reaction diffusion processes and evolution to harmonic maps initial boundary value problems are considered for the reaction diffusion equation u 1 varepsilon delta u varepsilon 1 f u with x in a domain omega in r and u in r first the asymptotic behavior of u is determined for varepsilon small when f u 0 on a connected manifold m of stable equilibrium points it is found that a tends rapidly to m being driven by reaction then u evolves slowly by diffusion restricted tom it tends ultimately to a limit that is a harmonic map of omega into m next the case where f u has stable equilibrium points on two manifolds m 1 and m 2 is treated in this case a front develops in omega it separates the regions where u is close to m 1 from the regions whereu is close to m 2 for f u v n u a boundary layer solution is constructed for u near the front and the velocity of the front is found to be proportional to the jump in v across it to leading order in varepsilon when v u has the same v

surfaces immersed in lie algebras obtained from the sigma models we study some geometrical aspects of two dimensional orientable surfaces arising from the study of sigma models to this aim we employ an identification of with the lie algebra by means of which we construct a generalized weierstrass formula for immersion of such surfaces the structural elements of the surface like its moving frame the gauss weingarten and the gauss codazzi ricci equations are expressed in terms of the solution of the model defining it further the first and second fundamental forms the gaussian curvature the mean curvature vector the willmore functional and the topological charge of surfaces are expressed in terms of this solution we present detailed implementation of these results for surfaces immersed in and lie algebras

pluriharmonic maps into kahler symmetric spaces and sym s formula a construction due to sym and bobenko recovers constant mean curvature surfaces in euclidean 3 space from their harmonic gauss maps we generalize this construction to higher dimensions and codimensions replacing the surface by a complex manifold and the sphere the target space of the gauss map by a kahler symmetric space of compact type with its standard embedding into the lie algebra mathfrak g of its transvection group thus we obtain a new class of immersed kahler submanifolds of mathfrak g and we derive their properties

stability of an elastic rod buckling into a soft wall the conjugate point theory of the calculus of variations is extended to apply to the buckling of an elastic rod in an external field we show that the operator approach presented by manning mannin

a dynamical system approach to the self similar large time behavior in scalar convection diffusion equations abstract we present a dynamical system method that provides both the existence of self similar solutions and the self similar large time behavior for convection diffusion equations in r n this method avoids a direct study of the elliptic problem related to the self similar profiles we concentrate our attention on the model example u t u a u 1 n u g x t in r n 0 u 0 u 0 l 1 r n l r n where a r n and g x t t 1 n 2 1h x formula with h l2 r n exp x 2 4 and l r n such that h y dy 0 and h y y dy being small enough in the natural similarity variables the self similar profiles become stationary solutions of a new convection diffusion equation by using lyapunov type arguments that rely in an essential manner on the l1 contraction property of the system we prove that those stationary solutions exists and that any trajectory converges to one of them the limit of any trajectory is completely determined by its mass which is conserved along the time in order to ensure the relative compactness of trajectories we work in the functional framework of the weighted sobolev spaces introduced by escobedo and kavian

the static deformation of a twisted elastic rod constrained to lie on a cylinder the cosserat director theory is used to formulate the problem of a long thin weightless rod constrained by suitable distributed forces to lie on a cylinder while being held by end tension and twisting moment applications of this problem are found for instance in the buckling of drill strings inside a cylindrical hole in the case of a rod of isotropic cross section the equilibrium equations can be reduced to those of a one degree of freedom oscillator in terms of the angle that the local tangent to the rod makes with the axis of the cylinder depending on the radius of the cylinder and the applied load the oscillator has several fixed points each of which corresponds to a different helical solution of the rod more complicated shapes are also possible and special attention is given to localized configurations described by homoclinic orbits of the oscillator heteroclinic saddle connections are found to play an important role in the post buckling behaviour by defining critical loads at which a straight rod may coil up into a helix

asymptotics for the burgers equation with pumping we consider the large time asymptotic behavior of solutions to the initial boundary value problem left u t e t u u x u xx 0 x in bf r t 0 u left 0 x right u 0 left x right x in bf r u left t x right rightarrow a pm x rightarrow pm infty t 0 right we find large time asymptotic formulas of solutions for three different cases 1 a 1 2 a 1 and 3 a 0

asymptotic decomposition of nonlinear dispersive wave equations with dissipation abstract provided 0 solutions of the generalized regularized long wave burgers equation u t u x p u x u xx u xxt 0 that begin with finite energy decay to zero as t becomes unboundedly large consideration is given here to the case where p vanishes at least cubically at the origin in this case solutions of may be decomposed exactly as the sum of a solution of the corresponding linear equation and a higher order correction term an explicit asymptotic form for the l 2 norm of the higher order correction is presented here the effect of the nonlinearity is felt only in the higher order term a similar decomposition is given for the generalized korteweg de vries burgers equation u t u x p u x u xx u xxx 0

