new approach to study splines by blossoming method and application to the construction of a bivariate c 1 quartic quasi interpolant this work is a contribution in the approximation theory for studying and analyzing piecewise polynomial functions splines which uses the blossoming approach some existing results in the literature are reformulated such as the smoothness conditions between polynomials of a spline by using the affinity property of the blossom some definitions of sub splines are proposed which can be very useful in the study and the construction of splines such as macro elements or quasi interpolants as an application of the proposed results a c 1 quartic spline quasi interpolant with optimal approximation order is defined without using any mask for smoothness or b spline basis numerical results are presented and compared with other methods given in the literature

the evolution fractional p laplacian equation in rn fundamental solution and asymptotic behaviour abstract we consider the natural time dependent fractional p laplacian equation posed in the whole euclidean space with parameters p 2 and s 0 1 fractional exponent we show that the cauchy problem for data in the lebesgue l q spaces is well posed and show that the solutions form a family of non expansive semigroups with regularity and other interesting properties as main results we construct the self similar fundamental solution for every mass value m and prove that general finite mass solutions converge towards that fundamental solution having the same mass and convergence holds in all l q spaces a number of additional properties and estimates complete the picture

a dynamically consistent method to solve nonlinear multidimensional advection reaction equations with fractional diffusion abstract this work is motivated by an extension of both the burgers fisher and the burgers huxley equations in multiple dimensions considering riesz fractional diffusion initial boundary conditions which are positive and bounded are imposed on a closed and bounded set and a finite difference method is proposed to approximate the solutions of the fractional model the methodology is a linear and implicit technique which is based on fractional centered differences we show in this manuscript that the method can be expressed in vector form using a minkowski matrix under suitable conditions the main properties of minkowski matrices are used then to establish the existence and the uniqueness of the solutions of the finite difference method as well as the capability of the technique to preserve the positivity and the boundedness additionally we show that the method is a consistent technique which is stable and convergent with first order of convergence in time and second order in space some illustrative simulations show that the scheme is capable of preserving the positivity and the boundedness of the numerical approximations

backward estimates for nonnegative solutions to a class of singular parabolic equations abstract in this paper we improve a recent result of the same authors recalde and vespri 2015 by using the same approach in a more tricky way more precisely we study a class of quasilinear singular parabolic equations with l coefficients whose prototypes are the p laplacian 2 n n 1 p 2 equations let us consider nonnegative solutions defined in r r n we recall that in recalde and vespri 2015 starting from the value attained in a point x 0 t 0 by the solution we proved sharp estimates in the stripe 1 e t 0 r n in this note we show that quite surprisingly sharp estimates hold also for the remote past i e also for the stripe 0 t 0 r n in the last section we briefly show how these results can be adapted to equations of porous medium type in the fast diffusion range i e n 2 n m 1

approximation orders and shape preserving properties of the multiquadric trigonometric b spline quasi interpolant the purpose of the paper is to derive some properties of the multiquadric trigonometric b spline quasi interpolant firstly the paper captures its approximation orders for high order derivatives based on the error estimate one can choose the shape parameter properly such that the quasi interpolant gives optimal approximations to high order derivatives moreover the approximation orders also show that from the theoretical point of view the considered quasi interpolant can be applied when the approximation of high order derivatives is needed i e numerical solution of some pdes construction of lyapunov function etc secondly the paper derives some shape preserving properties of the quasi interpolant these properties suggest that the quasi interpolant may be used in the geometric modeling cad cam for instance that requires shape preservation finally to illustrate the validity of the results some numerical examples are presented both theoretical and numerical results demonstrate that the quasi interpolant cannot only provide excellent approximations to high order derivatives but also preserve the shapes well

nonequilibrium fluctuations for a tagged particle in mean zero one dimensional zero range processes we prove a non equilibrium functional central limit theorem for the position of a tagged particle in mean zero one dimensional zero range process the asymptotic behavior of the tagged particle is described by a stochastic differential equation governed by the solution of the hydrodynamic equation

quasilinear elliptic boundary value problems on unbounded cylinders and a related mountain pass lemma sufficient conditions for the existence of a weak solution to quasilinear elliptic boundary value problems on unbounded cylinders are obtained

