t u t x and lim t t u x t x exist and are finite in other words blow up occurs only at the point x 0

a complete characterization of the discrete p laplacian parabolic equations with q nonlocal reaction with respect to the blow up property abstract in this paper we consider the following discrete p laplacian nonlinear parabolic equations with q nonlocal reaction u t x t p u x t y s u y t q 1 u y t x t s 0 u x t 0 x t s 0 u x 0 u 0 0 x s here s is a network with boundary s the goal of this paper is to characterize completely the parameters p 1 and q 0 to see when the solution blows up vanishes or exists globally indeed the blow up rates for the blow up solutions are derived

on the separation of the cohomology of universal coverings of 1 convex surfaces abstract we construct a 1 convex surface x such that its universal covering x has the property that h 1 x o x is not separated

tight closure and projective bundles abstract we formulate problems of tight closure theory in terms of projective bundles and subbundles this provides a geometric interpretation of such problems and allows us to apply intersection theory to them this yields new results concerning the tight closure of a primary ideal in a two dimensional graded domain

blow up for a porous medium equation with a localized source in this paper we investigate the blow up properties of the positive solutions to a localized porous medium equation v t dv m av p 1v q 1 x 0 t subject to homogeneous dirichlet condition and positive initial datum v 0 x under appropriate hypotheses we establish the local existence and obtain that in the case of p 1 q 1m the solution of the above problem blows up for large initial datum while it admits a global solution for small initial datum moreover for the special case p 1 0 q 1 m and a is large under an additional hypothesis on the initial datum we can also obtain the asymptotic behavior of the blow up solution

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

convergence acceleration algorithm via an equation related to the lattice boussinesq equation the molecule solution of an equation related to the lattice boussinesq equation is derived with the help of determinantal identities it is shown that this equation can for certain sequences be used as a numerical convergence acceleration algorithm numerical examples with applications of this algorithm are presented

analysis of explosion for nonlinear volterra equations this survey paper presents some analytical tools for determining the existence of a blow up solution for a wide class of nonlinear volterra integral equations the related phenomenon of quenching has also been investigated using similar techniques in both cases lower and upper bound estimates for the critical time as well as asymptotic solution behavior in the key limit are established this paper surveys recent results in the analysis of such problems and proposes problems for future study

diffusion models with blow up a number of physical situations including chemical reactions electrical heating and fluid flow give rise to nonlinear diffusion problems in this paper models are derived and results relating to the blow up of the solutions are given some of the proofs are outlined and the physical significance of the results is noted

complete blow up for degenerate semilinear parabolic equations abstract let t a q and x0 be constants with a 0 q 0 and 0 x q u t u xx f u x 0 t 0 0 u x 0 u 0 x 0 0 x a u 0 t u a t 0 0 where f and u0 are given functions we show that under certain conditions u blows up in a finite time and the set of blow up points is the entire interval 0 a

uniform blow up rate for parabolic equations with a weighted nonlocal nonlinear source abstract in this paper the blow up rate of solutions of semi linear reaction diffusion equations with a more complicated source term which is a product of nonlocal or localized source and weight function a x is investigated it is proved that the solutions have global blow up and that the rates of blow up are uniform in all compact subsets of the domain furthermore the blow up rate of u t is precisely determined

open sets with stein hypersurface sections in stein spaces let d c cn n 3 be an open set such that for any linear hyperplane h c cn the intersection h n d is stein according to a theorem of lelong 13 it then follows that d is itself stein using the solution of the levi problem by docquier grauert 7 this result can easily be generalized to open subsets in stein manifolds therefore it is natural to raise the following question see also 4

critical exponents and asymptotic estimates of solutions to parabolic systems with localized nonlinear sources this paper deals with two classes of parabolic systems with localized nonlinear sources the critical exponents as well as the estimates for blow up rates and boundary layer profiles are determined

on superheight conditions for the affineness of open subsets in this paper we consider the open complement u of a hypersurface y v a in an affine scheme x we study the relations between the affineness of u the intersection of y with closed subschemes the property that every closed surface in u is affine the property that every analytic closed surface is stein and the superheight of a defining ideal a

analytical and numerical aspects of certain nonlinear evolution equations ii numerical nonlinear schrodinger equation abstract various numerical methods are employed in order to approximate the nonlinear schrodinger equation namely i the classical explicit method ii hopscotch method iii implicit explicit method iv crank nicolson implicit scheme v the ablowitz ladik scheme vi the split step fourier method f tappert and vii pseudospectral fourier method fornberg and whitham comparisons between the ablowitz ladik scheme which was developed using notions of the inverse scattering transform and the other utilized schemes are obtained

