large time behavior of solutions for a system of nonlinear damped wave equations abstract we consider the cauchy problem of the semilinear damped wave system t 2 u u t u f u t 0 x r n u j 0 x a j x t u j 0 x b j x x r n where u t x u 1 t x u m t x with m 2 and j 1 m we show the asymptotic behavior of solutions under the sharp condition on the nonlinear exponents which is a natural extension of the results for the single nonlinear damped wave equations nishihara 2003 21 hayashi et al 2004 9 hosono and ogawa 2004 10 the proof is based on the l p l q type decomposition of the fundamental solutions of the linear damped wave equations into the dissipative part and hyperbolic part hosono and ogawa 2004 10 nishihara 2003 21

critical exponent for damped wave equations with nonlinear memory abstract we consider the cauchy problem in r n n 1 for a semilinear damped wave equation with nonlinear memory global existence and asymptotic behavior as t of small data solutions have been established in the case when 1 n 3 we also derive a blow up result under some positive data in any dimensional space

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

on the reversible quadratic polynomial vector fields on s 2 abstract we study a class of quadratic reversible polynomial vector fields on s 2 with 3 2 type reversibility we classify all isolated singularities and we prove the nonexistence of limit cycles for this class our study provides tools to determine the phase portrait for these vector fields

inference in dynamic models containing surprise variables abstract some new results on calculating moving average representation mar coefficients and their limiting distribution from estimated vector arma processes are presented the technique is applied to the problem of estimating the coefficients of unanticipated or surprise variables in a single equation for a multi period expectations horizon the method naturally conditions the expectations on all past values of the process and avoids the necessity of using two step regression procedures and adjusting the resulting standard errors

conditions for local reversibility abstract the paper follows from work by quispel and capel on local reversibility of a dynamical system the approach adopted is to write down conditions for a mapping to behave like an involution in the neighbourhood of a fixed point and then to combine mappings which satisfy these conditions thus producing a composite mapping which is locally reversible in the main this confirms the results obtained by quispel and capel although additional conditions were found in some cases

global existence of solutions for a weakly coupled system of semilinear damped wave equations abstract in this paper we consider the cauchy problem for a weakly coupled system of semilinear damped wave equations we prove the global existence of solutions for small data in the supercritical case for any space dimension we also give estimates of the weighted energy of solutions and in a special case we prove an almost optimal estimate moreover in the subcritical case we give an almost optimal estimate of the lifespan from both above and below

large time behavior of global solutions of the semilinear damped beam equation with slowly decaying data abstract we discuss the existence of the global solution and its asymptotic profile for the cauchy problem for the nonlinear damped beam equation we show that the equation admits a unique global solution with small slowly decaying data e g 1 x 2 k 2 0 k 1 moreover we show that the solution can be approximated by the solution of the linear heat equation with suitable data the proof is based upon the estimate for the evolution operator of the linearized equation in the homogeneous sobolev space

asymptotic behavior of solutions for the damped wave equation with slowly decaying data abstract we consider the cauchy problem for the damped wave equation u t t u u t u 1 u t x r r n and the heat equation t 1 t x r r n if the data is small and slowly decays likely c 1 1 x k n 0 k 1 then the critical exponent is c k 1 2 k n for the semilinear heat equation in this paper it is shown that in the supercritical case there exists a unique time global solution to the cauchy problem for the semilinear heat equation in any dimensional space r n whose asymptotic profile is given by 0 t x r n e x y 2 4 t 4 t n 2 c 1 1 y 2 k n 2 d y provided that the data 0 satisfies lim x x k n 0 x c 1 0 even in the semilinear damped wave equation in the supercritical case a time global solution u with the data u u t 0 x u 0 u 1 x is shown in low dimensional spaces r n n 1 2 3 to have the same asymptotic profile 0 t x provided that lim x x k n u 0 u 1 x c 1 0 those proofs are given by elementary estimates on the explicit formulas of solutions