global existence and finite time blow up for a class of semilinear wave equations on r n abstract in this paper we consider the initial value problem for a class of semilinear wave equations with damping and source terms under certain conditions we establish the global existence of solutions and the asymptotic behavior of the solutions by introducing an appropriate lyapunov functions moreover we examine the blow up in finite time when the initial data is sufficiently large

theory of extended solutions for fast diffusion equations in optimal classes of data radiation from singularities this paper is devoted to constructing a general theory of nonnegative solutions for the equation

fractional nonlinear degenerate diffusion equations on bounded domains part i existence uniqueness and upper bounds abstract we investigate quantitative properties of nonnegative solutions u t x 0 to the nonlinear fractional diffusion equation t u l f u 0 posed in a bounded domain x r n with appropriate homogeneous dirichlet boundary conditions as l we can use a quite general class of linear operators that includes the two most common versions of the fractional laplacian s 0 s 1 in a bounded domain with zero dirichlet boundary conditions but it also includes many other examples since our theory only needs some basic properties that are typical of linear heat semigroups the nonlinearity f is assumed to be increasing and is allowed to be degenerate the prototype is the power case f u u m 1 u with m 1 in this paper we propose a suitable class of solutions of the equation and cover the basic theory we prove existence uniqueness of such solutions and we establish upper bounds of two forms absolute bounds and smoothing effects as well as weighted l 1 estimates the class of solutions is very well suited for that work the standard laplacian case s 1 is included and the linear case m 1 can be recovered in the limit in a companion paper bonforte and vazquez in preparation we will complete the study with more advanced estimates like the upper and lower boundary behaviour and harnack inequalities for which the results of this paper are needed

a general method for constructing quasi interpolants from b splines a general method for constructing quasi interpolation operators based on b splines is developed given a b spline f in r s s 1 normalized by i z email protected i 1 the classical structure q f email protected i z email protected i f i for a quasi interpolation operator q is considered a minimization problem is derived from an estimate of the quasi interpolation error associated with q when lf is a linear combination of values of f at points in some neighbourhood of the support of f or a linear combination of values of f and some of its derivatives at some points in this set or a linear combination of weighted mean values of the function to be approximated that linear functional is defined to produce a quasi interpolant exact on the space of polynomials of maximal total degree included in the space spanned by the integer translates of f the solution of that minimization problem is characterized in terms of specific splines which do not depend on l but only on f

bifurcation problems for the laplacian in ⁿ in this paper we consider the bifurcation problem div ivulp 2vu ag x julp 2u f a x u in rn with p 1 we show that a continuum of positive solutions bifurcates out from the principal eigenvalue a1 of the problem div ivuip 2vu ag x ulp 2u here both functions f and g may change sign

a priori estimates for fractional nonlinear degenerate diffusion equations on bounded domains we investigate quantitative properties of the nonnegative solutions u t x geq 0 to the nonlinear fractional diffusion equation partial t u mathcal l u m 0 posed in a bounded domain x in omega subset mathbb r n with m 1 for t 0 as mathcal l we use one of the most common definitions of the fractional laplacian delta s 0 s 1 in a bounded domain with zero dirichlet boundary conditions we consider a general class of very weak solutions of the equation and obtain a priori estimates in the form of smoothing effects absolute upper bounds lower bounds and harnack inequalities we also investigate the boundary behaviour and we obtain sharp estimates from above and below in addition we obtain similar estimates for fractional semilinear elliptic equations either the standard laplacian case s 1 or the linear case m 1 are recovered as limits the method is quite general suitable to be applied to a number of similar problems

optimal lagrange interpolation by quartic c1 splines on triangulations we develop a local lagrange interpolation scheme for quartic c 1 splines on triangulations given an arbitrary triangulation d we decompose d into pairs of neighboring triangles and add diagonals to some of these pairs only in exceptional cases a few triangles are split based on this simple refinement of d we describe an algorithm for constructing lagrange interpolation points such that the interpolation method is local stable and has optimal approximation order the complexity for computing the interpolating splines is linear in the number of triangles for the local lagrange interpolation methods known in the literature about half of the triangles have to be split

point and differential c1 quasi interpolation on three direction meshes abstract in this paper we construct and analyse c 1 cubic and quartic quasi interpolating splines on type 1 triangulations approximating regularly distributed data without using minimal determining sets and without defining the approximating splines as linear combinations of compactly supported bivariate spanning functions in particular the c 1 cubic splines are directly determined by setting their bernstein bezier coefficients to appropriate combinations of the given data values without using prescribed derivatives at any point of the domain in such a way that the c 1 smoothness conditions are satisfied and approximation order three is guaranteed for smooth functions we also propose some numerical tests that confirm the theoretical results then from the above c 1 cubic splines we obtain c 1 quartic splines exact on p 3 achieving approximation order four the associated differential quasi interpolation operator involves the values of the first partial derivatives in its definition

