about the regularizing properties of the non cut off kac equation we prove in this work that under suitable assumptions the solution of the spatially homogeneous non cut off kac equation or of the spatially homogeneous non cut off 2d boltzmann equation with maxwellian molecules in the radial case becomes very regular with respect to the velocity variable as soon as the time is strictly positive

a stochastic particle numerical method for 3d boltzmann equations without cutoff using the main ideas of tanaka the measure solution pt t of a 3 dimensional spatially homogeneous boltzmann equation of maxwellian molecules without cutoff is related to a poisson driven stochastic differential equation using this tool the convergence to pt t of solutions ptl t of approximating boltzmann equations with cutoff is proved then a result of graham meleard is used and allows us to approximate ptl t with the empirical measure tl n t of an easily simulable interacting particle system precise rates of convergence are given a numerical study lies at the end of the paper

asymptotic of grazing collisions and particle approximation for the kac equation without cutoff the subject of this article is the kac equation without cutoff we first show that in the asymptotic of grazing collisions the kac equation can be approximated by a fokker planck equation the convergence is uniform in time and we give an explicit rate of convergence next we replace the small collisions by a small diffusion term in order to approximate the solution of the kac equation and study the resulting error we finally build a system of stochastic particles undergoing collisions and diffusion that we can easily simulate which approximates the solution of the kac equation without cutoff we give some estimates on the rate of convergence

global structure of self similar solutions in a semilinear parabolic equation abstract the aim of this paper is to investigate the structure of radial solutions for a semilinear elliptic equation u y u 2 u p 1 u p 1u 0 y rn which is related to forward self similar solutions of a semilinear parabolic equation vt v v p 1v t x 0 rn we study the existence uniqueness and parameter dependency of rapidly decaying solutions with a prescribed number of zeros

structure of positive radial solutions to scalar field equations with harmonic potential abstract we study a nonlinear elliptic equation with a harmonic potential u x 2 u u p 1 u 0 x r n e which is related to standing waves for a nonlinear schrodinger equation the existence of positive solution for e in some energy space can be shown by the standard variational method under n 1 n and 1 p n 2 n 2 however the uniqueness is a delicate problem in this paper noting that all positive solutions of e in the energy space are radially symmetric about the origin we study the structure of positive radial solutions to e and show the uniqueness of positive solution with finite energy when n 3

global existence nonexistence and asymptotic behavior of solutions for the cauchy problem of semilinear heat equations abstract in this paper we study the cauchy problem of semilinear heat equations by introducing a family of potential wells we first prove the invariance of some sets and isolating solutions then we obtain a threshold result for the global existence and nonexistence of solutions finally we discuss the asymptotic behavior of the solution

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 class of high order nonlocal operators we study a class of nonlocal operators that may be seen as high order generalizations of the well known nonlocal diffusion operators we present properties of the associated nonlocal functionals and nonlocal function spaces including nonlocal versions of sobolev inequalities such as the nonlocal poincare and nonlocal gagliardo nirenberg inequalities nonlocal characterizations of high order sobolev spaces in the spirit of bourgain brezis mironescu are provided applications of nonlocal calculus of variations to the well posedness of linear nonlocal models of elastic beams and plates are also considered

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

pseudodifferential operator with a nonanalytic symbol on a half line we study the initial boundary value problem for the nonlinear pseudodifferential equations with nonanalytic symbol we study traditionally important problems of a theory of nonlinear partial differential equations such as global in time existence of solutions to the initial boundary value problem and the asymptotic behavior of solutions for large time

convergence from boltzmann to landau processes with soft potential and particle approximations our aim in this paper is to show how a probabilistic interpretation of the boltzmann and landau equations gives a microscopic understanding of these equations we firstly associate stochastic jump processes with the boltzmann equations we consider then we renormalize these equations following asymptotics which make prevail the grazing collisions and prove the convergence of the associated boltzmann jump processes to a diffusion process related to the landau equation the convergence is pathwise and also implies a convergence at the level of the partial differential equations the best feature of this approach is the microscopic understanding of the transition between the boltzmann and the landau equations by an accumulation of very small jumps we deduce from this interpretation an approximation result for a solution of the landau equation via colliding stochastic particle systems this result leads to a monte carlo algorithm for the simulation of solutions by a conservative particle method which enables to observe the transition from boltzmann to landau equations numerical results are given

