scaling variables and asymptotic expansions in damped wave equations abstract we study the long time behavior of small solutions to the nonlinear damped wave equationeu u a u n u u u r 0 whereeis a positive not necessarily small parameter we assume that the diffusion coefficienta converges to positive limitsa as and that the nonlinearity n u u u vanishes sufficiently fast asu 0 introducing scaling variables and using various energy estimates we compute an asymptotic expansion of the solutionu in powers of 1 2as and we show that this expansion is entirely determined up to the second order by a linear parabolic equation which depends only on the limiting valuesa in particular this implies that the small solutions of the damped wave equation behave for large like those of the parabolic equation obtained by settinge 0

renormalization group and asymptotics of solutions of nonlinear parabolic equations we present a general method for studying long time asymptotics of nonlinear parabolic partial differential equations the method does not rely on a priori estimates such as the maximum principle it applies to systems of coupled equations to boundary conditions at infinity creating a front and to higher possibly fractional differential linear terms we present in detail the analysis for nonlinear diffusion type equations with initial data falling off at infinity and also for data interpolating between two different stationary solutions at infinity in an accompanying paper 5 the method is applied to systems of equations where some variables are slaved such as the complex ginzburg landau equation c 1994 john wiley sons inc

convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping we consider a model of hyperbolic conservation laws with damping and show that the solutions tend to those of a nonlinear parabolic equation time asymptotically the hyperbolic model may be viewed as isentropic euler equations with friction term added to the momentum equation to model gas flow through a porous media in this case our result justifies darcy s law time asymptotically our model may also be viewed as an elastic model with damping

upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation abstract for a smooth bounded domain r n n 3 and e a real parameter consider the hyperbolic equation eu tt u t u u g in with dirichlet boundary conditions under certain conditions on this equation has a compact attractor a e in h 0 1 l 2 for e 0 the parabolic equation also has a compact attractor which can be naturally embedded into a compact set a 0 in h 0 1 l 2 it is shown that for any neighborhood u of a 0 the set a e u for e small

the singular limit dynamics of semilinear damped wave equations on considere la dynamique qualitative d une equation d onde amortie semi lineaire a une dimension dans la limite singuliere correspondant a l annulation du coefficient de la derivee d ordre 2 en temps

long time asymptotic expansion for the damped semilinear wave equation the first initial boundary value problem is considered for the damped semilinear wave equation with the quadratic nonlinearity for small initial data and homogeneous boundary conditions its solution is constructed in the form of a series in the eigenfunctions of the laplace operator the long time asymptotic expansion is obtained which shows the nonlinear effects of amplitude and frequency multiplication the same results hold for the admissible initial data that are not small

an equivalence of h 1 norms for the simple exclusion process resolvent h 1 norms with respect to simple exclusion processes play an important role in many problems with respect to additive functionals tagged particles and hydrodynamics among other concerns here general translation invariant finite range simple exclusion processes with and without a distinguished particle are considered for the standard system of indistinguishable particles it is proved that the corresponding h 1 norms are equivalent in a sense to the h 1 norms of a nearest neighbor system the same result holds for systems with a distinguished particle in dimensions d 2 however in dimension d 1 this equivalence does not hold an application of the h 1 norm equivalence to additive functional variances is also given

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

continuity of non autonomous attractors for hyperbolic perturbation of parabolic equations abstract in this work we prove the lower and upper semicontinuity of pullback uniform and cocycle attractors for the non autonomous dynamical system given by hyperbolic equation on a bounded domain r 3 u t t u t u f t u for each 0 this equation has uniform pullback and cocycle attractors in h 0 1 l 2 and for 0 the limit parabolic equation u t u f 0 u has a global attractor a 0 in h 0 1 which can be naturally embedded into a compact set a 0 in h 0 1 l 2 we prove that all three types of non autonomous attractors converge both upper and lower semicontinuously to a 0 the study of the detailed structure of the non autonomous attractors under perturbation plays the crucial role in the arguments

large time behavior for convection diffusion equations in rn abstract we describe the large time behavior of solutions of the convection diffusion equation u t u a u q 1 u in 0 r n with a r n and q 1 1 n n 1 when q 1 1 n we prove that the large time behavior of solutions with initial data in l 1 r n is given by a uniparametric family of self similar solutions the relevant parameter is the mass of the solution that is conserved for all t our result extends to dimensions n 1 well known results on the large time behavior of solutions for viscous burgers equations in one space dimension the proof is based on la salle s invariance principle applied to the equation written in its self similarity variables when q 1 1 n the convection term is too weak and the large time behavior is given by the heat kernel in this case the result is easily proved applying standard estimates of the heat kernel on the integral equation related to the problem

