harnack inequalities for solutions of general second order parabolic equations and estimates of their holder constants on etudie les solutions u des equations lineaires lu d 0 u i j n a ij d i d j u i 1 n bid i u cu f et des equations non lineaires mu d 0 u i j n a ij d 0 d j u a u du 0 dans l espace de sobolev w 1 2 n 1 1 2 q on deduit une inegalite de harnack modifiee pour les solutions

multigrid solution of the two dimensional euler equations on unstructured triangular meshes a multigrid algorithm has been developed to accelerate the convergence of the euler equations to a steady state on unstructured triangular meshes the method operates on a sequence of coarse and fine unstructured meshes and assumes no relation exists between the various meshes a tree search algorithm is used to efficiently identify regions of overlap between coarse and fine grid cells the multigrid method is applied to two different discretizations of the euler equations and results about multi element airfoils are presented for both discretization schemes convergence is accelerated by an order of magnitude resulting in convergence rates comparable to those obtained with structured multigrid algorithms

entire solutions of time periodic fisher kpp equation on the half line abstract we study the entire solutions of the fisher kpp equation of the form u t u x x f t u on 0 with boundary condition u t 0 0 where f t u is a fisher kpp type of nonlinearity t periodic in t we obtain two types of entire solutions i for any c 2 a with a 1 t 0 t f u s 0 d s we obtain an entire solution u c t x connecting the periodic traveling wave solution c t x c t at t with the periodic solution v t x at t ii the second type of entire solution u t x connecting the solution of the initial problems of ode t f t at t and v t x at t with properties u x x 0 and u t p t where p t is the periodic solution of t f t

speed up of combustion fronts in shear flows this paper is concerned with the analysis of speed up of reaction diffusion advection traveling fronts in infinite cylinders with periodic boundary conditions the advection is a shear flow with a large amplitude and the reaction is nonnegative with either positive or zero ignition temperature the unique or minimal speeds of the traveling fronts are proved to be asymptotically linear in the flow amplitude as the latter goes to infinity solving an open problem from berestycki nonlinear pdes in condensed matter and reactive flows kluwer doordrecht 2003 the asymptotic growth rate is characterized explicitly as the unique or minimal speed of traveling fronts for a limiting degenerate problem and the convergence of the regular traveling fronts to the degenerate ones is proved for positive ignition temperatures under an additional hormander type condition on the flow

entire solutions of reaction advection diffusion equations with bistable nonlinearity in cylinders this paper deals with entire solutions and the interaction of traveling wave fronts of bistable reaction advection diffusion equation with infinite cylinders assume that the equation admits three equilibria two stable equilibria 0 and 1 and an unstable equilibrium it is well known that there are different wave fronts connecting any two of those three equilibria by considering a combination of any two of those different traveling wave fronts and constructing appropriate subsolutions and supersolutions we establish three different types of entire solutions finally we analyze a model for shear flows in cylinders to illustrate our main results

curved fronts in the belousov zhabotinskii reaction diffusion systems in r 2 in this paper we consider a diffusion system with the belousov zhabotinskii bz for short chemical reaction following brazhnik and tyson 4 and perez munuzuri et al 45 who predicted v shaped fronts theoretically and discovered v shaped fronts by experiments respectively we give a rigorous mathematical proof of their results we establish the existence of v shaped traveling fronts in r 2 by constructing a proper supersolution and a subsolution furthermore we establish the stability of the v shaped front in r 2 c 2018 elsevier inc all rights reserved

entire solutions in bistable reaction diffusion equations with nonlocal delayed nonlinearity this paper is concerned with entire solutions for bistable reaction diffusion equations with nonlocal delay in one dimensional spatial domain here the entire solutions are defined in the whole space and for all time t r assuming that the equation has an increasing traveling wave solution with nonzero wave speed and using the comparison argument we prove the existence of entire solutions which behave as two traveling wave solutions coming from both ends of the x axis and annihilating at a finite time furthermore we show that such an entire solution is unique up to space time translations and is liapunov stable a key idea is to characterize the asymptotic behavior of the solutions as t in terms of appropriate subsolutions and supersolutions in order to illustrate our main results two models of reaction diffusion equations with nonlocal delay arising from mathematical biology are considered