scaling variables and asymptotic profiles for the semilinear damped wave equation with variable coefficients abstract we study the asymptotic behavior of solutions for the semilinear damped wave equation with variable coefficients we prove that if the damping is effective and the nonlinearity and other lower order terms can be regarded as perturbations then the solution is approximated by the scaled gaussian of the corresponding linear parabolic problem the proof is based on the scaling variables and energy estimates

lp decay problem for the dissipative wave equation in odd dimensions abstract consider the cauchy problem in odd dimensions for the dissipative wave equation t u 0 in r 2 n 1 0 with u t u t 0 u 0 u 1 because the l 2 estimates and the l estimates of the solution u t are well known in this paper we pay attention to the l p estimates with 1 p 2 in particular p 1 of the solution u t for t 0 in order to derive l p estimates we first give the representation formulas of the solution u t t s t u 0 s t u 0 u 1 and then we directly estimate the exact solution s t g and its derivative t s t g of the dissipative wave equation with the initial data u 0 u 1 0 g in particular when p 1 and n 1 we get the l 1 estimate u t l 1 c e t 4 u 0 w n 1 u 1 w n 1 1 c u 0 l 1 u 1 l 1 for t 0

value at risk calculation through arch factor methodology proposal and comparative analysis abstract in this paper we put forward a new method to estimate value at risk var autoregressive conditional heteroskedastic arch factor which combines multivariate analysis with arch models firstly from a set of correlated portfolio risk factors we derive a smaller uncorrelated risk factors set by applying multivariate analysis secondly we use arch schemes to model uncorrelated factors historical behaviour thirdly we use the estimated models to predict future values for factors standard deviation from them var calculation is immediate in this way arch factor methodology overcomes the multivariate arch models drawbacks which in practice make these unworkable for var calculation purposes we apply the proposed methodology over a set of foreign exchange risk exposed portfolios obtaining better results than those reached when j p morgan s riskmetrics is used

critical exponent for nonlinear wave equations with frictional and viscoelastic damping terms abstract in this paper we study the cauchy problem for a nonlinear wave equation with frictional and viscoelastic damping terms our aim is to obtain the threshold to classify the global existence of solutions for small initial data or the finite time blow up of solutions under the growth condition of the nonlinearity

critical exponents for semilinear dissipative wave equations in rn abstract we shall present new critical exponents 1 2m n with m 1 2 to the cauchy problem utt u ut u p 1u with the initial data u 0 u 1 h 1 r n l m r n l 2 r n l m r n that is the small data global existence property can be derived to the cauchy problem above with power 1 2m n

prediction in dynamic models with time dependent conditional variances abstract this paper considers forecasting the conditional mean and variance from a single equation dynamic model with autocorrelated disturbances following an arma process and innovations with time dependent conditional heteroskedasticity as represented by a linear garch process expressions for the minimum mse predictor and the conditional mse are presented we also derive the formula for all the theoretical moments of the prediction error distribution from a general dynamic model with garch 1 1 innovations these results are then used in the construction of ex ante prediction confidence intervals by means of the cornish fisher asymptotic expansion an empirical example relating to the uncertainty of the expected depreciation of foreign exchange rates illustrates the usefulness of the results

optimal decay rate of solutions for cahn hilliard equation with inertial term in multi dimensions abstract the cauchy problem of the cahn hilliard equation with inertial term is considered based on green s function method together with energy estimates we get the global in time existence and optimal decay rate of solutions furthermore the viscous case is investigated in the last section

critical exponent for the cauchy problem to the weakly coupled damped wave system abstract in this paper we consider a system of weakly coupled semilinear damped wave equations we determine the critical exponent for any space dimensions our proof of the global existence of solutions for supercritical nonlinearities is based on a weighted energy method whose multiplier is appropriately modified in the case where one of the exponent of the nonlinear term is less than the so called fujita s critical exponent we also give estimates of the lifespan of solutions from above for subcritical nonlinearities

asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy we study the asymptotic time behavior of global smooth solutions to general entropy dissipative hyperbolic systems of balance law in m space dimensions under the shizuta kawashima condition we show that these solutions approach constant equilibrium state in the lp norm at a rate o t m 2 1 1 p as t tends to infty for p in min m 2 infty moreover we can show that we can approximate with a faster order of convergence theconservative part of the solution in terms of the linearized hyperbolic operator for m 2 and by a parabolic equation in the spirit of chapman enskog expansion the main tool is given by a detailed analysis of the green function for the linearized problem

