existence and stability of propagating fronts for an autocatalytic reaction diffusion system we study a one dimensional reaction diffusion system which describes an isothermal autocatalytic chemical reaction involving both a quadratic a b 2b and a cubic a 2b 3b autocatalysis the parameters of this system are in the ratio d d b d a of the diffusion constants of the reactant a and the autocatalyst b and the relative activity ii of the cubic reaction first for all values of d 0 and k greater than or equal to 0 we prove the existence of a family of propagating fronts or travelling waves describing the advance of the reaction in particular in the quadratic case k 0 we recover the results of billingham and needham phil trans r sec london a 334 1991 1 24 then if d is close to 1 and k is sufficiently small we prove using energy functionals that these propagating fronts are stable against small perturbations in exponentially weighted sobolev spaces this extends part of the results that are known for the scalar equation to which our system reduces when d 1 c 1998 elsevier science b v

anomalous spreading in a system of coupled fisher kpp equations abstract in this article we report on the curious phenomena of anomalous spreading in a system of coupled fisher kpp equations when a single parameter is set to zero the system consists of two uncoupled fisher kpp equations which give rise to traveling fronts propagating with the unique minimal kpp speed when the coupling parameter is nonzero various behaviors can be observed anomalous spreading occurs when one component of the system spreads at a speed significantly faster in the coupled system than it does in isolation while the speed of the second component remains unchanged we study these anomalous spreading speeds and show that they arise due to poles of the pointwise green s function corresponding to the linearization about the unstable homogeneous state these poles lead to anomalous spreading in the linearized system and come in two varieties one that persists and leads to anomalous spreading for the nonlinear system and one that does not we describe mechanisms leading to these two behaviors and prove that one class of poles are irrelevant as far as nonlinear wavespeed selection is concerned finally we show that the same mechanism can give rise to anomalous spreading even when the slower component does not spread

hopf bifurcation in a system of functional equations modeling the spread of an infectious disease f hoppensteadt and p waltman math biosci 9 1970 pp 71 91 12 1971 pp 133 145 describe a very general model for the spread of an infection in a population the model leads to a system of functional equations describing the evolution in time of the population sizes of four classes of individuals we show that a hopf bifurcation from the constant endemic level of infection occurs as a critical value of the length of temporary immunity is exceeded in addition sufficient conditions are obtained for the infection to die out and for an endemic level of infection to be stable

spreading speeds and traveling waves for periodic evolution systems abstract the theory of spreading speeds and traveling waves for monotone autonomous semiflows is extended to periodic semiflows in the monostable case then these abstract results are applied to a periodic system modeling man environment man epidemics a periodic time delayed and diffusive equation and a periodic reaction diffusion equation on a cylinder

global dynamics of a state dependent delay model with unimodal feedback abstract we obtain the global attractivity of nonnegative stationary states for a parameterized state dependent delay equation with unimodal feedback under certain mild conditions we show that the equation with unimodal type nonlinearity can generate rich dynamics as the parameter varies to be specific global attractivity of the positive stationary state is obtained in a set of nonnegative bounded continuous functions when the stationary state is less than a value implicitly determined by a condition on the unimodal feedback the general results of global attractivity are illustrated through two examples arising from population dynamics moreover hopf bifurcations are demonstrated in the examples when the positive stationary states lose global attractivity

on the gravitational attraction of a right rectangular prism with density varying with depth following a cubic polynomial juan garcia abdeslem 2005 geophysics 66 1793 1804 the author presents an interesting note that a cubic polynomial would be more appropriate than a quadratic polynomial to simulate the density depth variation of sedimentary rocks when the depth is confined to 7 km the author demonstrates this by simulating the density depth data from the green canyon offshore louisiana as presented by li 2001

nonlinear oscillations in epidemic models constant parameter epidemic models for a closed population described by integral and integrodifferential equations are analyzed to determine whether time delays can give rise to periodic oscillations delays can destabilize the steady state in one finite delay case and in one infinite delay case considered hopf bifurcation techniques are used to show the existence of locally asymptotically stable periodic solutions for certain parameter values inclusion of time delays in another related model does not alter the local asymptotic stability of the endemic steady state

