principal eigenvalues and the dirichlet problem for fully nonlinear elliptic operators abstract we study uniformly elliptic fully nonlinear equations of the type f d 2 u d u u x f x we show that convex positively 1 homogeneous operators possess two principal eigenvalues and eigenfunctions and study these objects we obtain existence and uniqueness results for nonproper operators whose principal eigenvalues in some cases only one of them are positive finally we obtain an existence result for nonproper isaac s equations

existence of multidimensional traveling waves in the transport of reactive solutes through periodic porous media we prove the existence of multidimensional traveling wave solutions to the scalar equation for the transport of solutes contaminants with nonlinear adsorption and spatially periodic convection diffusion adsorption coefficients under the assumption that the nonlinear adsorption function satisfies the lax and oleinik entropy conditions in the nondegenerate case we also prove the uniqueness of the traveling waves these traveling waves are analogues of viscous shock profiles they propagate with effective speeds that depend on the periodic porous media only up to their mean states and are given by an averaged rankine hugoniot relation this is a direct consequence of the fact that the transport equation is in conservation form we use the sliding domain method the continuation method spectral theory maximum principles and a priori estimates in the degenerate case the traveling waves are weak solutions of a degenerate parabolic equation and are only holder continuous we obtain them by taking suitable limits on the non degenerate traveling waves the uniqueness of the degenerate traveling waves is open

fujita type exponent for fully nonlinear parabolic equations and existence results abstract in this paper we prove that a class of parabolic equations involving a second order fully nonlinear uniformly elliptic operator has a fujita type exponent these exponents are related with an eigenvalue problem in all r n and play the same role in blow up theorems as the classical p 1 2 n introduced by fujita for the laplacian we also obtain some associated existence results

entire solutions of time periodic bistable reaction advection diffusion equations in infinite cylinders this paper is concerned with the existence and some qualitative properties of entire solutions for a reaction advection diffusion equation in infinite cylinders with time periodic bistable nonlinearity here an entire solution means a solution defined in the whole space and for all time t by the comparison principle coupled with the supersolution and subsolution technique it is proved that there exists an entire solution furthermore it is shown that such an entire solution is unique and liapunov stable unlike the reaction diffusion equation without advection the lack of symmetry between increasing and decreasing traveling fronts caused by the advection affects the construction of supersolutions and subsolutions

pulsating type entire solutions of monostable reaction advection diffusion equations in periodic excitable media abstract we establish the existence of pulsating type entire solutions of reaction advection diffusion equations with monostable nonlinearities in a periodic framework here the nonlinearities include the classic kpp case the pulsating type entire solutions are defined in the whole space and for all time t r by studying a pulsating traveling front connecting a constant unstable stationary state to a stable stationary state which is allowed to be a positive function we proved that there exist pulsating type entire solutions behaving as two pulsating traveling fronts coming from both directions and approaching each other the key techniques are to characterize the asymptotic behavior of the solutions as t in terms of appropriate subsolutions and supersolutions

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

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

solutions to an inhomogeneous equation involving infinity laplacian abstract we obtain existence and uniqueness results of viscosity solutions to the dirichlet boundary value problem for a nonlinear highly degenerate elliptic equation of the form 1 d u h u f where 0 h 2 and u denotes the so called infinity laplacian operator given by u i j 1 n u x i u x j 2 u x i x j we also give an asymptotic behavior of the viscosity solutions of two kinds of inhomogeneous infinity laplace equations near an isolated singular point

a tour of the theory of absolutely minimizing functions a detailed analysis of the class of absolutely minimizing functions in euclidean spaces and the relationship to the infinity laplace equation

convergence to travelling waves for solutions of a class of semilinear parabolic equations abstract in this paper we investigate the long time behaviour of the solutions of a class of semilinear parabolic equations in an infinite cylinder such equations arise in various fields such as biology or flame propagation theory we prove that the solutions whose initial data together with being increasing along the axis of the cylinder satisfy certain growth conditions at its ends will converge exponentially to travelling front solutions

on a controlled eigenvalue problem abstract we consider the problem of minimizing the asymptotic exit rate of a diffusion from a bounded domain and derive the associated hamilton jacobi bellman equation this turns out to be a nonlinear eigenvalue problem a verification theorem for optimal control is also provided

