existence and blow up for a degenerate parabolic equation with nonlocal source in this paper we investigate the positive solution of nonlinear degenerate equation ut up u au uq dx with dirichlet boundary condition conditions on the existence of global and blow up solution are given furthermore it is proved that there exist two positive constants c1 c2 such that c1 t t 1 p q maxx u x t c2 t t 1 p q

convergence of accelerated modulus based matrix splitting iteration methods for linear complementarity problem with an h matrix the theoretical analysis of the accelerated modulus based matrix splitting iteration methods for the solution of the large sparse linear complementarity problem is further studied the convergence conditions are presented by fully utilizing the h matrix property of the system matrix and the optimal iteration parameters in accelerated modulus based accelerated overrelaxation method are determined by minimizing the spectral radius of the iteration matrix numerical experiments further confirm the theoretical discussion and show that the proposed methods accelerate the convergence of the modulus based matrix splitting methods with less iteration steps and cpu time

further spectral properties of uniformly elliptic operators that include a non local term in this paper we consider the eigenvalues of a class of non local differential operators the real eigenvalues of certain examples of these operators have previously been plotted as functions of a fundamental parameter in the operator using appropriate computer software here we show how the complex eigenvalues of such operators may also be plotted we use the plots of both the real and the complex eigenvalues to motivate a number of new theoretical results regarding the structure of the spectrum which we subsequently prove under suitable hypotheses

positivity of solutions of elliptic equations with nonlocal terms in this paper we study a nonlocal problem for a second order partial differential equation which depends on a parameter n we prove the existence of critical values 0 and 0 such that for all and for all non negative right hand sides our problem has nonnegative solutions we obtain a formula for 0 the maximal possible value of and find the exact value of for spherical we also study the corresponding eigenvalue problem at the end of the paper we consider the application of our results to the nonlinear system describing the distribution of temperature and potential in a microsensor

quantization of chern simons gauge theory with complex gauge group the canonical quantization of chern simons gauge theory in 2 1 dimensions is generalized from the case in which the gauge group is a compact lie groupg to the case in which the gauge group is a complex lie groupg though the physical hilbert spaces become infinite dimensional in the latter case the quantization can be described as precisely as for compact gauge groups and using similar methods the special case in which the gauge group issl 2 gives a description of 2 1 dimensional quantum gravity with lorentz signature and positive cosmological constant or with euclidean signature and negative cosmological constant while it is not clear whether there is a 1 1 dimensional conformal field theory related to these 2 1 dimensional models there are natural computable candidates for the central charge and the conformal blocks of such a hypothetical theory

blowing up of solution for a nonlocal reaction diffusion problem in combustion theory abstract in a nonlocal reaction diffusion model in combustion theory the reaction function involves a physical parameter which is a measure of the strength of the reaction mechanism the purpose of this paper is to show the existence of a critical value such that for a unique global time dependent solution exists and converges to a steady state solution as t and for the solution blows up in finite time a characterization as well as upper and lower bounds of are given

blow up for a degenerate parabolic equation with a nonlocal source abstract in this paper 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 parabolic problem x q u t u xx 0 a u m x t d x ku n x t in 0 a 0 t where 0 t a 0 m n 1 k 0 are constants

the blow up rate for a degenerate parabolic equation with a non local source abstract in this paper we establish the local existence of the solution and the finite time blow up result for the equation v v m xx a l l v q dx x l l 0 moreover it is proved that the solution has global blow up and there exist constants c 1 c 2 such that c 1 t 1 1 q 1 max x l l v x c 2 t 1 1 q 1 where t 1 is the blow up time

global existence and asymptotic behavior of solutions to a nonlocal fisher kpp type problem abstract in this work we consider a nonlocal fisher kpp reaction diffusion problem with neumann boundary condition and nonnegative initial data in a bounded domain in r n n 1 with reaction term u 1 m t where m t is the total mass at time t with the help of poho aev s identity the non existence of nontrivial stationary solutions with dirichlet boundary conditions is being shown when 1 and the initial mass is greater than or equal to one the problem has nonnegative classical solutions while if the initial mass is less than one then the problem admits global solutions for n 1 2 with any 1 2 or n 3 with any 1 1 2 n moreover the asymptotic convergence to the solution of the heat equation is proved