sharp transition between extinction and propagation of reaction with x e r this equation has been extensively studied in mathematical physical and other literature starting with the pioneering works of fisher 7 and kol mogorov petrovskii piskunov 11 in these papers 1 2 was used to describe the propagation of advantageous genes in a population the main object of study in these and many subsequent works was the existence and stability of traveling fronts for 1 2 and 1 1 in recent years most of the results have been extended to in clude an advection term u vt in 1 1 and we refer to the reviews 2 16 for an extensive bibliography the above equations are used to model more than just population genetics phe nomena when f 0 0 for 9 g 0 1 then is a combustion non linearity and 1 1 1 2 model an exothermic chemical reaction in an infinite tube with a zero heat loss boundary in particular flame propagation in a premixed combustible gas without advection see zel dovich and frank kamenetskii 17 in this setting t is the normalized temperature taking values in 0 1 we note that 1 1 is usu ally obtained from a system involving both the temperature and the concentration of the reactants after the simplifying assumption of equal thermal and material diffusivities a special case of positive used particularly in chemical and biological lit erature is the kpp type with f 6 c 0 11 in combustion models the non linearity is often considered to be of arrhenius type with slow reaction rates at low temperatures modeled by f 0 e a e l 0 this case is often approximated

propagation phenomena in monostable integro differential equations acceleration or not abstract we consider the homogeneous integro differential equation t u j u u f u with a monostable nonlinearity f our interest is twofold we investigate the existence nonexistence of travelling waves and the propagation properties of the cauchy problem when the dispersion kernel j is exponentially bounded travelling waves are known to exist and solutions of the cauchy problem typically propagate at a constant speed 7 10 11 22 26 27 on the other hand when the dispersion kernel j has heavy tails and the nonlinearity f is nondegenerate i e f 0 0 travelling waves do not exist and solutions of the cauchy problem propagate by accelerating 14 20 27 for a general monostable nonlinearity a dichotomy between these two types of propagation behaviour is still not known the originality of our work is to provide such dichotomy by studying the interplay between the tails of the dispersion kernel and the allee effect induced by the degeneracy of f i e f 0 0 first for algebraic decaying kernels we prove the exact separation between existence and nonexistence of travelling waves this in turn provides the exact separation between nonacceleration and acceleration in the cauchy problem in the latter case we provide a first estimate of the position of the level sets of the solution

nonlocal dispersal cooperative systems acceleration propagation among species abstract we focus on the influences of nonlocal dispersal kernels on the spatial propagation in nonlocal dispersal cooperative systems with initial values having nonempty compact supports it is well known that the solution of a monostable nonlocal dispersal scalar equation spreads at a finite speed when the kernel is thin tailed and propagates by accelerating when the kernel has a heavy tail the fat tailed however in such systems we find that one species can propagate by accelerating although its dispersal kernel is thin tailed which is a new and interesting phenomenon more precisely we show that the spatial propagation of every species at large time is mainly determined by the tail of the maximum of their dispersal kernels which further implies that their spatial propagation is accelerated by each other because of the cooperation relation between them this gives us a new understanding of the cooperation relation in the spatial propagation of nonlocal dispersal cooperative systems

generalized transition waves and their properties in this paper we generalize the usual notions of waves fronts and propagation speeds in a very general setting these new notions which cover all usual situations involve uniform limits with respect to the geodesic distance to a family of hypersurfaces that are parametrized by time we prove the existence of new such waves for some time dependent reaction diffusion equations as well as general intrinsic properties some monotonicity properties and some uniqueness results for almost planar fronts the classification results which are obtained under some appropriate assumptions show the robustness of our general definitions 2012 wiley periodicals inc

global stability and wavefronts in a cooperation model with state dependent time delay abstract this article deals with a diffusive cooperative model with state dependent delay which is assumed to be an increasing function of the population density with lower and upper bounds for the cooperative dde system the positivity and boundedness of solutions are firstly given using the comparison principle of the state dependent delay equations obtained the stability criterion of model is analyzed both from local and global points of view when the diffusion is properly introduced the existence of traveling waves is obtained by constructing a pair of upper lower solutions and schauder s fixed point theorem calculating the minimum wave speed shows that the wave is slowed down by the state dependent delay finally the traveling wavefront solutions for large wave speed are also discussed and the fronts appear to be all monotone regardless of the state dependent time delay this is an interesting property since many findings are frequently reported that delay causes a loss of monotonicity with the front developing a prominent hump in some other delay models

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

domain decomposition method for the global dynamics of delay differential equations with unimodal feedback x t zkmx t c f x tkt with unimodal feedback is studied the existence of the global attractor is shown and bounds of the attractor are given we find attractive invariant intervals and give sufficient conditions that guarantee that all solutions enter the domain where f 0 is negative with respect to a positive equilibrium so the results for delayed monotone feedback can be applied to describe the asymptotic behaviour of solutions in particular the existence of heteroclinic orbits from the trivial equilibrium to a periodic orbit oscillating around the positive equilibrium is established numerical examples using nicholson s blowflies equation and the mackey glass equation are provided to illustrate the main results

