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

comparison of least squares and stochastic gradient lattice predictor algorithms using two performance criteria the least squares ls and stochastic gradient sg lattice prediction algorithms are compared using two different performance criteria these are a output mean squared error and b the accuracy of the autoregressive spectral estimate obtained from the mean values of the lattice coefficients assuming a stationayinput it is found that the second performance criterion is more sensitive than the first this spectral performance criterion is a measure of the accuracy of the estimatcd autoregressive model coefficients bias in the ls and sg coefficient estimates can cause significant deviation of the asymptotic spectral estimates from the actual input spectrum the similarly between the ls and sg lattice algorithms enables comparative simulations with analogous initial conditions and exponential weighting constants for both types of comparisons the ls algorithm offers a modest performance improvement over the sg algorithms simulated this improvement is more noticeable when the input is highly correlated it is also found that slight changes in the sg lattice algorithm may significantly affect its performance

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

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

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

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

small delay approximation of stochastic delay differential equations delay differential equations evolve in an infinite dimensional phase space in this paper we consider the effect of external fluctuations noise on delay differential equations involving one variable thus leading to univariate stochastic delay differential equations sdde s for small delays a univariate nondelayed stochastic differential equation approximating such a sdde is presented another approximation complementary to the first is also obtained using an average of the sdde s drift term over the delayed dynamical variable which defines a conditional average drift this second approximation is characterized by the fact that the diffusion term is identical to that of the original sdde for small delays our approach yields a steady state probability density and a conditional average drift which are in close agreement with numerical simulations of the original sdde we illustrate this scheme with the delayed linear langevin equation and a stochastic version of the delayed logistic equation the technique can be used with any type of noise and is easily generalized to multiple delays s1063 651x 99 08304 x

a small delay and correlation time limit of stochastic differential delay equations with state dependent colored noise we consider a general stochastic differential delay equation sdde with state dependent colored noises and derive its limit as the time delays and the correlation times of the noises go to zero the work is motivated by an experiment involving an electrical circuit with noisy delayed feedback an ornstein uhlenbeck process is used to model the colored noise the main methods used in the proof are a theorem about convergence of solutions of stochastic differential equations by kurtz and protter and a maximal inequality for sums of a stationary sequence of random variables by peligrad and utev

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

the block processing ftf adaptive algorithm a new block type adaptive filtering algorithm is presented this new block adaptive filter differs from the frequency domain block adaptive filters of ferrara 1980 and of clark mitra and parker 1980 in that the new method applies a deterministic time domain least squares criteria within each of the data blocks information is carried from block to block via a weighted initial condition this new block fast transversal filters bftf algorithm is a numerically stable algorithm and can also be used to perform efficient least squares system identification on any one data block in which case it shows a moderate computational advantage over the previous most efficient single data block algorithms of morf et al 1977 of marple 1981 and of kalouptsidis manolakis and carayannis 1984 1985 mechanisms for tracking and varying block length from block to block are also presented and evaluated finally we indicate how the new algorithm could be pipelined for maximum throughput with delay proportional to the number of parameters after computation of the sample correlation lags

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

a generalized multichannel least squares lattice algorithm based on sequential processing stages a generalized multichannel least squares ls lattice algorithm which is appropriate for multichannel adaptive filtering and estimation is presented in this paper it is shown that a muitichannel ls estimation algorithm with a different number of parameters to be estimated in each channel can be implemented by cascading lattice stages of nondescending dimension to form a generalized lattice structure a new realization of a multichannel lattice stage is also presented this realization employs only scalar operations and has a computational complexity of 0 p2 for each p channel lattice stage

arbitrarily high sampling rate adaptive filters adaptive filters have an inherent feedback from error signal back to the adaptation of the coefficients this represents a problem in high sampling rate realizations we demonstrate a realization of adaptive filters for which there is no theoretical limit on sampling rate in a given speed of hardware at the expense of additional hardware and latency our realization does not change the input output characteristics aside from finite precision effects and hence does not degrade the filter tracking capability the basis for our realization is the adaptive lattice filter which uses only local feedback in adapting reflection coefficients at each stage and for which the recursive portion of the adaptation algorithms in each stage is linear our realization is based on these two properties and the basic technique of look ahead computation several forms of our realization applying to different recursive least squares and stochastic gradient adaptation algorithms are described

a lattice algorithm for factoring the spectrum of a moving average process triangular decomposition of the semi infinite covariance matrix of a moving average process can be used as a spectral factorization technique an efficient lattice algorithm is derived for performing the necessary computations this technique is a special case of the fast cholesky decomposition of stationary covariance matrices the algorithm can be used to factor multichannel spectra to a desired degree of accuracy