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

existence of planar flame fronts in convective diffusive periodic media we prove the existence of planar travelling wave solutions in a reaction diffusion convection equation with combustion nonlinearity and self adjoint linear part in rn n 1 the linear part involves diffusion convection terms and periodic coefficients these travelling waves have wrinkled flame fronts propagating with constant effective speeds in periodic inhomogeneous media we use the method of continuation spectral theory and the maximum principle uniqueness and monotonicity properties of solutions follow from a previous paper these properties are essential to overcoming the lack of compactness and the degeneracy in the problem

bifurcation and optimal stochastic control abstract two questions concerning bifuraction theory and optimal stochastic control are considered first in a few examples we give the interpretation of a bifurcation in terms of optimal stochastic control next we introduce the analogue of the lowest eigenvalue for the nonlinear operator associated with the hamilton jacobi bellman equations of optimal stochastic control author

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

asymptotic speed of propagation and traveling wave solutions for a lattice integral equation abstract we investigate the asymptotic speed of propagation and monotone traveling wave solutions for a lattice integral equation which is an epidemic model while the population is distributed on one dimensional lattice z it is proved that the asymptotic speed of propagation c coincides with the minimal wave speed

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

a weighted eigenvalue problem of the degenerate operator associated with infinity laplacian abstract in this paper we study a weighted eigenvalue problem of the degenerate operator associated with infinity laplacian h u a x u h 1 u 0 in u 0 on where h 1 h u d u h 3 u is the h homogeneous infinity laplacian and a x is a positive continuous bounded function in we prove the existence of the principal eigenvalue and a corresponding positive eigenfunction the approach to the weighted eigenvalue problem is based on the maximum principle and when a parameter is less than the principal eigenvalue some existence and uniqueness results related to this problem are established

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

travelling wave fronts in reaction diffusion systems with spatio temporal delays this paper deals with the existence of travelling wave fronts in reaction diffusion systems with spatio temporal delays our approach is to use monotone iterations and a nonstandard ordering for the set of profiles of the corresponding wave system new iterative techniques are established for a class of integral operators when the reaction term satisfies different monotonicity conditions following this the existence of travelling wave fronts for reaction diffusion systems with spatio temporal delays is established finally we apply the main results to a single species diffusive model with spatio temporal delay and obtain some existence criteria of travelling wave fronts by choosing different kernels

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

message passing and moment closure for susceptible infected recovered epidemics on finite networks the message passing approach of karrer and newman phys rev e 82 016101 2010 is an exact and practicable representation of susceptible infected recovered dynamics on finite trees here we show that assuming poisson contact processes a pair based moment closure representation sharkey j math biol 57 311 2008 can be derived from their equations we extend the applicability of both representations and discuss their relative merits on arbitrary time independent networks as was shown for the message passing formalism the pair based moment closure equations also provide a rigorous lower bound on the expected number of susceptibles at all times

boundary layers and kpp fronts in a cellular flow we study an eigenvalue problem associated with a reaction diffusion advection equation of the kpp type in a cellular flow we obtain upper and lower bounds on the eigenvalues in the regime of a large flow amplitude a 1 it follows that the minimal pulsating traveling front speed c a satisfies the upper and lower bounds c1a1 4 c a c2a1 4 physically the speed enhancement is related to the boundary layer structure of the associated eigenfunction accordingly we establish an averaging along the streamlines principle for the unique positive eigenfunction

an eigenvalue approach to the risk sensitive control problem in near monotone case abstract risk sensitive control problem under near monotonicity condition is considered we prove existence of a solution to the corresponding hamilton jacobi bellman equation by an eigenvalue approach existence of an optimal control has also been proved

