stable self similar blowup in energy supercritical yang mills theory we consider the cauchy problem for an energy supercritical nonlinear wave equation that arises in 1 5 dimensional yang mills theory a certain self similar solution w 0 of this model is conjectured to act as an attractor for generic large data evolutions assuming mode stability of w 0 we prove a weak version of this conjecture namely that the self similar solution w 0 is nonlinearly stable phrased differently we prove that mode stability of w 0 implies its nonlinear stability the fact that this statement is not vacuous follows from careful numerical work by bizon and chmaj that verifies the mode stability of w 0 beyond reasonable doubt

stable soliton resolution for exterior wave maps in all equivariance classes abstract in this paper we consider finite energy l equivariant wave maps from r t x 1 3 r b 0 1 s 3 with a dirichlet boundary condition at r 1 and for all l n each such l equivariant wave map has a fixed integer valued topological degree and in each degree class there is a unique harmonic map which minimizes the energy for maps of the same degree we prove that an arbitrary l equivariant exterior wave map with finite energy scatters to the unique harmonic map in its degree class i e soliton resolution this extends the recent results of the first second and fourth authors on the 1 equivariant equation to higher equivariance classes and thus completely resolves a conjecture of bizon chmaj and maliborski who observed this asymptotic behavior numerically the proof relies crucially on exterior energy estimates for the free radial wave equation in dimension d 2 l 3 which are established in the companion paper 13

global existence of harmonic maps in minkowski space on considere la construction d une classe d applications harmoniques classiques globales de l espace de minkowski m 4 dans une variete de riemann complete a n dimensions

on uniqueness for the harmonic map heat flow in supercritical dimensions we examine the question of uniqueness for the equivariant reduction of the harmonic map heat flow in the energy supercritical dimension d 3 it is shown that generically singular data can give rise to two distinct solutions that are both stable and satisfy the local energy inequality we also discuss how uniqueness can be retrieved 2017 wiley periodicals inc

on global existence and scattering for the wave maps equation we prove global existence and scattering for the wave maps equation in n 1 dimensions n 2 3 for initial data which is small in the scale invariant homogeneous besov space 2 1 n 2 x 2 1 n 2 1 this result was proved in an earlier paper by the author for n 4

on delay differential equations with impulses abstract sufficient conditions are obtained respectively for the asymptotic stability of the trivial solution of x t ax t j 1 bjx tj t tj t tj and for the existence of a nonoscillatory solution conditions are also obtained for all solutions to be oscillatory the asymptotic behaviour of an impulsively perturbed delay logistic equation is investigated as an extension to a nonlinear equation

interaction of kink type solutions of the harmonic map equations in this article the interaction of kink type solutions as defined below in eqs 1 5 1 6 of the harmonic map equations is studied it is shown that if two kinks enter an interaction two kinks emerge from it as a result the shape of the emerging kinks is related to the shape of the original ones via a certain goursat problem

a geometrical theory for spiral waves in excitable media in this paper we develop a geometrical theory for waves in excitable reacting media using singular perturbation arguments and dispersion of traveling plane wave trains we derive an approximate theory of wave front propagation which has strong resemblance to the geometrical diffraction theory of high frequency waves in hyperbolic systems governed by the eikonal equation using this theory we study the effect of curvature on waves in excitable media specifically rotating spiral patterns in planar regions from this theory we are able to determine the frequency and wavelength for spiral patterns in excitable nonoscillatory media

a note on persistence and extinction of a randomized food limited logistic population model this paper addresses the issue of the asymptotic behavior for a non autonomous randomized food limited logistic population model several sufficient conditions are formulated and proved for p moment persistence and extinction of the population as well as in sense of almost sure results show that food limited assumption has an influence on the convergence rate of the solution to the equilibria for the deterministic and stochastic model some previously known results are improved numerical simulations are provided to support the results

dynamics of a discrete food limited population model with time delay abstract in this paper we consider a discrete food limited population model with time delay firstly the stability of the equilibrium of the system is investigated by analyzing the characteristic equation by choosing the time delay as a bifurcation parameter we prove that neimark sacker bifurcations occur when the delay passes a sequence of critical values then the explicit algorithm for determining the direction of the neimark sacker bifurcations and the stability of the bifurcating periodic solutions are derived finally some numerical simulations are given to verify the theoretical analysis