a general fractional porous medium equation we develop a theory of existence and uniqueness for the following porous medium equation with fractional diffusion u t 2 u m 1 u 0 x r n t 0 u x 0 f x x r n we consider data f l 1 r n and all exponents 0 0 existence and uniqueness of a weak solution is established for m m n n giving rise to an l 1 contraction semigroup in addition we obtain the main qualitative properties of these solutions in the lower range 0 m m existence and uniqueness of solutions with good properties happen under some restrictions and the properties are different from the case above m we also study the dependence of solutions on f m and moreover we consider the above questions for the problem posed in a bounded domain

the fractional laplacian in power weighted l p spaces integration by parts formulas and self adjointness abstract we consider the fractional laplacian operator s let s 0 1 on euclidean space and investigate the validity of the classical integration by parts formula that connects the l 2 r d scalar product between a function and its fractional laplacian to the nonlocal norm of the fractional sobolev space h s r d more precisely we focus on functions belonging to some weighted l 2 space whose fractional laplacian belongs to another weighted l 2 space we prove and disprove the validity of the integration by parts formula depending on the behaviour of the weight x at infinity the latter is assumed to be like a power both near the origin and at infinity the two powers being possibly different our results have direct consequences for the self adjointness of the linear operator formally given by 1 s the generality of the techniques developed allows us to deal with weighted l p spaces as well

existence of multiple solutions for a class of p q laplacian systems abstract we establish the existence and multiplicity of solutions for some p q laplacian systems we will show how the multiplicity of solutions change as and vary the results are proved by ekeland s variational principle the mountain pass theorem and the saddle point theorem

representation of solutions and large time behavior for fully nonlocal diffusion equations abstract we study the cauchy problem for a nonlocal heat equation which is of fractional order both in space and time we prove four main theorems i a representation formula for classical solutions ii a quantitative decay rate at which the solution tends to the fundamental solution iii optimal l 2 decay of mild solutions in all dimensions iv l 2 decay of weak solutions via energy methods the first result relies on a delicate analysis of the definition of classical solutions after proving the representation formula we carefully analyze the integral representation to obtain the quantitative decay rates of ii next we use fourier analysis techniques to obtain the optimal decay rate for mild solutions here we encounter the critical dimension phenomenon where the decay rate attains the decay rate of that in a bounded domain for large enough dimensions consequently the decay rate does not anymore improve when the dimension increases the theory is markedly different from that of the standard caloric functions and this substantially complicates the analysis finally we use energy estimates and a comparison principle to prove a quantitative decay rate for weak solutions defined via a variational formulation our main idea is to show that the l 2 norm is actually a subsolution to a purely time fractional problem which allows us to use the known theory to obtain the result

a hopf s lemma and a strong minimum principle for the fractional p laplacian abstract our propose here is to provide a hopf lemma and a strong minimum principle for weak supersolutions of p s u c x u p 2 u in where is an open set of r n s 0 1 p 1 c c and p s is the fractional p laplacian

non local gradient dependent operators abstract in this paper we study a general class of quasilinear non local equations depending on the gradient which arises from tug of war games we establish a c c 1 c 2 regularity theory for these equations the kind of regularity depending on the assumptions on the kernel and we construct different non local approximations of the p laplacian

increasing the approximation order of spline quasi interpolants abstract in this paper we show how by a very simple modification of bivariate spline discrete quasi interpolants we can construct a new class of quasi interpolants which have remarkable properties such as high order of regularity and polynomial reproduction more precisely given a spline discrete quasi interpolation operator q d which is exact on the space p m of polynomials of total degree at most m we first propose a general method to determine a new differential quasi interpolation operator q r d which is exact on p m r q r d uses the values of the function to be approximated at the points involved in the linear functional defining q d as well as the partial derivatives up to the order r at the same points from this result we then construct and study a first order differential quasi interpolant based on the c 1 cubic b spline on the equilateral triangulation with a hexagonal support when the derivatives are not available or extremely expensive to compute we approximate them by appropriate finite differences to derive new discrete quasi interpolants q d we estimate with small constants the quasi interpolation errors f q r d f and f q d f in the infinity norm finally numerical examples are used to analyze the performance of the method