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

global hopf bifurcation for differential equations with state dependent delay abstract we develop a global hopf bifurcation theory for a system of functional differential equations with state dependent delay the theory is based on an application of the homotopy invariance of s 1 equivariant degree using the formal linearization of the system at a stationary state our results show that under a set of mild conditions the information about the characteristic equation of the formal linearization with frozen delay can be utilized to detect the local hopf bifurcation and to describe the global continuation of periodic solutions for such a system with state dependent delay

speed of reaction diffusion fronts in spatially heterogeneous media the front speed problem for nonuniform reaction rate and diffusion coefficient is studied by using singular perturbation analysis the geometric approach of hamilton jacobi dynamics and the local speed approach exact and perturbed expressions for the front speed are obtained in the limit of large times for linear and fractal heterogeneities the analytic results have been compared with numerical results exhibiting a good agreement finally we reach a general expression for the speed of the front in the case of smooth and weak heterogeneities

propagation of fronts in the fisher kolmogorov equation with spatially varying diffusion the propagation of fronts in the fisher kolmogorov equation with spatially varying diffusion coefficients is studied using coordinate changes wkb approximations and multiple scales analysis we provide an analytic framework that describes propagation of the front up to the minimum of the diffusion coefficient we also present results showing the behavior of the front after it passes the minimum in each case we show that standard traveling coordinate frames do not properly describe front propagation last we provide numerical simulations to support our analysis and to show that around the minimum the motion of the front is arrested on asymptotically significant time scales

traveling wave solutions of diffusive lotka volterra equations a heteroclinic connection in we establish the existence of traveling wave solutions for a reaction dif fusion system based on the lotka volterra differential equation model of a predator and prey interaction the waves are of transition front type analogous to the solutions discussed by fisher and kolmogorov et al for a scalar reaction diffusion equation the waves discussed here are not necessarily monotone there is a speed c 0 such that for c c there is a traveling wave moving with speed c the proof uses a shooting argument based on the nonequivalence of a simply connected region and a nonsimply connected region together with a liapunov function to guarantee the existence of the traveling wave solution the traveling wave solution is equivalent to a heteroclinic orbit in 4 dimensional phase space

existence of periodic solutions for delay differential equations with state dependent delay abstract in this paper we study the existence of periodic solutions of a delay differential equation with delay depending indirectly on the state a fixed point problem related to a poincare operator is constructed and solved using an ejective fixed point theorem

the accommodation of traveling waves of fisher s type to the dynamics of the leading tail traveling wave solutions for equations of fisher s type are given by a connection from a saddle point to a node and their velocity for large times is known to be determined by the spatial decay rate of the initial conditions far ahead of the wave to understand this phenomenon in simple terms initial conditions are considered that correspond to a traveling wave but with an exponential decay rate far ahead of the wave that varies slowly in the region referred to as the leading tail the wave that results is supersonic in the sense that the speed of the core of the traveling wave is greater than the velocity of the characteristics or group velocity in the leading tail consequently the speed of the core adjusts to accommodate the slowly varying decay rate in the leading tail a generalization is also considered that includes fisher s equation for which traveling waves with a stable exponential tail corresponding to a node are supersonic

existence and stability traveling wave solutions for a system of social outbursts abstract we study the existence and stability of planar traveling wave solutions to the system of reaction diffusion equations u t 2 u x 2 u 1 u 1 e v 1 u x r n t 0 v t 2 v x 2 v 1 u p 1 x r n t 0 which was introduced in 8 as a model for the spatiotemporal dynamics of rioting activity and social tension the wave like spread of rioting activity observed in the 2005 french riots 12 motivates the study of these type of solutions which were numerically analyzed in 46 of particular interest is the qualitative difference in traveling wave solutions between the p 0 and p 0 case we present results for the existence and stability of these type of solutions for p 0 and provide the asymptotic approach rates of the traveling wave solutions to the steady states as x for the case of p 0 we study the stability of such solutions using evans functions and a combination of analytical and numerical methods to show irrefutable evidence of spectral stability

boundary layer phenomena for differential delay equations with state dependent time lags i in this paper we begin a study of the differential delay equation mathtype mtef 2 1 feaafiart1ev1aaatcvaufebsjuyzl2yd9gzlbvynv2caerbulwbln hiov2dgi1btfmbaexatlxbi9gbaerbd9wdylwzybitldharqqtubsr 4rnchbgeagqivu0je9sqqrpepc0xbbl8f4rqqrffpeea0jc9yq jc9 vqaqpepm0xbba9pwe9q8fs0 yqaqpepae9pg0firpeps0he9vr0 vr 0 vqpwqaaeaabigaciaacaqabeaadaqaaqaaaoqaaiabew7aljqadi hagaqbaiaacicacawg0bgaaiykaiabg2da9iabgkhitiaadihacagg oagaamidaiaacmcacqghrawkcawgmbgaaiikaiaadihacaggoagaam idaiabgkhitiaadkhacaggpagaaiykaiaacycacaqggagaaeiiaiaa dkhacqgh9aqpcawgybgaaiikaiaadihacaggoagaamidaiaacmcaca ggpaaaaa 5192 varepsilon x t x t f x t r r r x t we prove the existence of periodic solutions for 0 0 where 0 is an optimal positive number we investigate regularity and monotonicity properties of solutions x t which are defined for all t and of associated functions like t t r x t we begin the development of a poincare bendixson theory and phase plane analysis for such equations in a companion paper these results will be used to investigate the limiting profile and corresponding boundary layer phenomena for periodic solutions as approaches zero