uniform blow up rate for a nonlocal degenerate parabolic equations abstract in this paper we introduce a new method for investigating the rate of blow up of solutions of diffusion equations with nonlocal nonlinear reaction terms in some cases we prove that the solutions have global blow up and the rate of blow up is uniform in all compact subsets of the domain in each case the blow up rate of u t is precisely determined

blow up properties for a degenerate parabolic system with nonlinear localized sources abstract this paper deals with blow up properties for a degenerate parabolic system with nonlinear localized sources subject to the homogeneous dirichlet boundary conditions the main aim of this paper is to study the blow up rate estimate and the uniform blow up profile of the blow up solution our conclusions extend the results of l l du blow up for a degenerate reaction diffusion system with nonlinear localized sources j math anal appl 324 2006 304 320 at the end the blow up set and blow up rate with respect to the radial variable is considered when the domain is a ball

discrete approximations on functional classes for the integrable nonlinear schrodinger dynamical system a symplectic finite dimensional reduction approach abstract we investigate discretizations of the integrable discrete nonlinear schrodinger dynamical system and related symplectic structures we develop an effective scheme of invariant reducing the corresponding infinite system of ordinary differential equations to an equivalent finite system of ordinary differential equations with respect to the evolution parameter we construct a finite set of recurrent algebraic regular relations allowing to generate solutions of the discrete nonlinear schrodinger dynamical system and we discuss the related functional spaces of solutions finally we discuss the fourier transform approach to studying the solution set of the discrete nonlinear schrodinger dynamical system and its functional analytical aspects

global and blow up solutions to a p laplacian equation with nonlocal source this paper deals with a p laplacian equation ut div u p 2 u uq x t dx with null dirichlet boundary conditions in a bounded domain rn where p 2 q 1 under appropriate hypotheses we establish local theory of the solution and obtain that the solution either exists globally or blows up in finite time

recent results on blow up and quenching for nonlinear volterra equations in this survey paper the author examines nonlinear volterra integral equations of the second kind with solutions that blow up or quench the focus is on analytical results although a few words about numerical solutions for such equations are provided the integral equations arise in the mathematical modeling of thermal processes within a reactive diffusive medium the scope of this review is on the published literature between 1997 and 2005 serving as an update to a previous review by the same author

global existence and blow up for the degenerate and singular nonlinear parabolic system with a nonlocal source abstract the existence of a unique classical nonnegative solution is established and the sufficient conditions for the solution that exists globally or blows up in finite time are obtained for the degenerate and singular parabolic system u t x u x x 0 a u m x t v n x t d x v t x v x x 0 a v p x t u q x t d x in 0 a 0 t where t a 0 are constants and m n p q are positive real numbers furthermore under certain conditions it is proved that the blow up set of the solution is the entire interval 0 a and we also obtain the blow up rate under the condition

blow up behavior of positive solutions for a chemical fuel ignition device model blow up behavior of positive solutions of a semi linear parabolic system arising from thermal explosion which subject to the homogenous dirichlet boundary conditions is investigated in particular sufficient conditions for the solutions to blow up are obtained

existence and nonexistence of global solutions to the cauchy problem of a class of nonlocal convective reaction diffusion equations we study the existence and nonexistence of global solutions to the cauchy problem where u t x is a scalar function a rn a 0 p 0 q r 1 is a gradient operator the results obtained generalize the results of aguirre and escobedo j aguirre and m escobedo 1993 proc r soc edin a 123 433 60 which do not consider the nonlocal factor in the reaction term of the equation and also generalize the results of wang et al m x wang s wang and c h xie 1999 j partial diff eqs 12 201 11 which do not include the nonlinear convection term in the equation

blow up rate and profile for a degenerate parabolic system coupled via nonlocal sources this paper deals with a degenerate parabolic system with nonlocal nonlinear sources subject to homogeneous dirichlet condition we establish the critical exponent as well as the blow up rate and profile for the system it is interesting to observe that the blow up exponent is determined by the interaction between the nonlinear diffusion and the nonlinear reaction while the blow up rate is independent of the nonlinear diffusion the effects from the nonlocality of the sources are substatial