construction of type ii blowup solutions for the 1 corotational energy supercritical wave maps abstract we consider the energy supercritical wave maps from r d into the d sphere s d with d 7 under an additional assumption of 1 corotational symmetry the problem reduces to the one dimensional semilinear wave equation t 2 u r 2 u d 1 r r u d 1 2 r 2 sin 2 u we construct for this equation a family of c solutions which blow up in finite time via concentration of the universal profile u r t q r t where q is the stationary solution of the equation and the speed is given by the quantized rates t c u t t l l n l d 1 2 the construction relies on two arguments the reduction of the problem to a finite dimensional one thanks to a robust universal energy method and modulation techniques developed by merle raphael and rodnianski 49 for the energy supercritical nonlinear schrodinger equation then we proceed by contradiction to solve the finite dimensional problem and conclude using the brouwer fixed point theorem

characterization of large energy solutions of the equivariant wave map problem ii we consider 1 equivariant wave maps from bbb r 1 2 to bbb s 2 of finite energy we establish a classification of all degree one global solutions whose energies are less than three times the energy of the harmonic map q in particular for each global energy solution of topological degree one we show that the solution asymptotically decouples into a rescaled harmonic map plus a radiation term together with a companion article part i where we consider the case of finite time blow up this gives a characterization of all 1 equivariant degree 1 wave maps in the energy regime e q 3e q

scattering for wave maps exterior to a ball abstract we consider 1 equivariant wave maps from r t r x 3 b s 3 where b is a ball centered at 0 and b gets mapped to a fixed point on s 3 we show that 1 equivariant maps of degree zero scatter to zero irrespective of their energy for positive degrees we prove asymptotic stability of the unique harmonic maps in the energy class determined by the degree

uniqueness of weak solutions of 1 1 dimensional wave maps in this paper we prove that finite energy distributional solutions to the cauchy problem of wave maps from 1 1 dimensional minkowsky space to any complete riemannian manifold are unique

solution of lambda omega systems theta schemes and multigrid methods the numerical solution of a time dependent reaction diffusion lambda omega system discretized by theta schemes and finite differences is considered stability and accuracy of finite difference theta schemes for this system are established to solve the time implicit evolution equations a nonlinear multigrid method is applied the convergence properties of this solver are investigated considering a linearized lambda omega model

oscillation of solutions of second order mixed nonlinear differential equations under impulsive perturbations new oscillation criteria are obtained for second order forced mixed nonlinear impulsive differential equations of the form r t f a x q t f a x k 1nq k t f b k x e t t q ix q i a ix q i x q i b ix q i where f c s s c 1s and b 1 b 2 b m a b m 1 b n 0 if a 1 and the impulses are dropped then the results obtained by sun and wong y g sun j s w wong oscillation criteria for second order forced ordinary differential equations with mixed nonlinearities j math anal appl 334 2007 549 560 are recovered examples are given to illustrate the results

the defocusing energy supercritical nonlinear wave equation in three space dimensions we consider the defocusing nonlinear wave equation u tt u u p u 0 in the energy supercritical regime p 4 for even values of the power p we show that blowup or failure to scatter must be accompanied by blowup of the critical sobolev norm an equivalent formulation is that solutions with bounded critical sobolev norm are global and scatter the impetus to consider this problem comes from recent work of kenig and merle who treated the case of spherically symmetric solutions

on existence and scattering with minimal regularity for semilinear wave equations abstract we prove existence and scattering results for semilinear wave equations with low regularity data we also determine the minimal regularity that is needed to ensure local existence and well posedness and we give counterexamples to well posedness more specifically we show that equations of the type u u p with initial data u u t in r n 1 r n have a local solution if p n and we construct counterexamples if p n the existence results rely on mixed norm space time estimates of strichartz type