travelling fronts and entire solutions of the fisher kpp equation in ℝn this paper is devoted to time global solutions of the fisher kpp equation in n where f is a c 2 concave function on 0 1 such that f 0 f 1 0 and f 0 on 0 1 it is well known that this equation admits a finite dimensional manifold of planar travelling fronts solutions by considering the mixing of any density of travelling fronts we prove the existence of an infinite dimensional manifold of solutions in particular there are infinite dimensional manifolds of nonplanar travelling fronts and radial solutions furthermore up to an additional assumption a given solution u can be represented in terms of such a mixing of travelling fronts

entire solutions in a time delayed and diffusive epidemic model this paper is concerned with entire solutions in the following time delayed and diffusive epidemic model email protected tu 1 x t d email protected u 1 x t a 1 1u 1 x t a 1 2u 2 x t email protected tu 2 x t d email protected u 2 x t a 2 2u 2 x t gu 1x email protected where d i 0 i 1 2 using comparison arguments we establish the existence and its properties of entire solutions for the above system here an entire solution means a classical solution defined for all email protected r

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

nonplanar traveling fronts in reaction diffusion equations with combustion and degenerate fisher kpp nonlinearities abstract this paper is concerned with nonplanar traveling fronts in reaction diffusion equations with combustion nonlinearity and degenerate fisher kpp nonlinearity our study contains two parts in the first part we establish the existence of traveling fronts of pyramidal shape in r 3 and in the second part we establish the existence and stability of v shaped traveling fronts in r 2

stability of travelling fronts in a model for flame propagation part i linear analysis we investigate the stability of travelling wave solutions of the multidimensional thermodiffusive model for flame propagation with unit lewis number this model consists in a system of two nonlinear parabolic equations posed in an infinite cylinder with neumann boundary conditions in this paper we prove that every travelling wave solution of this model is linearly stable our tools are exponential decay estimates for solutions of elliptic equations in a cylinder and the maximum principle for parabolic equations

time integration algorithms for the two dimensional euler equations on unstructured meshes explicit and implicit time integration algorithms for the two dimensional euler equations on unstructured grids are presented both cell centered and cell vertex finite volume upwind schemes utilizing roe s approximate riemann solver are developed for the cell vertex scheme a four stage runge kutta time integration a four stage runge kutta time integration with implicit residual averaging a point jacobi method a symmetric point gauss seidel method and two methods utilizing preconditioned sparse matrix solvers are presented for the cell centered scheme a runge kutta scheme an implicit tridiagonal relaxation scheme modeled after line gaussseidel a fully implicit lower upper lu decomposition and a hybrid scheme utilizing both runge kutta and lu methods are presented

the uniqueness and asymptotic stability of pyramidal traveling fronts in the allen cahn equations abstract this paper studies the uniqueness and the asymptotic stability of a pyramidal traveling front in the three dimensional whole space for a given admissible pyramid we prove that a pyramidal traveling front is uniquely determined and that it is asymptotically stable under the condition that given perturbations decay at infinity for this purpose we characterize the pyramidal traveling front as a combination of planar fronts on the lateral surfaces moreover we characterize the pyramidal traveling front in another way that is we write it as a combination of two dimensional v form waves on the edges this characterization also uniquely determines a pyramidal traveling front

existence and convergence to a propagating terrace in one dimensional reaction diffusion equations we consider one dimensional reaction diffusion equations for a large class of spatially periodic nonlinearities including multistable ones and study the asymptotic behavior of solutions with heaviside type initial data our analysis reveals some new dynamics where the profile of the propagation is not characterized by a single front but by a layer of several fronts which we call a terrace existence and convergence to such a terrace is proven by using an intersection number argument without much relying on standard linear analysis hence on top of the peculiar phenomenon of propagation that our work highlights several corollaries will follow on the existence and convergence to pulsating traveling fronts even for highly degenerate nonlinearities that have not been treated before

location of the travelling wave for the kolmogorov equation under appropriate initial data solutions of the kolmogorov equation u t tfrac 1 2 u xx f u converge to travelling waves w x 21 2 as t in the case 21 2 a general formula for the asymptotic position of the wave is known as are formulas for certain cases of 21 2 the general formula for the case 21 2 given in 4 involves the behavior of the solution at earlier times and is typically not explicit enough for computation here we give an improvement of this formula not requiring such information the methodology involves use of sample path estimates for brownian bridge and manipulation of certain formulas from 4