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

adaptive tracking of multiple sinusoids whose power levels are widely separated the behavior of the gradient transversal filter lms the gradient lattice gl and the least squares lattice lsl when used to track multiple sinusoidal or narrow band components whose power levels are widely separated is investigated these approaches to the realization of the pth order one step linear predictor of the time series are recursive in time the lags of the instantaneous frequency estimates from their actual underlying values are of particular interest the frequency tracking characteristics of the lms gl and lsl algorithms are illustrated in several situations included are simulations of dual sinusoids undergoing a variety of frequency versus time dynamics and a formant tracking example of real speech

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

echo cancellation of voiceband data signals using recursive least squares and stochastic gradient algorithms the convergence properties of adaptive least squares ls and stochastic gradient sg algorithms are studied in the context of echo cancellation of voiceband data signals the algorithms considered are the sg transversal sg lattice ls transversal fast kalman and ls lattice it is shown that for the channel estimation problem considered here ls algorithms converge in approximately 2n iterations where n is the order of the filter in contrast both sg algorithms display inferior convergence properties due to their reliance upon statistical averages simulations are presented to verify this result and indicate that the fast kalman algorithm frequently displays numerical instability which can be circumvented by using the lattice structure finally the equivalence between an ls algorithm and a fast converging modified sg algorithm which uses a maximum length input data sequence is shown

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

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

spherical trigonometry yule s parcor identity and frls fully normalized lattice yule s parcor identity is recognized as the cosine law of spherical trigonometry the six parcors propagated by the fully normalized frls lattice filter are the cosines of the six elements of a spherical triangle and this lattice algorithm is one solution to a spherical triangle problem that arises naturally in navigation and astronomy exploiting this new geometric interpretation yields unnoticed and potentially useful recursions among frls quantities

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

a variable step vs adaptive filter algorithm in recent work a new version of an lms algorithm has been developed which implements a variable feedback constant for each weight of an adaptive transversal filter this technique has been called the vs variable step algorithm and is an extension of earlier ideas in stochastic approximation for varying the step size in the method of steepest descents the method may be implemented in hardware with only modest increases in complexity approx 15 percent over the lms widrow hoff algorithm it is shown that an upper bound for the convergence time is the classical mean square error time constant and examples are given to demonstrate that for broad signal classes both narrow band and broad band the convergence time is reduced by a factor of up to 50 in noise canceller applications for the proper selection of variable step parameters finally the vs algorithm is applied to an iir filter and simulations are presented for applications of the vs fir and iir adaptive filters

ito versus stratonovich white noise limits for systems with inertia and colored multiplicative noise we consider the dynamics of systems in the presence of inertia and colored multiplicative noise we study the limit where the particle relaxation time and the correlation time of the noise both tend to zero we show that the limiting equation for the particle position depends on the magnitude of the particle relaxation time relative to the noise correlation time in particular the limiting equation should be interpreted either in the ito or stratonovich sense with a crossover occurring when the two fast time scales are of comparable magnitude at the crossover the limiting stochastic differential equation is neither of ito nor of stratonovich type this means that after adiabatic elimination the governing equations have different drift fields leading to different physical behavior depending on the relative magnitude of the two fast time scales our findings are supported by numerical simulations

stochastic delay differential equations for genetic regulatory networks time delay is an important aspect in the modelling of genetic regulation due to slow biochemical reactions such as gene transcription and translation and protein diffusion between the cytosol and nucleus in this paper we introduce a general mathematical formalism via stochastic delay differential equations for describing time delays in genetic regulatory networks based on recent developments with the delay stochastic simulation algorithm the delay chemical master equation and the delay reaction rate equation are developed for describing biological reactions with time delay which leads to stochastic delay differential equations derived from the langevin approach two simple genetic regulatory networks are used to study the impact of intrinsic noise on the system dynamics where there are delays

on the identification of systems from data measurements using arma lattice models methods for the autoregressive moving average arma modeling of digital systems using a two channel autoregressive ar lattice are presented one method has the advantage that one half of the lattice parameters are zero when the system s excitation signal is white noise

convergence properties of an adaptive digital lattice filter convergence properties of a continuously adaptive digital lattice filter used as a linear predictor are investigated for both an unnormalized and a normalized gradient adaptation algorithm the parcor coefficient mean values and the output mean square error mse are approximated and a simple model is described which approximates these quantities as functions of time calculated curves using this model are compared with simulation results results obtained for a two stage lattice are then compared with the two stage least mean square lms transversal filter algorithm demonstrating that it is possible but unlikely for the transversal filter to converge faster than the analogous lattice filter

recursive least squares ladder estimation algorithms recursive least squares ladder estimation algorithms have attracted much attention recently because of their excellent convergence behavior and fast parameter tracking capability compared to gradient based algorithms we present some recently developed square root normalized exact least squares ladder form algorithms that have fewer storage requirements and lower computational requirements than the unnormalized ones a hilbert space approach to the derivations of magnitude normalized signal and gain recursions is presented the normalized forms are expected to have even better numerical properties than the unnormalized versions other normalized forms such as joint process estimators e g adaptive line enhancer and arma pole zero models will also be presented applications of these algorithms to fast or zero startup equalizers adaptive noise and echo cancellers non gaussian event detectors and inverse models for control problems are also mentioned