global asymptotics of solutions to the cauchy problem for the damped wave equation with absorption abstract we consider the cauchy problem for the damped wave equation with absorption u t t u u t u 1 u 0 t x r r n the behavior of u as t is expected to be same as that for the corresponding heat equation t 1 0 t x r r n in the subcritical case 1 c n 1 2 n there exists a similarity solution w b t x with the form t 1 1 f x t depending on b lim x x 2 1 f x 0 our first aim is to show the decay rates u t l 2 u t l 1 u t l 2 o t 1 1 n 4 t 1 1 n 2 1 t 1 1 1 2 n 4 provided that the initial data without initial data size restriction spatially decays with reasonable polynomial order the decay rates are sharp in the sense that they are same as those of the similarity solution the second aim is to show that the gauss kernel is the asymptotic profile in the supercritical case which has been shown in case of one dimensional space by hayashi kaikina and naumkin n hayashi e i kaikina p i naumkin asymptotics for nonlinear damped wave equations with large initial data preprint 2004 we show this assertion in two and three dimensional space to prove our results both the weighted l 2 energy method and the explicit formula of solutions will be employed the weight is an improved one originally developed in y todorova b yordanov critical exponent for a nonlinear wave equation with damping j differential equations 174 2001 464 489

invariant manifolds for parabolic partial differential equations on unbounded domains in this paper finite dimensional invariant manifolds for nonlinear parabolic partial differential equations of the form

asymptotic behavior of solutions of quasilinear hyperbolic equations with linear damping we consider the asymptotic behavior of the solution of quasilinear hyperbolic equation with linear dampingvtt a vx x vt 0 x t r 0 subsequent to k nishihara j differential equations131 1996 171 188 in that article the system with dampingvt ux 0 ut p v x u p v 0 was treated and the convergence rates to the diffusion wave by l hsiao and t p liu comm math phys 143 1992 599 605 and l hsiao and t p liu chinese ann math ser b14 1993 465 480 were improved for initial data v0 u0 x satisfying v0 u0 v0 0 and v0 x v0 dx 0 this system is reduced to the quasilinear hyperbolic equation with linear dampingvtt p v0 vx p v0 x vt 0 and the decay rates v t l o t 1 2 vx t l o t 1 vt t l o t 3 2 were obtained provided that v vt t 0 h3 h2 is small and inl1 l1 from these decay rates vis expected to behave as a solution to the parabolic equation the aims of this paper are to seek the asymptotic profile x t satisfying t l o t 1 2 x t l o t 1 t t l o t 3 2 and to show that the solution of satisfies v t l o t 1 v x t l o t 3 2 v t t l o t 2 the proof is based on the elementary energy method and the green function method for the parabolic equation

resonant motion of a spherical pendulum abstract the weakly nonlinear resonant response of a damped spherical pendulum length l damping ratio natural frequency 0 to the planar displacement e l cos t e 1 of its point of suspension is examined in a four dimensional phase space in which the coordinates are slowly varying amplitudes of a sinusoidal motion the loci of equilibrium points and the corresponding bifurcation points in this space are determined the control parameters are 2 e 2 3 and v 2 2 2 0 e 2 3 2 if v within which no stable equilibrium points exist as v decreases through the upper bound a hopf bifurcation point of this interval the motion in the phase space becomes periodic and then following a period doubling cascade chaotic there may be alternating sub intervals of chaotic and periodic motion the chaotic trajectories in the phase space appear to lie on fractal attractors

large time behavior and lp lq estimate of solutions of 2 dimensional nonlinear damped wave equations abstract we show the asymptotic behavior of the solution to the cauchy problem of the two dimensional damped wave equation it is shown that the solution of the linear damped wave equation asymptotically decompose into a solution of the heat and wave equations and the difference of those solutions satisfies the lp lq type estimate this is a two dimensional generalization of the three dimensional result due to nishihara math z 244 2003 631 to show this we use the fourier transform and observe that the evolution operators of the damped wave equation can be approximated by the solutions of the heat and wave equations by using the lp lq estimate we also discuss the asymptotic behavior of the semilinear problem of the damped wave equation with the power nonlinearity u u our result covers the whole super critical case 1 where the 1 is well known as the fujita exponent when n 2