strauss exponent for semilinear wave equations with scattering space dependent damping abstract it has been asserted that the semilinear wave equation with scattering damping admits the same strauss critical exponent as the classical semilinear wave equation in this paper we devote to showing that this assertion holds for the space dependent damping wave equation when the decay rate of the space dependent variable coefficients of the damping is greater than 2 in particular we derive the lifespan upperbound estimates for both sub critical and critical exponents moreover when the nonlinear term is of derivative type we also obtain the same glassey exponent and the same upper bound lifespan as that of the corresponding problem for wave equation without damping both results show the hyperbolicity of the equation from the blow up aspect

time reversible and equivariant pitchfork bifurcation abstract in this note we describe a new phenomenon in steady state bifurcations of time reversible and equivariant vector fields when the eigenspace associated with a pair of imaginary eigenvalues that passes through zero onto the real line is separable from the nullspace associated with the fixed zeros of the spectrum we prove that a pitchfork bifurcation occurs

lifespan of solutions to the damped wave equation with a critical nonlinearity abstract in the present paper we study a lifespan of solutions to the cauchy problem for semilinear damped wave equations dw t 2 u u t u f u t x 0 t e r n u 0 x e u 0 x x r n t u 0 x e u 1 x x r n where n 1 f u u p 1 u or u p p 1 e 0 is a small parameter and u 0 u 1 is a given initial data the main purpose of this paper is to prove that if the nonlinear term is f u u p and the nonlinear power is the fujita critical exponent p p f 1 2 n then the upper estimate to the lifespan is estimated by t e exp c e p for all e 0 1 and suitable data u 0 u 1 without any restriction on the spatial dimension our proof is based on a test function method utilized by zhang 35 we also prove a sharp lower estimate of the lifespan t e to dw in the critical case p p f

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

prediction tests for structural stability abstract three groups of test statistics for testing structural stability are compared the statistics are based on predictions from univariate time series models the first group ignores the specification and estimation variability the second group ignores only the specification uncertainty while the statistics in the third group take into account both specification and estimation variability although corresponding statistics in the three groups have identical asymptotic distributions their small sample distributions differ markedly it is suggested to use the statistics in the third group in practice

dissipative structure and diffusion phenomena for doubly dissipative elastic waves in two space dimensions abstract in this paper we study the cauchy problem for doubly dissipative elastic waves in two space dimensions where two different damping terms consist of friction or structural damping we derive energy estimates and diffusion phenomena with different assumptions on initial data particularly we find a new threshold for the influence from different damping on diffusion phenomena additionally optimal decay estimates of solution from the above and the below in a weighted l 1 framework are obtained

a class of reversible quadratic polynomial vector fields on s2 abstract we study a class of quadratic reversible polynomial vector fields on s 2 we classify all the centers of this class of vector fields and we characterize its global phase portrait

stationarity of stable power garch processes we present conditions for strict stationarity of power garch processes whose innovations are described by a heavy tailed and possibly asymmetric stable paretian distribution the results generalize those of bougerol and picard j econom 52 1992 115 who derived analogous conditions for standard i e power two garch processes with finite variance innovations

on the choice of garch parameters for efficient modelling of real stock price dynamics we propose two different methods for optimal choice of garch 1 1 parameters for the efficient modelling of stock prices by using a particular return series using as an example stock return data for intel corporation we vary parameters to fit the average volatility as well as fourth linked to kurtosis of data and eighth statistical moments and observe pure convergence of our simulated eighth moment to the stock data results indicate that fitting higher order moments of a return series might not be an optimal approach for choosing garch parameters in contrast the simulated exponent of the fourier spectrum decay is much less noisy and can easily fit the corresponding decay of the empirical fourier spectrum of the used return series of intel stock allowing us to efficiently define all garch parameters we compare the estimates of garch parameters obtained by fitting price data fourier spectra with the ones obtained from standard software packages and conclude that the obtained estimates here are deeper in the stability region of parameters thus the proposed method of using fourier spectra of stock data to estimate garch parameters results in a more robust and stable stochastic process but with a shorter characteristic autocovariance time