hopf bifurcations in a reaction diffusion population model with delay effect abstract a reaction diffusion population model with a general time delayed growth rate per capita is considered the growth rate per capita can be logistic or weak allee effect type from a careful analysis of the characteristic equation the stability of the positive steady state solution and the existence of forward hopf bifurcation from the positive steady state solution are obtained via the implicit function theorem where the time delay is used as the bifurcation parameter the general results are applied to a food limited population model with diffusion and delay effects as well as a weak allee effect population model

on a stochastic logistic equation with impulsive perturbations a stochastic logistic model with impulsive perturbations is proposed and investigated first we give a new definition of a solution of an impulsive stochastic differential equation isde which is more convenient for use than the existing one using this definition we show that our model has a global and positive solution and obtain its explicit expression then we establish the sufficient conditions for extinction non persistence in the mean weak persistence persistence in the mean and stochastic permanence of the solution the critical value between weak persistence and extinction is obtained in addition the limit of the average in time of the sample path of the solution is estimated by two constants afterwards the lower growth rate and the upper growth rate of the solution are estimated finally sufficient conditions for global attractivity are established

mixed problems for fully nonlinear hyperbolic equations 1 i n the summation convention is used here i j n f 2 c l r is a bounded domain with smooth boundary f v the outer normal f2t 0 t x f2 and ft 0 t x f 0 t oo f and g satisfy some natural conditions which assure hyperbolicity the main result is concerned with classical solutions for small t assume that uoe ws l f2 n and ul e ws f2 n where s is an integer bigger than 1 n 2 this implies uo e c2 0 n under suitable assumptions the following holds

stable rotating waves in two dimensional discrete active media the existence and stability of stable rotating solutions spiral waves in a discrete system of coupled phase models is proven monotone methods are used to obtain existence and qualitative features of the solutions an application of fenichel s results for singular perturbation implies the existence for a one variable model for excitability numerical comparisons with the phase model and a realistic membrane model are made a numerically computed hopf bifurcation shows the existence of the wobbling core observed in spatially continuous models

global well posedness scattering and blow up for the energy critical focusing non linear wave equation we study the energy critical focusing non linear wave equation with data in the energy space in dimensions 3 4 and 5 we prove that for cauchy data of energy smaller than the one of the static solution w which gives the best constant in the sobolev embedding the following alternative holds if the initial data has smaller norm in the homogeneous sobolev space h1 than the one of w then we have global well posedness and scattering if the norm is larger than the one of w then we have break down in finite time

endpoint strichartz estimates we prove an abstract strichartz estimate which implies previously unknown endpoint strichartz estimates for the wave equation in dimension n 4 and the schrodinger equation in dimension n 3 three other applications are discussed local existence for a nonlinear wave equation and strichartz type estimates for more general dispersive equations and for the kinetic transport equation

oscillation properties of nonlinear impulsive delay differential equations and applications to population models comparison theorem and explicit sufficient conditions are obtained for oscillation and nonoscillation of solutions of nonlinear impulsive delay differential equations which can be utilized to population dynamic models our results in this paper generalize and improve several known results

ill posedness for one dimensional wave maps at the critical regularity we show that the wave map equation in r 1 1 is in general ill posed in the critical space 1 2 and the besov space 1 2 1 2 the problem is attributed to the bad behavior of the one dimensional bilinear expression d 1 fdg in these spaces

picone s formula for linear non selfadjoint impulsive differential equations in this paper we derive a picone type formula for second order linear non selfadjoint impulsive differential equations having fixed moments of impulse actions and obtain a wirtinger type inequality a leighton type comparison theorem and a sturm picone comparison theorem for such equations moreover several oscillation criteria are also derived as applications

dynamics of high order bam neural networks with and without impulses the main aim of this paper is to study the existence and global exponential stability of periodic solution for high order bidirectional associative memory bam neural networks with and without impulses easily verifiable sufficient conditions are established the method is based on coincidence degree theory as well as a priori estimates and lyapunov functional it is shown that the convergence characteristics of periodic solution for the impulsive system are preserved by the corresponding nonimpulsive system with some restriction imposed on the impulse effect numerical simulation results are given to support the theoretical predictions