asymptotically compatible schemes and applications to robust discretization of nonlocal models many problems in nature being characterized by a parameter are of interest both with a fixed parameter value and with the parameter approaching an asymptotic limit numerical schemes that are convergent in both regimes offer robust discretizations which can be highly desirable in practice the asymptotically compatible schemes studied in this paper meet such objectives for a class of parametrized problems an abstract mathematical framework is established rigorously here together with applications to the numerical solution of both nonlocal models and their local limits in particular the framework can be applied to nonlocal diffusion models and a general state based peridynamic system parametrized by the horizon radius recent findings have exposed the risks associated with some discretizations of nonlocal models when the horizon radius is proportional to the discretization parameter thus it is desirable to develop asymptotically compatible schemes for such models so as to offer robust numerical dis

front propagation into unstable states universal algebraic convergence towards uniformly translating pulled fronts fronts that start from a local perturbation and propagate into a linearly unstable state come in two classes pulled fronts and pushed fronts the term pulled front expresses that these fronts are pulled along by the spreading of linear perturbations about the unstable state accordingly their asymptotic speedv equals the spreading speed of perturbations whose dynamics is governed by the equations linearized about the unstable state the central result of this paper is the analysis of the convergence of asymptotically uniformly traveling pulled fronts towardsv we show that when such fronts evolve from sufficiently steep initial conditions which initially decay faster than e x forx 1 t hey have auniversal relaxation behavioras timet 1 the velocity of a pulled front always relaxes algebraically like v t d v 3 2t c 3 p d d 2 t 3 2 c o 1 t 2 the parametersv and d are determined through a saddle point analysis from the equation of motion linearized about the unstable invaded state this front velocity is independent of the precise value of the front amplitude which one tracks to measure the front position the interior of the front is essentially slaved to the leading edge and develops universally as x t d8v t x rt dt 0 v t 0 c o 1 t 2 where8v x vt is a uniformly translating front solution with velocity v v our result which can be viewed as a general center manifold result for pulled front propagation is derived in detail for the well known nonlinear diffusion equation of type t d 2 x c 3 where the invaded unstable state is d 0 even for this simple case the subdominant t 3 2 term extends an earlier result of bramson our analysis is then generalized to more general sets of partial differential equations with higher spatial or temporal derivatives to pdes with memory kernels and also to difference equations such as those that occur in numerical finite difference codes our universal result for pulled fronts thus implies independence i of the level curve which is used to track the front position ii of the precise nonlinearities iii of the precise form of the linear operators in the dynamical equation and iv of the precise initial conditions as long as they are sufficiently steep the only remainders of the explicit form of the dynamical equation are the nonlinear solutions 8v and the three saddle point parameters v and d as our simulations confirm all our analytical predictions in every detail it can be concluded that we have a complete analytical understanding of the propagation mechanism and relaxation behavior of pulled fronts if they are uniformly translating for t 1 an immediate consequence of the slow algebraic relaxation is that the standard moving boundary approximation breaks down for weakly curved pulled fronts in two or three dimensions in addition to our main result for pulled fronts we also discuss the propagation and convergence of fronts emerging from

regularity for a local nonlocal transmission problem we formulate and study an elliptic transmission like problem combining local and nonlocal elements let mathbb r n be separated into two components by a smooth hypersurface on one side of a function satisfies a local second order elliptic equation on the other it satisfies a nonlocal one of lower order in addition a nonlocal transmission condition is imposed across the interface which has a natural variational interpretation we deduce the existence of solutions to the corresponding dirichlet problem and show that under mild assumptions they are holder continuous following the method of de giorgi the principal difficulty stems from the lack of scale invariance near which we circumvent by deducing a special energy estimate which is uniform in the scaling parameter we then turn to the question of optimal regularity and qualitative properties of solutions and show that in the case of constant coefficients and flat they satisfy a kind of transmission condition where the ratio of fractional conormal derivatives on the two sides of the interface is fixed a perturbative argument is then given to show how to obtain regularity for solutions to an equation with variable coefficients and smooth interface throughout we pay special attention to a nonlinear version of the problem with a drift given by a singular integral operator of the solution which has an interpretation in the context of quasigeostrophic dynamics