entire solutions originating from multiple fronts of an epidemic model with nonlocal dispersal and bistable nonlinearity abstract this paper is concerned with the entire solutions of a nonlocal dispersal epidemic model which arises from the spread of fecally orally transmitted diseases under bistable assumptions it is well known that this model has three different types of traveling wave fronts the annihilating front and merging front entire solutions originating from two fronts of the system have also been constructed in 38 we first prove the uniqueness liapunov stability and continuous dependence on shift parameters of annihilating front entire solutions obtained in 38 a positive time derivative estimate for such entire solution is also obtained then we establish the existence of two different types of entire solutions merging three different fronts furthermore we show that these entire solutions are global lipschitz continuous with respect to the spatial variable x to the best of our knowledge it is the first time that the entire solutions originating from three fronts of diffusion systems have been constructed

transition semi wave solutions of reaction diffusion equations with free boundaries abstract in this paper we define the transition semi wave solution cf definition 1 1 of the following reaction diffusion equation with free boundaries 1 u t u x x f t x u t r x h t u t h t 0 t r h t u x t h t t r in the homogeneous case i e f t x u f u under the hypothesis f u c 1 0 1 f 0 f 1 0 f 1 0 f u 0 for u 1 we prove that the semi wave connecting 1 and 0 of 1 is unique provided it exists furthermore we prove that any bounded transition semi wave connecting 1 and 0 is exactly the semi wave in the cases where f is kpp fisher type and almost periodic in time space i e f t x u u c t u resp u a x u with c t resp a x being almost periodic applying totally different method we also prove any bounded transition semi wave connecting the unique almost periodic positive solution of u t u c t u resp u x x u a x u 0 and 0 is exactly the unique almost periodic semi wave of 1 finally we provide an example of the heterogeneous equation to show the existence of the transition semi wave without any global mean speeds

spreading vanishing dichotomy in a diffusive logistic model with a free boundary ii abstract we study the diffusive logistic equation with a free boundary in higher space dimensions and heterogeneous environment such a model may be used to describe the spreading of a new or invasive species with the free boundary representing the expanding front for simplicity we assume that the environment and the solution are radially symmetric in the special case of one space dimension and homogeneous environment this free boundary problem was investigated in du and lin 2010 10 we prove that the spreading vanishing dichotomy established in du and lin 2010 10 still holds in the more general and ecologically realistic setting considered here moreover when spreading occurs we obtain best possible upper and lower bounds for the spreading speed of the expanding front when the environment is asymptotically homogeneous at infinity these two bounds coincide our results indicate that the asymptotic spreading speed determined by this model does not depend on the spatial dimension

entire solutions for a multi type sis nonlocal epidemic model in r or z abstract in this paper we study entire solutions for a multi type sis nonlocal epidemic model in r or z the existence and asymptotic behavior of spatially independent solutions are first established some new entire solutions are then constructed by combining traveling wave fronts with different speeds and a spatially independent solution from the viewpoint of epidemiology the results provide some new spread ways of the epidemic

existence and uniqueness of entire solutions for a reaction diffusion equation abstract we consider entire solutions of u t u xx f u i e solutions that exist for all x t r 2 where f 0 f 1 0 f 0 in particular we are interested in the entire solutions which behave as two opposite wave fronts of positive speed s approaching each other from both sides of the x axis and then annihilating in a finite time in the case f 1 0 we show that such entire solution exists and is unique up to space time translations in the case f 1 0 we derive two families of such entire solutions in the first family one cannot be any space time translation of the other yet all entire solutions in the second family only differ by a space time translation

existence and global stability of traveling curved fronts in the allen cahn equations abstract this paper is concerned with existence and stability of traveling curved fronts for the allen cahn equation in the two dimensional space by using the supersolution and the subsolution we construct a traveling curved front and show that it is the unique traveling wave solution between them our supersolution can be taken arbitrarily large which implies some global asymptotic stability for the traveling curved front

