canonical bases of invariant polynomials for the irreducible reflection groups of types e6 e7 and e8 abstract given a rank n irreducible finite reflection group w the w invariant polynomial functions defined in r n can be written as polynomials of n algebraically independent homogeneous polynomial functions p 1 x p n x called basic invariant polynomials their degrees are well known and typical of the given group w the polynomial p 1 x has the lowest degree equal to 2 it has been proved that it is possible to choose all the other n 1 basic invariant polynomials in such a way that they satisfy a certain system of differential equations including the laplace equations p a x 0 a 2 n and so are harmonic functions bases of this kind are called canonical explicit formulas for canonical bases of invariant polynomials have been found for all irreducible finite reflection groups except for those of types e 6 e 7 and e 8 those for the groups of types e 6 e 7 and e 8 are determined in this article

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

the qualitative behavior of solutions of a nonlinear difference equation abstract this paper is concerned with the qualitative behavior of solutions to the difference equation x n 1 p qx n k 1 x n n 0 1 2 where the initial conditions x k x 1 x0 are non negative k 1 2 3 and the parameters p q are non negative we start by establishing the periodicity the character of semicycles the global stability and the boundedness of the above mentioned equation we also present solutions that have unbounded behavior it is worth to mention that this difference equation is a special case of an open problem was introduced by m r s kulenovic and g ladas dynamics of second order rational difference equations with open problems and conjectures chapman and hall crc boca raton 2002 several computational examples are given to support our theoretical discussions the presented numerical tests represent different types of qualitative behavior of solutions to our nonlinear difference equation

fast fourier transform method for partial differential equations case study the 2 d diffusion equation abstract the solution of time dependent partial differential equations using discrete i e finite difference versions of those equations traditionally has been done by marching forward in time one time unit per step the authors here extend earlier results to higher dimensions focusing on the solution of the discrete diffusion or heat equation in 2 d and show that the solution may be found using fast fourier transform fft techniques that treat the solution as the output of a linear filter this technique is analogous to an integral method for the differential equation and is related to the green s function of the difference equation it allows the solution to be determined with a single step computation that does not require the solution at intermediate time levels the number of numerical operations is essentially independent of the time level at which the solution is desired the results of numerical examples are given and computation times are shown to be much faster than those of the traditional method for all but small time levels

improved conditions for single point blow up in reaction diffusion systems abstract we study positive blowing up solutions of the system u t u v p v t v u q as well as of some more general systems for any p q 1 we prove single point blow up for any radially decreasing positive and classical solution in a ball this improves on previously known results in 3 directions i no type i blow up assumption is made and it is known that this property may fail ii no equidiffusivity is assumed i e any 0 is allowed iii a large class of nonlinearities f u v g u v can be handled which need not follow a precise power behavior as a side result we also obtain lower pointwise estimates for the final blow up profiles

supersolutions for a class of nonlinear parabolic systems abstract in this paper by using scalar nonlinear parabolic equations we construct supersolutions for a class of nonlinear parabolic systems including t u u v p x t 0 t v v u q x t 0 u v 0 x t 0 u x 0 v x 0 u 0 x v 0 x x where p 0 q 0 is a possibly unbounded smooth domain in r n and both u 0 and v 0 are nonnegative and locally integrable functions in the supersolutions enable us to obtain optimal sufficient conditions for the existence of the solutions and optimal lower estimates of blow up rate of the solutions

on the higher order rational recursive sequence xn axn k bx n 3k the main purpose of the current paper is to prove that every positive solution of the delay difference equationx n ax n k bx n 3 k where a b 0 the initial conditions x 3 k 1 x 3 k 2 x 0 0 and k 1 2 3 converges eventually to a period k solution we also give the results of computational examples to support our theoretical discussion

on the global behavior of higher order recursive sequences our basic purpose in this paper is to establish some conditions for the global attractivity and convergence of the kth order recursive sequencex n f x n 1 x n 2 x n k n 1 2 3 we will show that if f is a real valued function with a convex basin w in r k which is differentiable with bounded differential then f has an equilibrium point our final comments are about informative examples

