a numerical model for the boltzmann equation with applications to micro flows given an integer lattice email protected r d we define g as the orthogonal group leaving l invariant starting from a basic kinetic model on g we construct a collision operator on l which keeps all the essential features of the classical boltzmann collision operator for a particular 3d lattice we demonstrate the suitability of this discrete model for the numerical simulation of rarefied flows for several examples e g in the context of micro flows we find a good qualitative and quantitative agreement of our simulation results with test data

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

complete blow up after tmax for the solution of a semilinear heat equation soit un sous ensemble ouvert borne de r n a frontiere lisse on considere le probleme u t u f u dans o t u 0 sur o t u x o u 0 x x ou f r r est localement de lipschitz non decroissante et f o 0

a computational strategy for the regularized 13 moment equations with enhanced wall boundary conditions the challenge of modeling low speed rarefied gas flow in the transition regime is well known in this paper we propose a numerical solution procedure for the regularized 13 moment equations within a finite volume framework the stress and heat flux equations arising in the method of moments are transformed into the governing equations for the stress and heat flux deviators based on their first order approximation to model confined flows a complete set of wall boundary conditions for the 13 moment equations are derived based on the maxwell wall boundary model this has been achieved by expanding the molecular distribution function to fourth order accuracy in hermite polynomials empirical correction factors are introduced into the boundary conditions and calibrated against direct simulation monte carlo data the numerical predictions obtained from the regularized 13 moment equations and the navier stokes fourier equations are compared with data generated using the direct simulation monte carlo method for planar couette flow for a range of wall velocities and knudsen numbers 0 012 1 0 the results indicate that the regularized 13 moment equations are in good qualitative agreement with the direct simulation monte carlo data the results also highlight limitations that are caused by the use of a first order expansion of the third moment

rate of convergence for one dimensional quasilinear parabolic problem and its applications abstract based on a comparison principle we derive an exponential rate of convergence for solutions to the initial boundary value problem for a class of quasilinear parabolic equations in one space dimension we then apply the result to some models in population dynamics and image processing

refined asymptotics for the blowup of ut δu up this work is concerned with positive blowing up solutions of the semilinear heat equation ut u up in rn our main contribution is a sort of center manifold analysis for the equation in similarity variables leading to refined asymptotics for u in a backward space time parabola near any blowup point we also explore a connection between the asymptotics of u and the local geometry of the blowup set

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

a particle system with explosions law of large numbers for the density of particles and the blow up time consider a system of independent random walks in the discrete torus with creation annihilation of particles and possible explosion of the total number of particles in finite time rescaling space and rates for diffusion creation annihilation of particles we obtain a strong law of large numbers for the density of particles in the supremum norm the limiting object is a classical solution to the semilinear heat equation t u xx u f u if f u u p 1 p 3 we also obtain a law of large numbers for the explosion time

asymptotic and limiting profiles of blowup solutions of the nonlinear schrodinger equation with critical power this paper is a sequel to previous ones 38 39 41 we continue the study of the blowup problem for the nonlinear schrodinger equation with critical power nonlinearity nsc we introduce a new idea to prove the existence of a blowup solution in h1 n without any weight condition and reduce the problem to a kind of variational problem our new method refines the previous results concerning the asymptotic and limiting profiles of blowup solutions for a certain class of initial data the blowup solution behaves like a finite superposition of zero energy h1 bounded global in time solutions of nsc these singularities stay in a bounded region in n and one can see that the so called shoulder emerges outside these singularities as suggested by some numerical computations see e g 26 we investigate the asymptotic behavior of zero energy global in time solutions of nsc and find that such a solution behaves like a multisoliton however it is not an assemblage of free particles the solitons interact with each other 1999 john wiley sons inc

structural stability for time periodic one dimensional parabolic equations denote q w x 0 l x r the nonlinear term f is assumed to satisfy the following conditions f l f q r is of class c f 2 f t l x u f t x u for all te w xe o l and uer define a self adjoint operator a in x l2 0 l by a t bxx for e d a h2 0 l a ha o l for each c1 e r let a be the fractional power of a and x d a be the fractional power space equipped with the graph norm ii 11 xn 1111 11x iia i ii x throughout this section we fix a c1 1 by the above assumptions f l f 2 for any u0 e x there exists a unique local in time classical solution u t x u t x of

