multiple positive solutions of boundary value problems for systems of nonlinear second order differential equations this paper is concerned with boundary value problems for systems of nonlinear second order differential equations under the suitable conditions the existence and multiplicity of positive solutions are established by using abstract fixed point theorems

discretization of the multiscale semiconductor boltzmann equation by diffusive relaxation schemes in this paper we derive diffusive relaxation schemes for the linear semiconductor boltzmann equation that work in both the kinetic and diffusive regimes similar to our earlier approach for multiscale transport equations we use the even and odd parity formulation of the kinetic equation and then reformulate it into the diffusive relaxation system drs in order to handle the implicit anisotropic collision term efficiently we utilize a suitable power series expansion based on the wild sum which yields a time discretization uniformly stable with any desired order of accuracy yet is explicitly solvable with the correct drift diffusion limit the velocity discretization is done with the gauss hermite quadrature rule equivalent to a moment expansion method asymptotic analysis and numerical experiments show that the schemes have the usual advantages of a diffusive relaxation scheme for multiscale transport equations and are asymptotic preserving

existence and multiplicity of non negative solutions for systems of perturbed hammerstein integral equations abstract we study the existence of positive solutions for a system of perturbed hammerstein integral equations our setting is quite general and covers a wide class of systems of boundary value problems the main ingredient in our theory is the classical fixed point index theory for compact maps

radial solutions and phase separation in a system of two coupled schrodinger equations we consider the nonlinear elliptic system left begin aligned delta u u u 3 beta v 2u 0 quad rm in mathbb b delta v v v 3 beta u 2v 0 quad rm in mathbb b u v 0 quad rm in mathbb b quad u v 0 quad rm on partial mathbb b end aligned right where n leqq 3 and mathbb b subset mathbb r n is the unit ball we show that for every beta leqq 1 and k in mathbb n the above problem admits a radially symmetric solution u v such that u v changes sign precisely k times in the radial variable furthermore as beta to infty after passing to a subsequence u w and v w uniformly in mathbb b where w w w has precisely k nodal domains and is a radially symmetric solution of the scalar equation w w w3 0 in mathbb b w 0 on partial mathbb b within a hartree fock approximation the result provides a theoretical indication of phase separation into many nodal domains for bose einstein double condensates with strong repulsion

a new asymptotic preserving scheme based on micro macro formulation for linear kinetic equations in the diffusion limit we propose a new numerical scheme for linear transport equations it is based on a decomposition of the distribution function into equilibrium and nonequilibrium parts we also use a projection technique that allows us to reformulate the kinetic equation into a coupled system of an evolution equation for the macroscopic density and a kinetic equation for the nonequilibrium part by using a suitable time semi implicit discretization our scheme is able to accurately approximate the solution in both kinetic and diffusion regimes it is asymptotic preserving in the following sense when the mean free path of the particles is small our scheme is asymptotically equivalent to a standard numerical scheme for the limit diffusion model a uniform stability property is proved for the simple telegraph model various boundary conditions are studied our method is validated in one dimensional cases by several numerical tests and comparisons with previous asymptotic preserving schemes

an asymptotic induced scheme for nonstationary transport equations in the diffusive limit an asymptotic induced scheme for nonstationary transport equations with the diffusion scaling is developed the scheme works uniformly for all ranges of mean free paths it is based on the asymptotic analysis of the diffusion limit of the transport equation a theoretical investigation of the behavior of the scheme in the diffusion limit is given and an approximation property is proven moreover numerical results for different physical situations are shown and the uniform convergence of the scheme is established numerically

uniformly accurate diffusive relaxation schemes for multiscale transport equations many transport equations such as the neutron transport radiative transfer and transport equations for waves in random media have a diffusive scaling that leads to the diffusion equations in many physical applications the scaling parameter mean free path may differ in several orders of magnitude from the rarefied regimes to the hydrodynamic diffusive regimes within one problem and it is desirable to develop a class of robust numerical schemes that can work uniformly with respect to this relaxation parameter in an earlier work jin pareschi and toscani siam j numer anal 35 1998 pp 2405 2439 we handled this numerical problem for discrete velocity kinetic models by reformulating the system into a form commonly used for a relaxation scheme for conservation laws jin and xin comm pure appl math 48 1995 pp 235 277 such a reformulation allows us to use the splitting technique for relaxation schemes to design a class of implicit yet explicitly implementable schemes that work with high resolution uniformly with respect to the relaxation parameter in this paper we show that such a numerical technique can be applied to a large class of transport equations with continuous velocities when one uses the even and odd parities of the transport equation

critical exponents for the blowup of solutions with sign changes in a semilinear parabolic equation ii abstract the blowup of solutions of the cauchy problem formula is studied let k be the set of functions on r which change sign k times it is shown that for p k 1 2 k 1 k 0 1 2 any solution with u 0 k blows up in finite time if 1 p p k whereas a global solution with u 0 k exists if p p k this is an extension of our previous result 17 in which a fast decay condition was imposed on initial data it is also shown in this paper that if u0 decays more slowly than x 2 p 1 as x then the solution blows up in finite time regardless of the number of sign changes