nonexistence of positive solutions to a semilinear elliptic system and blow up estimates for a reaction diffusion system abstract we get simple conditions under which the elliptic system u u p 1 v q 1 v u p 2 v q 2 in r n n 3 with p i q i 1 i 1 2 has no positive radially symmetric solutions then by using this nonexistence result we establish blow up estimates for reaction diffusion system of fujita type u t u u p 1 v q 1 v t v u p 2 v q 2 with the homogeneous dirichlet boundary value condition

dynamics of a higher order rational difference equation abstract our aim in this paper is to investigate the dynamics of difference equation y n 1 p qy n 1 y n k n 0 1 2 where the initial conditions y k y 1 y0 are non negative k 1 2 3 and the parameters p and q are non negative we study characteristics such as the global stability the boundedness of positive solutions and the character of semicycles of the above mentioned difference equation in particular our results solve the open problem introduced by kulenovic and ladas in their monograph m r s kulenovic g ladas dynamics of second order rational difference equations with open problems and conjectures chapman and hall crc boca raton 2002

blowup estimates for a semilinear reaction diffusion system this paper deals with the blowup estimates of positive solutions for a semilinear reaction diffusion system ut u u vp vt v uqv with null dirichlet boundary conditions the upper and lower bounds of blowup rates are obtained

a bound for global solutions of semilinear heat equations we show that global positive solutions of the initial boundary value problem foru t u u p are bounded provided thatp 1 is subcritical our bound depends only on sup norm of the initial data and is useful to classify initial data by the asymptotic behavior of the solutions as time tend to infinity

the oscillatory character of the recursive sequence xn 1 α βxn k 1a bxn 2k 1 abstract the aim of this work is to generalize techniques of difference equations in order to apply them to a wider class we start by establishing the boundedness the oscillatory character and the global stability of the equation x n 1 x n k 1 a bx n 2 k 1 n 0 1 2 with positive parameters k 1 2 and nonnegative initial conditions

blow up set of type i blowing up solutions for nonlinear parabolic systems we consider the blow up problem for systems of nonlinear parabolic inequalities and establish a criterion for the location of the blow up set our criterion enables us to obtain sufficient conditions for the boundedness of the blow up set and no boundary blow up for type i blowing up solutions

orbit spaces of compact coregular simple lie groups with 2 3 and 4 basic polynomial invariants effective tools in the analysis of invariant potentials the coregular real representations of the compact simple lie groups are classified and the equalities and inequalities defining their orbit spaces are determined in the case of 2 3 and 4 basic polynomial invariants the results are obtained making use of the classification of the complex coregular simple lie groups given by g w schwarz and of the determination of all the allowable metric matrices for q dimensional q 4 orbit spaces of compact coregular linear groups recently obtained by the present authors the results are used to determine in a rigorous way the minima of two so 10 and e6 invariant higgs potentials

liouville theorems universal estimates and periodic solutions for cooperative parabolic lotka volterra systems we consider positive solutions of cooperative parabolic lotka volterra systems with equal diffusion coefficients in bounded and unbounded domains the systems are complemented by the dirichlet or neumann boundary conditions under suitable assumptions on the coefficients of the reaction terms these problems possess both global solutions and solutions which blow up in finite time we show that any solution u v defined on the time interval 0 t satisfies a universal estimate of the form u x t v x t c 1 t 1 t t 1 where c does not depend on x t u v t in particular this bound guarantees global existence and boundedness for threshold solutions lying on the borderline between blow up and global existence moreover this bound yields optimal blow up rate estimates for solutions which blow up in finite time our estimates are based on new liouville type theorems for the corresponding scaling invariant parabolic system and require an optimal restriction on the space dimension n n 5 as an application we also prove the existence of time periodic positive solutions if the coefficients are time periodic our approach can also be used for more general parabolic systems