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

an extension of dickstein s small lambda theorem for finite time blowup abstract in this paper we prove for 1 p 1 1 n 1 l 1 r n with r n 0 and c 0 r n w 1 1 r n with r n 0 such that j that there exists 0 such that the solution u of the equation u t u u p 1 u with u 0 blows up in finite time for all 0 this extends a similar result of dickstein who treated the case r n 0 and 1 p 1 2 n

radial symmetry of self similar solutions for semilinear heat equations abstract the symmetry properties of positive solutions of the equation u 1 2 x u 1 p 1 u u p 0 in r n where n 2 p n 2 n was studied it was proved that u must be radially symmetric about the origin provided u x o x 2 p 1 as x and that there exist non radial solutions u satisfying lim sup x x 2 p 1 u x 0

sign changing stationary solutions and blowup for the nonlinear heat equation in a ball in this paper we prove that if phi is a radially symmetric sign changing stationary solution of the nonlinear heat equation u t delta u u alpha u quad quad quad quad quad in the unit ball of mathbb r n n 3 with dirichlet boundary conditions then the solution of nlh with initial value lambda phi blows up in finite time if 1 0 is sufficiently small and if is subcritical and sufficiently close to 4 n 2

a posteriori error analysis of finite element method for linear nonlocal diffusion and peridynamic models abstract in this paper we present some results on a posteriori error analysis of finite element methods for solving linear nonlocal diffusion and bond based peridynamic models in particular we aim to propose a general abstract frame work for a posteriori error analysis of the peridynamic problems a posteriori error estimators are consequently prompted the reliability and efficiency of the estimators are proved connections between nonlocal a posteriori error estimation and classical local estimation are studied within continuous finite element space numerical experiments 1d are also given to test the theoretical conclusions

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

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

on a l functional derivative estimate relating to the cauchy problem for scalar semi linear parabolic partial differential equations with general continuous nonlinearity in this paper we consider a l infty functional derivative estimate for the first spatial derivative of bounded classical solutions u mathbb r times 0 t to mathbb r to the cauchy problem for scalar semi linear parabolic partial differential equations with a continuous nonlinearity f mathbb r to mathbb r and initial data u 0 mathbb r to mathbb r of the form sup x in mathbb r u x x t leq mathcal f t f u 0 u forall t in 0 t here mathcal f t mathcal a t to mathbb r is a functional as defined in textsection 1 we establish that the functional derivative estimate is non trivially sharp by constructing a sequence f n 0 u n where for each n in mathbb n u n mathbb r times 0 t to mathbb r is a solution to the cauchy problem with zero initial data and nonlinearity f n mathbb r to mathbb r and for which sup x in mathbb r u x n x t geq alpha 0 with lim n to infty left inf t in 0 t left sup x in mathbb r u x n cdot t mathcal f t f n 0 u n right right 0

an asymptotically compatible formulation for local to nonlocal coupling problems without overlapping regions abstract in this paper we design and analyze an explicit partitioned procedure for a 2d dynamic local to nonlocal ltn coupling problem based on a new nonlocal robin type transmission condition the nonlocal subproblem is modeled by the nonlocal heat equation with a finite horizon parameter characterizing the range of nonlocal interactions and the local subproblem is described by the classical heat equation we consider a heterogeneous system where the local and nonlocal subproblems present different physical properties and employ no overlapping region between the two subdomains we first propose a new generalization of classical local neumann type condition by converting the local flux to a correction term in the nonlocal model and show that the proposed neumann type boundary formulation recovers the local case as o 2 in the l norm we then extend the nonlocal neumann type boundary condition to a robin type boundary condition and develop a local to nonlocal coupling formulation with robin dirichlet transmission conditions to stabilize the explicit coupling procedure and to achieve asymptotic compatibility the choice of the coefficient in the robin condition is obtained via amplification factor analysis for the discretized system with coarse grids employing a high order meshfree discretization method in the nonlocal solver and a linear finite element method in the local solver the selection of optimal robin coefficients are verified with numerical experiments on heterogeneous and complicated domains with the developed optimal coupling strategy we numerically demonstrate the coupling framework s asymptotic convergence to the local limit with an o o h rate when there is a fixed ratio between the horizon size and the spatial discretization size h