a mathematical model for spatially varying extracellular matrix alignment orientation of extracellular matrix fibers in the skin is a keyingredient of tissue appearance and function and differences in fiber alignment are one of the main distinctions between scar tissue and normal skin in this paper the authors develop a mathematical model for alignment of collagen fibers and the fibroblast cells that remodel them the model extends previous work in which spatial variation was excluded numerical simulations of the model are presented which show spatial variations in alignment over long transients but with spatiallyuniform behavior in the long term this is investigated further via asymptotic analysis using the angular diffusion coefficient as a small parameter this method enables calculation of the form of the steadystate orientation peaks observed numerically by considering behavior at large times the rate of approach to these peaks is shown to be exponential extension of this analysis to the spatially varying model confirms that long time behavior will be spatiallyuniform except in one special and biologicallyunrealistic case the authors conclude that the spatiallyvary ing alignment patterns observed in skin are in fact slow transients and biological implications of the modeling are discussed

comparison theorems and variable speed waves for a scalar reaction diffusion equation this paper concerns the reaction diffusion equation u t u xx u 2 1 u previous numerical solutions of this equation have demonstrated various different types of wave front solutions generated by different initial conditions in this paper the authors use a phase plane form of comparison theorems for partial differential equations pdes to confirm analytically these numerical results in particular they show that initial conditions with an exponentially decaying tail evolve to the unique exponentially decaying travelling wave while initial conditions with algebraically decaying tails evolve either to an algebraically decaying travelling wave or to the exponentially decaying wave or to a perpetually accelerating wave dependent upon the exact form of the decay of the initial conditions we then focus on the case of accelerating waves and investigate their form in more detail by approximating the full equation in this case with a hyperbolic pde which we solve using the method of characteristics we use this approximate solution to derive a leading order approximation to the wave speed

absolute and convective instabilities of waves on unbounded and large bounded domains abstract instabilities of nonlinear waves on unbounded domains manifest themselves in different ways an absolute instability occurs if the amplitude of localized wave packets grows in time at each fixed point in the domain in contrast convective instabilities are characterized by the fact that even though the overall norm of wave packets grows in time perturbations decay locally at each given point in the unbounded domain wave packets are convected towards infinity in experiments as well as in numerical simulations bounded domains are often more relevant we are interested in the effects that the truncation of the unbounded to a large but bounded domain has on the aforementioned in stability properties of a wave these effects depend upon the boundary conditions that are imposed on the bounded domain we compare the spectra of the linearized evolution operators on unbounded and bounded domains for two classes of boundary conditions it is proved that periodic boundary conditions reproduce the point and essential spectrum on the unbounded domain accurately spectra for separated boundary conditions behave in quite a different way firstly separated boundary conditions may generate additional isolated eigenvalues secondly the essential spectrum on the unbounded domain is in general not approximated by the spectrum on the bounded domain instead the so called absolute spectrum is approximated that corresponds to the essential spectrum on the unbounded domain calculated with certain optimally chosen exponential weights we interpret the difference between the absolute and the essential spectrum in terms of the convective behavior of the wave on the unbounded domain in particular it is demonstrated that the stability of the absolute spectrum implies convective instability of the wave but that convectively unstable waves can destabilize under domain truncation the theoretical predictions are compared with numerical computations

a numerical perspective on traveling wave solutions in a system for rioting activity abstract motivated by the wave like dynamics of rioting activity that occurred during the 2005 french riots we analyze numerically the existence and qualitative properties of traveling wave solutions to a system of reaction diffusion equations which was proposed to model the spatio temporal dynamics of rioting activity and social tension in 6 with the use of auto and matlab we classify the various regions separating the number of homogeneous steady states solutions and analyze their stability which turns out to be very rich we discover a zoology of traveling wave solutions displaying a variety of interesting characteristics through this analysis we can make conclusions about the factors leading to different characteristics which have been observed in different riots for example the analysis provides evidence that the 2005 french riots were tension inhibitive where rioting helps release tension versus tension enhansing where rioting slows down the release of tension