blow up set for a semilinear heat equation with small diffusion abstract we consider the blow up problem for a semilinear heat equation t u u u p in 0 t u x t 0 on 0 t if u x 0 x 0 in where is a domain in r n n 1 0 p 1 and t 0 in this paper under suitable assumptions on we prove that if the family of the solutions u satisfies a uniform type i blow up estimate with respect to then the solution u blows up only near the maximum points of the initial datum for any sufficiently small 0 this is proved without any conditions on the exponent p and the domain such as n 2 p n 2 and the convexity of the domain

blow up problems for a semilinear heat equation with large diffusion abstract we consider the blow up problem of a semilinear heat equation u t d u u p in 0 t d u x t 0 on 0 t d u x 0 x 0 in where is a bounded smooth domain in r n t d 0 d 0 and p 1 we study the blow up time the location of the blow up set and the blow up profile of the blow up solution for sufficiently large d in particular we prove that for almost all initial data if d is sufficiently large then the solution blows up only near the maximum points of the orthogonal projection of the initial data from l 2 onto the second neumann eigenspace

green s function and stability of a linear partial difference scheme abstract this paper is concerned with a linear partial difference equation which arises from discretizing a heat and a first order wave equations a green s function for this equation is derived various representation theorems monotonicity and oscillatory properties of the green s functions are obtained then by means of the representation theorems we derive several stability criteria for arbitrary solutions of this equation

bounds for solutions of a six point partial difference scheme abstract this paper is concerned with a class of partial difference equations which arise from discretising the one dimensional heat and the first order wave equations explicit bounds are found for their solutions and subexponential solutions are considered using the green s function we derive a stability criteria for arbitrary solutions of this equation

blowup stability of solutions of the nonlinear heat equation with a large life span abstract we study the cauchy problem for the nonlinear heat equation u t u u p 1 u in r n the initial data is of the form u 0 where c 0 r n is fixed and 0 we first take 1 p p f where p f is the fujita critical exponent and c 0 r n l 1 r n with nonzero mean we show that u t blows up for small extending the h fujita blowup result for sign changing solutions next we consider 1 p p s where p s is the sobolev critical exponent and x decaying as x at infinity where p 1 2 we also prove that u t blows up when is small extending a result of t lee and w ni for both cases the solution enjoys some stable blowup properties for example there is single point blowup even if is not radial

flat bases of invariant polynomials and p matrices of e7 and e8 let g be a compact group of linear transformations of a euclidean space v the g invariant c functions can be expressed as c functions of a finite basic set of g invariant homogeneous polynomials sometimes called an integrity basis the mathematical description of the orbit space v g depends on the integrity basis too it is realized through polynomial equations and inequalities expressing rank and positive semidefiniteness conditions of the p matrix a real symmetric matrix determined by the integrity basis the choice of the basic set of g invariant homogeneous polynomials forming an integrity basis is not unique so it is not unique the mathematical description of the orbit space too if g is an irreducible finite reflection group saito et al commun algebra 8 373 1980 characterized some special basic sets of g invariant homogeneous polynomials that they called flat they also found explicitly the flat basic sets of invariant homogeneous polynomials of all the irreducible finite reflection g

convergence asymptotic periodicity and finite point blow up in one dimensional semilinear heat equations abstract we consider the initial value problem for the semilinear heat equation u t u xx f u t 0 x l t 0 under the dirichlet the neumann or the periodic boundary conditions we show that each solution whether it exists globally for t 0 or blows up in a finite time possesses an asymptotic profile in a certain sense and tends to this profile as time increases in the special case where f u t t f u t for some t 0 among other things the above statement is interpreted as saying that any bounded global solution converges as t to a time t periodic solution having some specific spatial structure in the case where the solution blows up in a finite time say at t t 0 assuming simply that f is a smooth function satisfying some growth conditions and that the initial data is a nonconstant bounded function we prove that the blow up set is a finite set and that lim t t 0 u x t x exists with being a smooth function having at most finitely many singular points