antimaximum principle in rn local versus global we study the antimaximum principle for the p laplacian defined on the whole set rn with an indefinite weight function we show that a local version of this principle holds but that the global version does not hold

on an indefinite semilinear elliptic problem on rn abstract we are dealing with the problem u x h x u x g x u p x for x r n u d 1 2 r n u 0 where is a real parameter n 2 h and g are a changing sign functions and 1 n 2 n 2 under suitable assumptions and by combining the global bifurcation result of rabinowitz j funct anal 7 1971 485 513 with a priori estimates of positive solutions we prove the existence of a continuum of positive solutions bifurcating from 1 h and 1 h the two principal eigenvalues of multiplicity one of the associated linear problem

uniqueness and properties of distributional solutions of nonlocal equations of porous medium type abstract we study the uniqueness existence and properties of bounded distributional solutions of the initial value problem for the anomalous diffusion equation t u l u 0 here l can be any nonlocal symmetric degenerate elliptic operator including the fractional laplacian and numerical discretizations of this operator the function r r is only assumed to be continuous and nondecreasing the class of equations include nonlocal generalized porous medium equations fast diffusion equations and stefan problems in addition to very general uniqueness and existence results we obtain stability l 1 contraction and a priori estimates we also study local limits continuous dependence and properties and convergence of a numerical approximation of our equations

optimization problems in partially ordered hilbert resolution spaces recent studies hove introduced the concept of a partially ordered hilbert resolution space as a means of modelling the behaviour of multidimensional systems the present study utilizes this framework to formulate and solve some typical optimization problems the resultant solutions provide design criteria for multidimensional systems and highlight both the similarities and the differences between the single dimensional and multidimensional cases

transformations of self similar solutions for porous medium equations of fractional type abstract we consider four different models of nonlinear diffusion equations involving fractional laplacians and study the existence and properties of classes of self similar solutions such solutions are an important tool in developing the general theory we introduce a number of transformations that allow us to map complete classes of solutions of one equation into those of another one thus providing us with a number of new solutions as well as interesting connections special attention is paid to the property of finite propagation

near best quasi interpolants associated with h splines on a three direction mesh spline quasi interpolants with optimal approximation orders and small norms are useful in several applications in this paper we construct the so called near best discrete and integral quasi interpolants based on h splines i e b splines with regular hexagonal supports on the uniform three directional mesh of the plane these quasi interpolants are obtained so as to be exact on some space of polynomials and to minimize an upper bound of their infinite norms which depend on a finite number of free parameters we show that this problem has always a solution which is not unique in general concrete examples of these types of quasi interpolants are given in the two last sections

positivity local smoothing and harnack inequalities for very fast diffusion equations abstract we investigate qualitative properties of local solutions u t x 0 to the fast diffusion equation t u u m m with m 1 corresponding to general nonnegative initial data our main results are quantitative positivity and boundedness estimates for locally defined solutions in domains of the form 0 t with r d they combine into forms of new harnack inequalities that are typical of fast diffusion equations such results are new for low m in the so called very fast diffusion range precisely for all m m c d 2 d the boundedness statements are true even for m 0 while the positivity ones cannot be true in that range

global bifurcation results for a fractional equation in rn abstract in this paper we study the existence of positive solution for the following class of fractional equation p s u g x f u in r n 0 u x 1 x r n lim x u x 0 where n 2 s s 0 1 0 f r r is a continuous function g r n r is a bounded continuous function that changes sign and s u is the fractional laplacian the main tools used are a theorem due to crandall and rabinowitz 14 and the global bifurcation theorem due to rabinowitz 23

global bifurcation of the p laplacian in rn abstract we study global bifurcation for the following nonlinear equation div a x u p 2 u 0 g x u p 2 u f x u in r n satisfying certain conditions on a g and f when 0 is not an eigenvalue of the above divergence form it is based on a bifurcation result on noncompact components of solutions for nonlinear operator equations