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

some remarks on the smoluchowski kramers approximation according to the smoluchowski kramers approximation solution qt of the equation mu ddot q t mu b q t mu dot q t mu sigma q t mu dot w t q 0 q dot q p where dot w t is the white noise converges to the solution of equation dot q t b q t sigma q t dot w t q 0 q as 0 many asymptotic problems for the last equation were studied in recent years we consider relations between asymptotics for the first order equation and the original second order equation homogenization large deviations and stochastic resonance approximation of brownian motion wt by a smooth stochastic process stationary distributions are considered

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

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

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

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

fast recursive least squares transversal filters for adaptive filtering fast transversal filter ftf implementations of recursive least squares rls adaptive filtering algorithms are presented in this paper substantial improvements in transient behavior in comparison to stochastic gradient or lms adaptive algorithms are efficiently achieved by the presented algorithms the true not approximate solution of the rls problem is always obtained by the ftf algorithms even during the critical initialization period first n iterations of the adaptive filter this true solution is recursively calculated at a relatively modest increase in computational requirements in comparison to stochastic gradient algorithms factor of 1 6 to 3 5 depending upon application additionally the fast transversal filter algorithms are shown to offer substantial reductions in computational requirements relative to existing fast rls algorithms such as the fast kalman algorithms of morf ljung and falconer 1976 and the fast ladder lattice algorithms of morf and lee 1977 1981 they are further shown to attain steady state unnormalized or improve upon first n initialization steps the very low computational requirements of the efficient rls solutions of carayannis manolakis and kalouptsidis 1983 finally several efficient procedures are presented by which to ensure the numerical stability of the transversal filter algorithms including the incorporation of soft constraints into the performance criteria internal bounding and rescuing procedures and dynamic range increasing square root normalized variations of the transversal filters

convergence analysis of ladder algorithms for ar and arma models abstract a class of exact fast algorithms originally introduced in the signal processing area is provided by the so called recursive least squares ladder forms the many nice numerical and structural properties of these algorithms have made them a very powerful alternative in a large variety of applications yet the convergence properties of the algorithms have not received the necessary attention this paper gives an asymptotic analysis of two ladder algorithms designed for autoregressive ar and autoregressive moving average arma models convergence is studied based on the stability properties of an associated differential equation it is shown that the convergence conditions obtained for the algorithms parallel those known for prediction error methods and for a particular type of pseudo linear regression

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

a new maximal inequality and invariance principle for stationary sequences we derive a new maximal inequality for stationary sequences under a martingale type condition introduced by maxwell and woodroofe ann probab 28 2000 713 724 then we apply it to establish the donsker invariance principle for this class of stationary sequences a markov chain example is given in order to show the optimality of the conditions imposed

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

tracking model of an adaptive lattice filter for a linear chirp fm signal in noise the paper studies the behavior of the partial correlation parcor coefficients and the output misadjustment of the stochastic gradient adaptive lattice filter in response to a complex linear chirp fm signal in white gaussian noise analytic expressions for the optimal parcor coefficients of the filter are derived analytic as well as iterative models for a three stage filter are also derived the analytic expressions show that the tracking and convergence properties of the filter are separate phenomena simulation results also show that the spectral contents of the parcor coefficients for the stochastic gradient update algorithm consist of a stationary and a linearly swept component a single stage model is developed to explain this behavior finally output misadjustment plots for the filter show that an optimum value for the forgetting factor can be obtained to minimize the misadjustment but the value required to achieve local minimum misadjustment varies with each stage of the filter it is shown that in applications where the input has a high signal to noise ratio snr the misadjustment decreases rapidly at each successive stage thus implying that relatively short filter lengths are sufficient to provide effective tracking

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

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

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

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

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

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

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

the smoluchowski kramers limit of stochastic differential equations with arbitrary state dependent friction we study a class of systems of stochastic differential equations describing diffusive phenomena the smoluchowski kramers approximation is used to describe their dynamics in the small mass limit our systems have arbitrary state dependent friction and noise coefficients we identify the limiting equation and in particular the additional drift term that appears in the limit is expressed in terms of the solution to a lyapunov matrix equation the proof uses a theory of convergence of stochastic integrals developed by kurtz and protter the result is sufficiently general to include systems driven by both white and ornstein uhlenbeck colored noises we discuss applications of the main theorem to several physical phenomena including the experimental study of brownian motion in a diffusion gradient

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

convergence models for lattice joint process estimators and least squares algorithms a simple model characterizing the convergence properties of an adaptive digital lattice filter using gradient algorithms has been reported 1 this model is extended to the least mean square lms lattice joint process estimator 5 and to the least squares ls lattice and fast kalman algorithms 9 16 the models in each case are compared with computer simulation the single stage lms lattice analysis presented in 1 is also applied to the ls lattice results indicate that for stationary inputs the lms lattice and ls algorithms exhibit similar behavior