analytical and numerical aspects of certain nonlinear evolution equations i analytical abstract nonlinear partial difference equations are obtained which have as limiting forms the nonlinear schrodinger korteweg devries and modified korteweg devries equations these difference equations have a number of special properties they are constructed by methods related to the inverse scattering transform they can be used as a basis for numerical schemes to the associated nonlinear evolution equations experiments have shown that they compare very favorably with other known numerical methods papers ii iii in paper ii the ablowitz ladik scheme for the nonlinear schrodinger equation is compared to other known numerical schemes and generally proved to be faster than all utilized finite difference schemes but somewhat slower than the finite fourier pseudospectral methods in paper iii a proposed scheme for the korteweg de vries equation proved to be faster than both the finite difference and finite fourier methods already considered

on the open immersion problem we give two examples of complex spaces on which global holomorphic functions separate points and give local coordinates and they cannot be realized as open subsets of stein spaces at the same time we notice that these examples are open subsets of stein schemes a notion introduced by grauert math z 81 377 391 1963 in the context of complex schemes we notice that by contracting a nori string one obtains a complex scheme and not a complex space the covering spaces of 1 convex surfaces are divided in two categories those that have an envelope of holomorphy and those that do not more interesting are those in the second category and they correspond to covering spaces for singularities which in the desingularization with normal crossings contain cycles in the exceptional set

uniform blow up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source in this paper we introduce a new method for investigating the rate and profile of blow up of solutions of diffusion equations with nonlocal nonlinear reaction terms for large classes of equations we prove that the solutions have global blow up and that the rate of blow up is uniform in all compact subsets of the domain this results in a flat blow up profile except for a boundary layer whose thickness vanishes astapproaches the blow up timet in each case the blow up rate of u t is precisely determined furthermore in many cases we derive sharp estimates on the size of the boundary layer and on the asymptotic behavior of the solution in the boundary layer the size of the boundary layer then decays liket t and the solutionu t x behaves like u t d x t tin the boundary layer wheredis the distance to the boundary some fujita type critical exponents results are also given for the cauchy problem

the hartogs extension theorem on n 1 complete complex spaces employing morse theory for the global control of monodromy and the method of analytic discs for local extension we establish a version of the global hartogs extension theorem in a singular setting for every domain d of an n 1 complete normal complex space x of pure dimension n 2 and for every compact set k in d such that d k is connected holomorphic or meromorphic functions in d k extend holomorphically or meromorphically to d normality is an unvavoidable assumption for holomorphic extension but we show that meromorphic extension holds on a reduced globally irreducible not necessarily normal x of pure dimension n 2 provided that the regular part of d k is connected

total versus single point blow up of solutions of a semilinear parabolic equation with localized reaction abstract in this paper we study the initial boundary value problem of a semilinear parabolic equation with localized reaction p u t u u p u q x t x t b 0 t u x t 0 x t b 0 t u x 0 u 0 x x b where b is a unit ball in r n x b and p q 0 for the case x 0 we completely classify blow up solutions of p into total blow up cases and single point blow up cases according to the values p and q moreover we give the blow up rates of solutions near the blow up time for the other case x 0 we show total blow up and single point blow up of solutions of p in some cases depending on p and q

coupled diffusion systems with localized nonlinear reactions this paper deals with the blowup rate and profile near the blowup time for the system of diffusion equations uit ui upii 1 x0 t i 1 k uk 1 u1 in 0 t with boundary conditions ui 0 on 0 t we show that the solution has a global blowup the exact rate of the blowup is obtained and we also derive the estimate of the boundary layer and on the asymptotic behavior of the solution in the boundary layer

a class of non linear non classical parabolic equations abstract in this paper we deal with a class of non classical parabolic equations which arise from some practical problems such as the identification of unknown parameters free boundary problems etc one of the characteristics of this kind of equation is that the maximum principle is no longer valid we will first study the local solvability for a general equation of this kind via an auxiliary problem and the schauder fixed point principle then some a priori estimates are derived to establish the global existence of the solution moreover uniqueness and continuous dependence of the solution are proved finally some examples are given to illustrate the applications of the theory

blow up for a degenerate reaction diffusion system with nonlinear localized sources abstract this paper deals with the global existence and blow up of nonnegative solution of the degenerate reaction diffusion system with nonlinear localized sources involved a product with local terms we investigate the influence of localized sources and local terms on global existence and blow up for this system moreover we establish the precise blow up estimates finally for the special case p 1 p 2 0 we show the blow up set is whole region and the uniform blow up profiles are obtained these extend a resent work of chen and xie in y chen c xie blow up for a porous medium equation with a localized source appl math comput 159 2004 79 93 which considered the single equation with localized sources

