existence and symmetries of solutions in besov morrey spaces for a semilinear heat wave type equation abstract this paper considers a semilinear integro differential equation of volterra type which interpolates semilinear heat and wave equations global existence of solutions is showed in spaces of besov type based in morrey spaces namely besov morrey spaces our initial data is larger than the previous works and our results provide a maximal existence class for semilinear interpolated heat wave equation some symmetries self similarity and asymptotic behavior of solutions are also investigated in the framework of besov morrey spaces

critical exponent for damped wave equations with nonlinear memory abstract we consider the cauchy problem in r n n 1 for a semilinear damped wave equation with nonlinear memory global existence and asymptotic behavior as t of small data solutions have been established in the case when 1 n 3 we also derive a blow up result under some positive data in any dimensional space

nonexistence of global solutions for a fractional differential problem in this paper we study the non existence of non trivial global solutions for a fractional differential inequality containing a lower order fractional derivative and a polynomial source term on the right hand side non existence of non trivial global solutions result is proved in an appropriate space by means of the test function method the range of values of the exponent involved there ensuring non existence is found to depend only on the lower order derivative this is in agreement with the integer order case numerical examples are given to illustrate the results

on the moments of the integrated geometric brownian motion abstract this note demonstrates how the divided differences characterization for the moments of the integrated geometric brownian process arises naturally from the solution to their differential equations the characterization was introduced by baxter and brummelhuis in their paper baxter and brummelhuis 2011 where they demonstrate its consistency with the results found by oshanin et al 1993 here we demonstrate its validity with more general expressions presented in other papers and explore the divided differences characterization of these moments further

decomposition of weighted triebel lizorkin and besov spaces on the ball weighted triebel lizorkin and besov spaces on the unit ball b d in r d with weights w x 1 x 2 1 2 0 are introduced and explored a decomposition scheme is developed in terms of almost exponentially localized polynomial elements needlets and it is shown that the membership of a distribution to the weighted triebel lizorkin or besov spaces can be determined by the size of the needlet coefficients f in appropriate sequence spaces localized bases and frames allow to decompose functions and distributions in terms of building blocks of simple nature and have numerous advantages over other means of representation in particular they enable one to encode smoothness and other norms in terms of the coefficients of the decompositions meyer s wavelets 10 and the transform of frazier and jawerth 5 7 provide such building blocks for decomposition of triebel lizorkin and besov spaces in the classical case on r d the aim of this article is to develop similar tools for decomposition of weighted triebel lizorkin and besov spaces on the unit ball b d in r d d 1 with weights w x 1 x 2 1 2 0 where x is the euclidean norm of x b d these include lp b d w the hardy spaces hp b d w and weighted sobolev spaces for our purpose we develop localized frames which can be viewed as an analog of the transform of frazier and jawerth on b d

decomposition of besov and triebel lizorkin spaces on the sphere abstract a discrete system of almost exponentially localized elements needlets on the n dimensional unit sphere s n is constructed it shown that the needlet system can be used for decomposition of besov and triebel lizorkin spaces on the sphere as an application of besov spaces on s n a jackson estimate for nonlinear m term approximation from the needlet system is obtained

local and global solvability of a class of semilinear parabolic equations pierre baras laborutoire imag tour des mathkmatiques b p 68 38402 saint martin d heres cedex france robert kersner computer and automation institute hungarian academy of sciences p 0 box 63 h 1502 budapest hungary received february 11 1986 revised september 5 1986

convergence analysis of moving finite element methods for space fractional differential equations most existent papers have focused on the fixed mesh methods for solving space fractional differential equations however since some classes of space fractional differential equations may have singular or even finite time blowup solutions it is highly needed to develop adaptive mesh methods to solve these problems in this paper the moving finite element methods are studied for a class of time dependent space fractional differential equations the convergence theories of the methods are derived with l 2 norm and numerical examples are provided to support the theoretical results to simplify the analysis a fractional ritz projection operator is introduced and the error estimation of the projection is derived under the moving mesh setting

fractional calculus and continuous time finance in this paper we present a rather general phenomenological theory of tick by tick dynamics in financial markets many well known aspects such as the levy scaling form follow as particular cases of the theory the theory fully takes into account the non markovian and non local character of financial time series predictions on the long time behaviour of the waiting time probability density are presented finally a general scaling form is given based on the solution of the fractional diffusion equation

the role of critical exponents in blow up theorems the sequel abstract in 27 fujita showed that for positive solutions the initial value problem in r n for u t u u p with p 1 exhibited the following behavior if p p c 1 2 n then the initial value problem does not have any nontrivial non negative solution existing on r n 0 a global solution whereas if p p c there exist global small data positive solutions as well as solutions which are non global we call such a result a blow up theorem of fujita type in 50 levine discussed the various theorems of this type that had appeared in the literature prior to 1990 in this paper we revisit the literature since 1990

global weak solutions to a spatio temporal fractional landau lifshitz bloch equation abstract the present paper deals with global existence of weak solutions of a time space fractional landau lifshitz bloch equation involving the weak caputo derivative and a fractional laplacian we use faedo galerkin method with some commutator estimates in order to prove global existence of weak solutions for the model the uniqueness is also discussed in a special one dimensional case