the evolution of travelling waves in the weakly hyperbolic generalized fisher model in this paper we consider an initial boundary value problem for a weakly hyperbolic generalized fisher equation of order p 0 we show that for each p 0 the large time structure of the solution involves the evolution of a propagating wavefront however we demonstrate that the case p 1 is a bifurcation point in the sense that for p 1 the wavefront is of reaction diffusion type while for 0 p

growup of solutions for a semilinear heat equation with supercritical nonlinearity abstract we consider a cauchy problem for a semilinear heat equation p u t u u p in r n 0 u x 0 u 0 x 0 in r n let v be the radially symmetric singular steady state of p it is proved that if p n 2 n 1 n 4 2 n 1 and n 11 then for each nonnegative even integer n there exists a radially symmetric global solution u n of p with n intersections with v such that t a n u n t 1 as t for some a n 0 depending on n the exact value of a n is also given we show that a 0 is the optimal upper bound of growup rate for solutions below v

on a singular initial boundary value problem for a reaction diffusion equation arising from a simple model of isothermal chemical autocatalysis in this paper we examine the evolution that occurs when a localized input of an autocatalyst b is introduced into an expanse of a reactant a the reaction is autocatalytic of order p so a rightarrow b at rate k a b p with rate constant k we examine the case when 0 p 1 with p geq 1 having been examined by needham merkin phil trans r soc lond a 337 261 274 1991 in particular we show that the fully reacted state is not achieved as t rightarrow infty via the propagation of a travelling wavefront as for p geq 1 but is approached uniformly in space as t rightarrow infty

existence and regularity of a solution of a kac equation without cutoff using the stochastic calculus of variations we consider a generalized kac equation without cutoff with which we associate a non standard nonlinear stochastic differential equation we adapt the techniques in bichteler and jacod 2 to prove that the law of a solution of the stochastic differential equation has a density which is a solution of the kac equation the initial law is very general we only assume it has second order moments and is not the dirac mass at 0 we thus generalize the analytical results of existence of a solution of this equation if we furthermore assume existence of all moments for the initial law we obtain as a corollary using the proof in desvillettes 6 that the density is smooth we prove a slightly better regularity result under more stringent assumptions using the stochastic calculus of variations adapting the methods in 1

regularization properties of the 2d homogeneous boltzmann equation without cutoff we consider the 2 dimensional spatially homogeneous boltzmann equation for hard potentials we assume that the initial condition is a probability measure that has some exponential moments and is not a dirac mass we prove some regularization properties for a class of very hard potentials the solution instantaneously belongs to hr for some r in 1 2 depending on the parameters of the equation our proof relies on the use of a well suited malliavin calculus for jump processes

nonlinear evolution equations with a fractional derivative on a half line abstract we study the initial boundary value problem on a half line for the nonlinear evolution equations with a fractional derivative we study traditionally important problems of a theory of nonlinear partial differential equations such as global in time existence of solutions to the initial boundary value problem and the asymptotic behavior of solutions for large time

non uniqueness of solutions to the cauchy problem for semilinear heat equations with singular initial data the cauchy problem for semilinear heat equations with singular initial data is studied where n 2 0 is a parameter and a 0 a 0 we show that when p n 2 n and n 2 p n 2 there exists a positive constant such that the problem has two positive self similar solutions and with if and no positive self similar solutions if furthermore for each fixed and in l rn as 0 where w0 is a non unique solution to the problem with zero initial data which is constructed by haraux and weissler

probability metrics and uniqueness of the solution to the boltzmann equation for a maxwell gas we consider a metric for probability densities with finite variance on d and compare it with other metrics we use it for several applications both in probability and in kinetic theory the main application in kinetic theory is a uniqueness result for the solution of the spatially homogeneous boltzmann equation for a gas of true maxwell molecules