a sequence of order relations encoding heteroclinic connections in scalar parabolic pde abstract we address the problem of heteroclinic connections in the attractor of dissipative scalar semilinear parabolic equations type display u t u xx f x u u x 0 on a bounded interval with neumann conditions introducing a sequence of order relations we prove a new and simple criterion for the existence of heteroclinic connections using only information about nodal properties of solutions to the stationary ode problem this result also allows for a complete classification of possible attractors in terms of the permutation of the equilibria given by their order at the two boundaries of the interval

efficient asymptotic preserving ap schemes for some multiscale kinetic equations many kinetic models of the boltzmann equation have a diffusive scaling that leads to the navier stokes type parabolic equations as the small scaling parameter approaches zero in practical applications it is desirable to develop a class of numerical schemes that can work uniformly with respect to this relaxation parameter from the rarefied kinetic regimes to the hydrodynamic diffusive regimes an earlier approach in s jin l pareschi and g toscani siam j numer anal 35 1998 pp 2405 2439 reformulates such systems into the common hyperbolic relaxation system by jin and xin for hyperbolic conservation laws used to construct the relaxation schemes and then uses a multistep time splitting method to solve the relaxation system here we observe that the combination of the two time split steps may yield hyperbolic parabolic systems that are more advantageous in both stability and efficiency for numerical computations we show that such an approach yields a class of asymptotic preserving ap schemes which are suitable for the computation of multiscale kinetic problems we use the goldstein taylor and carleman models to illustrate this approach

on the asymptotic preserving property of the unified gas kinetic scheme for the diffusion limit of linear kinetic models the unified gas kinetic scheme ugks of k xu et al 2010 37 originally developed for multiscale gas dynamics problems is applied in this paper to a linear kinetic model of radiative transfer theory while such problems exhibit purely diffusive behavior in the optically thick or small knudsen regime we prove that ugks is still asymptotic preserving ap in this regime but for the free transport regime as well moreover this scheme is modified to include a time implicit discretization of the limit diffusion equation and to correctly capture the solution in case of boundary layers contrary to many ap schemes this method is based on a standard finite volume approach it does neither use any decomposition of the solution nor staggered grids several numerical tests demonstrate the properties of the scheme

asymptotic behavior of zeros of solutions for parabolic equations abstract let p 1 0 and l r l 1 r change its sign finite times this paper is concerned with a cauchy problem formula define the set of zeros of a solution u by z t x r u x t 0 for t 0 in the case of 0 we show that the set z t is contained in ct ct for large t 0 with some c 0 and that this order of t is best possible when 0 we also give estimates of z t for global solutions and prove that z t k k for all t 0 t with some k 0 for each blowup solution where t is the blowup time

slow decay of solutions in a semilinear dissipative parabolic equation this paper is concerned with a cauchy problem p u t u x x u p 1 u in r x 0 u x 0 u 0 x in r where p 1 and u0 e l r a solution u of p is said to decay fast as t if limt tl p 1u x t 0 uniformly in r and to decay slowly as t otherwise we prove that if u0 x does not decay faster than x q with some q 0

a strong unique continuation theorem for parabolic equations the unique continuation is best understood for second order elliptic operators the classical paper by carleman 7 established the strong unique continuation theorem for second order elliptic operators which need not have analytic coefficients this was remarkable since it indicates that even a non analytic solution of an elliptic equation can behave somewhat in an analytic manner the powerful technique he used the so called carleman weighted inequality has played a central role in later developments in 1950 s aronszajn 4 and cordes 11 generalized his result to higher dimensions in recent years this classical subject caught new attention from a great number of people much efforts have been made to relax the smoothness hypotheses on the coefficients see 6 17 43 1 22 a seminal paper by jerison and kenig 26 has shown the unique continuation property for operators of the form c x with c l 2 loc where n 3 is the space dimension further improvements have been carried out in considering other classes of coefficients fefferman phong class and kato class 42 8 12 29 in extending the result to operators with first derivative terms and variable leading coefficients 46 47 51 52 and etc see 25 15 16 28 5 27 and the references therein for second order linear parabolic operators with time independent coefficients the strong unique continuation property along with detailed estimates on the hausdorff measures of nodal sets was reduced in 32 33 to the previously established elliptic counterparts the reduction from time independent parabolic

fractional diffusion limit for collisional kinetic equations this paper is devoted to diffusion limits of linear boltzmann equations when the equilibrium distribution function is a maxwellian distribution it is well known that for an appropriate time scale the small mean free path limit gives rise to a diffusion equation in this paper we consider situations in which the equilibrium distribution function is a heavy tailed distribution with infinite variance we then show that for an appropriate time scale the small mean free path limit gives rise to a fractional diffusion equation

multiple blow up for a porous medium equation with reaction the present paper is concerned with the cauchy problem left begin array ll partial t u delta u m u p quad rm in mathbb r n times 0 infty u x 0 u 0 x geq 0 quad rm in mathbb r n end array right with p m 1 a solution u with bounded initial data is said to blow up at a finite time t if lim rm sup t nearrow t u t l infty mathbb r n infty for n 3 we obtain in a certain range of values of p weak solutions which blow up at several times and become bounded in intervals between these blow up times we also prove a result of a more technical nature proper solutions are weak solutions up to the complete blow up time