analytical solutions to fisher s equation with time variable coefficients abstract this paper provides analytical solutions to the generalized fisher equation with a class of time varying diffusion coefficients to accomplish this we use the painleve property for partial differential equations as defined by weiss in 1983 in the painleve property for partial differential equations this was first done for the variable coefficient fisher s equation by ogun and kart in 2007 we build on this work finding additional solutions with a weaker restriction on the trial solution we also use the same technique to find solutions to fisher s equation with time dependent coefficients for both diffusion and nonlinear terms lastly we compute specific solutions to illustrate their behaviors

the speed of propagation for kpp type problems ii general domains this paper is devoted to nonlinear propagation phenomena in general unbounded domains of rn for reaction diffusion equations with kolmogorov petrovsky piskunov kpp type nonlinearities this article is the second in a series of two and it is the follow up of the paper 7 which dealt which the case of periodic domains this paper is concerned with general domains and we give various definitions of the spreading speeds at large times for solutions with compactly supported initial data we study the relationships between these new notions and analyze their dependence on the geometry of the domain and on the initial condition some a priori bounds are proved for large classes of domains the case of exterior domains is also discussed in detail lastly some domains which are very thin at infinity and for which the spreading speeds are infinite are exhibited the construction is based on some new heat kernel estimates in such domains

the influence of fractional diffusion in fisher kpp equations we study the fisher kpp equation where the laplacian is replaced by the generator of a feller semigroup with power decaying kernel an important example being the fractional laplacian in contrast with the case of the standard laplacian where the stable state invades the unstable one at constant speed we prove that with fractional diffusion generated for instance by a stable levy process the front position is exponential in time our results provide a mathematically rigorous justification of numerous heuristics about this model

the fisher kpp problem with doubly nonlinear fast diffusion abstract the famous fisher kpp reaction diffusion model combines linear diffusion with the typical fisher kpp reaction term and appears in a number of relevant applications it is remarkable as a mathematical model since in the case of linear diffusion it possesses a family of travelling waves that describe the asymptotic behaviour of a wide class of solutions 0 u x t 1 of the problem posed in the real line the existence of propagation wave with finite speed has been confirmed in the cases of slow and pseudo linear doubly nonlinear diffusion too see audrito and vazquez 2016 we investigate here the corresponding theory with fast doubly nonlinear diffusion and we find that general solutions show a non tw asymptotic behaviour and exponential propagation in space for large times finally we prove precise bounds for the level sets of general solutions even when we work in with spacial dimension n 1 in particular taking spatial logarithmic scale we show that the location of the positive level sets is approximately linear for large times this represents a strong departure from the linear case in which the location of the level sets is not purely linear but presents the celebrated logarithmic deviation for large times

nonlinear stability of planar traveling waves in a chemotaxis model of tumor angiogenesis with chemical diffusion abstract we consider a simplified chemotaxis model of tumor angiogenesis described by a keller segel system on the two dimensional infinite cylindrical domain x y r s where s is the circle of perimeter 0 the domain models a virtual channel where newly generated blood vessels toward the vascular endothelial growth factor will be located the system is known to allow planar traveling wave solutions of an invading type in this paper we establish the nonlinear stability of these traveling invading waves when chemical diffusion is present if is sufficiently small the same result for the corresponding system in one dimension was obtained by li li wang 2014 17 our result solves the problem remained open in chae choi kang lee 2018 3 at which only linear stability of the planar traveling waves was obtained under certain artificial assumption

traveling wave solutions of a mathematical model for tumor encapsulation the formation of a capsule of dense fibrous extracellular matrix around a solid tumor is a key prognostic indicator in a wide range of cancers however the cellular mechanisms underlying capsule formation remain unclear one hypothesized mechanism is the expansive growth hypothesis which suggests that a capsule may form by the rearrangement of existing extracellular matrix without new matrix production a mathematical model was recently proposed to study the implications of this hypothesis perumpanani sherratt and norbury nonlinearity 10 1997 pp 1599 1614 the model consists of conservation equations for tumor cells and extracellular matrix and exhibits traveling wave solutions in which a pulse of extracellular matrix corresponding to a capsule moves in parallel with the advancing front of the tumor in this paper the author presents a detailed study of traveling wave behavior in the model deriving conditions for the existence of traveling waves and their key properties numerical meth