travelling wave solutions to monostable reaction diffusion systems of mixed monotone type theorems concerning the existence and stability of travelling wave solutions to mono stable reaction diffusion systems of mixed monotone type are established these results are applied to an autocatalytic system which is a particular case in the class of monostable systems of mixed monotone type a series of numerical experiments is conducted to simulate formation and propagation of waves

critical exponents of fujita type for inhomogeneous parabolic equations and systems abstract we consider the large time behavior of sign changing solutions of inhomogeneous parabolic equations and systems for example for u t u u p w x in r n 0 t we show the following if n 3 and r n w x dx 0 and 1 p n n 2 then all solutions blow up in finite time while if p n n 2 there are both global and nonglobal solutions we show by example that global solutions exist for all p 1 and w satisfying r n w x dx n 1 2 and r n w x dx 0 no solution can exist for all time extensions of the above result to various geometries and some other problems are indicated

the evolution of travelling waves in fractional order autocatalysis with decay ii the initial boundary value problem in this paper we examine an initial boundary value problem for a singular reaction diffusion equation arising from a model of isothermal chemical autocatalysis with termination we consider the case where the reaction and termination orders m and n respectively are fractional that is 0 0 k being a parameter which measures the relative strength of the termination step to that of the autocatalysis we show that solutions approach the equilibrium state 1 k for k 1 we demonstrate that the problem exhibits solutions which decay to zero with contracting support in finite t finally for m n we show that traveling waves with semi infinite support occ

the asymptotic critical wave speed in a family of scalar reaction diffusion equations abstract we study traveling wave solutions for the class of scalar reaction diffusion equations u t 2 u x 2 f m u where the family of potential functions f m is given by f m u 2 u m 1 u for each m 1 real there is a critical wave speed c crit m that separates waves of exponential structure from those which decay only algebraically we derive a rigorous asymptotic expansion for c crit m in the limit as m this expansion also seems to provide a useful approximation to c crit m over a wide range of m values moreover we prove that c crit m is c smooth as a function of m 1 our analysis relies on geometric singular perturbation theory as well as on the blow up technique and confirms the results obtained by means of asymptotic methods in d j needham a n barnes reaction diffusion and phase waves occurring in a class of scalar reaction diffusion equations nonlinearity 12 1 1999 41 58 t p witelski k ono t j kaper critical wave speeds for a family of scalar reaction diffusion equations appl math lett 14 1 2001 65 73

the evolution to localized and front solutions in a non lipschitz reaction diffusion cauchy problem with trivial initial data abstract in this paper we establish the existence of spatially inhomogeneous classical self similar solutions to a non lipschitz semi linear parabolic cauchy problem with trivial initial data specifically we consider bounded solutions to an associated two dimensional non lipschitz non autonomous dynamical system for which we establish the existence of a two parameter family of homoclinic connections on the origin and a heteroclinic connection between two equilibrium points additionally we obtain bounds and estimates on the rate of convergence of the homoclinic connections to the origin

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

asymptotic behavior and attractors for reaction diffusion equations in unbounded domains abstract in this paper we give general and flexible conditions for a reaction diffusion equation to be dissipative in an unbounded domain the functional setting is based on standard lebesgue and sobolev lebesgue spaces we show how the reaction and diffusion mechanisms have to work together to obtain the asymptotic compactness of solutions and therefore the existence of the compact attractor in particular cases our results allow us to improve some previous known results

a pure jump markov process associated with smoluchowski s coagulation equation the aim of the present paper is to construct a stochastic process whose law is the solution of the smoluchowski s coagulation equation we introduce first a modified equation dealing with the evolution of the distribu tion q t dx of the mass in the system the advantage we take on this is that we can perform an unified study for both continuous and discrete models the integro partial differential equation satisfied by q t t 0 can be interpreted as the evolution equation of the time marginals of a markov pure jump process at this end we introduce a nonlinear poisson driven stochastic differential equation related to the smoluchowski equation in the following way if x t satisfies this stochastic equation then the law of x t satisfies the modified smoluchowski equation the nonlinear process is richer than the smoluchowski equation since it provides historical information on the particles existence uniqueness and pathwise behavior for the solution of this sde are studied finally we prove that the nonlinear process x can be obtained as the limit of a marcus lushnikov procedure 1 introduction the coagulation model governs various phenomena as for example polymerization aggregation of colloidal particles formation of stars and planets behavior of fuel mixtures in engines etc smoluchowski s coagulation equation models the dynamic of such phenomena and describes the evolution of a system of clusters which coalesce in order to form bigger clusters each cluster is identified by its size the only mechanism taken into account is the coalescence of two clusters other effects as multiple coagulation are neglected we assume also that the rate of these reactions depends on the sizes of clusters involved in the coagulation denoting by n k t the nonnegative concentration of clusters of size k at time t the discrete