blow up solutions of second order differential inequalities with a nonlinear memory term abstract we consider the cauchy problem for second order nonlinear ordinary differential inequalities with a nonlinear memory term we obtain blow up results under some conditions on the initial data we also give an application to a semilinear hyperbolic equation in a bounded domain

decomposition of triebel lizorkin and besov spaces in the context of laguerre expansions abstract a pair of dual frames with almost exponentially localized elements needlets are constructed on r d based on laguerre functions it is shown that the triebel lizorkin and besov spaces induced by laguerre expansions can be characterized in terms of respective sequence spaces that involve the needlet coefficients

sub exponentially localized kernels and frames induced by orthogonal expansions the aim of this paper is to construct sup exponentially localized kernels and frames in the context of classical orthogonal expansions namely expansions in jacobi polynomials spherical harmonics orthogonal polynomials on the ball and simplex and hermite and laguerre functions

global nonexistence for the cauchy problem of some nonlinear reaction diffusion systems we first consider the parabolic equationut 2um a x uq f t x u p w t x t 0 x rn where 2 is the 2 fractional power of the laplacian which for 0 2 stands for impurities and f t x and w t x are given nonnegative functions and find its critical exponent then we consider the criticality for five systems of strongly coupled parabolic equations two of them with nonlinear convective terms our results answer positively some open problems raised recently by bandle deng levine and zhang

constructing cubature formulae by the method of reproducing kernel we examine the method of reproducing kernel for constructing cubature formulae on the unit ball and on the triangle in light of the compact formulae of the reproducing kernels that are discovered recently several new cubature formulae are derived

decomposition of spaces of distributions induced by tensor product bases abstract rapidly decaying kernels and frames needlets in the context of tensor product jacobi polynomials are developed based on several constructions of multivariate c cutoff functions these tools are further employed to the development of the theory of weighted triebel lizorkin and besov spaces on 1 1 d it is also shown how kernels induced by cross product bases can be constructed and utilized for the development of weighted spaces of distributions on products of multidimensional ball cube sphere or other domains

heat kernel based decomposition of spaces of distributions in the framework of dirichlet spaces classical and nonclassical besov and triebel lizorkin spaces with complete range of indices are developed in the general setting of dirichlet space with a doubling measure and local scale invariant poincare inequality this leads to a heat kernel with small time gaussian bounds and holder continuity which play a central role in this article frames with band limited elements of sub exponential space localization are developed and frame and heat kernel characterizations of besov and triebel lizorkin spaces are established this theory in particular allows the development of besov and triebel lizorkin spaces and their frame and heat kernel characterization in the context of lie groups riemannian manifolds and other settings

fractional diffusion equation on fractals three dimensional case and scattering function a fractional equation for diffusion in isotropic and homogeneous fractal structures is discussed it generalizes the fractional diffusion equation valid for d dimensional euclidean systems the asymptotic behaviour of the probability density function is obtained exactly analytical expressions are derived for the scattering and relaxation functions which can be studied by x ray and neutron scattering experiments on fractals

fractal burgers equations abstract the paper studies local and global in time solutions to a class of multidimensional generalized burgers type equations with a fractional power of the laplacian in the principal part and with general algebraic nonlinearity such equations naturally appear in continuum mechanics our results include existence uniqueness regularity and asymptotic behavior of solutions to the cauchy problem as well as a construction of self similar solutions the role of critical exponents is also explained

abstract fractional cauchy problems with almost sectorial operators abstract of concern are the cauchy problems for linear and semilinear time fractional evolution equations involving in the linear part a linear operator a whose resolvent satisfies the estimate of growth 1 0 in a sector of the complex plane which occurs when one considers for instance the partial differential operators in the limit domain of dumb bell with a thin handle or in the space of holder continuous functions by constructing a pair of families of operators in terms of the generalized mittag leffler type functions and the resolvent operators associated with a for the first time and a deep analysis on the properties for these families we obtain the existence and uniqueness of mild solutions and classical solutions to the cauchy problems moreover we present three examples to illustrate the feasibility of our results

global existence and blow up of solutions of the cauchy problem for a time fractional diffusion system abstract we prove sharp blow up and global existence results for the cauchy problem of a time fractional diffusion system when initial values u 0 0 and v 0 0 the critical exponent of this problem is determined

approximation of smooth functions on compact two point homogeneous spaces estimates of kolmogorov n widths dn bpr lq and linear n widths n bpr lq 1 q of sobolev s classes bpr r 0 1 p on compact two point homogeneous spaces ctphs are established for part of p q 1 1 sharp orders of dn bpr lq or n bpr lq were obtained by bordin et al j funct anal 202 2 2003 307 in this paper we obtain the sharp orders of dn bpr lq and n bpr lq for all the remaining p q our proof is based on positive cubature formulas and marcinkiewicz zygmund type inequalities on ctphs

nonexistence of global solutions for a class of two time nonlinear evolution equations nonexistence results of global solutions for a class of ultraparabolic equations and systems involving the fractional laplacian operator are derived the proofs are based on the choice of a suitable test function in the weak formulation of the considered problems the considered equations and systems have their applications in diffusion theory in porous media