existence and nonexistence of time global solutions to damped wave equation on half line on the half line r 0 infinity the initial boundary value problems with null dirichlet boundary for both the semilinear heat equation and damped wave equation are considered the critical exponent p c n k of semilinear term for the existence and nonexistence about the semilinear heat equation on the halved space d n d k r k rn k is given by p c n k 1 2 n k j appl math phys 39 1988 135 149 arch rational mech anal 109 1990 63 71 since the damped wave equation is expected to be close to the heat equation j differential equations 191 2003 445 469 math z 244 2003 631 649 the critical exponent for the semilinear damped wave equation is expected to be same as that of the semilinear heat equation however there is no blow up result on the halved space for the damped wave equation in this paper the exponent pc l 1 2 is shown to be critical for the existence and nonexistence of time global solution to both the semilinear heat equation and damped wave equation on the half line r together with the derivation of the blow up time for the proof the explicit formulas of solutions are used in a similar fashion to those in li and zhou discrete continuous dynamic systems 1 1995 503 520 c 2005 elsevier ltd all rights reserved

exact representation of the asymptotic drift speed and diffusion matrix for a class of velocity jump processes abstract this paper examines a class of linear hyperbolic systems which generalizes the goldstein kac model to an arbitrary finite number of speeds v i with transition rates i j under the basic assumptions that the transition matrix is symmetric and irreducible and the differences v i v j generate all the space the system exhibits a large time behavior described by a parabolic advection diffusion equation the main contribution is to determine explicit formulas for the asymptotic drift speed and diffusion matrix in term of the kinetic parameters v i and i j establishing a complete connection between microscopic and macroscopic coefficients it is shown that the drift speed is the arithmetic mean of the velocities v i the diffusion matrix has a more complicate representation based on the graph with vertices the velocities v i and arcs weighted by the transition rates i j the approach is based on an exhaustive analysis of the dispersion relation and on the application of a variant of the kirchoff s matrix tree theorem from graph theory

a damped hyperbolic equation on thin domains for a damped hyperbolic equation in a thin domain in r3 over a bounded smooth domain in r2 it is proved that the global attractors are upper semicontinuous it is shown also that a global attractor exists in the case of the critical sobolev exponent

diffusive stability of spatial periodic solutions of the swift hohenberg equation we are interested in the nonlinear stability of the eckhaus stable equilibria of the swift hohenberg equation on the infinite line with respect to small integrable perturbations the difficulty in showing stability for these stationary solutions comes from the fact that their linearizations possess continuous spectrum up to zero the nonlinear stability problem is treated with renormalization theory which allows us to show that the nonlinear terms are irrelevant i e that the fully nonlinear problem behaves asymptotically as the linearized one which is under a diffusive regime

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

decay properties of solutions to the cauchy problem for the damped wave equation with absorption abstract we consider the cauchy problem for the damped wave equation with absorption u t t u u t u p 1 u 0 t x r r n the behavior of u as t is expected to be the same as that for the corresponding heat equation t p 1 0 t x r r n which has the similarity solution w a t x with the form t 1 p 1 f x t depending on a lim x x 2 p 1 f x 0 provided that p is less than the fujita exponent p c n 1 2 n in this paper as a first step if 1 p p c n and the data u 0 u 1 x decays exponentially as x without smallness condition the solution is shown to decay with rates as t u t l 2 u t l p 1 u t l 2 o t 1 p 1 n 4 t 1 p 1 n 2 p 1 t 1 p 1 1 2 n 4 those of which seem to be reasonable because the similarity solution w a t x has the same decay rates as for the proof the weighted l 2 energy method will be employed with suitable weight similar to that in todorova and yordanov y todorova b yordanov critical exponent for a nonlinear wave equation with damping j differential equations 174 2001 464 489

geometric stability analysis for periodic solutions of the swift hohenberg equation in this paper we describe invariant geometrical structures in the phase space of the swift hohenberg equation in a neighborhood of its periodic stationary states we show that in spite of the fact that these states are only marginally stable i e the linearized problem about these states has continuous spectrum extending all the way up to zero there exist finite dimensional invariant manifolds in the phase space of this equation which determine the long time behavior of solutions near these stationary solutions in particular using this point of view we obtain a new demonstration of schneider s recent proof that these states are nonlinearly stable