global positivity estimates and harnack inequalities for the fast diffusion equation abstract we investigate local and global properties of positive solutions to the fast diffusion equation u t u m in the range d 2 d m 1 corresponding to general nonnegative initial data for the cauchy problem posed in the whole euclidean space r d we prove sharp local positivity estimates weak harnack inequalities and elliptic harnack inequalities we use them to derive sharp global positivity estimates and a global harnack principle for the mixed initial and boundary value problem posed in a bounded domain of r d with homogeneous dirichlet condition we prove weak and elliptic harnack inequalities our work shows that these fast diffusion flows have regularity properties comparable and in some senses better than the linear heat flow

initial trace of positive solutions to fractional diffusion equation with absorption in this paper we prove the existence of an initial trace t u of any positive solution u of the semilinear fractional diffusion equation h t u u f t x u 0 in r times r n where n 1 where the operator with 0 1 is the fractional laplacian and f r times r n times r r is a caratheodory function satisfying f t x u u 0 for all t x u r times r n times r we define the regular set of the trace t u as an open subset of r u r n carrying a nonnegative radon measive u such that lim t 0 ru u t x x dx ru d c 2 0 r u and the singular set s u r n r u as the set points a such that lim sup t 0 b a u t x dx 0 we study the reverse problem of constructing a positive solution to h with a given initial trace s where s r n is a closed set and is a positive radon measure on r r n s and develop the case f t x u t u p where 1 and p 1

antimaximum principle in exterior domains abstract we consider the antimaximum principle for the p laplacian in the exterior domain p u k x u p 2 u h x in b 1 c u 0 on b 1 where p is the p laplace operator with p 1 is the spectral parameter and b 1 c is the exterior of the closed unit ball in r n with n 1 the function h is assumed to be nonnegative and nonzero however the weight function k is allowed to change its sign for k in a certain weighted lebesgue space we prove that the antimaximum principle holds locally a global antimaximum principle is obtained for h with compact support for a compactly supported k with n 1 and p 2 we provide a necessary and sufficient condition on h for the global antimaximum principle in the course of proving our results we also establish the boundary regularity of solutions of certain boundary value problems

optimal bivariate c1 cubic quasi interpolation on a type 2 triangulation in a guessab o nouisser g schmeisser multivariate approximation by a combination of modified taylor polynomials j comput appl math 196 2006 162 179 a general method is proposed to increase the approximation order of approximation operators in this work by using these enhancement techniques we introduce and study new schemes based on a c 1 spline quasi interpolant on a type 2 triangulation they are designed for approximating real valued functions defined on r 2 the proposed method is based on the following idea from a discrete quasi interpolant defined by the quadratic box spline exact on p 2 and by judiciously choosing the first order taylor coefficients we derive a cubic differential quasi interpolant yielding optimal approximation order in addition when the derivatives are not available or are extremely expensive to compute we approximate them by finite difference approximations having the desired accuracy to derive a new class of discrete quasi interpolants as an essential difference to some of the existing methods we only use the given data values and then we do not modify the original triangulation finally we present some numerical tests which confirm the efficiency of the newly quasi interpolant and demonstrate good visual quality in particular we compare it with a differential quasi interpolant done by lai m j lai approximation order from bivariate c 1 cubics on a four directional mesh is full comput aided geom design 11 2 1994 215 223 which is also exact on p 3 but uses third order partial derivatives

on the bifurcation of large amplitude solutions for a system of wave and beam equations we consider the existence of solutions for linearly coupled system of wave and beam equation with a sublinear perturbation the main concept is the matrix spectrum which is a natural extension of the usual spectrum using the standard change of degree argument we shall find necessary and sufficient conditions for the existence of asymptotic bifurcation with respect to the matrix spectrum both in one parameter and four parameter cases obviously the results can be modified for more general systems of partial differential equations

harnack estimates at large sharp pointwise estimates for nonnegative solutions to a class of singular parabolic equations abstract in this paper we deal with quasilinear singular parabolic equations with l coefficients whose prototypes are the p laplacian 2 n n 1 p 2 equations in this range of the parameters we are in the so called fast diffusion case extending a recent result ragnedda et al 2013 we are able to prove harnack estimates at large i e starting from the value attained in a point by the solution we are able to give explicit and sharp pointwise estimates from below by using the barenblatt solutions in the last section we briefly show how these results can be adapted to equations of porous medium type in the fast diffusion range i e n 2 n m 1

nonequilibrium scaling limit for a tagged particle in the simple exclusion process with long jumps we prove an invariance principle for a tagged particle in a simple exclusion process out of equilibrium the scaling limit is a time inhomogeneous process of independent increments related to the solution of a fractional heat equation