estimates for the blowup time of a combustion model with nonlocal heat sources abstract in this paper we establish the lower and upper bounds for the blowup time of the solution to a system of parabolic equations with nonlocal nonlinearities which appears in a model of combustion different techniques are applied to deal with the problems in different space dimensions we also discuss non simultaneous blowup phenomenon of the solution

integrable discretization of time and its application on the fourier pseudospectral method to the korteweg de vries equation this paper presents a new integrable discretization of the korteweg de vries kdv equation different from other discrete analogues we discretize the variable time and obtain an integrable differential difference system this system has the original kdv equation as a standard limit when the step size tends to zero the main idea is based on hirota s bilinear method and backlund transformation and can be applied to other integrable systems by applying the fourier pseudospectral method to the space variable we derive a new numerical scheme for the kdv equation numerical results are found to agree with the exact solution and the first five conservation quantities are preserved quite well

blow up behavior for a nonlinear heat equation with a localized source in a ball in this paper we consider the initial boundary value problem of a semilinear parabolic equation with local and non local localized reactions in a ball ut u up uq x t in b r where p q 0 b r x rn x 1 there exist blow up solutions of this problem for large initial data we treat the radially symmetric and one peak non negative solution u x t u r t r x of this problem we give the complete classification of total blow up phenomena and single point blow up phenomena according to p and q i if q 1 or p q 2 then single point blow up occurs whenever solutions blow up ii if 1 p q both phenomena total blow up and single point blow up occur depending on the initial data iii if p 1 q total blow up occurs whenever solutions blow up iv if max p q 1 every solution exists globally in time

periodic solutions of the kdv equation in this paper we construct a large family of special solutions of the kdv equation which are periodic in x and almost periodic in t these solutions lie on n dimensional tori very likely they are dense among all solutions the special solutions are characterized variationally they minimize fn u subject to the constraints fj u aj j 1 n 1 here fj denote the remarkable sequence of conserved functionals discovered by kruskal and zabusky the above minimum problem was originally suggested by them in exploring the manifold of solutions of this minimum problem we make essential use of gardner s discovery that these functionals are in involution with respect to a suitable poisson bracket gardner greene kruskal and miura have shown that the eigenvalues of the schrodinger operator are conserved functionals if the potential is a function of t and satisfies the kdv equation in section 6 a new set of conserved quantities is constructed which serve as a link between the eigenvalues of the schrodinger operator and the fj another result in section 6 is a slight sharpening of an earlier result of the author and j moser for the special solutions constructed above all but 2n 1 eigenvalues of the schrodinger operator are double the simplest class of special solutions n 1 are cnoidal waves in an appendix m hyman describes the results of computing numerically the next simplest case n 2 these calculations show that the shape of these solutions recurs exactly after a finite time in a shifted position the theory verifies this fact

boundedness of global solutions for a porous medium system with moving localized sources abstract this paper deals with a class of porous medium systems with moving localized sources u t u r 1 u a f v x 0 t t v t v r 2 v b g u x 0 t t with homogeneous dirichlet boundary conditions it is shown that under certain conditions solutions of the above system blow up in finite time for large a and b or large initial data while there exist global positive solutions to the above system for small a and b or small initial data moreover in the one dimensional space case it is also shown that all global positive solutions of the above problem are uniformly bounded

a reaction diffusion system with mixed type coupling this paper deals with asymptotic behavior of solutions to a reaction diffusion system coupled via localized and local sources u t du v p x t t v t dv u q both the initial boundary problem with null dirichlet boundary condition and the cauchy problem are considered to study the interaction between the two kinds of sources for the initial boundary problem we prove that the nonglobal solutions blow up everywhere in the bounded domain with uniform blow up profiles in addition it is interesting to observe that the cauchy problem admits an infinity fujita exponent namely the solutions blow up under any nontrivial and nonnegative initial data whenever pq 1 all these imply that the blow up behavior of solutions is governed by the localized source for the two problems with mixed type coupling

neumann problem for reaction diffusion systems with nonlocal nonlinear sources abstract this paper considers the neumann problem for several types of systems with nonlocal nonlinear terms we first give the blow up conditions and then for the blow up solution we establish the precise blow up estimates and show the blow up set is the whole region