global attractivity in a food limited population model with impulsive effects abstract sufficient conditions are obtained for the global attractivity of the positive equilibrium of the delay logistic equation with impulsive effect n t r t n t k n p t k c t n p t n k n k exp i k ln n k k p where r t c t are continuous functions p k are positive constants and k k 1 denotes the sequence of impulsive time which assumed to be strictly increasing some special cases are also considered

persistence of periodic patterns for perturbed biological oscillators abstract we apply the perturbation method and the implicit function theorem to study the persistence of periodic patterns for two perturbed general biological oscillators one weakly coupled ordinary differential equation and one reaction diffusion system the proof relies also on the formal adjoint equation theory and the lyapunov schmidt method for the perturbed reaction diffusion model spatial temporal patterns such as rotating waves are shown to exist numerical simulations are presented to illustrate our analytical results

perron s theorem for linear impulsive differential equations with distributed delay in this paper it is shown that under a perron condition trivial solution of linear impulsive differential equation with distributed delay is uniformly asymptotically stable

rotating spiral waves in λ ω systems on circular domains abstract systems are reaction diffusion systems whose reaction kinetics admit a stable limit cycle it is known that systems can possess various types of solutions among them spiral waves are the most fascinating pattern however the effects of the diffusivity the sizes of the domains and the reaction kinetics on spiral waves are largely unknown in this paper we investigate how these quantities affect the properties of m armed spiral waves in a generalized class of system on a circular disk with no flux boundary condition first we derive a criterion for the existence of m armed spiral waves specifically we show that m armed spiral waves do not exist for d 0 r 2 j m 2 while for d 0 0 r 2 j m 2 there exists an m armed spiral wave if the twist parameter q is small here d is the diffusivity for the system r is the radius of the circular disk 0 is the value of the function a at a 0 and j m is the first positive zero of the first derivative of the bessel function of the first kind of order m we also show that the critical diffusivity d 0 r 2 j m 2 is a bifurcation point next we use the numerical simulation to show that for small twist parameter the rotational frequency increases with increasing domain size while for large twist parameter the dependence of the rotational frequency on the domain size is not monotonic moreover small circular domains may change the properties of spiral waves drastically these numerical results are in contrast to those in excitable media finally the stability of spiral waves is investigated numerically

environmental periodicity and time delays in a food limited population model abstract sufficient conditions are obtained for the existence of a globally attracting positive periodic solution of the food limited population system modelled by the equation n t r t k t n t m k t c t r t n t m where m is a nonnegative integer and k r c are continuous positive periodic functions of period

oscillation and stability of linear impulsive delay differential equations the main results of this paper are that the oscillation and the stability of a linear impulsive delay differential equation are equivalent respectively to the oscillation and the stability of a corresponding linear delay differential equation without impulses the results of this paper can be used to improve and derive some of the known results in the literature

on harmonic maps from r1 1 into hilbert loop groups motivated by the works of gu pure appl math 33 727 1980 ginibre and velo ann phys 142 393 1982 on classical solutions of a two dimensional model the global existence of harmonic maps from the minkowski plane r1 1 into hilbert loop groups is proved the construction of the harmonic maps strongly depends on some geometric properties of the loop groups

profile decompositions for wave equations on hyperbolic space with applications the goal for this paper is twofold our first main objective is to develop bahouri gerard type profile decompositions for waves on hyperbolic space recently such profile decompositions have proved to be a versatile tool in the study of the asymptotic dynamics of solutions to nonlinear wave equations with large energy with an eye towards further applications we develop this theory in a fairly general framework which includes the case of waves on hyperbolic space perturbed by a time independent potential our second objective is to use the profile decomposition to address a specific nonlinear problem namely the question of global well posedness and scattering for the defocusing energy critical semi linear wave equation on three dimensional hyperbolic space possibly perturbed by a repulsive time independent potential using the concentration compactness rigidity method introduced by kenig and merle we prove that all finite energy initial data lead to a global evolution that scatters to linear waves as t rightarrow pm infty this proof serves as a blueprint for the arguments in the work 45 by the authors where we study the asymptotic behavior of large energy equivariant wave maps on the hyperbolic plane