inhomogeneous infinity laplace equation abstract we present the theory of the viscosity solutions of the inhomogeneous infinity laplace equation x i u x j u x i x j 2 u f in domains in r n we show existence and uniqueness of a viscosity solution of the dirichlet problem under the intrinsic condition f does not change its sign we also discover a characteristic property which we call the comparison with standard functions property of the viscosity sub and super solutions of the equation with constant right hand side applying these results and properties we prove the stability of the inhomogeneous infinity laplace equation with nonvanishing right hand side which states the uniform convergence of the viscosity solutions of the perturbed equations to that of the original inhomogeneous equation when both the right hand side and boundary data are perturbed in the end we prove the stability of the well known homogeneous infinity laplace equation x i u x j u x i x j 2 u 0 which states the viscosity solutions of the perturbed equations converge uniformly to the unique viscosity solution of the homogeneous equation when its right hand side and boundary data are perturbed simultaneously

the infinity laplacian aronsson s equation and their generalizations the infinity laplace equation u 0 arose originally as a sort of euler lagrange equation governing the absolute minimizer for the l variational problem of minimizing the functional ess sup u du the more general functional ess sup u f x u du leads similarly to the so called aronsson equation a f u 0 in this paper we show that these pde operators and various interesting generalizations also appear in several other contexts seemingly quite unrelated to l variational problems including two person game theory with random order of play rapid switching of states in control problems etc the resulting equations can be parabolic and inhomogeneous equation types precluded in conventional l variational problems

the eigenvalue problem the euler lagrange equation of the nonlinear rayleigh quotient left int omega nabla u p dx right bigg left int omega u p dx right is div left nabla u p 2 nabla u right lambda p p u p 2 u where lambda p p is the minimum value of the quotient the limit as p to infty of these equations is found to be max left lambda infty frac nabla u x u x delta infty u x right 0 where the constant lambda infty lim p to infty lambda p is the reciprocal of the maximum of the distance to the boundary of the domain

on viscosity solutions of fully nonlinear equations with measurable ingredients we study fully nonlinear uniformly elliptic equations with measurable ingredients caffarelli s recent work on w2 p estimates for viscosity solutions has led to significant progress in this area here we present a unified treatment of this theory based on an appropriate notion of viscosity solution for instance it is shown that strong solutions are viscosity solutions that viscosity solutions are twice differentiable a e and that the pointwise derivatives satisfy the equation a e an important consequence of our approach is the possibility of passage to various kinds of limits in fully nonlinear equations this extends results of this type due to evans and krylov our work is to some extent expository the main purpose being to provide an easily accessible set of tools and techniques to study equations with measurable ingredients 1996 john wiley sons inc

tug of war and the infinity laplacian we consider a class of zero sum two player stochastic games called tug of war and use them to prove that every bounded real valued lipschitz function f on a subset y of a length space x admits a unique absolutely minimal am extension to x i e a unique lipschitz extension u x for which lip u u lip u u for all open u x y

uniqueness of lipschitz extensions minimizing the sup norm of the gradient in this paper we examine the problem of minimizing the sup norm of the gradient of a function with prescribed boundary values geometrically this can be interpreted as finding a minimal lipschitz extension due to the weak convexity of the functional associated to this problem solutions are generally nonunique by adopting g aronsson s notion of absolutely minimizing we are able to prove uniqueness by characterizing minimizers as the unique solutions of an associated partial differential equation in fact we actually prove a weak maximum principle for this partial differential equation which in some sense is the euler equation for the minimization problem this is significantly difficult because the partial differential equation is both fully nonlinear and has very degenerate ellipticity to overcome this difficulty we use the weak solutions of m g crandall and p l lions also known as viscosity solutions in conjunction with some arguments using integration by parts

on the regularity theory of fully nonlinear parabolic equations ii recently m crandall and p l lions 3 developed a very successful method for proving the existence of solutions of nonlinear second order partial differential equations their method called the theory of viscosity solutions also applies to fully nonlinear equations in which even the second order derivatives can enter in nonlinear fashion solutions produced by the viscosity method are guaranteed to be continuous but not necessarily smooth here we announce smoothness results for viscosity solutions our methods extend those of 1 we obtain krylov safonov i e c estimates 8 c 1 schauder c and w estimates for viscosity solutions of uniformly parabolic equations in general form the results can be viewed as a priori estimates on the classical c solutions our method produces in particular regularity results for a broad new array of nonlinear heat equations including the bellman equation 6