damped wave equation in the subcritical case abstract we study large time asymptotics of small solutions to the cauchy problem for the one dimensional nonlinear damped wave equation 1 v tt v t v xx v 1 0 x r t 0 v 0 x v 0 x v t 0 x v 1 x in the sub critical case 2 3 2 we assume that the initial data v 0 1 x 1 v 1 l l 1 a a 0 1 where l 1 a l 1 l 1 a a l 1 x 1 x 2 also we suppose that the mean value of initial data r v 0 x v 1 x dx 0 then there exists a positive value such that the cauchy problem 1 has a unique global solution v t x c 0 l l 1 a satisfying the following time decay estimate v t l c t 1 for large t 0 here 2 3 2

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

decay estimates for the damped wave equation with weighted odd initial data abstract in this paper we consider the cauchy problem to the damped wave equation several authors have shown the diffusive structure of the problem which has been represented by l p l q estimates we show the new l p l q estimates to the problem when initial data are odd and belong to certain weighted function space these l p l q estimates imply the new decay estimates for the above cauchy problem moreover we apply the above decay estimates to the cauchy problem for the semilinear wave equation with power nonlinearities lower than fujita critical exponent we show the existence of the time global solutions when the initial data are small and odd

the lp lq estimates of solutions to one dimensional damped wave equations and their application to the compressible flow through porous media we first obtain the lp lq estimates of solutions to the cauchy problem for one dimensional damped wave equation vtt vxx vt 0 v vt t 0 v0 v1 x x t r r corresponding to that for the parabolic equation t xx 0 t 0 v0 v1 x the estimates are shown by v t e t 2v0 t v0 t 2lp ct 121q 1p 1 v0 v1 lq t 1 etc for 1 q p to show the explicit formula of the damped wave equation will be used to apply the estimates to nonlinear problems is the second aim we will treat the system of a compressible flow through porous media the solution is expected to behave as the diffusion wave which is the solution to the porous media equation due to the darcy law when the initial data has the same constant state at a sharp lp convergence rate for p 2 has been recently obtained by nishihara proc roy soc edinburgh sect a 133a 2003 1 20 by choosing a suitably located diffusion wave we will show the l1 convergence applying

exact predictive densities for linear models with arch disturbances abstract it is shown how exact predictive densities may be formed in the arch linear model by means of monte carlo integration with importance sampling several improvements in computational efficiency over earlier implementations of this procedure are developed including use of the exact likelihood function rather than an asymptotic approximation to construct the importance sampling distribution and antithetic acceleration of convergence a numerical approach to the formulation of posterior odds ratios and the combination of non nested models is also introduced these methods are applied to daily quotations of closing stock prices forecasts are formulated using linear models arch linear models and an integrated model constructed from the posterior probabilities of the respective models the use of the exact predictive density in a decision theoretic context is illustrated by deriving the optimal day to day portfolio adjustments of a trader with constant relative risk aversion

optimal predictions of powers of conditionally heteroscedastic processes in conditionally heteroskedastic models the optimal prediction of powers or logarithms of the absolute process has a simple expression in terms of the volatility process and an expectation involving the independent process a standard procedure for estimating this prediction is to estimate the volatility by gaussian quasi maximum likelihood qml in a first step and to use empirical means based on rescaled innovations to estimate the expectation in a second step this paper proposes an alternative one step procedure based on an appropriate non gaussian qml estimation of the model and establishes the asymptotic properties of the two approaches their performances are compared for finite order garch models and for the infinite arch for the standard garch p q and the asymmetric power garch p q it is shown that the are of the estimators only depends on the prediction problem and some moments of the independent process an application to indexes of major stock exchanges is proposed

symmetric singularities of reversible vector fields in dimension three abstract for a large class of reversible vector fields on the three dimensional space we present all the topological types and their respective normal forms of the symmetric singularities of codimension 0 1