convergence in a quasilinear parabolic equation with neumann boundary conditions abstract consider the problem u t a u x u x x f u x x 1 t 0 u x 1 t k t u 1 t t 0 where k are smooth functions which are periodic in both t and u brunovsk et al proved in their paper brunovsk et al 1992 8 that if a time global solution u is bounded then it converges to a periodic solution we prove that if u is unbounded from above then it converges to a periodic traveling wave v x t c t in case k k t or k k u where v is a time periodic function and c 0 in addition the periodic traveling wave is unique up to space shifts or time shifts it is stable and asymptotically stable the average traveling speed c and the instantaneous speed v t c are also studied

blow up for a non local diffusion problem with neumann boundary conditions and a reaction term abstract in this paper we study the blow up problem for a non local diffusion equation with a reaction term u t x t j x y u y t u x t d y u p x t we prove that non negative and non trivial solutions blow up in finite time if and only if p 1 moreover we find that the blow up rate is the same as the one that holds for the ode u t u p that is lim t t t t 1 p 1 u t 1 p 1 1 p 1 next we deal with the blow up set we prove single point blow up for radially symmetric solutions with a single maximum at the origin as well as the localization of the blow up set near any prescribed point for certain initial conditions in a general domain with p 2 finally we show some numerical experiments which illustrate our results

construction of solutions with exactly k blow up points for the schrodinger equation with critical nonlinearity we consider the nonlinear schrodinger equation i partial u mathord left vphantom i partial u partial t right kern nulldelimiterspace partial t delta u left u right 4 mathord left vphantom 4 n right kern nulldelimiterspace n uandu left 0 right varphi left right 1 whereu 0 t n for any given pointsx1 x2 x k in n we construct a solution of eq 1 u t which blows up in a finite timet at exactlyx1 x2 x k in addition we describe the precise behavior of the solutionu t whent t at the blow up points x1 x2 x k and in n x1 x2 x k

scale induced closure for approximations of kinetic equations the order of magnitude method proposed by struchtrup phys fluids 16 11 3921 3934 2004 is a new closure procedure for the infinite moment hierarchy in kinetic theory of gases taking into account the scaling of the moments the scaling parameter is the knudsen number kn which is the mean free path of a particle divided by the system size

blow up of solutions of nonlinear heat equations on considere des solutions de u t u xx f u pour a 0 u a t 0 pour t 0 u x 0 x pour a x a ou f u est une fonction superlineaire continument differentiable

convergence for degenerate parabolic equations abstract we prove that any bounded global solution to a degenerate parabolic problem in one spatial dimension converges to a unique stationary state

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

blow up of solutions with sign changes for a semilinear diffusion equation abstract this paper is concerned with the initial boundary value problem formula with the dirichlet neumann or periodic boundary condition here 0 is a parameter and f is an odd function of u satisfying f 0 0 and some convexity condition let z u be the number of times of sign changes for u c 0 1 it is shown that there exists an increasing sequence of positive numbers k k 0 1 2 such that any solution with z u 0 k blows up in finite time if k and there exists a global solution with z u 0 k if k

symmetry of the blow up set of a porous medium type equation we study the blow up set of a porous medium type equation with source under some technical conditions we prove that if the blow up set is a bounded smooth region then it must be a ball with a certain radius this problem can be reduced to a sublinear elliptic equation coupled with an overdetermined boundary condition roughly speaking the overdetermined boundary condition forces the domain to be a ball because the nonlinear term is sublinear and then non lipschitz many difficulties arise if one wants to use the moving plane method to reach the goal in particular the hopf boundary lemma is not applicable to this problem instead we investigate various related problems in a half space and a problem in the first quadrant of the entire space and then use the symmetry results obtained for these problems to overcome the obstacles encountered 1995 john wiley sons inc