asymptotic spreading in heterogeneous diffusive excitable media abstract we establish propagation and spreading properties for nonnegative solutions of nonhomogeneous reaction diffusion equations of the type t u a t x u q t x u f t x u with compactly supported initial conditions at t 0 here a q f have a general dependence in t r and x r n we establish properties of families of propagation sets which are defined as families of subsets s t t 0 of r n such that lim inf t inf x s t u t x 0 the aim is to characterize such families as sharply as possible in particular we give some conditions under which 1 a given path t t 0 where t r n forms a family of propagation sets or 2 one can find such a family with s t x r n x r t and lim t r t this second property is called here complete spreading furthermore in the case q 0 and inf t x r r n f u t x 0 0 as well as under some more general assumptions we show that there is a positive spreading speed that is r t can be chosen so that lim inf t r t t 0 in the general case we also show the existence of an explicit upper bound c 0 such that lim sup t r t t c on the other hand we provide explicit examples of reaction diffusion equations such that for an arbitrary e 0 any family of propagation sets s t t 0 has to satisfy s t x r n x e t for large t in connection with spreading properties we derive some new uniqueness results for the entire solutions of this type of equations lastly in the case of space time periodic media we develop a new approach to characterize the largest propagation sets in terms of eigenvalues associated with the linearized equation in the neighborhood of zero

global well posedness and uniform boundedness of urban crime models one dimensional case abstract in this paper we study a pde system which was proposed by short et al 2008 42 to describe the spatial temporal dynamics in urban criminal activity by establishing a user friendly integral inequality we prove the global existence uniqueness and uniform boundedness of the classical solution to this system in one dimension space our result extends the local well posedness by rodriguez and bertozzi 2010 38 to global

sharp thresholds between finite spread and uniform convergence for a reaction diffusion equation with oscillating initial data abstract we investigate the large time dynamics of solutions of multi dimensional reaction diffusion equations with ignition type nonlinearities we consider solutions which are in some sense locally persistent at large time and initial data which asymptotically oscillate around the ignition threshold we show that as time goes to infinity any solution either converges uniformly in space to a constant state or spreads with a finite speed uniformly in all directions furthermore the transition between these two behaviors is sharp with respect to the period vector of the asymptotic profile of the initial data we also show the convergence to planar fronts when the initial data are asymptotically periodic in one direction

stability of fronts and transient behaviour in kpp systems we consider a system of two reaction diffusion equations with the kolmogorov petrovsky piskunov kpp type nonlinearity which describes propagation of pressure driven flames it is known that the system admits a family of travelling wave solutions parameterized by their velocity in this paper we show that these travelling fronts are stable under the assumption that perturbations belong to an appropriate weighted l 2 space we also discuss an interesting meta stable pattern the system exhibits in certain cases

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

yamabe flow steady solitons and type ii singularities abstract we study the convergence of complete non compact conformally flat solutions to the yamabe flow to yamabe steady solitons we also prove the existence of type ii singularities which develop at either a finite time t or as t

accelerating invasions along an environmental gradient abstract we consider a population structured by a space variable and a phenotypical trait submitted to dispersion mutations growth and nonlocal competition this population is facing an environmental gradient the optimal trait for survival depends linearly on the spatial variable the survival or extinction depends on the sign of an underlying principal eigenvalue we investigate the survival case when the initial data satisfies a so called heavy tail condition in the space trait plane under these assumptions we show that the solution propagates in the favorable direction of survival by accelerating we derive some precise estimates on the location of the level sets corresponding to the total population in the space variable regardless of their traits our analysis also reveals that the orientation of the initial heavy tail is of crucial importance

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

spreading speeds and traveling waves for abstract monostable evolution systems abstract this paper is devoted to the development of the theory of spreading speeds and traveling waves for abstract monostable evolution systems with spatial structure under appropriate assumptions we show that the spreading speeds coincide with the minimal wave speeds for monotone traveling waves in the positive and negative directions then we use this theory to study the spatial dynamics of a parabolic equation in a periodic cylinder with the dirichlet boundary condition a reaction diffusion model with a quiescent stage a porous medium equation in a tube and a lattice system in a periodic habitat

the evolution of traveling waves in a simple isothermal chemical system modeling quadratic autocatalysis with strong decay abstract in this paper we study a reaction diffusion system for an isothermal chemical reaction scheme governed by a quadratic autocatalytic step a b 2 b and a decay step b c where a b and c are the reactant the autocatalyst and the inner product respectively previous numerical studies and experimental evidences demonstrate that if the autocatalyst is introduced locally into this autocatalytic reaction system where the reactant a initially distributes uniformly in the whole space then a pair of waves will be generated and will propagate outwards from the initial reaction zone one crucial feature of this phenomenon is that for the strong decay case the formation of waves is independent of the amount of the autocatalyst b introduced into the system it is this phenomenon of kpp type which we would like to address in this paper to study the propagation of reactant and autocatalyst analytically we first use the tail behavior of waves to construct a pair of generalized super sub solutions for the approximate system of the autocatalytic reaction system note that the autocatalytic reaction system does not enjoy comparison principle together with a family of truncated problems we can establish the existence of a family of traveling waves with the minimal speed second we use this pair of generalized super sub solutions to show that the propagation of waves is fully determined by the rate of decay of the initial data at infinity in the sense of aronson weinberger formulation which in turn confirms the aforementioned numerical and experimental results