bifurcations of lattice structures equilibrium configurations of a model for a crystal in one spatial dimension with anharmonic and up to third neighbour harmonic interactions are related to orbits in a two or four dimensional space under a nonlinear symplectic mapping in this way a connection is given with dynamical systems with two or three degrees of freedom depending on the parameters the orbits are smooth or stochastic their characterisation in terms of their fixed points and their ljapunov exponents is given the mapping shows a large number of bifurcations of which one can distinguish various types and these are discussed

reversing symmetries in dynamical systems dynamical systems may possess in addition to symmetries that leave the equations of motion invariant reversing symmetries that invert the equations of motion such dynamical systems are called weakly reversible some consequences of the existence of reversing symmetries for dynamical systems with discrete time mappings are discussed a reversing symmetry group is introduced and it is shown that every weakly reversible mapping l can be decomposed into two mappings k0 and k1 of the same order 2l limit l to infinity included such that k02 k12 i some applications are discussed briefly

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

interval forecasting an analysis based upon arch quantile estimators abstract in this paper we explore techniques for obtaining interval forecasts based on estimated time series models for processes which may exhibit autoregressive conditional heteroskedasticity arch to deal with the available variety of possible interval forecasts we propose a method for combining these forecasts based on quantile regression techniques our approach is practical rather than theoretical with attention focused directly on obtaining interval forecasts for two u s time series a measure of unemployment and a treasury bill rate we evaluate the performance of our procedures using a variety of diagnostics we find interval estimates which perform reasonably well judged by both in sample and out of sample criteria our experience suggests that a certain amount of care is required in order to obtain useful forecasts

asymptotic expansion of solutions to the nonlinear dissipative equation with the anomalous diffusion abstract the initial value problem for the dissipative equation with the extra force term is considered the dissipative effect on this equation is given by the fractional laplacian which describes the anomalous diffusion the main goal of this paper is to give an estimate on the difference between solutions and their asymptotic expansion as the space and the time variables tend to infinity furthermore as an application the large time behavior of solutions to some concrete problems are discussed

large time behavior of solutions for the nonlinear wave equation with viscosity in odd dimensions abstract we study the dissipation of solutions of the cauchy problem for the nonlinear dissipative wave equation in odd multi spatial dimensions pointwise estimates of the time asymptotic shape of the solutions are obtained and shown to exhibit the generalized huygens principle our approach is based on the detailed analysis of the green function of the linearized system this is used to study the coupling of nonlinear diffusion waves

generalized autoregressive conditional heteroskedasticity a natural generalization of the arch autoregressive conditional heteroskedastic process introduced in engle 1982 to allow for past conditional variances in the current conditional variance equation is proposed stationarity conditions and autocorrelation structure for this new class of parametric models are derived maximum likelihood estimation and testing are also considered finally an empirical example relating to the uncertainty of the inflation rate is presented

on the initial value problem of the semilinear beam equation with weak damping ii asymptotic profiles abstract we study the initial value problem for some semilinear damped beam equation in takeda and yoshikawa submitted for publication 9 unique global existence of a decaying solution for the problem and the smoothing effect of the solution was shown in this article we shall give the asymptotic profiles of the solution as a result we observe that the decay estimates are optimal moreover considering the higher order expansion of the solution we observe more detailed information such as the contribution of the nonlinear term to the solution as t

global existence of solutions to 1 d euler equations with time dependent damping abstract in this paper we study the 1 d isentropic euler equations with time dependent damping our damping decays at a speed of order 1 with respect to time which is a little weaker than the linear one under our assumption we will prove the global existence of solutions to the euler system and obtain l 2 and l estimates for the solutions our approach is based on some detailed energy estimates

damped wave equation with a critical nonlinearity in higher space dimensions abstract we study the cauchy problem for nonlinear damped wave equations with a critical defocusing power nonlinearity u 2 n u where n denotes the space dimension for n 1 2 3 global in time existence of small solutions was shown in 4 in this paper we generalize the results to any spatial dimension via the method of decomposition of the equation into the high and low frequency components under the assumption that the initial data are small and decay rapidly at infinity furthermore we present a sharp time decay estimate of solutions with a logarithmic correction