on minimal sets of scalar parabolic equations with skew product structures skew product semi flow i t x x y x x y which is generated by ut uxx f y t ix u ux t 0 0 x 1 y e y d or n boundary conditions is considered where x is an appropriate subspace of h2 0 1 y r is a minimal flow with compact phase space it is shown that a minimal set e c x x y of ilt is an almost 1 1 extension of y that is set yo y e y i card e c p 1 y i is a residual subset of y where p x x yy is the natural projection consequently if y r is almost periodic minimal then any minimal set e c x x y of ilt is an almost automorphic minimal set it is also proved that dynamics of it is closed in the category of almost automorphy that is a minimal set e c x x y of fit is almost automorphic minimal if and only if y r is almost automorphic minimal asymptotically almost periodic parabolic equations and certain coupled parabolic systems are discussed examples of nonalmost periodic almost automorphic minimal sets are provided

convergence in time periodic quasilinear parabolic equations in one space dimension abstract we consider time periodic quasilinear parabolic equations in the domain t x r 2 0 x r t t 0 where the right boundary r t of the spatial interval is a positive function which might be periodic or asymptotically periodic or a function tending to infinity or infinity we show that in the first case that is r t is a periodic function any bounded solution of the equations converges as t to a periodic one in the other three cases any positive bounded solution converges as t to a nonnegative periodic one using such a result we study the long time dynamics of the initial boundary value problem on the half line as well as the stefan free boundary problem of a general heterogeneous reaction diffusion equation also we use the convergence result to study the long time dynamics of the initial boundary value problem for a time periodic mean curvature flow equation

existence and stability of time periodic solutions in a model for spherical flames with time periodic heat losses the problem under investigation is the study of the effect of periodicity when time periodic heat losses are added in a model for spherical flames introduced by g joulin in the present context the flame either quenches in finite or infinite time or its radius converges to an upper stable time periodic solution or to a lower unstable time periodic solution

orthogonal polynomials asymptotics and heun equations the painleve equations arise from the study of hankel determinants generated by moment matrices whose weights are expressed as the product of classical weights multiplied by suitable deformation factors usually dependent on a time variable t from ladder operators see a magnus j comput appl math 57 1 2 215 237 1995 one finds second order linear ordinary differential equations for associated orthogonal polynomials with coefficients being rational functions the painleve and related functions appear as the residues of these rational functions we will be interested in the situation when n the order of the hankel matrix and also the degree of the polynomials pn x orthogonal with respect to the deformed weights gets large we show that the second order linear differential equations satisfied by pn x are particular cases of heun equations when n is large in some sense monic orthogonal polynomials generated by deformed weights mentioned below are solutions of a variety of heun equations heun equations are of considerable importance in mathematical physics and in the special cases they degenerate to the hypergeometric and confluent hypergeometric equations in this paper we look at three types of weights the jacobi type the laguerre type and the weights deformed by the indicator function of a b x and the step function x in particular we consider the following jacobi type weights 1 1 x 1 x e tx x 0 1 t 0 1 2 x 1 x e t x x 0 1 t 0 1 3 1 x2 1 k2x2 x 1 1 0 k2 0 1 the laguerre type weights 2 1 x x t e x x 0 t 0 2 2 x e x t x x 0 t 0 and another type of deformation when the classical weights are multiplied by a b x or x 3 1 e x2 1 a a x x r a 0 3 2 1 x2 1 a a x x 1 1 a 0 1 0 3 3 x e x a b x t x 0 t 0 a 0 a b 0 the weights mentioned above were studied in a series of papers related to the deformation of classical weights