existence uniqueness and qualitative properties of positive solutions of quasilinear elliptic equations we study the following quasilinear elliptic equation p pu x a x up 1 b x g u 0in rn where p 1 a b l rn b g 0 b 0 and lloc rn infrn we provide a sharp criterion in term of generalized principal eigenvalues for existence nonexistence of positive solution of p in suitable classes of functions uniqueness result for p in those classes is also derived under additional conditions on we further show that i either for every 0 nonexistence phenomenon occurs ii or there exists a threshold value 0 in the sense that for every 0 existence and uniqueness phenomenon occurs and for every nonexistence phenomenon occurs in the latter case we study the limits as 0 and of the sequence of positive solutions of p our results are new even in the case p 2

principal eigenvalue of a very badly degenerate operator and applications abstract in this paper we define and investigate the properties of the principal eigenvalue of the singular infinity laplace operator u d 2 u d u d u d u d u this operator arises from the optimal lipschitz extension problem and it plays the same fundamental role in the calculus of variations of l functionals as the usual laplacian does in the calculus of variations of l 2 functionals our approach to the eigenvalue problem is based on the maximum principle and follows the outline of the celebrated work of berestycki nirenberg and varadhan h berestycki l nirenberg s r s varadhan the principal eigenvalue and maximum principle for second order elliptic operators in general domains comm pure appl math 47 1 1994 47 92 in the case of uniformly elliptic linear operators as an application we obtain existence and uniqueness results for certain related non homogeneous problems and decay estimates for the solutions of the evolution problem associated to the infinity laplacian

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

impact of the infectious period on epidemics the duration of the infectious period is a crucial determinant of the ability of an infectious disease to spread we consider an epidemic model that is network based and non markovian containing classic kermack mckendrick pairwise message passing and spatial models as special cases for this model we prove a monotonic relationship between the variability of the infectious period with fixed mean and the probability that the infection will reach any given subset of the population by any given time for certain families of distributions this result implies that epidemic severity is decreasing with respect to the variance of the infectious period the striking importance of this relationship is demonstrated numerically we then prove with a fixed basic reproductive ratio r 0 a monotonic relationship between the variability of the posterior transmission probability which is a function of the infectious period and the probability that the infection will reach any given subset of the population by any given time thus again even when r 0 is fixed variability of the infectious period tends to dampen the epidemic numerical results illustrate this but indicate the relationship is weaker we then show how our results apply to message passing pairwise and kermack mckendrick epidemic models even when they are not exactly consistent with the stochastic dynamics for poissonian contact processes and arbitrarily distributed infectious periods we demonstrate how systems of delay differential equations and ordinary differential equations can provide upper and lower bounds respectively for the probability that any given individual has been infected by any given time

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

pairwise approximation for sir type network epidemics with non markovian recovery we present the generalized mean field and pairwise models for non markovian epidemics on networks with arbitrary recovery time distributions first we consider a hyperbolic partial differential equation pde system where the population of infective nodes and links are structured by age since infection we show that the pde system can be reduced to a system of integro differential equations which is analysed analytically and numerically we investigate the asymptotic behaviour of the generalized model and provide an implicit analytical expression involving the final epidemic size and pairwise reproduction number as an illustration of the applicability of the general model we recover known results for the exponentially distributed and fixed recovery time cases for gamma and uniformly distributed infectious periods new pairwise models are derived theoretical findings are confirmed by comparing results from the new pairwise model and explicit stochastic network simulation a major benefit of the generalized pairwise model lies in approximating the time evolution of the epidemic

eigenvalue and dirichlet problem for fully nonlinear operators in non smooth domains we study the maximum principle the existence of eigenvalue and the existence of solution for the dirichlet problem relative to operators which are fully nonlinear elliptic but presenting some singularity or degeneracy which are similar to those of the p laplacian we consider the equations in bounded domains which only satisfy the exterior cone condition

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

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

landesman lazer type results for second order hamilton jacobi bellman equations we study the boundary value problem f d2u du u x u f x u in u 0on where the second order differential operator f is of hamilton jacobi bellman type f is sub linear in u at infinity and rn is a regular bounded domain we extend the well known landesman lazer conditions to study various bifurcation phenomena taking place near the two principal eigenvalues associated to the differential operator we provide conditions under which the solution branches extend globally along the eigenvalue gap we also present examples illustrating the results and hypotheses

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