solution of a nonlinear heat equation with arbitrarily given blow up points we consider the equation where i u is scalar valued and p 1 it has been proven that if u t blows up at time t the blow up points are finite in number and located in i our aim is to prove that this result is optimal that is for any given points x1 xk in i there is a solution u such that its blow up points are exactly x1 xk

blow up for a semilinear parabolic equation with large diffusion on rn ii we are concerned with the cauchy problem for a semilinear heat equation p tu d u u p 1u x rn t 0 u x 0 x x rn where d 0 p 1 n 3 0 and l rn l1 rn 1 x 2dx in the paper of fujishima and ishige 2011 8 the authors of this paper studied the behavior of the blow up time and the blow up set of the solution of p as d for the case rn x dx 0 in this paper as a continuation of fujishima and ishige 2011 8 we consider the case rn x dx 0 and study the behavior of the blow up time and the blow up set of the solution of p as d the behavior in the case rn x dx 0 is completely different from the one in the case rn x dx 0

determination of nodes in numerical integration rules using difference equation in this paper the difference equations will be employed to produce a polynomial which its roots are the nodes of gauss quadrature rules in this procedure we obtain a system of nonlinear equations according to the concepts of precision degree of integration and the moments of order k hence the resulted polynomials obtained by difference equations have their roots as the nodes finally some examples are given to show the application of the procedure

determination of the curvature of the blow up set and refined singular behavior for a semilinear heat equation we consider u x t a solution of ut u u p 1u which blows up at some time t 0 where u rn 0 t r p 1 and n 2 p n 2 under a nondegeneracy condition we show that the mere hypothesis that the blow up set s is continuous and n 1 dimensional implies that it is c2 in particular we compute the n 1 principal curvatures and directions of s moreover a much more refined blow up behavior is derived for the solution in terms of the newly exhibited geometric objects refined regularity for s and refined singular behavior of u near s are linked through a new mechanism of algebraic cancellations that we explain in detail

dynamics of xn 1 xn 2k 1xn 2k 1 αxn 2l abstract we investigate the global stability of all positive solutions of the difference equation x n 1 x n 2 k 1 x n 2 k 1 x n 2 l where is a positive real number k 1 2 l 0 1 and the initial conditions are positive real numbers when x 1 1 is globally asymptotically stable also when 1 we show that this equation possesses solutions that converge to the two cycle 0 1 0 1

the asymptotic behavior of the solutions of a discrete reaction diffusion equation abstract by means of bihari type inequalities we derive sufficient conditions for solutions of a discrete reaction diffusion equation to be bounded or to converge to zero asymptotic representation of solutions are also derived our results yield estimates and explicit attractive regions for the solutions

boundedness and blow up for a semilinear reaction diffusion system abstract we consider the semilinear parabolic system s u t u p t u q where x rnn 1 t 0 and p q are positive real numbers at t 0 nonnegative continuous and bounded initial values u0 v0 x are prescribed the corresponding cauchy problem then has a nonnegative classical and bounded solution u t x v t x in some strip st 0 t r n 0 t sup t 0 u v remain bounded in s t we show in this paper that if 0 t so that solutions can be continued for all positive times when pq 1 and 1 pq 1 n 2 with max p q one has t for every nontrivial solution u v t is then called the blow up time of the solution under consideration finally if 1 pq 1 n 2 both situations coexist since some nontrivial solutions remain bounded in any strip s while others exhibit finite blow up times

on the recursive sequence xn 1 α βxn k 1 γxn 2k 1bxn k 1 cxn 2k 1 abstract our aim in this paper is to investigate the global asymptotic stability of all positive solutions of the higher order nonlinear difference equation x n 1 x n k 1 x n 2 k 1 bx n k 1 cx n 2 k 1 n 0 1 2 where b c and are positive k 1 2 3 and the initial conditions x 2 k 1 x 1 x 0 are positive real numbers we show that the unique positive equilibrium of the equation is globally asymptotically stable and has some basins that depend on certain conditions posed on the coefficients our concentration is on invariant intervals the character of semicycles and the boundedness of the above mentioned equation our final comments are about informative examples