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

positive solutions of a system of second order boundary value problems involving first order derivatives via r n monotone matrices abstract in this paper we study the existence and multiplicity of positive solutions for the system of second order boundary value problems for nonlinear ordinary differential equations depending on first order derivatives u i f i t u 1 u 1 u n u n u i 0 u i 1 0 i 1 n here n 2 f i c 0 1 r 2 n r r 0 based on a priori estimates achieved by using integral inequalities and r n monotone matrices we use fixed point index theory to establish our main results for the above problem

transition operators of diffusions reduce zero crossing if u t x is a solution of a one dimensional parabolic secondorder linear partial differential equation pde then it is known that under suitable conditions the number of zero crossings of the function u t decreases that is does not increase as time t increases such theorems have applications to the study of blow up of solutions of semilinear pde time dependent sturm liouville theory curve shrinking problems and control theory we generalise the pde results by showing that the transition operator of a possibly time inhomogenous one dimensional diffusion reduces the number of zero crossings of a function or even suitably interpreted a signed measure our proof is completely probabilistic and depends in a transparent manner on little more than the sample path continuity of diffusion processes

simplicity of zeros in scalar parabolic equations soit le systeme u t u xx f x u x 0 1 u t 0 u t 1 0 f x 0 0 u 0 x u 0 x soit t 0 u t a seulement des zeros simples soit z u t le nombre de zeros de u 0 on suppose z u 0 fini alors est ouvert et dense dans l intervalle de temps maximal d existence de u t

the zero set of a solution of a parabolic equation on etudie l ensemble nul d une solution u t x de l equation u t a x t u xx b x t u x c x t u sous des hypotheses tres generales sur les coefficients a b et c

numerical schemes for kinetic equations in the anomalous diffusion limit part i the case of heavy tailed equilibrium in this work we propose some numerical schemes for linear kinetic equations in the anomalous diffusion limit when the equilibrium distribution function is a maxwellian distribution it is well known that for an appropriate time scale the small mean free path limit gives rise to a diffusion type equation however when a heavy tailed distribution is considered another time scale is required and the small mean free path limit leads to a fractional anomalous diffusion equation our aim is to develop numerical schemes for the original kinetic model which works for the different regimes without being restricted by stability conditions of standard explicit time integrators starting from some numerical schemes for the diffusion asymptotics their extension to the anomalous diffusion limit is then studied in this case it is crucial to capture the effect of the large velocities of the heavy tailed equilibrium so that some important transformations of the schemes derived for the diffusion asymptotics are ne

movement of hot spots over unbounded domains in rn abstract we consider the initial boundary value problems of the heat equation over unbounded domains in r n and study the movement of hot spots of the solution for bounded nonnegative initial data having compact support

an asymptotic preserving scheme for linear kinetic equation with fractional diffusion limit we present a new asymptotic preserving scheme for the linear boltzmann equation which under appropriate scaling leads to a fractional diffusion limit our scheme rests on novel micro macro decomposition to the distribution function which splits the original kinetic equation following a reshuffled hilbert expansion as opposed to classical diffusion limit a major difficulty comes from the fat tail in the equilibrium which makes the truncation in velocity space depending on the small parameter our idea is while solving the macro micro part in a truncated velocity domain truncation only depends on numerical accuracy to incorporate an integrated tail over the velocity space that is beyond the truncation and its major component can be precomputed once with any accuracy such an addition is essential to drive the solution to the correct asymptotic limit numerical experiments validate its efficiency in both kinetic and fractional diffusive regimes

volume preserving mean curvature flow of revolution hypersurfaces in a rotationally symmetric space in an ambient space with rotational symmetry around an axis which include the hyperbolic and euclidean spaces we study the evolution under the volume preserving mean curvature flow of a revolution hypersurface m generated by a graph over the axis of revolution and with boundary in two totally geodesic hypersurfaces tgh for short requiring that for each time t 0 the evolving hypersurface m t meets such tgh orthogonally we prove that a the flow exists while m t does not touch the axis of rotation b throughout the time interval of existence b1 the generating curve of m t remains a graph and b2 the averaged mean curvature is double side bounded by positive constants c the singularity set if non empty is finite and lies on the axis d under a suitable hypothesis relating the enclosed volume to the n volume of m we achieve long time existence and convergence to a revolution hypersurface of constant mean curvature