damped wave equation with a critical nonlinearity we study large time asymptotics of small solutions to the cauchy problem for nonlinear damped wave equations with a critical nonlinearity 2 t u t u u u 1 2 n 0 x r n t 0 u 0 x eu 0 x t u 0 x eu 1 x x r n where e 0 and space dimensions n 1 2 3 assume that the initial data uo h 0 n h 0 u 1 e h 1 0 n h 1 where n 2 weighted sobolev spaces are h l m e l2 m l x l 2 1 x 2 also we suppose that 2 n 0 u 0 x dx 0 where then we prove that there exists a positive e 0 such that the cauchy problem above has a unique global solution u c 0 oo h 0 satisfying the time decay property u t e g t x e t lp ce 1 2 n g 1 n 2 t n 2 1 1 p for all t 0 1 p where e 0 e 0

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

numerically efficient solution of dense linear system of equations arising in a class of electromagnetic scattering problems in this paper we present an efficient technique for solving a dense complex symmetric linear system of equations arising in the method of moments mom formulation to illustrate the application of the method we consider a finite array of scatterers which gives rise to a large number of unknowns the solution procedure utilizes preconditioned transpose free qmr ptfqmr iterations and computes the matrix vector products by employing a compressed impedance matrix the compression is achieved by reduced rank representation of the off diagonal blocks based on a partial qr decomposition which is followed by an iterative refinement both the preconditioning and the compression steps take advantage of the block structure of the matrix the convergence of the iterative procedure is investigated and the performance of the proposed algorithm is compared to that achieved by other schemes the effectiveness of the preconditioner and the degree of matrix compression are quantified finite arrays of variable shape and sizes are considered and it is demonstrated that the ability to solve large problems using this technique enables one to evaluate the edge effects in the finite array such array is basically flat and periodic but the algorithm is still efficient when variation with strict periodicity or flatness exists

recursive identification method for piecewise arx models a sparse estimation approach this paper deals with the identification of nonlinear systems using piecewise linear models by means of a sparse overparameterization this challenging problem is turned into a convex optimization problem the proposed method uses a likelihood based methodology which adaptively penalizes model complexity and directly leads to a recursive implementation in this sparse estimation approach the tuning of user parameters is avoided and the computational complexity is kept linear in the number of data samples numerical examples with both simulated and experimental data are presented and the results are compared with previously published methods

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

asymptotic behaviors of solutions for viscoelastic wave equation with space time dependent damping term abstract in this paper we consider a viscoelastic wave equation with an absorbing term and space time dependent damping term based on the weighted energy method and by assuming that the kernel decaying exponentially we obtain the l 2 decay rates of the solutions more precisely we show that the decay rates are the same as those obtained in lin et al 2010 15 for the semilinear wave equation with absorption term

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

a transpose free quasi minimal residual algorithm for non hermitian linear systems the biconjugate gradient method bcg for solving general non hermitian linear systems ax b and its transpose free variant the conjugate gradients squared algorithm cgs both typically exhib

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

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

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

the role of critical exponents in blow up theorems the sequel abstract in 27 fujita showed that for positive solutions the initial value problem in r n for u t u u p with p 1 exhibited the following behavior if p p c 1 2 n then the initial value problem does not have any nontrivial non negative solution existing on r n 0 a global solution whereas if p p c there exist global small data positive solutions as well as solutions which are non global we call such a result a blow up theorem of fujita type in 50 levine discussed the various theorems of this type that had appeared in the literature prior to 1990 in this paper we revisit the literature since 1990

global existence large time behavior and life span of solutions of a semilinear parabolic cauchy problem we investigate the behavior of the solution u x t of where 1 1 n 2 xi 2 is the laplace operator p 1 is a constant t 0 and is a nonnegative bounded continuous function in r n the main results are for the case when the initial value has polynomial decay near x assuming 1 x 75a with a 0 various questions of global in time existence and nonexistence large time behavior or life span of the solution u x t are answered in terms of simple conditions on a p and the space dimension n

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

asymptotic behavior of solutions to the generalized kdv burgers equation with slowly decaying data abstract we consider the asymptotic behavior of the global solutions to the initial value problem for the generalized kdv burgers equation it is known that the solution to this problem converges to a self similar solution to the burgers equation called a nonlinear diffusion wave in this paper we derive the optimal asymptotic rate to the nonlinear diffusion wave when the initial data decays slowly at spatial infinity in particular we investigate that how the change of the decay rate of the initial value affects the asymptotic rate to the nonlinear diffusion wave

decay property of solutions for damped wave equations with space time dependent damping term abstract we consider the cauchy problem for the damped wave equation with space time dependent potential b t x and absorbing semilinear term u 1 u here b t x b 0 1 x 2 2 1 t with b 0 0 0 and 0 1 using the weighted energy method we can obtain the l 2 decay rate of the solution which is almost optimal in the case c n 1 2 n combining this decay rate with the result that we got in the paper j lin k nishihara j zhai l 2 estimates of solutions for damped wave equations with space time dependent damping term j differential equations 248 2010 403 422 we believe that c n is a critical exponent note that when 0 c n coincides to the fujita exponent f n 1 2 n the new points include the estimate in the supercritical exponent and for not necessarily compactly supported data

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

l2 estimates of solutions for damped wave equations with space time dependent damping term abstract in this paper we consider the damped wave equation with space time dependent potential b t x and absorbing semilinear term u 1 u here b t x b 0 1 x 2 2 1 t with b 0 0 0 and 0 1 based on the local existence theorem we obtain the global existence and the l 2 decay rate of the solution by using the weighted energy method the decay rate coincides with the result of nishihara k nishihara decay properties for the damped wave equation with space dependent potential and absorbed semilinear term preprint in the case of 0 and coincides with the result of nishihara and zhai k nishihara j zhai asymptotic behaviors of time dependent damped wave equations preprint in the case of 0