complex predator invasion waves in a holling tanner model with nonlocal prey interaction abstract we consider predator invasions for the nonlocal holling tanner model predators are introduced in a small region adjacent to an extensive predator free region in its simplest form an invasion front propagates into the predator free region with a predator prey coexistence state displacing the predator free state however patterns may form in the wake of the invasion front due to instability of the coexistence state the coexistence state can be subject to either oscillatory or cellular instability depending on parameters furthermore the oscillatory instability can be either at zero wave number or finite wave number in addition the unstable predator free state can be subject to additional cellular instabilities when the extent of the nonlocality is sufficiently large we perform numerical simulations that demonstrate that the invasion wave may have a complex structure in which different spatial regions exhibit qualitatively different behaviors these regions are separated by relatively narrow transition regions that we refer to as fronts we also derive analytic approximations for the speeds of the fronts and find qualitative and quantitative agreement with the results of computations

stability of planar traveling waves in a keller segel equation on an infinite strip domain abstract we consider a simplified model of tumor angiogenesis described by a keller segel equation on the two dimensional domain x y r s where s is the circle of perimeter it is known that the system allows planar traveling wave solutions of an invading type in case that is sufficiently small we establish the nonlinear stability of traveling wave solutions in the absence of chemical diffusion if the initial perturbation is sufficiently small in some weighted sobolev space when chemical diffusion is present it can be shown that the system is linearly stable lastly we prove that any solution with our front condition eventually becomes planar under certain regularity conditions

on the nonlinear diffusion equation of kolmogorov petrovsky and piscounov abstract recently bramson proved a theorem that classifies the initial data under which solutions of the k p p equation converge to the appropriate travelling waves in this paper a simplified proof is given by using maximal principles instead of his brownian motion approach the regularity condition on the forcing term is also weakened

gravity inversion of an interface above which the density contrast varies exponentially with depth mapping of an interface above which the density contrast varies exponentially with depth as is common at the basement surface of sedimentary basins is efficiently achieved by a theoretically precise gravity method which can be applied to either profile data or twodimensional data the contrast in mass above the interface is modeled by an array of vertical rectangular prisms with density contrasts varying exponentially with depth gravity anomalies due to the prisms are calculated in the wavenumber domain and then converted to the space domain the precision of the inverse numerical fourier transform in this procedure is significantly increased by a shift sampling technique based on the discrete fourier deviation equation depth to the interface is determined by iterative adjustment of the vertical extent of the prisms in accordance with observed gravity anomaly data the basement surface of the los angeles basin california calculated by this method closely duplicates the published configuration based

spreading speeds for a two species competition diffusion system abstract in this paper spreading properties of a competition diffusion system of two equations are studied this system models the invasion of an empty favorable habitat by two competing species each obeying a logistic growth equation such that any coexistence state is unstable if the two species are initially absent from the right half line x 0 and the slowest one dominates the fastest one on x 0 then the latter will invade the right space at its fisher kpp speed and will be replaced by or will invade the former depending on the parameters at a slower speed thus the system forms a propagating terrace linking an unstable state to two consecutive stable states

fast propagation for reaction diffusion cooperative systems abstract this paper deals with the spatial propagation for reaction diffusion cooperative systems it is well known that the solution of a reaction diffusion equation with monostable nonlinearity spreads at a finite speed when the initial condition decays to zero exponentially or faster and propagates fast when the initial condition decays to zero more slowly than any exponentially decaying function however in reaction diffusion cooperative systems a new possibility happens in which one species propagates fast although its initial condition decays exponentially or faster the fundamental reason is that the growth sources of one species come from the other species simply speaking we find a new interesting phenomenon that the spatial propagation of one species is accelerated by the other species this is a unique phenomenon in reaction diffusion systems we present a framework of fast propagation for reaction diffusion cooperative systems

a group analysis via weak equivalence transformations for a model of tumour encapsulation a symmetry reduction of a pdes system describing the expansive growth of a benign tumour is obtained via a group analysis approach the presence in the model of three arbitrary functions suggests the use of lie symmetries by using the weak equivalence transformations an invariant classification is given which allows us to reduce the initial pdes system to an odes system numerical simulations show a realistic enough description of the physical process