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

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

numbers of zeros on invariant manifolds in reaction diffusion equations on considere l equation a une dimension u t u xx f x u t 0 0 x 1 avec les conditions aux limites de dirichlet u t 0 t 1 0 ou f 0 1 r r est bc 1 c k k 1

on the affine heat equation for non convex curves 1998 american mathematical society first published in journal of the american mathematical society vol 11 no 3 july 1998 published by the american mathematical society

long time behavior of almost periodically forced parabolic equations on the circle abstract we study the long time behavior of the skew product semiflow generated by scalar reaction diffusion equation on the circle with almost periodic forcing u t u x x f t u u x t 0 x s 1 r 2 z where f t u u x is uniformly almost periodic in t almost periodic environmental forcing exhibits the external effects which are roughly but not exactly periodic contrary to the time periodic cases for which any limit set can be embedded into a periodically forced circle flow we show that for almost periodic forcing the problem that whether can be embedded into an almost periodically forced circle flow is strongly related to the dimension of the center space v c associated with on the one hand if dim v c 1 then is either spatially inhomogeneous or spatially homogeneous and moreover any spatially inhomogeneous can be embedded into an almost periodically forced circle flow on the other hand when dim v c 1 it is shown that the above embedding property cannot hold anymore these reveal that for such system there are essential differences between time periodic forcing and non periodic forcing

blow up of sign changing solutions for a one dimensional nonlinear diffusion equation abstract this paper is concerned with a spatially inhomogeneous semilinear diffusion equation on a bounded interval under the neumann or dirichlet boundary conditions assuming that the nonlinearity satisfies a rather general growth condition we consider the blow up and global existence of sign changing solutions it is shown that for some nonnegative integer k depending on the linearized operator at a trivial solution the solution blows up in finite time if an initial value changes its sign at most k times whereas there exist stationary solutions with more than k zeros the proof is based on the intersection number principle combined with a topological method

the eigenvalue equation on the eguchi hanson space we consider the eigenvalue equation for the laplace beltrami operator acting on scalar functions on the noncompact eguchi hanson space the corresponding differential equation is reducible to a confluent heun equation with ince symbol 0 2 12 we construct approximations for the eigenfunctions and their asymptotic scattering phases with the help of the liouville green approximation wkb furthermore for specific discrete eigenvalues obtained by a continued t fraction we construct the solution by the frobenius method and determine its scattering phase by a monodromy computation

instabilities in the chapman enskog expansion and hyperbolic burnett equations it is well known that the classical chapman enskog procedure does not work at the level of burnett equations the next step after the navier stokes equations roughly speaking the reason is that the solutions of higher equations of hydrodynamics burnett s etc become unstable with respect to short wave perturbations this problem was recently attacked by several authors who proposed different ways to deal with it we present in this paper one of possible alternatives first we deduce a criterion for hyperbolicity of burnett equations for the general molecular model and show that this criterion is not fulfilled in most typical cases then we discuss in more detail the problem of truncation of the chapman enskog expansion and show that the way of truncation is not unique the general idea of changes of coordinates based on analogy with the theory of dynamical systems leads finally to nonlinear hyperbolic burnett equations hbes without using any information beyond the classical burnett equations it is proved that hbes satisfy the linearized h theorem the linear version of the problem is studied in more detail the complete chapman enskog expansion is given for the linear case a simplified proof of the slemrod identity for burnett coefficients is also given

global blow up of a nonlinear heat equation solutions to semilinear parabolic equations of the form u t u f u x in which blow up at some finite time t are investigated for slowly growing functions f for nonlinearities such as f s 2 s ln 2 s 1 b with 0 u becomes infinite throughout as t t it is alsofound that for marginally more quickly growing functions e g f s 2 s ln 2 s 2 u is unbounded on some subset of which has positive measure and is unbounded throughout if is a small enough region

a symptotic behaviour for a numerical approximation of a parabolic problem with blowing up solutions abstract in this paper we study the asymptotic behaviour of a semidiscrete numerical approximation for u t u xx u p in a bounded interval 0 1 with dirichlet boundary conditions we focus in the behaviour of blowing up solutions we find that the blow up rate for the numerical scheme is the same as for the continuous problem also we find the blow up set for the numerical approximations and prove that it is contained in a neighbourhood of the blow up set of the continuous problem when the mesh parameter is small enough