decay of solutions of some nonlinear wave equations abstract we study the large time behaviour of solutions to the initial value problem for the korteweg de vries equation and for the regularized long wave equation with a dissipative term appended using energy estimates a maximum principle and a transformation of cole hopf type sharp rates of temporal decay of certain norms of the solution are obtained

large time behavior of the vorticity of two dimensional viscous flow and its application to vortex formation we consider the cauchy problem for the two dimensional vorticity equation we show that the solution behaves like a constant multiple of the gauss kernel having the same total vorticity as time tends to infinity no particular structure of initial data 0 x 0 is assumed except the restriction that the reynolds numberr 0 dx v is small wherev is the kinematic viscosity applying a time dependent scale transformation we show a stability of burgers vortex which physically implies formation of a concentrated vortex

harmonic maps into lie groups classical solutions of the chiral model on considere deux aspects de la structure algebrique de 7b m l espace des applications harmoniques d un domaine a 2 dimensions simplement connexe dans un groupe de lie reel g r la forme reelle d un groupe complexe g on construit une representation du groupe boucle 7b a s 1 g r sur 7b m correspondant a l algebre de lie de kacmoody des deformations infinitesimales on decrit l action sur 7b m et l action d un sous groupe sur l espace des applications harmoniques dans des gaussmanniens dans une seconde partie on se restreint a une theorie qui s applique seulement quand est un domaine riemannien a 2 dimensions simplement connexe et le groupe est g r u n

asymptotics in the critical case for whitham type equations abstract consider the cauchy problem for nonlinear dissipative evolution equations u t n u u l u 0 x r t 0 u 0 x u 0 x x r where l is the linear pseudodifferential operator l u f x l u and the nonlinearity is a quadratic pseudodifferential operator n u u f x r a t y u t y u t y d y u f x u is direct fourier transformation let the initial data u 0 h 0 h 0 1 2 are sufficiently small and have a non zero total mass m u 0 x d x 0 here h n m l 2 x m i x n x l 2 is the weighted sobolev space then we prove that the main term of the large time asymptotics of solutions in the critical case is given by the self similar solution defined uniquely by the total mass m of the initial data

asymptotic profiles for covection diffusion equations with variable diffusion abstract we investigate the large time behavior of solutions of the convection diffusion equation u t div a x u d u q 1 u d r n in 0 r n with integrable initial data u0 x we take a x 1 b x 0 with b smooth and decaying to zero fast enough as x when q 1 1 n it is known that the solutions behave in a first approximation like the solutions of the head equation taking the same initial data as t we show here the influence of the nonlinear term and the variable diffusion in the large time behavior by obtaining the second term in the asymptotic development of solutions as t

uniform decay rated for parabolic conservations laws on etablit des taux de decroissance uniformes optimaux pour des solutions du probleme de cauchy avec donnees grandes pour des lois de conservation paraboliques scalaires a plusieurs dimensions d espace

large time behaviour of the solutions of a semilinear parabolic equation in rn abstract the asymptotic behaviour as t tends to of the solution of u t u u u p 1 0 in r n r p 1 was studied it was proved that the behaviour depends strongly on the sign of n 2 n p and also on the rate of decay of the admissible initial data u 0 x as x tends to

isoperimetric conjugate points with application to the stability of dna minicircles a conjugate point test determining an index of the constrained second variation in one dimensional isoperimetric calculus of variations problems is described the test is then implemented numerically to determine stability properties of equilibria within a continuum mechanics model of dna minicircles

source solutions and asymptotic behavior in conservation laws abstract we study the uniqueness of the solutions to the scalar conservation law u t u x 0 when the initial datum is a finite measure the case of a dirac mass is particularly emphasized it is shown how it provides a description of the asymptotic behavior of the solutions initiated by an arbitrary integrable function this behavior is proved to depend on one parameter in the case when is odd while it depends on two when is convex

decay estimates for the solutions of some nonlinear evolution equations abstract we study decay estimates for the solutions and their derivatives to the initial value problems for some generalized nonlinear evolution equations which have lower order diffusion terms