riesz transforms and fractional integration for orthogonal expansions on spheres balls and simplices abstract this paper studies the hardy littlewood sobolev hls inequality and the riesz transforms for fractional integration associated to weighted orthogonal polynomial expansions on spheres balls and simplexes with weights being invariant under a general finite reflection group on r d the sharp index for the validity of the hls inequality is determined and the l p boundedness of the riesz transforms is established in particular our results extend a classical inequality of muckenhoupt and stein on conjugate ultraspherical polynomial expansions our idea is based on a new decomposition of the dunkl laplace beltrami operator on the sphere and some sharp asymptotic estimates of the weighted christoffel functions

blow up phenomena for a nonlinear time fractional heat equation in an exterior domain abstract we consider the nonlinear time fractional heat equation c d 0 u t x u t x u t x p t x 0 d c subject to the initial condition u 0 x u 0 x x d c and the boundary condition u t x f x t x 0 d where d b 0 1 is the closed unit ball in r n n 3 d c is its complement p 1 u 0 l l o c 1 d c u 0 0 f l 1 d d f x d x 0 and c d 0 u is the caputo fractional derivative of u of order 0 1 with respect to the time variable t using a test function approach we obtain the critical exponent of the considered problem in the sense of fujita

the critical exponent for an ordinary fractional differential problem abstract we consider the cauchy problem for an ordinary fractional differential inequality with a polynomial nonlinearity with variable coefficient a nonexistence result is proved and the critical exponent separating existence from nonexistence is found this is proved in the absence of any regularity assumptions

the error in linear interpolation at the vertices of a simplex a formula for the error in multivariate quasi interpolation which reproduces the linear polynomials is given from it sharp pointwise l infty bounds for the error in linear interpolation interpolation by linear polynomials to function values at the vertices of a simplex are obtained the corresponding envelope theorem giving the optimal recovery of functions is discussed

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

life span of blowing up solutions to the cauchy problem for a time space fractional diffusion equation abstract in this paper we study the blow up and global existence of solutions to the following time space fractional diffusion problem 0 c d t u x t 2 u x t 0 j t u p 1 u x t x r n t 0 u x 0 u 0 x c 0 r n where 0 1 0 1 0 2 p 1 0 j t denotes the left riemann liouville fractional integral of order 0 c d t is the caputo fractional derivative of order and 2 stands for the fractional laplacian operator of order 2 we show that if p 1 n then every nonnegative solution blows up in finite time and if p 1 n and u 0 l q c r n is sufficiently small where q c n p 1 then the problem has global solutions finally we give an upper bound estimate of the life span of blowing up solutions

boundedness and global nonexistence for a nonlinear convective parabolic system abstract a two component parabolic system with convective terms is studied it is shown that if the forces on the right hand side are no worse than quadratic in the variables then the solution remains bounded for all time it is also proved that if the forces are allowed to grow faster than quadratic then finite time blow up is to be expected

nonlinear hamiltonian equations with fractional damping existence and uniqueness are proved for a class of hamiltonian nonlinear differential equations of second order with a caputo fractional derivative of order 0 2 for noninteger the rate of convergence of solutions to a critical point is algebraic for 0 1 a non negative dissipation function is constructed for 2 the equation is hamiltonian the solution is an analytic function of for 0 1 and for 1 2 numerical studies of nonlinear pendulum and the duffing equation with a fractional derivative term replacing the usual damping show striking differences between ordinary differential and fractional differential equations

critical exponents of fujita type for certain evolution equations and systems with spatio temporal fractional derivatives this paper is concerned with establishing necessary or sufficient conditions for the existence of solutions to evolution equations with fractional derivatives in space and time the fujita exponent is determined then these results are extended to systems of reaction diffusion equations our new results shed lights on important practical questions

on the critical exponent for nonlinear schrodinger equations without gauge invariance in exterior domains abstract we obtain the critical exponent for the dirichlet exterior problem i u t u u p in 0 d c u t x f x in 0 d u 0 x g x in d c where u is a complex valued function d b 0 1 is the closed unit ball in r n n 3 d c is its complement p 1 c 0 f 0 f l 1 d c and g l l o c 1 d c c more precisely we show that if 1 p p n n 2 and re im d c g x h x d x 0 and re re d f x d s x 0 or im re d c g x h x d x 0 and im im d f x d s x 0 where h is a certain harmonic function then the considered problem possesses no global weak solution if p p then the problem admits global solutions for some f and g

functionals of exponential brownian motion and divided differences we provide a surprising new application of classical approximation theory to a fundamental asset pricing model of mathematical finance specifically we calculate an analytic value for the correlation coefficient between the exponential brownian motion and its time average and we find that the use of divided differences greatly elucidates formulae providing a path to several new results as applications we find that this correlation coefficient is always at least 1 2 and via the hermite genocchi integral relation demonstrate that all moments of the time average are certain divided differences of the exponential function we also prove that these moments agree with the somewhat more complex formulae obtained by oshanin and yor