the reversibility and the center problem abstract in this work we study the narrow relation between reversibility and the center problem and also between reversibility and the integrability problem it is well known that an analytic system having either a non degenerate or nilpotent center at the origin is analytically reversible or orbitally analytically reversible respectively in this paper we prove the existence of a smooth map that transforms an analytic system having a degenerate center at the origin with either an analytic first integral or a c inverse integrating factor into a reversible linear system after rescaling the time moreover if the degenerate center has an analytic or a c reversing symmetry then the transformed system by the map also has a reversing symmetry from the knowledge of a first integral near the center we give a procedure to detect reversing symmetries

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

the pointwise estimates of solutions for semilinear dissipative wave equation in multi dimensions abstract in this paper we study the global in time existence and the pointwise estimates of solutions to the cauchy problem for the dissipative wave equation in multi dimensions using the fixed point theorem we obtain the global existence of the solution in addition the pointwise estimates of the solution are obtained by the method of the green function furthermore we obtain the l p 1 p convergence rate of the solution

l p l q decay estimates for dissipative linear hyperbolic systems in 1d abstract given a b r n n we consider the cauchy problem for partially dissipative hyperbolic systems having the form t u a x u b u 0 with the aim of providing a detailed description of the large time behavior sharp l p l q estimates for 1 q p are established for the distance between the solution to the system and a time asymptotic profile where the profile is the superposition of diffusion waves and exponentially decaying waves they show that the solution to the system decays to the diffusion waves 1 2 faster than the diffusive decay rate 1 q 1 p 2 while the exponential decaying waves are negligible for large time in particular under a symmetry property it decays 1 faster than the proof is based on a complex interpolation argument once l r estimates for the fundamental solution to the system in the frequency space and fourier multiplier estimates are accomplished

remark on the blow up of solutions for the semilinear wave equation with overdamping term abstract we consider the cauchy problem for the wave equation with overdamping and semilinear source terms p u t t u b t u t n u t x r r n u u t 0 x u 0 u 1 x x r n with b t b 0 t 1 b 0 0 1 and n u u p 1 u p 1 ikeda and wakasugi 8 have recently showed that when n u c u p for any p 1 there is a global in time solution to p for suitable small data and that when n u u p the local in time solution blows up within a finite time for suitable large data to show the blow up result their method seems to be not applicable to our semilinear term our aim is to show the blow up of solutions for suitable large data by the method much different from theirs

on the initial and terminal value problem for a class of semilinear strongly material damped plate equations abstract we study the semilinear strongly damped plate equation by considering its two different problems for initial value problem we prove the local well posedness and blow up results of solution for the problem with polynomial nonlinear source terms for terminal value problem given the ill posedness in the sense of hadamard we propose a regularization method for the problem with logarithmic nonlinearities under the a priori assumption on the exact solution belonging to a gevrey space we propose the fourier truncation method to stabilize the ill posed problem also by establishing some stability estimates of logarithmic type in l q space

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

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

modeling of financial data comparison of the truncated levy flight and the arch 1 and garch 1 1 processes we compare our results on empirical analysis of nancia l data with simulations of two stochastic models of the dynamics of stock market prices the two models are i the truncated l evy igh t recently introduced by us and ii the arch 1 and garch 1 1 processes we n d that the tlf well describes the scaling and its breakdown observed in empirical data while it is not able to properly describe the uctuation s of volatility empirically detected the arch 1 and garch 1 1 models are able to describe the probability density function of price changes at a given time horizon but both fail to describe the scaling properties of the pdfs for short time horizons c 1998 elsevier science b v all rights reserved

the reversible umbilic bifurcation abstract hamiltonian systems with several degrees of freedom regularly lead to the investigation of bifurcating equilibria in reduced one degree of freedom systems this paper concerns equilibria with vanishing linearisation a co dimension two phenomenon in the reversible context under appropriate transversality conditions such equilibria have versal unfoldings related to the elliptic and hyperbolic umbilic catastrophes this has applications to gyrostat motion and also helps to explain the dynamics defined by the normal form of the henon heiles system the occurring unfoldings turn out to be versal even in the general reversible context of not necessarily hamiltonian systems