general solutions of a three level partial difference equation abstract this paper is concerned with a linear partial difference equation which includes the well known dufort frankel multilevel difference scheme for the heat equation by introducing green s functions for this equation we obtain via a novel formal approach an explicit formula for all its solutions given exponential initial conditions separable solutions are also found as applications we derive several stability criteria for the solutions of this equation

single point blow up for parabolic systems with exponential nonlinearities and unequal diffusivities abstract we study positive blowing up solutions of systems of the form u t 1 u e p v v t 2 v e q u with 1 2 0 and p q 0 we prove single point blow up for large classes of radially decreasing solutions this answers a question left open in a paper of friedman and giga 1987 where the result was obtained only for the equidiffusive case 1 2 and the proof depended crucially on this assumption

global stability of yn 1 p qyn ryn k1 yn in this paper we investigate the global stability of difference equation y n 1 p qy n ry n k1 y n n 0 1 2 where the initial conditions y k y 1 y 0 are non negative email protected 1 2 3 and the parameters p q and r are non negative we show that when rq 1 and k is odd then this equation possesses unbounded solutions

oscillation and asymptotic behavior of a class of higher order nonlinear recursive sequences abstract we investigate the global attractivity the periodic behavior and the invariant interval with the semicycles character of solutions of the equation x n 1 x n 2 k 1 bx n k 1 cx n 2 k 1 n 0 1 2 for all admissible non negative values of the parameters b c and the initial conditions x 2 k 1 x 1 x 0 we show that the solutions exhibit a periodic attitude depending on the parameters of the family of the above mentioned difference equation in comparison with each other

stability criteria for two partial difference equations abstract this paper is concerned with two classes of linear partial difference equations with constant coefficients explicit conditions are derived which are sufficient and or necessary for these equations to have stable solutions

global stability of a higher order rational recursive sequence abstract in this paper we study the qualitative behavior of solutions of the class of delay difference equation x n 1 x n k 1 x n 2 k 1 a bx n k 1 n 0 1 2 where the initial conditions x 2 k 1 x 1 x 0 are positive k 1 2 and the parameters a b are positive our concentration is on invariant intervals and the global stability of the above mentioned equation we obtain sufficient conditions for the global attractivity of all positive solutions about the zero and positive equilibrium points with basins that depend on specific conditions posed on the coefficients furthermore the oscillation and the character of semicycles about the positive equilibrium are thoroughly discussed it is also illustrated that for some special case of parameters the solution will be either a period 2 solution or will converge to the equilibrium point or will have unbounded solutions

on the recursive sequence xn 1 α βxn k 1a bxn k 1 cxn 2k 1 abstract in this paper we study the global asymptotic stability invariant intervals the character of semicycles and the boundedness of all positive solutions of the higher order nonlinear difference equation x n 1 x n k 1 a bx n k 1 cx n 2 k 1 n 0 1 where a b c and are positive k 1 2 3 and the initial conditions x 2k 1 x 1 x0 are positive real numbers we show that the unique positive equilibrium of the equation is globally asymptotically stable finally we present some informative examples

modulation theory for the blowup of vector valued nonlinear heat equations this paper is concerned with the blowup of solutions of the nonlinear vector valued heat equation ut u u p 1u u 0 u0 where u x t u1 x t um x t is a vector valued function from rn 0 t to rm and 1 p 3n 8 3n 4 working with the equation in similarity variables and using modulation theory and ideas from center manifold theory we obtain the asymptotic behavior of u in a backward space time parabola near any blowup point

dynamics of a non linear difference equation abstract our aim in this paper is to investigate the dynamics of difference equation y n 1 y n py n k y n q n 0 1 2 where the initial conditions y k y 1 y0 are non negative k 1 2 3 and the parameters p and q are non negative we study characteristics such as periodicity invariant interval the character of semicycles the global stability and the boundedness of positive solutions of the above mentioned difference equation in particular our results solve the open problem introduced by kulenovic and ladas in their monograph dynamics of second order rational difference equations with open problems and conjectures chapman and hall crc boca raton 2002