the fisher kpp problem with doubly nonlinear diffusion abstract the famous fisher kpp reaction diffusion model combines linear diffusion with the typical kpp reaction term and appears in a number of relevant applications in biology and chemistry it is remarkable as a mathematical model since it possesses a family of travelling waves that describe the asymptotic behaviour of a large class solutions 0 u x t 1 of the problem posed in the real line the existence of propagation waves with finite speed has been confirmed in some related models and disproved in others we investigate here the corresponding theory when the linear diffusion is replaced by the slow doubly nonlinear diffusion and we find travelling waves that represent the wave propagation of more general solutions even when we extend the study to several space dimensions a similar study is performed in the critical case that we call pseudo linear i e when the operator is still nonlinear but has homogeneity one with respect to the classical model and the pseudo linear case the slow travelling waves exhibit free boundaries

on some fourth order semilinear elliptic problems in r n the existence of two solutions for a fourth order semilinear elliptic problem involving critical growth from the viewpoint of sobolev embedding was established the basic tools used in the analysis were the mountain pass theorem constrained minimization and concentration compactness principle the existence of positive principle eigenvalues for the corresponding linear elliptic problem was also investigated

algorithms for fast multidimensional signal extraction abstract in recent studies operator factorization techniques have been developed for problems associated with multidimensional control and image processing in particular the special factorization of gohberg and krein and the schur coleski factorization have both been extended to the multivariate setting the present study focuses on computational aspects of the earlier results also of interest are implications for designing computer architectures which facilitate high speed online computation of the operator factors

fractional cahn hilliard allen cahn and porous medium equations abstract we introduce a fractional variant of the cahn hilliard equation settled in a bounded domain r n and complemented with homogeneous dirichlet boundary conditions of solid type i e imposed in the whole of r n after setting a proper functional framework we prove existence and uniqueness of weak solutions to the related initial boundary value problem then we investigate some significant singular limits obtained as the order of either of the fractional laplacians appearing in the equation is let tend to 0 in particular we can rigorously prove that the fractional allen cahn fractional porous medium and fractional fast diffusion equations can be obtained in the limit finally in the last part of the paper we discuss existence and qualitative properties of stationary solutions of our problem and of its singular limits

uniqueness of very weak solutions for a fractional filtration equation abstract we prove existence and uniqueness of distributional bounded solutions to a fractional filtration equation in r d with regards to uniqueness it was shown even for more general equations in 22 that if two bounded solutions u w of 1 1 satisfy u w l 1 r d 0 t then u w we obtain here that this extra assumption can in fact be removed and establish uniqueness in the class of merely bounded solutions for nonnegative initial data we first show that a minimal solution exists and then that any other solution must coincide with it a similar procedure is carried out for sign changing solutions as a consequence distributional solutions have locally finite energy

global bifurcation results for a semilinear elliptic equation on all of mathbb r n we prove the existence of positive solutions for the semilinear elliptic equation u x g x f u x 0 u 1 for x r lim x u x 0 which arises in population genetics under the hypotheses that n 3 4 5 and g lies in l n 2 ir n we establish the existence of a principal eigenvalue 1 for the corresponding linearized problem and making use of the asymptotic properties of solutions and local and global bifurcation theory prove the existence of a continuum of solutions lying in the space d 1 2 extending from 1 to

a fractional porous medium equation abstract we develop a theory of existence uniqueness and regularity for the following porous medium equation with fractional diffusion u t 1 2 u m 1 u 0 x r n t 0 u x 0 f x x r n with m m n 1 n n 1 and f l 1 r n an l 1 contraction semigroup is constructed and the continuous dependence on data and exponent is established nonnegative solutions are proved to be continuous and strictly positive for all x r n t 0

linking and multiplicity results for the p laplacian on unbounded cylinders abstract we consider the p laplacian problem formula on unbounded cylinders rn m rn n m 2 where pu div u p 2 u is a constant in a certain range and a ln p l is nonnegative a 0 using the principle of symmetric criticality existence and multiplicity are proved under suitable conditions on a and f

angular factorization of matrices let q denote a selfadjoint matrix or operator of finite rank the factorization of i q into the form i q i w d i w is considered the well known case where w is constrained to be lower triangular while d is diagonal is extended to quadrangular and other generalized canonical forms the case where q is not selfadjoint is also developed applications for these results may be found in the realm of multivariate stochastic approximation and signal extraction