asymptotic behavior of strongly monotone time periodic dynamical processes with symmetry given a strongly monotone discrete time dynamical system tn x x n z in an open and order convex subset x of a separable strongly ordered banach space v and a compact connected metrizable group g whose action on x is monotone and commutes with t we prove under some reasonable additional hypotheses that the limit set x of every stable point x x consists of symmetric points only i e g w w for all w x and g g moreover the set of all unstable points is contained in a union of at most countably many lipschitz manifolds of codimension one in v where each manifold is invariant under t and the action of g this result is applied to the time periodic spatially independent irreducible cooperative system of n reaction diffusion equations u t d t u f t u for t x r1 rn with spatially periodic boundary conditions in rn and with an initial distribution u0 which is continuous and satisfies the boundary conditions if u0 is stable then u0 contains spatially constant functions only and the dynamics on u0 is given by the irreducible cooperative system of n odes dudt f t u for t r1 here t is the poincare map if n 1 then u0 is a single fixed point of t if n 2 then v0 is a single fixed point of t for each v0 u0

sharp upper bound for lifespan of solutions to some critical semilinear parabolic dispersive and hyperbolic equations via a test function method abstract this paper is concerned with the blowup phenomena for initial boundary value problem 0 1 t 2 u x t u x t a x t u x t u x t p x t c 0 t u x t 0 x t c 0 t u x 0 e f x x c t u x 0 e g x x c where c is a cone like domain in r n n 2 defined as c int r r n r 0 with a connected open set in s n 1 with smooth boundary if n 1 then we only consider two cases c 0 and c r here a x is a non zero coefficient of t u which could be complex valued and space dependent is a fixed constant and e 0 is a small parameter the constants 0 1 switch the parabolicity and hyperbolicity of the problem 0 1 the result proposes an argument for sharp upper lifespan estimates of solutions to 0 1 by a test function method based on mitidieri and pokhozhaev 2001 the crucial idea is to introduce an ordinary differential inequality with a parameter as a variable

convergence to periodic solutions in autonomous reaction diffusion systems in one space dimension we consider the autonomous reaction diffusion system ddt uv uv f uv uv vu t 0 0 x 1 with dirichlet boundary conditions where f c1 r2 r and is positive we show a poincare bendixson theorem which means that all solutions converge either to zero or to a periodic orbit furthermore we determine stability properties of the periodic orbits

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

remark on upper bound for lifespan of solutions to semilinear evolution equations in a two dimensional exterior domain abstract in this paper we consider the following initial boundary value problem with the power type nonlinearity u p with 1 p 2 in a two dimensional exterior domain 0 1 t 2 u x t u x t e i t u x t u x t p x t 0 t u x t 0 x t 0 t u x 0 e f x x t u x 0 e g x x where is given by x r 2 x 1 2 2 c and 0 1 switches the parabolicity dispersivity and hyperbolicity remark that 2 1 2 n is well known as the fujita exponent if p 2 then there exists a small global in time solution of 0 1 for some initial data small enough see ikehata 11 and if p 2 then global in time solutions cannot exist for any positive initial data see ogawa takeda 22 and lai yin 14 the result is that for given initial data f g h 0 1 l 2 satisfying f g log x l 1 with some requirement the solution blows up at finite time and moreover the upper bound for lifespan of solutions to 0 1 is given as the following double exponential type when p 2 lifespan u exp exp c e 1 the crucial idea is to use test functions which approximates the harmonic function log x satisfying dirichlet boundary condition and the technique modified from 9

generalized spheroidal wave equation and limiting cases we find sets of solutions to the generalized spheroidal wave equation gswe or equivalently to the confluent heun equation each set is constituted by three solutions one given by a series of ascending powers of the independent variable and the others by series of regular and irregular confluent hypergeometric functions for a fixed set the solutions converge over different regions of the complex plane but present series coefficients proportional to each other these solutions for the gswe afford solutions to a double confluent heun equation by a taking limit process due to leaver e w leaver j math phys 27 1238 1986 another procedure called whittaker ince limit b d figueiredo j math phys 46 113503 2005 provides solutions in series of powers and bessel functions for two other equations with a different type of singularity at infinity in addition new solutions are obtained for the whittaker hill and mathieu equations f m arscott proc r soc edinburg a67 265 1967 b