the evolution of travelling waves in fractional order autocatalysis with decay i permanent form travelling waves this paper considers a model isothermal autocatalytic reaction scheme with termination when the reaction orders m n are fractional i e 0 n and k in 0 k c where kc depends upon m and n it is further established that these wavefronts are of excitable rather than fisher kolmogorov type and have only semi infinite support that is the reaction is totally dormant ahead of the wavefront these features are in contrast to the case when m n geq 1 as studied by merkin needham and scott proc roy soc london ser a 424 1989 pp 187 209 merkin and needham j engrg math 23 1989 pp 343 356 and needham and merkin phil trans roy soc london ser a 337 1991 pp 261 274 it should be noted that the corresponding reaction diffusion equations are singular in the sense that the reaction and decay terms are not lipsch

critical exponents for the decay rate of solutions in a semilinear parabolic equation this paper is concerned with the cauchy problem a solution u is said to decay fast if t 1 p 1 u rightarrow 0 as t rightarrow infty uniformly in r and is said to decay slowly otherwise for each nonnegative integer k let lambda k be the set of uniformly bounded functions on r which change sign k times and let p k 1 be defined by p k 1 2 k 1 it is shown that any nontrivial bounded solution with u 0 in lambda k decays slowly if 1 p p k whereas there exists a nontrivial fast decaying solution with u 0 in lambda k if p gep k

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

linear behaviour of solutions of a superlinear heat equation we give sufficient conditions which guarantee that solutions of a superlinear heat equation decay to zero at the same rate as the solutions of the linear heat equation with the same initial data this improves a previous result of lee and ni by showing that this behaviour holds for a significantly larger and rather tightly defined class of solutions

variational problems related to self similar solutions of the heat equation on demontre la compacite de certains plongements de sobolev sous de bonnes conditions sur les poids k x exp x c 2 r n r on considere des methodes variationnelles pour la resolution d equations elliptiques

on the cauchy problem for boltzmann equations global existence and weak stability we study the large data cauchy problem for boltzmann equations with general collision kernels we prove that sequences of solutions which satisfy only the physically natural a priori bounds converge weakly in l to a solution from this stability result we deduce global existence of a solution to the cauchy problem our method relies upon recent compactness results for velocity averages a new formulation of the boltzmann equation which involves nonlinear normalization and an analysis of subsolutions and supersolutions it allows us to overcome the lack of strong a priori estimates and define a meaningful collision operator for general configurations

coupling of nonlocal and local continuum models by the arlequin approach summary the objective of this work is to develop and apply the arlequin framework to couple nonlocal and local continuum mechanical models a mechanically based model of nonlocal elasticity which involves both contact and long range forces is used for the fine scale description in which nonlocal interactions are considered to have non negligible effects classical continuum mechanics only involving local contact forces is introduced for the rest of the structure where these nonlocal effects can be neglected both models overlap in a coupling subdomain called the gluing area in which the total energy is separated into nonlocal and local contributions by complementary weight functions a weak compatibility is ensured between kinematics of both models using lagrange multipliers over the gluing area the discrete formulation of this specific arlequin coupling framework is derived and fully described the validity and limits of the technique are demonstrated through two dimensional numerical applications and results are compared against those of the fully nonlocal elasticity method copyright 2011 john wiley sons ltd

