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

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

a distributional version of functional equations and their stabilities abstract generalizing the results in j math anal appl 286 2003 177 186 j math anal appl 295 2004 107 114 arch math to appear j math anal appl 299 2004 578 586 that consider the hyers ulam stability problems of several functional equations in the spaces of the schwartz tempered distributions and the fourier hyperfunctions we consider the stability problems of the functional equations in the space of distributions

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

the asymptotic behavior for gaseous ignition models abstract the blowup asymptotics for classical solutions of ut u eu g t are analyzed including the nonlocal case where g has a functional dependence of the form utdx the method of analyzing the evolution of the blowup singularity is a dynamical systems approach using center manifold techniques and analyzing an limit set for a corresponding hamilton jacobi equation

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

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

a class of conservative orthogonal spline collocation schemes for solving coupled klein gordon schrodinger equations abstract a class of discrete time orthogonal spline collocation schemes for solving coupled klein gordon schrodinger equations with initial and boundary conditions are considered these schemes are constructed by using piecewise cubic hermite interpolations in space combined with finite difference methods in time it is proved that the schemes have the conservation laws of discrete energy and possess second order accuracy in maximum norm and fourth order in l 2 norm for time and space respectively the conservation laws and rate of convergence are verified in numerical experiments moreover the propagations of solitary waves and collisions of two head on solitary waves are also well simulated

uniform blow up rate for diffusion equations with localized nonlinear source abstract in this short paper we investigate blow up rate of solutions of reaction diffusion equations with localized reactions we prove that the solutions have a global blow up and the rate of blow up is uniform in all compact subsets of the domain

global existence and blowup of a nonlocal problem in space with free boundary abstract this paper concerns a double fronts free boundary problem for the reaction diffusion equation with a nonlocal nonlinear reaction term in space for such a problem we mainly study the blowup property and global existence of the solutions our results show that if the initial value is sufficiently large then the blowup occurs while the global fast solution exists for a sufficiently small initial data and the intermediate case with a suitably large initial data gives the existence of the global slow solution

stability of functional equations of drygas in the space of schwartz distributions in this paper we reformulate and prove the stability theorems of s m jung and p k sahoo s m jung p k sahoo stability of a functional equation of drygas aequationes math 64 2002 263 273 in the spaces of generalized functions such as the schwartz distributions and tempered distributions

on the stability of functional equations in banach spaces the paper is devoted to some results on the problem of s m ulam for the stability of functional equations in banach spaces the problem was posed by ulam 60 years ago

single point blowup for nonlocal parabolic problems abstract the formation of shear bands in materials has important implications to a variety of physical processes these bands are observed in very thin zones and are generally regarded as a precursor to material failure shear band formation is caused by the heat generated in regions with highest strain rate with insufficient time for diffusion of this heat a localized thermal softening of the material occurs which enhances plastic flow in a thin zone this adiabatic strain localization can be modelled as nonlinear thermally activated reaction diffusion equations this leads to a class of nonlocal parabolic problems and their associated time independent steady state counterparts a discussion of the models and an analysis of the problems will be presented

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

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

a nonlocal sturm liouville eigenvalue problem a nonlocal eigenvalue problem of the form u a x u bu u with homogeneous dirichlet boundary conditions is considered where b is a rank one bounded linear operator and x belongs to some bounded interval on the real line the behaviour of the eigenvalues is studied using methods of linear perturbation theory in particular some results are given which ensure that the spectrum remains real a sturm type comparison result is obtained finally these results are applied to the study of some nonlocal reaction diffusion equations

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

uniform blow up profiles for nonlinear and nonlocal reaction diffusion equations abstract in this paper we investigate the blow up properties of positive solutions to the following integro parabolic equations u t u e u 1 e u d x where 0 and 1 and x r n x r for the radially symmetric and non increasing initial data we give a complete classification in terms of global and single point blow up according to the parameters and moreover the blow up rates are also determined in each case particularly for the special case and n 2 it will be proved that the blow up rate at x 0 is faster than that at the other point x 0 this seems to be a new phenomenon for this kind of problem

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

global existence and convergence to a singular steady state for a semilinear heat equation with certain initial and boundary conditions the solution u to the semilinear heat equation u t u f u where f is a positive superlinear function and is the supremum of the open spectrum for the steady state problem w f w 0 is found to exist for all time and to be unbounded moreover u approaches w a singular steady state as tends to infinity

uniform blow up profile for a degenerate parabolic system with nonlocal source in this paper we establish the local existence of the solution and the finite time blow up result for the following system ut ur1 u a updx forx ohm t 0 ut ur2 u b uqdx forx ohm t 0 where p q 1 and 0 r1 r2 1 moreover it is proved that the solution has global blow up and limt t t t p 1 r2 yu x t 1 r1 11 r1 a a y p 1 r2 y q 1 r1 py xa p 1 r2 b1 r2 y limt t t t q 1 r1 yu x t 1 r2 11 r2 b a y q 1 r1 y p 1 r2 qy xb q 1 r1 a1 r1 y uniformly on compact subsets of where pq 1 r1 1 r2 and t is the blow up time

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

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

stability of jensen equations in the space of generalized functions abstract making use of heat kernel we prove stabilities of the jensen and jensen pexider equations in a space of generalized functions like the spaces of tempered distributions and fourier hyperfunctions

stability of functional equations in the spaces of distributions and hyperfunctions making use of the fundamental solution of the heat equation we prove the stability theorems of quadratic functional equation and d alembert equation in the spaces of schwartz distributions and sato hyperfunctions

asymptotic analysis for a localized nonlinear diffusion equation in this paper we study a localized nonlinear diffusion equation u t du m l 1u p l 2u q 0 t subject to null dirichlet boundary condition with p q 0 max p q m 1 and l 1 l 2 0 we investigate interactions among the localized and local sources nonlinear diffusion with the zero boundary value condition to establish blow up rates and uniform blow up profiles of solutions under different dominations in addition as results of the interactions of multiple nonlinearities the blow up sets of solutions namely total versus single point blow up of solutions are also determined

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

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

on the global existence and finite time blow up of shadow systems abstract shadow systems are often used to approximate reaction diffusion systems when one of the diffusion rates is large in this paper we study the global existence and blow up phenomena for shadow systems our results show that even for these fundamental aspects there are serious discrepancies between the dynamics of the reaction diffusion systems and that of their corresponding shadow systems

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

the stability of cauchy equations in the space of schwartz distributions abstract we reformulate the stability theorem of d h hyers in the schwartz tempered distributions and prove that every additive tempered distribution can be approximated by a linear function we also consider the superstability of the cauchy equation f x y f x f y

blow up properties for a degenerate parabolic system with nonlinear localized sources in this paper we investigate the initial boundary problem of a degenerate parabolic system with nonlinear localized sources we classify the blow up solutions into global blow up cases and single point blow up cases according to the values of m n p i q i furthermore we obtain the uniform blow up profiles of solutions for the global blow up case finally we give some numerical examples to verify the results these extend and generalize a recent work of one of the authors l du blow up for a degenerate reaction diffusion systems with nonlinear localized sources j math anal appl 324 2006 304 320 which only considered uniform blow up profiles under the special case p 1 p 2 0