continuity of the temperature in boundary heat control problems abstract motivated by the boundary heat control problems formulated in the book of duvaut and lions we study a boundary stefan problem and a boundary porous media problem we prove continuity of the solution with the appropriate modulus we also extend the results to the fractional order case and to the anomalous diffusion problems

existence and uniqueness of an attractive nonlinear diffusion system we establish the existence and uniqueness of an attractive fractional coupled system such a system has applications in biological populations of cells we confirm that the fractional system under consideration admits a global solution in the sobolev space the solution is shown to be unique the technique is founded on analytic method of the fixed point theory and the fractional differential operator is scrutinized from the view of the riemann liouville differential operator finally we illustrate some entropy fractional differential inequalities regarding the solution of the system

spectrum of the fractional p laplacian in rn and decay estimate for positive solutions of a schrodinger equation abstract in this paper we prove the existence of unbounded sequence of eigenvalues for the fractional p laplacian with weight in r n we also show a nonexistence result when the weight has positive integral in addition we show some qualitative properties of the first eigenfunction including a sharp decay estimate finally we extend the decay result to the positive solutions of a schrodinger type equation

superconvergent local quasi interpolants based on special multivariate quadratic spline space over a refined quadrangulation abstract in this paper we first recall some results concerning the construction and the properties of quadratic b splines over a refinement of a quadrangulation of a planar domain introduced recently by lamnii et al then we introduce the b spline representation of hermite interpolant in the special space s 2 1 0 of any polynomial or any piecewise polynomial over refined quadrangulation of after that we use this b representation for constructing several superconvergent discrete quasi interpolants the new results that we present in this paper are an improvement and a generalization of those developed in the above cited paper

existence of a global attractor for semilinear dissipative wave equations on rn abstract we consider the semilinear hyperbolic problem utt ut x u f u x x r n t 0 with the initial conditions u x 0 u0 x and ut x 0 u1 x in the case where n 3 and x 1 g x lies in ln 2 r n the energy space x 0 d 1 2 r n l2g r n is introduced to overcome the difficulties related with the noncompactness of operators which arise in unbounded domains we derive various estimates to show local existence of solutions and existence of a global attractor in x 0 the compactness of the embedding d 1 2 r n l2g r n is widely applied

multivariate approximation by a combination of modified taylor polynomials we study an approximation of a multivariate function f by an operator of the form i 1n r f xi x i x where 1 n are certain basis functions and r f xi x are modified taylor polynomials of degree r expanded at xi the modification is such that the operator has highest degree of algebraic precision in the univariate case this operator was investigated by xuli multinode higher order expansions of a function j approx theory 124 2003 242 253 special attention is given to the case where the basis functions are a partition of unity of linear precision for this setting we establish two types of sharp error estimates in the two dimensional case we show that this operator gives access to certain classical interpolation operators of the finite element method in the case where 1 n are multvariate bernstein polynomials we establish an asymptotic representation for the error as n

global bifurcation results for semilinear elliptic equations on rn the fredholm case abstract we prove the existence of a continuum of positive solutions for the semilinear elliptic equation u x g x f u x 0 u x r n lim x u x 0 which arises in population genetics under the hypotheses that n 3 and the weight g changes sign being negative and away from zero at after establishing the existence of a simple positive principal eigenvalue 1 for the corresponding linearized problem we prove the existence of a continuum of solutions lying in the space r h 2 extended from 1 to to complete this task we state a new version of the global bifurcation theory for nonlinear fredholm noncompact operators and prove the compactness of the solution set of the problem

construction techniques for multivariate modified quasi interpolants with high approximation order in this paper we propose several approximations of a multivariate function by quasi interpolants on non uniform data and we study their properties in particular we characterize those that preserve constants via the partition of unity approach as one of the main results we show how by a very simple modification of a given quasi interpolant it is possible to construct new quasi interpolants with remarkable properties we also provide some results regarding bivariate c 2 quintic spline quasi interpolation finally numerical tests are presented to show the approximation power of these quasi interpolants