convergence of solutions to the equation of quasi static approximation of viscoelasticity with capillarity the long time behavior of solutions to the quasi steady approximation of the equation of viscoelasticity with capillarity is investigated we consider the equation with a number of boundary conditions including such that the set of equilibria is necessarily at least one dimensional we prove that any solution converges to an equilibrium point we admit some kind of initial data with infinite energy we use tools based on the assumption that the nonlinear term is real analytic

moment closure hierarchies for kinetic theories this paper presents a systematicnonperturbative derivation of a hierarchy of closed systems of moment equations corresponding to any classical kinetic theory the first member of the hierarchy is the euler system which is based on maxwellian velocity distributions while the second member is based on nonisotropic gaussian velocity distributions the closure proceeds in two steps the first ensures that every member of the hierarchy is hyperbolic has an entropy and formally recovers the euler limit the second involves modifying the collisional terms so that members of the hierarchy beyound the second also recover the correct navier stokes behavior this is achieved through the introduction of a generalization of the bgk collision operator the simplest such system in three spatial dimensions is a 14 moment closure which also recovers the behavior of the grad 13 moment system when the velocity distributions lie near local maxwellians the closure procedure can be applied to a general class of kinetic theories

symmetry and convergence properties for non negative solutions of nonautonomous reaction diffusion problems nonautonomous parabolic equations of the form u t u f u t on a symmetric domain are considered using the moving hyperplane method it is proved that any bounded nonnegative solution symmetrises as t this is then used to show that for nonlinearities periodic in t any non negative bounded solution approaches a periodic solution

on a simple maximum principle technique applied to equations on the circle abstract for a class of general quasilinear equations on s 1 we show that by a very simple maximum principle technique as long as the solution stays finite all of its derivatives also remain finite some specific examples are given under suitable assumptions we also derive exponential decay of the derivatives of the solution

exponentially improved asymptotic solutions of ordinary differential equations ii irregular singularities of rank one re expansions are found for the optimal remainder terms in the well known asymptotic series solutions of homogeneous linear differential equations of the second order in the neighbourhood of an irregular singularity of rank one the re expansions are in terms of generalized exponential integrals and have greater regions of validity than the original expansions as well being considerably more accurate and providing a smooth interpretation of the stokes phenomenon they are also of strikingly simple form in addition explicit asymptotic expansions for the higher coefficients of the original asymptotic solutions are obtained

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

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

the dynamics of rotating waves in scalar reaction diffusion equations on etudie l attracteur compact maximal pour l equation u t u xx f u u x avec des conditions aux limites periodiques on montre qu un ensemble limite contient une onde de rotation

comparison of analysis techniques for the lattice boltzmann method we show that the chapman enskog expansion can be viewed as a special instance of a general expansion procedure which also encompasses other methods like the regular error expansion and multi scale techniques and that any two expansions which properly describe the lattice boltzmann solution necessarily coincide up to higher order terms for a model problem both the regular error expansion and the chapman enskog expansion are carried out it turns out that the classical chapman enskog method leads to an unstable equation at super burnett order in a parameter regime for which the underlying lattice boltzmann algorithm is stable however our approach naturally allows us to consider variants of the super burnett equation which do not suffer from instabilities the article concludes with a detailed comparison of the chapman enskog and the regular error expansion

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

moment closure approximations of the boltzmann equation based on varphi divergences this paper is concerned with approximations of the boltzmann equation based on the method of moments we propose a generalization of the setting of the moment closure problem from relative entropy to varphi divergences and a corresponding closure procedure based on minimization of varphi divergences the proposed description encapsulates as special cases grad s classical closure based on expansion in hermite polynomials and levermore s entropy based closure we establish that the generalization to divergence based closures enables the construction of extended thermodynamic theories that avoid essential limitations of the standard moment closure formulations such as inadmissibility of the approximate phase space distribution potential loss of hyperbolicity and singularity of flux functions at local equilibrium the divergence based closure leads to a hierarchy of tractable symmetric hyperbolic systems that retain the fundamental structural properties of the boltzmann equation