entire solutions of nonlocal dispersal equations with monostable nonlinearity in space periodic habitats abstract this paper is concerned with the new types of entire solutions other than traveling wave solutions of nonlocal dispersal equations with monostable nonlinearity in space periodic habitats we first establish the existence and properties of spatially periodic solutions connecting two steady states then new types of entire solutions are constructed by combining the rightward and leftward pulsating traveling fronts with different speeds and a spatially periodic solution finally for a class of special heterogeneous reaction we further establish the uniqueness of entire solutions and the continuous dependence of such an entire solution on parameters such as wave speeds and the shifted variables in other words we build a five dimensional manifold of solutions and the traveling wave solutions are on the boundary of the manifold

asymptotic behavior of traveling fronts and entire solutions for a periodic bistable competition diffusion system abstract this paper is concerned with a time periodic competition diffusion system u t u x x u r 1 t a 1 t u b 1 t v t 0 x r v t d v x x v r 2 t a 2 t u b 2 t v t 0 x r where u t x and v t x denote the densities of two competing species d 0 is some constant r i t a i t and b i t are t periodic continuous functions under suitable conditions it has been confirmed by bao and wang 2013 2 that this system admits periodic traveling fronts connecting two stable semi trivial t periodic solutions p t 0 and 0 q t associated to the corresponding kinetic system in the present work we first investigate the asymptotic behavior of periodic bistable traveling fronts with non zero speeds at infinity by a dynamical approach combined with the two sided laplace transform method with these asymptotic properties we then obtain some key estimates as a result by applying the super and subsolutions techniques as well as the comparison principle we establish the existence and various qualitative properties of the so called entire solutions defined for all time and the whole space which provides some new spreading ways other than periodic traveling waves for two strongly competing species interacting in a heterogeneous environment

how travelling waves attract the solutions of kpp type equations we consider in this paper a general reaction diffusion equation of the kpp kolmogorov petrovskii piskunov type posed on an infinite cylinder such a model will have a family of pulsating waves of constant speed larger than a critical speed c the family of all supercritical waves attract a large class of initial data and we try to understand how we describe in this paper the fate of an initial datum trapped between two supercritical waves of the same velocity the solution will converge to a whole set of translates of the same wave and we identify the convergence dynamics as that of an effective drift around which an effective diffusion process occurs in several nontrivial particular cases we are able to describe the dynamics by an effective equation

implicit method for the computation of unsteady flows on unstructured grids an implicit method for the computation of unsteady flows on unstructured grids is presented following a finite difference approximation for the time derivative the resulting nonlinear system of equations is solved at each time step by using an agglomeration multigrid procedure the method allows for arbitrarily large time steps and is efficient in terms of computational effort and storage inviscid and viscous unsteady flows are computed to validate the procedure the issue of the mass matrix which arises with vertex centered finite volume schemes is addressed the present formulation allows the mass matrix to be inverted indirectly a mesh point movement and reconnection procedure is described that allows the grids to evolve with the motion of bodies as an example of flow over bodies in relative motion flow over a multi element airfoil system undergoing deployment is computed

entire solutions in nonlocal dispersal equations with bistable nonlinearity abstract we consider entire solutions of nonlocal dispersal equations with bistable nonlinearity in one dimensional spatial domain a two dimensional manifold of entire solutions which behave as two traveling wave solutions coming from both directions is established by an increasing traveling wave front with nonzero wave speed furthermore we show that such an entire solution is unique up to space time translations and liapunov stable a key idea is to characterize the asymptotic behaviors of the solutions as t in terms of appropriate subsolutions and supersolutions we have to emphasize that a lack of regularizing effect occurs

transition fronts for the fisher kpp equation this paper is concerned with transition fronts for reaction diffusion equations of the fisher kpp type basic examples of transition fronts connecting the unstable steady state to the stable one are the standard traveling fronts but the class of transition fronts is much larger and the dynamics of the solutions of such equations is very rich in the paper we describe the class of transition fronts and we study their qualitative dynamical properties in particular we characterize the set of their admissible asymptotic past and future speeds and their asymptotic profiles and we show that the transition fronts can only accelerate we also classify the transition fronts in the class of measurable superpositions of standard traveling fronts