conjugate points revisited and neumann neumann problems the theory of conjugate points in the calculus of variations is reconsidered with a perspective emphasizing the connection to finite dimensional optimization the object of central importance is the spectrum of the second variation operator analogous to the eigenvalues of the hessian matrix in finite dimensions with a few basic properties of this spectrum one can gain a new perspective on the classic result that stability requires the lack of conjugate points furthermore we show how the spectral perspective allows the extension of the conjugate point approach to variants of the classic problems in the literature such as problems with neumann neumann boundary conditions

the weak solutions to the evolution problems of harmonic maps on demontre l existence de solutions faibles globales a des problemes d evolution des applications harmoniques de varietes de riemann compactes sans bord dans des spheres

temporal asymptotic behavior of solutions of the benjamin ono burgers equation abstract we study certain similarity solutions of the benjamin ono burgers bob equation and their role in the asymptotic behavior of the general solution for small initial data in l 1 r we prove that a solution of the bob equation exists in bc r l 1 r and depends continuously on its initial data results about existence uniqueness regularity and spatial asymptotics of solutions of a similarity reduction of the bob equation are proved furthermore the solutions of the bob equation are proved to converge as t to appropriate similarity solutions faster than the typically sharp rate of decay of the solutions

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

asymptotic behaviour of nonlinear parabolic equations with critical exponents a dynamical systems approach we investigate the large time behaviour of solutions to the nonlinear heat conduction equation with absorption ut u 1 u in q rn 0 e with n 1 0 and critical absorption exponent 1 2n the initial function u x 0 0 is assumed to be integrable nonnegative and compactly supported we prove that u converges as t to a unique self similar function which is a contracted version of one of the asymptotic profiles of the nonabsorptive problem ut u 1 the same for any initial data the cornerstone of the proof is a result about limits of infinite dimensional asymptotical dynamical systems combining this result with an asymptotic evaluation of the mass function as well as typical pde estimates gives the behaviour of e for large times similar unusual asymptotic behaviour is obtained for the equation ut div du du u with same conditions on and u x 0 and critical value for 1 2 n

pluriharmonic maps of maximal rank if f m p is a pluriharmonic map of a complex manifold m into a compact symmetric space p then the image of any holomorphic tangent space t xm under df x is a flat subspace of t p cp where p f x thus the maximal dimension of flat subspaces of t p cp other than t p p and t p p if p is hermitian is a bound for the rank of any non holomorphic pluriharmonic map into p we give examples of pluriharmonic maps of maximal rank when p is a complex grassmannian in fact we classify all those maps under an additional condition isotropy

constrained euler buckling an interplay of computation and analysis we consider elastic buckling of an inextensible beam with hinged ends and fixed end displacements confined to the plane and in the presence of rigid frictionless sidewalls which constrain overall lateral displacements we formulate the geometrically nonlinear euler problem and develop global search and path following algorithms to find equilibria in various classes satisfying different contact patterns and hence boundary conditions we derive complete analytical results for the case of line contacts with the sidewalls and partial results for point contact and mixed cases the analysis is essential to understanding the numerical results for in contrast to the unconstrained problem we find a very rich bifurcation structure with the cardinality of branches growing exponentially with mode number

entropy solutions for diffusion convection equations with partial diffusivity we consider the cauchy problem for the following scalar conservation law with partial viscosity ut a u 0 f u x y e r n t o the existence of solutions is proved by the vanishing viscosity method by introducing a suitable entropy condition we prove uniqueness of solutions this entropy condition is inspired by the entropy criterion introduced by kruzhkov for hyperbolic conservation laws but it takes into account the effect of diffusion introduction we present a theory of existence and uniqueness for suitable entropy solutions of a nonlinear equation which describes the combined effects of diffusion and convection of matter it has the form 0 1 ut axu ayf u posed in the space rn n 2 denoted by the variable z x y x e rn y e r the main characteristic of this equation is that it has mixed parabolichyperboolic type due to the directional separation of the diffusion and convection effects while matter is convected along the y axis it is simultaneously diffused along all orthogonal directions our interest in the model is not merely academic we have encountered it while studying in evz the asymptotic behavior of a standard nonlinear diffusion convection model of the form 0 2 ut azu ayf u in the case f u ululq l for 1 q n 1 n notice that azu axu o2u in fact certain self similar solutions of equation 0 1 will give the large time behaviour of the nonnegative solutions of 0 2 having finite mass the unique characterization of solutions of 0 1 is essential in establishing the asymptotic behaviour of 0 2 the comparison of both equations suggests the name reduced equation for 0 1 the existence of solutions of equation 0 1 can be obtained by the classical method of adding a vanishing artificial viscosity in other terms a diffusion in received by the editors december 21 1992 1991 mathematics subject classification primary 35l65 35k65 35a05