global existence and blow up solutions for doubly degenerate parabolic system with nonlocal source abstract this paper deals with the following nonlocal doubly degenerate parabolic system u t div u m p 2 u m a u 1 x t v 1 x t d x v t div v n q 2 v n b u 2 x t v 2 x t d x with null dirichlet boundary conditions in a smooth bounded domain r n where m n 1 p q 2 i i 0 i 1 2 and a b 0 are positive constants we first get the non existence result for a related elliptic systems of non increasing positive solutions secondly by using this non existence result blow up estimates for above non newton polytropic filtration systems with the homogeneous dirichlet boundary value conditions are obtained then under appropriate hypotheses we establish local theory of the solutions and prove that the solution either exists globally or blows up in finite time in the special case 1 n q 1 2 2 m p 1 1 we also give a criterion for the solution to exist globally or blow up in finite time which depends on a b and x x as defined in the main results

an iteration method for nonlinear complementarity problems abstract in this paper we propose a preconditioned modulus based iteration method for nonlinear complementarity problems the convergence analysis of the proposed method is presented numerical examples are given to illustrate the efficiency of the proposed method

non semisimple and complex gaugings of n 16 supergravity maximal and non maximal supergravities in three dimensions allow for a large variety of semisimple chern simons gauge groups in this paper we analyze non semisimple and complex gauge groups that satisfy the pertinent consistency relations for a maximal n 16 gauged supergravity to exist we give a general procedure how to generate non semisimple gauge groups from known admissible semisimple gauge groups by a singular boost within e8 8 examples include the theories with gauge group so 8 t28 that describe the reduction of iia iib supergravity on the seven sphere in addition we exhibit two strange embeddings of the complex gauge group open image in new window into real e8 8 and prove that both can be consistently gauged we discuss the structure of the associated scalar potentials as well as their relation to those of d 4 gauged supergravities

on global attractor for m laplacian parabolic equation with local and nonlocal nonlinearity abstract in this paper we study the long time behavior of solutions for m laplacian parabolic equation u t m u a x u u f 0 u k y u y t d y g x in 0 with the initial data u x 0 u 0 x l q q 1 and zero boundary condition in two cases for a x a 0 0 and a x 0 are considered we obtain the existence and l p estimate of global attractor a in l p for any p max 1 q the attractor a is in fact a bounded set in w 0 1 m l if a x a 0 0 in and a is bounded in w 0 1 m l p if a x 0 in

new general solutions to linear fredholm integro differential equations and their applications on solving the boundary value problems abstract this paper introduces the n general solution to linear fredholm integro differential equations and sets its properties the conditions for existence of classical general solution and solvability criteria for the equation are provided necessary and sufficient conditions for solvability of linear boundary value problems are established algorithms for construction of the n general solutions are obtained this paper also proposes approximate and numerical methods to solve the boundary value problems

two step modulus based matrix splitting iteration method for a class of nonlinear complementarity problems abstract in this paper we reformulate the nonlinear complementarity problem as an implicit fixed point equation we establish a modulus based matrix splitting iteration method based on the implicit fixed point equation and prove its convergence theorem under suitable conditions furthermore we propose a two step modulus based matrix splitting iteration method which may achieve higher computing efficiency we can obtain many matrix splitting iteration methods by suitably choosing the matrix splittings and the parameters the proposed methods can be regarded as extensions of the methods for linear complementarity problem numerical experiments are presented to show the effectiveness of the proposed methods

global existence and divergence of critical solutions of a non local parabolic problem in ohmic heating process abstract we investigate the behaviour of some critical solutions of a non local initial boundary value problem for the equation u t u f u f u d x 2 r n n 1 2 under specific conditions on f there exists a such that for each 0 there corresponds a unique steady state solution and u u x t is a global in time bounded solution which tends to the unique steady state solution as t uniformly in x whereas for there is no steady state and if then u blows up globally here we show that when a n 1 1 1 and f s 0 f s s 0 or b n 2 b 0 1 and f s e s the solution u u x t is global in time and diverges in the sense u t as t moreover it is proved that this divergence is global i e u x t as t for all x the asymptotic form of divergence is also discussed for some special cases

properties of non simultaneous blow up solutions in nonlocal parabolic equations abstract this paper deals with blow up solutions in parabolic equations coupled via nonlocal nonlinearities subject to homogeneous dirichlet conditions firstly some criteria on non simultaneous and simultaneous blow up are given including four kinds of phenomena i the existence of non simultaneous blow up ii the coexistence of non simultaneous and simultaneous blow up iii any blow up must be simultaneous iv any blow up must be non simultaneous next total versus single point blow up are classified completely moreover blow up rates are obtained for both non simultaneous and simultaneous blow up solutions