traveling waves with multiple and nonconvex fronts for a bistable semilinear parabolic equation we construct new examples of traveling wave solutions to the bistable and balanced semilinear parabolic equation in input amssym bbb r n 1 our first example is that of a traveling wave solution with two non planar fronts that move with the same speed our second example is a traveling wave solution with a nonconvex moving front to our knowledge no existence results of traveling fronts with these type of geometric characteristics have been previously known our approach explores a connection between solutions of the semilinear parabolic pde and eternal solutions to the mean curvature flow in input amssym bbb r n 1

existence of pulsating waves of advection reaction diffusion equations of ignition type by a new method abstract the goal of this paper is to study the advection reaction diffusion equation of ignition type u t u q x y u f u xin j x xin existence of planar flame fronts in convective diffusive periodic media arch ration mech anal 121 1992 205 233 berestycki and hamel h berestycki f hamel front propagation in periodic excitable media comm pure appl math 55 2002 949 1032 proved the existence of pulsating waves using the theory of degenerate elliptic equations our goal is to give an alternative and rather natural proof of the same result first we transform the problem so that the existence of pulsating waves is equivalent to the existence of 1 periodic in time solutions of a nonlinear parabolic equation with 1 periodic in time coefficients next we prove the existence of such periodic solutions using a continuation method based on the implicit function theorem

asymptotic behavior of traveling fronts and entire solutions for a nonlocal monostable equation abstract this paper is concerned with the asymptotic behavior of traveling fronts and entire solutions for a nonlocal monostable equation the asymptotic behavior of traveling fronts is obtained using ikehara s theorem by the sub super solution method the existence of entire solutions is obtained they behave as two opposite wave fronts approaching each other from either side of the x axis and then annihilating in a finite time by an entire solution we mean one which is defined over the whole space for all time t r

entire solutions for nonlocal dispersal equations with spatio temporal delay monostable case abstract this paper deals with entire solutions for a general nonlocal dispersal monostable equation with spatio temporal delay i e solutions that are defined in the whole space and for all time t r we first derive a particular model for a single species and show how such systems arise from population biology then we construct some new types of entire solutions other than traveling wave solutions and equilibrium solutions of the equation under consideration with quasi monotone and non quasi monotone nonlinearities various qualitative properties of the entire solutions are also investigated in particular the relationship between the entire solutions and the traveling wave fronts which they originated is considered our main arguments are based on the comparison principle the method of super and sub solutions and the construction of auxiliary control systems

long time behaviour of heat flow global estimates and exact asymptotics we obtain global upper and lower bounds on the heat kernel of an elliptic second order differential operator which become sharp in certain long time and large space asymptotics we prove a generalization of aronson s gaussian bounds which identifies correctly an effective drift for heat flow in the case of periodic coefficients we give variational characterizations of the effective conductivity which is then made to appear in heat kernel bounds these results are for heat kernels with measurable coefficients for differentiable coefficients we prove tighter estimates in which the rate of homogenization is known to be optimal

entire solutions of diffusive lotka volterra system abstract this work is concerned with the existence of entire solutions of the diffusive lotka volterra competition system on the real line we prove the existence of some entire solutions that are asymptotic as t to a traveling wave consisting of a single species depending on system parameters these entire solutions evolve into two ore more stacked invasion waves as t our results cover both the weak and strong competition case in the weak competition case the existence of a class of entire solutions that forms a 4 dimensional manifold is proved

entire solutions in monostable reaction advection diffusion equations in infinite cylinders abstract this paper is concerned with entire solutions of a monostable reaction advection diffusion equation in infinite cylinders without the condition f u f 0 by constructing a quasi invariant manifold we prove that there exist two classes of entire solutions furthermore we show that one class of such entire solutions is unique up to space and time translation

pulsating fronts and front like entire solutions for a reaction advection diffusion competition model in a periodic habitat abstract this paper is devoted to the study of pulsating fronts and pulsating front like entire solutions for a reaction advection diffusion model of two competing species in a periodic habitat under certain assumptions the competition system admits a leftward and a rightward pulsating fronts in the bistable case in this work we construct some other types of entire solutions by interacting the leftward and rightward pulsating fronts some of these entire solutions behave as the two pulsating fronts approaching each other from both sides of the x axis which turn out to be unique and liapunov stable 2 dimensional manifolds of solutions furthermore the leftward and rightward pulsating fronts are on the boundary of these 2 dimensional manifolds the others behave as the two pulsating fronts propagating from one side of the x axis the faster one then invades the slower one as t these kinds of pulsating front like entire solutions then provide some new spreading ways other than pulsating fronts for two strongly competing species interacting in a heterogeneous habitat