gravity interpretation using fourier transforms and simple geometrical models with exponential density contrast the decrease of density contrast in sedimentary basins can often be approximated by an exponential function theoretical fourier transforms are derived for symmetric trapezoidal vertical fault vertical prism syncline and anticline models this is desirable because there are no equivalent closed form solutions in the space domain for these models combined with an exponential density contrast these transforms exhibit characteristic minima maxima and zero values and hence graphical methods have been developed for interpretation of model parameters after applying end corrections to improve the discrete transforms of observed gravity data the transforms are interpreted for model parameters this method is first tested on two synthetic models then applied to gravity anomalies over the san jacinto graben and los angeles basin

propagation in a kinetic reaction transport equation travelling waves and accelerating fronts in this paper we study the existence and stability of travelling wave solutions of a kinetic reaction transport equation the model describes particles moving according to a velocity jump process and proliferating thanks to a reaction term of monostable type the boundedness of the velocity set appears to be a necessary and sufficient condition for the existence of positive travelling waves the minimal speed of propagation of waves is obtained from an explicit dispersion relation we construct the waves using a technique of sub and supersolutions and prove their weak stability in a weighted l2 space in case of an unbounded velocity set we prove a superlinear spreading it appears that the rate of spreading depends on the decay at infinity of the velocity distribution in the case of a gaussian distribution we prove that the front spreads as t3 2

bistable traveling waves around an obstacle we consider traveling waves for a nonlinear diffusion equation with a bistable or multistable nonlinearity the goal is to study how a planar traveling front interacts with a compact obstacle that is placed in the middle of the space n as a first step we prove the existence and uniqueness of an entire solution that behaves like a planar wave front approaching from infinity and eventually reaching the obstacle this causes disturbance on the shape of the front but we show that the solution will gradually recover its planar wave profile and continue to propagate in the same direction leaving the obstacle behind whether the recovery is uniform in space is shown to depend on the shape of the obstacle 2008 wiley periodicals inc

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

stabilization in a state dependent model of turning processes we consider a two degree of freedom model for turning processes which involves a system of differential equations with state dependent delay depending on process parameters e g spindle speed depth of cut the cutting tool can exhibit unwanted vibrations resulting in a nonsmooth surface of the workpiece in this paper we propose a feedback law to stabilize the turning process for a large range of system parameters the feedback law introduces a generic nonhyperbolic stationary point into the model which generates the main technical challenge of this work we establish the stability equivalence between the differential equations with state dependent delay and a corresponding nonlinear system with the delay fixed at its stationary value then we show the stability of that nonlinear system with constant delay by computing its normal form finally we obtain conditions on system parameters which guarantee the stability of the state dependent delay model at the nonhyperbolic stationary point

large time behavior of solutions for degenerate p degree fisher equation with algebraic decaying initial data abstract this paper is concerned with the large time behavior of solutions to the one dimensional degenerate p degree fisher equation where the initial data are assumed to be asymptotically front like and to decay to zero non exponentially at one end by applying sub super solution method we first prove the lyapunov stability of all the wave fronts with noncritical speeds in some polynomially weighted space further by using semigroup estimates sub super solution method and the nonlinear stability of the traveling waves we obtain the asymptotic estimates of the solutions which indicate that the solution still moves like a wave front and the asymptotic spreading speed of the level set of the solution can be finite or infinite which is solely determined by the detailed decaying rate of the initial data

accelerating propagation in a recursive system arising from seasonal population models with nonlocal dispersal abstract the system of interest is defined by iterations of the poincare map of a time periodic population model with symmetric nonlocal dispersal and periodic time delay under the kpp type setting we show that the system admits the spreading speed c 0 that coincides with the minimal speed of traveling waves where c if the nonlocal dispersal kernel j is exponentially bounded and c if j is exponentially unbounded further the latter case gives rises to an accelerating propagation that the level set of solutions with compactly supported initial values locate in between two exponential functions in time

stability of travelling fronts in a model for flame propagation part ii nonlinear stability this paper is concerned with the nonlinear stability of the travelling wave solutions of the multidimensional thermodiffusive model for flame propagation with unit lewis number the model consists in a semilinear parabolic equation in an infinite cylinder with neumann boundary conditions we prove that any solution which is initially close to a travelling wave will converge to a translate of that wave

fast propagation for kpp equations with slowly decaying initial conditions abstract this paper is devoted to the analysis of the large time behavior of solutions of one dimensional fisher kpp reaction diffusion equations the initial conditions are assumed to be globally front like and to decay at infinity towards the unstable steady state more slowly than any exponentially decaying function we prove that all level sets of the solutions move infinitely fast as time goes to infinity the locations of the level sets are expressed in terms of the decay of the initial condition furthermore the spatial profiles of the solutions become asymptotically uniformly flat at large time this paper contains the first systematic study of the large time behavior of solutions of kpp equations with slowly decaying initial conditions our results are in sharp contrast with the well studied case of exponentially bounded initial conditions