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

a quasilinear reaction diffusion system coupled via nonlocal sources in this paper we establish the critical exponent and the blow up rate for a quasilinear reaction diffusion system with nonlocal nonlinear sources and inner absorptions subject to homogeneous dirichlet conditions and nonnegative initial data it is found that the critical exponent is determined by the interaction among all the six nonlinear exponents from all the three kinds of the nonlinearities while the blow up rate is independent of the nonlinear diffusion exponents due to the effects of the coupled nonlocal sources two kinds of characteristic algebraic systems are introduced to get simple descriptions for the critical exponent and the blow up rate respectively we prove moreover that the blow up could be global due to the nonlocality of the nonlinear sources

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

blow up for a degenerate reaction diffusion system with nonlinear nonlocal sources this paper investigates the global existence and blow up of nonnegative solution of the systemu email protected m u p email protected wv q 1dx v email protected n v p email protected wu q 2dx x t email protected 0 t with homogeneous dirichlet boundary conditions where email protected r n is a bounded domain with smooth boundary email protected m n 1 p 1 p 2 q 1 q 2 0 the results depend crucially on the number p i q i m n the domain w and the initial data u 0 x v 0 x moreover we obtain the blow up rate of the blow up solution under some appropriate hypotheses

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

two level schwarz method for solving variational inequality with nonlinear source terms in this paper we extend the two level schwarz method to solve the variational inequality problems with nonlinear source terms and establish a convergence theorem the method converges within finite steps with an appropriate initial point the numerical results show that the methods are efficient

three dimensional quantum gravity chern simons theory and the a polynomial we study three dimensional chern simons theory with complex gauge group sl 2 which has many interesting connections with three dimensional quantum gravity and geometry of hyperbolic 3 manifolds we show that in the presence of a single knotted wilson loop in an infinite dimensional representation of the gauge group the classical and quantum properties of such theory are described by an algebraic curve called the a polynomial of a knot using this approach we find some new and rather surprising relations between the a polynomial the colored jones polynomial and other invariants of hyperbolic 3 manifolds these relations generalize the volume conjecture and the melvin morton rozansky conjecture and suggest an intriguing connection between the sl 2 partition function and the colored jones polynomial

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

convergence of relaxed parallel multisplitting methods abstract multisplitting methods are parallel methods for the solution of a linear system ax b it has been observed that the convergence of multisplitting methods is often improved if some kind of relaxation is used we investigate two different variants of relaxed multisplitting methods if a is an h matrix these methods converge if the relaxation parameter is from an interval 0 0 with 0 1

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

the utilization of total mass to determine the switching points in the symmetric boundary control problem with a linear reaction term abstract the authors study the problem u t u x x u 0 x 1 t 0 u x 0 0 and u 0 t u 1 t t where t u 0 for t 2 k t t 2 k 1 and t 0 for t 2 k 1 t t 2 k 2 k 0 1 2 with t 0 0 and the sequence t k is determined by the equations 0 1 u x t k d x m for k 1 3 5 and 0 1 u x t k d x m for k 2 4 6 and where 0 m m note that the switching points t k k 1 2 3 are unknown existence and uniqueness are demonstrated theoretical estimates of the t k and t k 1 t k are obtained and numerical verifications of the estimates are presented the case of u x 0 t u x 1 t t is also considered and analyzed

on a degenerate nonlocal parabolic equation with variable source abstract we study the dirichlet problem for the degenerate nonlocal parabolic equation u t div a u l 2 2 u b u l 2 2 u q x t 2 u f in q t where q t 0 t t 0 r n n 2 is a bounded domain with a sufficiently smooth boundary a b are real valued functions defined on r and q x t is a measurable function in q t with values in an interval q q 1 we show that under appropriate conditions on a b q the problem has a global in time solution and find conditions of uniqueness sufficient conditions for extinction in finite time or strict positivity of u t l 2 for all t 0 are found in the case f 0 the decay rates for u t l 2 are derived

on the convergence of modulus based matrix splitting iteration methods for a class of nonlinear complementarity problems with h matrices abstract a class of modulus based iteration methods for the nonlinear complementarity problem are presented by reformulating the complementarity problem to an implicit fixed point equation in order to efficiently implement these methods practical iteration schemes based on matrix splitting and the choices of the parameters are presented and well studied moreover we extend the convergence theory from the h compatible splitting to h splitting when the matrix is an h matrix which results in more choices for the parameters numerical experiments are given to verify the theoretical results and illustrate the efficiency of the proposed methods