local existence and uniqueness for a semilinear accretive wave equation abstract we study local existence and uniqueness in the phase space h h 1 r n of the solution of the semilinear wave equation u t t u u t u t p 1 for p 1

lp lq estimates of solutions to the damped wave equation in 3 dimensional space and their application it has been asserted that the damped wave equation has the diffusive structure as t in this paper we consider the cauchy problem in 3 dimensional space for the linear damped wave equation and the corresponding parabolic equation and obtain the lp lq estimates of the difference of each solution which represent the assertion precisely explicit formulas of the solutions are analyzed for the proof the second aim is to apply the lp lq estimates to the semilinear damped wave equation with power nonlinearity if the power is larger than the fujita exponent then the time global existence of small weak solution is proved and its optimal decay order is obtained

predictions of multivariate autoregressive moving average models summary a simple formula for multiperiod predictions of multivariate autoregressive moving average models is derived the formula is explicitly given as a function of suitably defined parameter matrices and an observation vector and includes simpler models as its special cases the asymptotic prediction mean squared error is also derived for predictions with estimated parameters

non existence of weak solutions to nonlinear damped wave equations in exterior domains abstract we consider the initial boundary value problem of the nonlinear damped wave equation in an exterior domain t 2 u u t u u p t 0 x u 0 x u 0 x t u 0 x u 1 x x u 0 t 0 x when 1 p 1 2 n and the initial data u 0 u 1 h 0 1 l 2 having compact support we prove the non existence of non negative global solutions of the above problem we employ the kaplan fujita h fujita on the blowing up of solutions of the cauchy problem for u t u u 1 j sci univ tokyo sec i 13 1966 109 124 s kaplan on the growth of solutions quasi linear parabolic equations comm pure appl math 16 1963 305 330 method to avoid the difficulty of the reflection from the boundary

energy decay for linear dissipative wave equations in exterior domains abstract in earlier works 2 12 we have shown the uniform decay of the local energy of the damped wave equation in exterior domain when the damper is spatially localized near captive rays in order to have uniform decay of the total energy the damper has also to act at space infinity in this work we establish uniform decay of both the local and global energies the rates of decay turn out to be the same as those for the heat equation which shows that an effective damper at space infinity strengthens the parabolic structure in the equation

common volatility and correlation clustering in asset returns abstract we present a new multivariate framework for the estimation and forecasting of the evolution of financial asset conditional correlations our approach assumes return innovations with time dependent covariances a cholesky decomposition of the asset covariance matrix with elements written as sines and cosines of spherical coordinates allows for modelling conditional variances and correlations and guarantees its positive definiteness at each time t as in christodoulakis and satchell christodoulakis g a satchell s e 2002 correlated arch corrarch modelling the time varying conditional correlation between financial asset returns european journal of operational research 139 2 350 369 correlation is generated by conditionally autoregressive processes thus allowing for an autocorrelation structure for correlation our approach allows for explicit out of sample forecasting and is consistent with stylized facts as time varying correlations and correlation clustering co movement between correlation coefficients correlation and volatility as well as between volatility processes co volatility the latter two are shown to depend on correlation and volatility persistence empirical evidence on a trivariate model using monthly data from dow jones industrial nasdaq composite and the 3 month us treasury bill yield supports our theoretical arguments

wave equations with strong damping in hilbert spaces abstract we study the cauchy problem for abstract dissipative equations in hilbert spaces generalizing wave equations with strong damping terms in r n or exterior domains our main result is a generalized diffusion phenomenon the long time asymptotics of strongly damped wave equations is a combination of solutions of diffusion and wave equations in particular we obtain sharp decay estimates the proofs rely on the energy method in the fourier space and its generalization based on the spectral theorem for self adjoint operators

singularities of reversible vector fields abstract for a large class of reversible vector fields on the plane we present all the topological types and their respective normal forms of the symmetric singularities of codimensions 0 1 and 2