lyapunov type inequalities for second order forced mixed nonlinear impulsive differential equations in this paper we present some new lyapunov and hartman type inequalities for second order forced impulsive differential equations with mixed nonlinearities x t p t x t 1 x t q t x t 1 x t f t t i x t p i x t 1 x t q i x t 1 x t f i t i where p q f are real valued functions pi qi fi are real sequences and 0 1 2 no sign restrictions are imposed on the potential functions p q and the forcing term f and the sequences pi qi fi the inequalities obtained generalize and complement the existing results for the special cases of this equation in the literature

stability switches and hopf bifurcations of an isolated population model with delay dependent parameters the dynamical behaviors of an isolated population model involving delay dependent parameters are investigated it is shown that the positive equilibrium switches from being stable to unstable and then back to stable as the delay increases and the hopf bifurcation occurs finite times between the two critical values of stability changes which can be analytically determined moreover the bifurcating periodic solutions are expressed analytically in an approximate form by the perturbation approach and floquet technique the direction and stability of the bifurcating periodic solutions are also determined finally the validity of the results is shown by the consistency with the numerical simulations

conditional global existence and scattering for a semi linear skyrme equation with large data we study a generalization of energy super critical wave maps due to adkins and nappi that can also be viewed as a simplified version of the skyrme model these are maps from 1 3 dimensional minkowski space that take values in the 3 sphere and it follows that every finite energy adkins nappi wave map has a fixed topological degree which is an integer here we initiate the study of the large data dynamics for adkins nappi wave maps by proving that there is no type ii blow up in the class of maps with topological degree zero in particular any degree zero map whose critical norm stays bounded must be global in time and scatter to zero as t to pm infty

nondispersive radial solutions to energy supercritical non linear wave equations with applications in this paper we establish optimal pointwise decay estimates for non dispersive compact radial solutions to non linear wave equations in 3 dimensions in the energy supercritical range as an application we show for the full energy supercritical range in the defocusing case that if the scale invariant sobolev norm of a radial solution remains bounded in its maximal interval of existence then the solution must exist for all times and scatter

high frequency approximation of solutions to critical nonlinear wave equations this work is devoted to the description of bounded energy sequences of solutions to the equation 1 u u 4 0 in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i up to remainder terms small in energy norm and in every strichartz norm the proof relies on scattering theory for 1 and on a structure theorem for bounded energy sequences of solutions to the linear wave equation in particular we infer the existence of an a priori estimate of strichartz norms of solutions to 1 in terms of their energy

global well posedness and scattering for the defocusing energy supercritical cubic nonlinear wave equation abstract in this paper we consider the defocusing cubic nonlinear wave equation u t t u u 2 u 0 in the energy supercritical regime in dimensions d 6 with no radial assumption on the initial data we prove that if a solution satisfies an a priori bound in the critical homogeneous sobolev space throughout its maximal interval of existence that is u l t h x s c h x s c 1 then the solution is global and it scatters our analysis is based on the methods of the recent works of kenig and merle 2008 21 and killip and visan 2010 26 27 treating the energy supercritical nonlinear schrodinger and wave equations

spatially structured activity in synaptically coupled neuronal networks i traveling fronts and pulses we consider traveling front and pulse solutions to a system of integro differential equations used to describe the activity of synaptically coupled neuronal networks in a single spatial dimension our first goal is to establish a series of direct links between the abstract nature of the equations and their interpretation in terms of experimental findings in the cortex and other brain regions this is accomplished first by presenting a biophysically motivated derivation of the system and then by establishing a framework for comparison between numerical and experimental measures of activity propagation speed our second goal is to establish the existence of traveling pulse solutions using more rigorous methods two techniques are presented the first a shooting argument reduces the problem from finding a specific solution to an integro differential equation system to finding any solution to an ode system the second a singular perturbation argument provides a construction of traveling pulse solutions un

the existence of spiral waves in an oscillatory reaction diffusion system rotating waves are proven to exist on the unit disk for an oscillatory reaction diffusion equation with neuman boundary conditions the method of proof relies on a two parameter shooting argument for the ensemble frequency and the radial derivative of the magnitude numerical solutions indicate that the waves are stable if the diffusion is sufficiently small it is also shown that these solutions cease to exist for large diffusion the origin of the rotating waves is discussed key words reaction diffusion equations spiral waves oscillatory media