integrable semi discretizations of the reduced ostrovsky equation based on our previous work on the reduced ostrovsky equation j phys a math theor 45 355203 we construct its integrable semi discretizations since the reduced ostrovsky equation admits two alternative representations one being its original form the other the differentiated form the short wave limit of the degasperis procesi equation two semi discrete analogues of the reduced ostrovsky equation are constructed possessing the same n loop soliton solution the relationship between these two versions of semi discretizations is also clarified

the blowup property of solutions to some diffusion equations with localized nonlinear reactions abstract in this paper we investigate the blowup property of solutions to the equation u t u u x 0 t where x0 is a fixed point in the domain we show that under certain conditions the solution blows up in finite time moreover we prove that the set of all blowup points is the whole region furthermore the growth rate of solutions near the blowup time is also derived finally the results are generalized to the following nonlocal reaction diffusion equation u t u u dx

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

travelling wave solutions for the mkdv sine gordon and the mkdv sinh gordon equations by using a variable separated ode method in this work we formally derive exact travelling wave solutions for the modified kdv sine gordon mkdv sine gordon equation and the modified kdv sinh gordon equation mkdv sinh gordon the proposed analysis depends mainly on a variable separated ode method two distinct sets of exact solitary wave solutions that possess distinct physical structures are formally derived for each equation the derived solutions include others results and introduce entirely new solutions

uniform blow up rate for diffusion equations with nonlocal nonlinear source abstract in this paper we investigate the blow up rate of solutions of diffusion equations with nonlocal nonlinear reaction terms for large classes of equations we prove that the solutions have global blow up and that the rate of blow up is uniform in all compact subsets of the domain in each case the blow up rate of u t is precisely determined

boundedness of global solutions for a nonlinear degenerate parabolic porous medium system with localized sources abstract this paper deals with nonlinear degenerate parabolic porous medium system with localized sources it is shown that under certain conditions solutions of the equation blow up in finite time for large a and b or large initial data while there exist global positive solutions for small a and b or small initial data moreover it is also shown that all global positive solutions of the above equation are uniformly bounded these extend a recent work of chen et al in y chen q liu h gao boundedness of global solutions of a porous medium equation with a localized source nonlinear anal 64 2006 2168 2182 which considered the single equation with localized source

integrable discretizations and self adaptive moving mesh method for a coupled short pulse equation in the present paper integrable semi discrete and fully discrete analogues of a coupled short pulse csp equation are constructed the key to the construction are the bilinear forms and determinant structure of the solutions of the csp equation we also construct n soliton solutions for the semi discrete and fully discrete analogues of the csp equations in the form of casorati determinants in the continuous limit we show that the fully discrete csp equation converges to the semi discrete csp equation then further to the continuous csp equation moreover the integrable semi discretization of the csp equation is used as a self adaptive moving mesh method for numerical simulations the numerical results agree with the analytical results very well

critical exponents for non simultaneous blow up in a localized parabolic system abstract the paper deals with the radially symmetric solutions of u t u u m x t v n 0 t v t v u p 0 t v q x t subject to null dirichlet boundary conditions for the blow up classical solutions we propose the critical exponents for non simultaneous blow up by determining the complete and optimal classification for all the non negative exponents i there exist initial data such that u v blows up alone if and only if m p 1 q n 1 which means that any blow up is simultaneous if and only if m p 1 q n 1 ii any blow up is u v blowing up with v u remaining bounded if and only if m p 1 q n 1 m p 1 q n 1 iii both non simultaneous and simultaneous blow up may occur if and only if m p 1 q n 1 moreover we consider the blow up rate and set estimates which were not obtained in the previously known work for the same model

blow up solutions and global solutions for a class of quasilinear parabolic equations with robin boundary conditions the type of problem under consideration isu email protected email protected a u b x u g x t f u indx 0 t email protected email protected x t u 0 email protected dx 0 t u x 0 u 0 x 0 ind where d is a smooth bounded domain of r n by constructing an auxiliary function and using hopf s maximum principles on it existence theorems of blow up solutions upper bound of blow up time upper estimates of blow up rate existence theorems of global solutions and upper estimates of global solutions are given under suitable assumptions on a b f g s and initial data u 0 x the obtained results are applied to some examples in which a b f g and s are power functions or exponential functions