dynamics of interfaces in the fisher kpp equation for slowly decaying initial data abstract this paper is concerned with the behavior of solutions of the fisher kpp equation when initial data decay slowly in space first we show by a comparison technique that the motion of an interface or a thin transition layer can be approximated as a level set of a first order pde of hamilton jacobi type then by the method of characteristics it is shown that various but not all motion of interfaces can be observed by taking initial data appropriately finally we study the large time behavior of solutions especially the existence of similarly expanding interfaces

entire solutions of the kpp equation this paper deals with the solutions defined for all time of the kpp equation ut uxx f u 0 0 f 0 1 0i n 0 1 and f 0 s f 0 0 in 0 1 this equation admits infinitely many traveling wave type solutions increasing or decreasing in x i t also admits solutions that depend only on t in this paper we build four other manifolds of solutions one is 5 dimensional one is 4 dimensional and two are 3 dimensional some of these new solutions are obtained by considering two traveling waves that come from both sides of the real axis and mix furthermore the traveling wave solutions are on the boundary of these four manifolds c 1999 john wiley sons inc

symmetry of traveling wave solutions to the allen cahn equation in mathbb r 2 in this paper we prove even symmetry of monotone traveling wave solutions to the balanced allen cahn equation in the entire plane related results for the unbalanced allen cahn equation are also discussed

entire solutions of a lattice vector disease model in this article the existence and qualitative properties of entire solutions for a lattice vector disease model is considered these entire solutions are constructed by the combinations of traveling wave solutions and the spatial independent solution of the model which reveal some new transmission forms of the disease

the stefan problem for the fisher kpp equation abstract we study the fisher kpp equation with a free boundary governed by a one phase stefan condition such a problem arises in the modeling of the propagation of a new or invasive species with the free boundary representing the propagation front in one space dimension this problem was investigated in du and lin 2010 11 and the radially symmetric case in higher space dimensions was studied in du and guo 2011 10 in both cases a spreading vanishing dichotomy was established namely the species either successfully spreads to all the new environment and stabilizes at a positive equilibrium state or fails to establish and dies out in the long run moreover in the case of spreading the asymptotic spreading speed was determined in this paper we consider the non radially symmetric case in such a situation similar to the classical stefan problem smooth solutions need not exist even if the initial data are smooth we thus introduce and study the weak solution for a class of free boundary problems that include the fisher kpp as a special case we establish the existence and uniqueness of the weak solution and through suitable comparison arguments we extend some of the results obtained earlier in du and lin 2010 11 and du and guo 2011 10 to this general case we also show that the classical aronson weinberger result on the spreading speed obtained through the traveling wave solution approach is a limiting case of our free boundary problem here

entire solutions in monostable reaction diffusion equations with delayed nonlinearity abstract entire solutions for monostable reaction diffusion equations with nonlocal delay in one dimensional spatial domain are considered a comparison argument is employed to prove the existence of entire solutions which behave as two traveling wave solutions coming from both directions some new entire solutions are also constructed by mixing traveling wave solutions with heteroclinic orbits of the spatially averaged ordinary differential equations and the existence of such a heteroclinic orbit is established using the monotone dynamical systems theory key techniques include the characterization of the asymptotic behaviors of solutions as t in term of appropriate subsolutions and supersolutions two models of reaction diffusion equations with nonlocal delay arising from mathematical biology are given to illustrate main results

traveling curved fronts of bistable lotka volterra competition diffusion systems in r 3 this paper is concerned with the existence and other qualitative properties of three dimensional traveling curved fronts for bistable lotka volterra competition diffusion systems in r 3 it is first shown that there exists a traveling front of convex polyhedral shape to such systems which is asymptotically stable then by taking the limits of such solutions as lateral surfaces go to infinity one proves the existence as well as the uniqueness of a three dimensional traveling curved front for any given g c s 1 with min 0 2 g 0