weak solutions and development of singularities of the su 2 σ model pour toutes donnees initiales d energie finie le modele sigma o k a des solutions faibles globales en chaque dimension d espace temps

synchrony stability and firing patterns in pulse coupled oscillators we study non trivial firing patterns in small assemblies of pulse coupled oscillatory maps we find conditions for the existence of waves in rings of coupled maps that are coupled bi directionally we also find conditions for stable synchrony in general all to all coupled oscillators surprisingly we find that for maps that are derived from physiological data the stability of synchrony depends on the number of oscillators we describe rotating waves in two dimensional lattices of maps and reduce their existence to a reduced system of algebraic equations which are solved numerically

the effect of boundaries on the asymptotic wavenumber of spiral wave solutions of the complex ginzburg landau equation abstract in this paper we consider an oscillatory medium whose dynamics are modeled by the complex ginzburg landau equation in particular we focus on n armed spiral wave solutions of the complex ginzburg landau equation in a disk of radius d with homogeneous neumann boundary conditions it is well known that such solutions exist for small enough values of the twist parameter q and large enough values of d we investigate the effect of boundaries on the rotational frequency of the spirals which is an unknown of the problem uniquely determined by the parameters d and q we show that there is a threshold in the parameter space where the effect of the boundary on the rotational frequency switches from being algebraic to exponentially weak we use the method of matched asymptotic expansions to obtain explicit expressions for the asymptotic wavenumber as a function of the twist parameter and the domain size for small values of q

on existence of a globally attractive periodic solution of impulsive delay logarithmic population model in this paper it is shown that a logarithmic population model which is governed by impulsive delay differential equation has a globally attractive periodic solution

harmonic maps in three dimensional minkowski space with image near geodesics shatah i sh has constructed global weak solutions of the nonlinear sigma model in h1 2 for arbitrary dimension moreover for the case of maps from four dimensional minkowski space r 3 1 to sphere s 3 he has found an example of smooth symmetric initial data for which the corresponding solution develops singularities on the second order derivatives in finite time showing that the large initial data may contribute the formation of singularities to avoid this situation sideris i si proved the existence of global smooth solution from r 3 1 into an n dimensional complete riemannian manifold n as a perturbation of a special class of harmonic maps geodesic geometrically the solution lies in a small tubular region of a given geodesic on n instead of a small neighborhood of a point on n therefore the initial data is partly small the purpose of this paper is to extend sideris s result to the case where domain is the minkowski space r 2 1 in this case the existence of smooth solution of small amplitude was proved by kovalyov ko but ours is more general since the parameter along the geodesic may vary freely formally a harmonic map from the minkowski space r 2 1 into an n dimensional complete riemannian manifold n h is described as a critical point r 2 1 n of the energy integral

scattering for h 1 2 bounded solutions to the cubic defocusing nls in 3 dimensions we show that if a solution of the defocusing cubic nls in 3d remains bounded in the homogeneous sobolev norm of order 1 2 in its maximal interval of existence then the interval is infinite and the solution scatters no radial assumption is made

the wave maps equation the wave maps equation has become a very popular topic in recent years the aim of these expository notes is to present a non technical survey of the ideas and methods which have proved useful in the study of wave maps leading up to the latest results the remaining open problems are also stated and explained

no blow up to a variational wave equation in liquid crystals we establish the global existence of smooth solutions to the cauchy problem for a system of variational wave equations in one space dimension modeling a type of nematic liquid crystals that has equal splay and bend coefficients

boundary equations and regularity theory for geometric variational systems with neumann data we study boundary regularity of maps from two dimensional domains into manifolds which are critical with respect to a generic conformally invariant variational functional and which at the boundary intersect perpendicularly with a support manifold for example harmonic maps or h surfaces with a partially free boundary condition in the interior it is known by the celebrated work of riviere that these maps satisfy a system with an antisymmetric potential from which one can derive the interior regularity of the solution avoiding a reflection argument we show that these maps satisfy along the boundary a system of equations which also exhibits a nonlocal antisymmetric potential that combines information from the interior potential and the geometric neumann boundary condition we then proceed to show boundary regularity for solutions to such systems