a markov process associated with a boltzmann equation without cutoff and for non maxwell molecules tanaka 18 showed a way to relate the measure solution pt t of a spatially homogeneous boltzmann equation of maxwellian molecules without angular cutoff to a poisson driven stochastic differential equation pt is the flow of time marginals of the solution of this stochastic equation in the present paper we extend this probabilistic interpretation to much more general spatially homogeneous boltzmann equations then we derive from this interpretation a numerical method for the concerned boltzmann equations by using easily simulable interacting particle systems

gelfand shilov and gevrey smoothing effect for the spatially inhomogeneous non cutoff kac equation abstract we consider the spatially inhomogeneous non cutoff kac s model of the boltzmann equation we prove that the cauchy problem for the fluctuation around the maxwellian distribution enjoys gelfand shilov regularizing properties with respect to the velocity variable and gevrey regularizing properties with respect to the position variable

quasi nonlocal coupling of nonlocal diffusions we developed a new self adjoint consistent and stable coupling strategy for nonlocal diffusion models inspired by the quasi nonlocal atomistic to continuum method for crystalline solids the proposed coupling model is coercive with respect to the energy norms induced by the nonlocal diffusion kernels as well as the l 2 norm and it satisfies the maximum principle a finite difference approximation is used to discretize the coupled system which inherits the property from the continuous formulation furthermore we design a numerical example that shows the discrepancy between the fully nonlocal and fully local diffusions whereas the result of the coupled diffusion agrees with that of the fully nonlocal diffusion

extended weak maximum principles for parabolic partial differential inequalities on unbounded domains in this paper we establish extended maximum principles for solutions to linear parabolic partial differential inequalities on unbounded domains where the solutions satisfy a variety of growth dec

global similarity solutions to a class of semilinear parabolic equations existence bifurcations and asymptotics this paper examines the existence structure and bifurcation in the parameters n and p of a class of global positive similarity solutions to the semilinear parabolic equation u t n 2 u u p with p 1 and n 1 here n 2 is the n dimensional laplace operator these are related to the existence of geometric threshold in the p th order autocatalytic or chain branching reactions

well posedness and qualitative behaviour of a semi linear parabolic cauchy problem arising from a generic model for fractional order autocatalysis in this paper we examine a semi linear parabolic cauchy problem with non lipschitz nonlinearity which arises as a generic form in a significant number of applications specifically we obtain a well posedness result and examine the qualitative structure of the solution in detail the standard classical approach to establishing well posedness is precluded owing to the lack of lipschitz continuity for the nonlinearity here existence and uniqueness of solutions is established via the recently developed generic approach to this class of problem meyer needham 2015 the cauchy problem for non lipschitz semi linear parabolic partial differential equations london mathematical society lecture note series vol 419 which examines the difference of the maximal and minimal solutions to the problem from this uniqueness result the approach of meyer needham allows for development of a comparison result which is then used to exhibit global continuous dependence of solutions to the problem on a suitable initial dataset the comparison and continuous dependence results obtained here are novel to this class of problem this class of problem arises specifically in the study of a one step autocatalytic reaction which is schematically given by a b at rate apbq where a and b are the concentrations of a and b respectively with 0 p q 1 and well posedness for this problem has been lacking up to the present

on two signed solutions to a second order semi linear parabolic partial differential equation with non lipschitz nonlinearity abstract in this paper we establish the existence of a 1 parameter family of spatially inhomogeneous radially symmetric classical self similar solutions to a cauchy problem for a semi linear parabolic pde with non lipschitz nonlinearity and trivial initial data specifically we establish well posedness for an associated initial value problem for a singular two dimensional non autonomous dynamical system with non lipschitz nonlinearity additionally we establish that solutions to the initial value problem converge algebraically to the origin and oscillate as

on a decay property of solutions to the haraux weissler equation abstract we give a sufficient condition that non radial h 1 solutions to the haraux weissler equation should belong to the weighted sobolev space h 1 r n where is the weight function exp x 2 4 our result provides in some sense a connection between the solutions obtained by ode method and those by variational approach in the space h 1 r n

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

global existence and blowup for sign changing solutions of the nonlinear heat equation abstract in this paper given 0 2 n we prove the existence of a function with the following properties the solution of the equation u t u u u on r n with the initial condition u 0 is global on the other hand the solution with the initial condition u 0 blows up in finite time if 0 is either sufficiently small or sufficiently large