analytic solutions of some nonlinear diffusion equations on considere u t u u 2 x n u 0 avec des fonctions initiales non constantes est un domaine borne de r n a frontiere lisse

a limit of the confluent heun equation and the schrodinger equation for an inverted potential and for an electric dipole we re examine and extend a group of solutions in series of bessel functions for a limiting case of the confluent heun equation and then apply such solutions to the one dimensional schrodinger equation with an inverted quasiexactly solvable potential as well as to the angular equation for an electron in the field of a point electric dipole for the first problem we find finite and infinite series solutions which are convergent and bounded for any value of the independent variable for the angular equation we also find expansions in series of jacobi polynomials

classification of singularities for blowing up solutions in higher dimensions consider the cauchy problem p u t u u p when x r n t 0 n 1 u x 0 u 0 x when x r n where p 1 and u 0 x is a continuous nonnegative and bounded function it is known that under fairly general assumptions on u 0 x the unique solution of p u x t blows up in a finite time by which we mean that lim sup sup u x t t t x r n in this paper we shall assume that u x t blows up at x 0 t t and derive the possible asymptotic behaviours of u x t as x t 0 t under general assumptions on the blow up rate

model reduction of kinetic equations by operator projection by a further study of the mechanism of the hyperbolic regularization of the moment system for the boltzmann equation proposed in cai et al commun math sci 11 2 547 571 2013 we point out that the key point is treating the time and space derivative in the same way based on this understanding a uniform framework to derive globally hyperbolic moment systems from kinetic equations using an operator projection method is proposed the framework is so concise and clear that it can be treated as an algorithm with four inputs to derive hyperbolic moment systems by routine calculations almost all existing globally hyperbolic moment systems can be included in the framework as well as some new moment systems including globally hyperbolic regularized versions of grad s ordered moment systems and a multi dimensional extension of the quadrature based moment system

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

integral inequalities and global solutions of semilinear evolution equations a criterion for the nonexplosion of solutions to semilinear evolution equations on banach spaces is proved the result is obtained by applying a modification of the bihari type inequality to the case of a weakly singular nonlinear integral inequality

boundary conditions for regularized 13 moment equations for micro channel flows boundary conditions are the major obstacle in simulations based on advanced continuum models of rarefied and micro flows of gases in this paper we present a theory how to combine the regularized 13 moment equations derived from boltzmann s equation with boundary conditions obtained from maxwell s kinetic accommodation model while for the linear case these kinetic boundary conditions suffice we need additional conditions in the non linear case these are provided by the bulk solutions obtained after properly transforming the equations while keeping their asymptotic accuracy with respect to boltzmann s equation after finding a suitable set of boundary conditions and equations a numerical method for generic shear flow problems is formulated several test simulations demonstrate the stable and oscillation free performance of the new approach

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

critical exponent for the bipolar blowup in a semilinear parabolic equation abstract the blowup of solutions of the initial boundary value problem formula is studied where is a cone inrn we say that a solutionuexhibits bipolar blowup if formula for somet 1 for the bipolar blowup in the following sense if 1 p then the bipolar blowup does not occur for any sufficiently small initial data the value ofp is expressed in terms of the dimensionnand the second dirichlet eigenvalue of the laplace beltrami operator on sn 1 in the case of r we definek polar blowup and determine critical exponents for thek polar blowup fork 1 2 3

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

on the quenching behavior of the solution of a semilinear parabolic equation where 0 e 0 0 6 u0 0 there exists an e b 0 such that u quenches at finite time t if e e b we say that a is a quenching point for u if there exists a sequence x t with x a and t r t such that u x t 1 as n co throughout this paper we assume that u quenches at finite time t and that u0 is smooth and satisfies

static large deviations for a reaction diffusion model we consider the superposition of a symmetric simple exclusion dynamics speeded up in time with a spin flip dynamics in a one dimensional interval with periodic boundary conditions we prove the large deviations principle for the empirical measure under the stationary state we deduce from this result that the stationary state is concentrated on the stationary solutions of the hydrodynamic equation which are stable