reversing k symmetries in dynamical systems we generalize the concept of reversing symmetries of a dynamical system i e we study dynamical systems that possess symmetry properties only if considered on a proper time scale in particular considering dynamical systems with discrete time the kth iterate of a map may possess more reversing symmetries than the map itself in this way the concepts of reversing symmetries and reversing symmetry groups are generalized to reversing k symmetries and reversing k symmetry groups furthermore a method is studied for finding orbits that are k symmetric with respect to reversing k symmetries firstly an existing method for finding orbits that are symmetric with respect to one reversing symmetry is extended to the case of more than one reversing symmetry and secondly a generalization of this method to the case of reversing k symmetries is introduced some physically relevant examples of dynamical systems possessing reversing k symmetries are discussed briefly

on l1 estimates for solutions of dissipative wave equations abstract we consider the cauchy problem for the dissipative wave equation t u 0 in r n 0 with u t u t 0 u 0 u 1 in r n where t t 2 t x is the dissipative wave operator with laplacian x we derive the representation formulas in any dimension of the solution u u x t and then we obtain the l 1 estimates of the solution u t for any t 0 without any weighted conditions on the initial data u 0 u 1

global existence of solutions for semilinear damped wave equations in rn with noncompactly supported initial data abstract we consider the n dimensional cauchy problem in r n for a semilinear damped wave equation with a power type nonlinearity u p for a noncompactly supported initial data which has a small energy we shall derive a global in time existence result in the case when the power of the nonlinear term satisfies 1 2 n p n n 2 this generalizes a previous result due to todorova yordanov j differential equations 174 2001 464 489 which dealt with a solution in the framework of compactly supported initial data

global existence and nonexistence of solutions for a system of nonlinear damped wave equations abstract we consider the cauchy problem for the system of semilinear damped wave equations with small initial data t 2 u 1 u 1 t u 1 u k p 1 t 0 x r n t 2 u 2 u 2 t u 2 u 1 p 2 t 0 x r n t 2 u k u k t u k u k 1 p k t 0 x r n u j 0 x a j x t u j 0 x b j x x r n 1 j k we show that a critical exponent which classifies the global existence and the finite time blow up of solutions indeed coincides with the one to a corresponding semilinear heat systems with small data the proof of the global existence is based on the l p l q estimates of fundamental solutions for linear damped wave equations k nishihara l p l q estimates of solutions to the damped wave equation in 3 dimensional space and their application math z 244 2003 631 649 k marcati p nishihara the l p l q estimates of solutions to one dimensional damped wave equations and their application to compressible flow through porous media j differential equations 191 2003 445 469 t hosono t ogawa large time behavior and l p l q estimate of 2 dimensional nonlinear damped wave equations j differential equations 203 2004 82 118 t narazaki l p l q estimates for damped wave equations and their applications to semilinear problem j math soc japan 56 2004 585 626 and the blow up is shown by the fujita kaplan zhang method q zhang a blow up result for a nonlinear wave equation with damping the critical case c r acad sci paris 333 2001 109 114 f sun m wang existence and nonexistence of global solutions for a nonlinear hyperbolic system with damping nonlinear anal 66 12 2007 2889 2910 t ogawa h takeda non existence of weak solutions to nonlinear damped wave equations in exterior domains nonlinear anal 70 10 2009 3696 3701

on the initial value problem of the semilinear beam equation with weak damping i smoothing effect abstract this paper is devoted to the study of the initial value problem for some semilinear damped beam equation we shall prove the unique global existence of a small solution for the equation and also obtain a certain smoothing effect of the solution our method is based on the expansion formula for the propagator to decompose into the dissipative part and the dispersive part to control the dispersive part without regularity loss we apply the stationary phase method developed by van der corput as a result we observe the smoothing effect of the solution which does not occur on the solution for damped wave equations

movement of time delayed hot spots in euclidean space we investigate the shape of the solution of the cauchy problem for the damped wave equation in particular we study the existence location and number of spatial maximizers of the solution studying the shape of the solution of the damped wave equation we prepare a decomposed form of the solution into the heat part and the wave part moreover as its another application we give l p l q estimates of the solution