uniqueness of entropy solutions to fractional conservation laws with fully infinite speed of propagation our goal is to study the uniqueness of bounded entropy solutions for a multidimensional conservation law including a non lipschitz convection term and a diffusion term of nonlocal porous medium type the nonlocality is given by a fractional power of the laplace operator for a wide class of nonlinearities the l 1 contraction principle is established despite the fact that the finite infinite speed of propagation alibaud jee 2007 cannot be exploited in our framework existence is deduced with perturbation arguments the method of proof adapted from andreianov maliki nodea 2010 requires a careful analysis of the action of the fractional laplacian on truncations of radial powers

infinite speed of propagation and regularity of solutions to the fractional porous medium equation in general domains we study the positivity and regularity of solutions to the fractional porous me dium equations ut sum 0 in 0 for m 1 and s 0 1 with dirichlet boundary data u 0 in 0 n and nonnegative initial condition u 0 u0 0 our first result is a quantitative lower bound for solutions that holds for all positive times t 0 as a consequence we find a global harnack principle stating that for any t 0 solutions are comparable to ds m where d is the distance to this is in sharp contrast with the local case s 1 where the equation has finite speed of propagation after this we study the regularity of solutions we prove that solutions are classical in the interior c in x and c1 in t and establish a sharp cxs m regularity estimate up to the boundary our methods are quite general and can be applied to wider classes of nonlocal parabolic equations of the form ut f u 0 in both in bounded and unbounded domains 2016 wiley periodicals inc

the dirichlet problem for the fractional p laplacian evolution equation abstract we consider a model of fractional diffusion involving a natural nonlocal version of the p laplacian operator we study the dirichlet problem posed in a bounded domain of r n with zero data outside of for which the existence and uniqueness of strong nonnegative solutions is proved and a number of quantitative properties are established a main objective is proving the existence of a special separate variable solution u x t t 1 p 2 f x called the friendly giant which produces a universal upper bound and explains the large time behaviour of all nontrivial nonnegative solutions in a sharp way moreover the spatial profile f of this solution solves an interesting nonlocal elliptic problem we also prove everywhere positivity of nonnegative solutions with any nontrivial data a property that separates this equation from the standard p laplacian equation

estimates at infinity for positive solutions to a class of p laplacian problems in rn abstract in this paper we consider a class of logistic type problems for the p laplacian in the whole space using minimization we prove existence of a positive solution and its behavior at infinity we also consider questions of uniqueness and sharp estimates at infinity

an elliptic equation with concave and convex nonlinearities we obtain infinitely many solutions for the problem pu g x u p 2 u k x u q 2u h x u s 2u in rn where n 3 1 q p s our result extends those of tonkes topology methods nonlinear anal 13 1999 251 in two aspects we don t require s p and the requirements on the weight h also be relaxed

the cauchy problem for ut δum when 0 m 1 on considere le probleme de cauchy pour l equation de diffusion non lineaire u t u u m 1 0 sur 0 r n u 0 u 0 quand 0 m 1 on montre qu il existe une solution temporelle globale pour une fonction localement integrable u 0 on montre l unicite dans cette classe

quantitative local and global a priori estimates for fractional nonlinear diffusion equations abstract we establish quantitative estimates for solutions u t x to the fractional nonlinear diffusion equation t u s u m 0 in the whole range of exponents m 0 0 s 1 the equation is posed in the whole space x r d we first obtain weighted global integral estimates that allow to establish existence of solutions for classes of large data in the core of the paper we obtain quantitative pointwise lower estimates of the positivity of the solutions depending only on the norm of the initial data in a certain ball the estimates take a different form in three exponent ranges slow diffusion good range of fast diffusion and very fast diffusion finally we show existence and uniqueness of initial traces

optimal existence and uniqueness theory for the fractional heat equation abstract we construct a theory of existence uniqueness and regularity of solutions for the fractional heat equation t u s u 0 0 s 1 posed in the whole space r n with data in a class of locally bounded radon measures that are allowed to grow at infinity with an optimal growth rate we consider a class of nonnegative weak solutions and prove that there is an equivalence between nonnegative data and solutions which is given in one direction by the representation formula in the other one by the initial trace we review many of the typical properties of the solutions in particular we prove optimal pointwise estimates and new harnack inequalities

on the dynamics of a strongly singular parabolic equation abstract in this paper we study the semiflow defined by a semilinear parabolic equation in which both the diffusion and the reaction term present strong order of singularity at the origin we justify the existence of a global branch of nontrivial equilibrium solutions for subcritical nonlinearities the approach based on the critical caffarelli kohn nirenberg inequality follows the arguments of 21 22