the monotone convergence rate of the parallel nonlinear aor method abstract a new comparison theorem about the parallel nonlinear aor method 1 is set up which describes in detail the influence of either the multisplitting of the coefficient matrix or the pair of the relaxation parameters on the convergence rate of this method

fixed point approach for complementarity problems abstract in this paper we use the fixed point technique to suggest a new unified and general algorithm for computing the approximate solution of a nonlinear complementarity problem of finding u such that u 0 tu a u 0 u tu a u 0 where t is a continuous mappping from r n into itself and a is a non linear transformation from r n into itself this algorithm includes many existing algorithms for complementarity problems as special cases convergence properties are also discussed and analyzed

a mathematical analysis of blowup for thermal reactions the spatially nonhomogeneous case the equations describing the induction period process for a super critical high activation energy thermal explosion in a bounded domain are studied a formal proof based on comparison techniques is used to show that the temperature perturbation becomes unbounded as t to t b leqq infty for values of the frank kamenetski parameter delta greater than the critical value upper and lower bound estimates for t b are found by using specified comparison equations a comparison of these bounds with values of t b obtained from numerical solutions of the basic equations shows that the estimates provide an excellent prediction of the escape time when delta is greater than 2 3 times the critical value for smaller values of delta where heat loss is more significant a better comparison equation is required

on convergence of the modulus based matrix splitting iteration method for horizontal linear complementarity problems of h matrices abstract horizontal linear complementarity problem has wide applications such as in mechanical and electrical engineering structural mechanics piecewise linear system telecommunication systems and so on in this paper we focus on the convergence conditions of the modulus based matrix splitting iteration method proposed recently for solving horizontal linear complementarity problems by the proposed theorems the assumptions on the matrix splitting and the system matrices are weakened and the convergence domain is enlarged numerical examples are presented to show the improvement

on logistic diffusion equations with nonlocal interaction terms abstract this paper is concerned with logistic diffusion equations with nonlocal interaction terms appearing in population biology we intend to study effects of nonlocal terms and discuss the similarity and difference between local problems and nonlocal problems mainly the stationary problem is investigated for a certain class of nonlocal terms a constructive approach is proposed to look for positive stationary solutions and the unique existence of such a positive solution is established the analysis of stationary solutions depends on the spectrum for the linearized operator around the stationary solution however the linearized operator contains a nonlocal term which makes the spectral analysis delicate and difficult putting some additional assumptions we will derive the asymptotic stability of the unique positive solution and furthermore its global attractivity finally it will be seen that some arguments are valid to show the unique existence of a positive stationary solution for a considerably general class of nonlocal terms

a unified framework for the construction of various matrix multisplitting iterative methods for large sparse system of linear equations abstract a unified framework for the construction of various synchronous and asynchronous parallel matrix multisplitting iterative methods suitable to the simd and mimd multiprocessor systems respectively is presented and its convergence theory is established under rather weak conditions these afford general method models and systematical convergence criterions for studying the parallel iterations in the sense of matrix multisplitting in addition how the known parallel matrix multisplitting iterative methods can be classified into this new framework and what novel ones can be generated by it are shown in detail

modulus based matrix splitting iteration methods for a class of nonlinear complementarity problem some modulus based matrix splitting iteration methods for a class of nonlinear complementarity problem are presented and convergence analyses of the methods are given numerical experiments confirm the theoretical analysis and show that the proposed methods are efficient

asymptotic behaviour of nonlocal reaction diffusion equations abstract the existence of a global attractor in l 2 is established for a reaction diffusion equation on a bounded domain in r d with dirichlet boundary conditions where the reaction term contains an operator f l 2 l 2 which is nonlocal and possibly nonlinear existence of weak solutions is established but uniqueness is not required compactness of the multivalued flow is obtained via estimates obtained from limits of galerkin approximations in contrast with the usual situation these limits apply for all and not just for almost all time instants

parallel nonlinear aor method and its convergence abstract we set up a class of parallel nonlinear aor method in the sense of matrix multi splitting for solving the large scale system of nonlinear equations ax x b with a l rn nonsingular b rn and rn rn being continuously diagonal the global as well as the monotone convergence of this method is proved