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

the l2 boundedness of pseudodifferential operators on donne une nouvelle demonstration du theoreme de calderon vaillancourt on obtient la l 2 continuite de a x d si le symbole a x satisfait a certaines conditions

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

on the critical nonlinear damped wave equation with large initial data abstract we study the nonlinear damped wave equation 0 1 u t t u t u u u x r n t 0 u 0 x u 0 x u t 0 x u 1 x x r n in the critical case of 2 n our aim is to prove the large time asymptotic formulas for solutions of the cauchy problem 0 1 without any restriction on the size of the initial data and on the spatial dimension

the inverse problem for distributive lattices abstract the inverse problem givena in vl n b in vl m andb in vl m find all a in vl n such that a a b b where l is distributive lattice denotes max min composition vl n denotes the set of all vector with n member on l and x a1 a2 an b1 b2 bn ai bj n m is investigated the necessary and sufficient condition for the existence of a solution is shown and an analytical solution is given

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

refined asymptotic profiles for a semilinear heat equation we study the large time behavior of the solutions of the cauchy problem for a semilinear heat equation partial t u delta u f x t u quad rm in bf r n times 0 infty quad u x 0 varphi x quad rm in bf r n quad quad rm p where f in c bf r n times 0 infty times bf r and varphi in l 1 bf r n 1 x k dx with k 0 assume that u is a solution of p satisfying f x t u x t le c 1 t a u x t quad x t in bf r n times 0 infty for some constants c 0 and a 1 then it is well known that the solution u behaves like the heat kernel in this paper we give the k 2 th order asymptotic expansion of the solution u and reveal the relationship between the asymptotic profile of the solution u and the nonlinear term f here k is the integer satisfying k 1 k k

on fuzzy bilinear equations abstract when a1 and a2 are m n fuzzy matrices x is an n 1 fuzzy matrix and is the max min composition the equations a1 x a2 x are called fuzzy bilinear equations the purpose of this present note is to solve these equations and determine the greatest solution

eigen fuzzy sets and fuzzy relations abstract when r is a fuzzy relation between the elements of a finite set x the fuzzy sets a of x such that r a a max min composition are called eigen fuzzy sets the main result of this paper is the determination of the greatest eigen fuzzy set associated with a given fuzzy relation and we give three methods illustrated by an example we then state that the greatest eigen fuzzy set associated with r the transitive closure of r is exactly the one associated with r finally we describe how to obtain all fuzzy relations keeping invariant a given fuzzy set

profile decompositions and blowup phenomena of mass critical fractional schrodinger equations abstract we study under the radial symmetry assumption the solutions to the fractional schrodinger equations of critical nonlinearity in r 1 d d 2 with levy index 2 d 2 d 1 2 we first prove the linear profile decomposition and then apply it to investigate the properties of the blowup solutions of the nonlinear equations with mass critical hartree type nonlinearity

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

nonlinear fractional schrodinger equations in one dimension abstract we consider the question of global existence of small smooth and localized solutions of a certain fractional semilinear cubic nls in one dimension i t u u c 0 u 2 u c 1 u 3 c 2 u u 2 c 3 u 3 x x 1 2 where c 0 r and c 1 c 2 c 3 c this model is motivated by the two dimensional water wave equation which has a somewhat similar structure in the eulerian formulation in the case of irrotational flows we show that one cannot expect linear scattering even in this simplified model more precisely we identify a suitable nonlinear logarithmic correction and prove global existence and modified scattering of solutions

fractional nonlinear schrodinger equation of order α 0 1 abstract we consider the cauchy problem for the fractional nonlinear schrodinger equation i t u 1 x u u 2 u t 0 x r u 0 x u 0 x x r where r the order of the fractional derivative 0 1 we obtain the large time asymptoitic behavior of solutions which has a logarithmic phase modifications for large time comparing with the linear problem

long range scattering for nonlinear schrodinger equations in one space dimension we consider the scattering problem for the nonlinear schrodinger equation in 1 1 dimensions open image in new window where x r 0 r p 3 we show that modified wave operators for exist on a dense set of a neighborhood of zero in the lebesgue spacel2 r or in the sobolev spaceh1 r the modified wave operators are introduced in order to control the long range nonlinearity u 2u

global analysis of a model for capillary water waves in two dimensions in this paper we prove a global regularity result for a quadratic quasilinear model associated to the water waves system with surface tension and no gravity in dimension 2 the capillary waves system the model we consider here retains most of the difficulties of the full capillary water waves system including the delicate time resonance structure and modified scattering it is slightly simpler however at the technical level and our goal here is to present our method in this simplified situation the analysis of the full system is presented in a preprint by the authors 2014 2016 wiley periodicals inc

fractional schrodinger equation some properties of the fractional schrodinger equation are studied we prove the hermiticity of the fractional hamilton operator and establish the parity conservation law for fractional quantum mechanics as physical applications of the fractional schrodinger equation we find the energy spectra of a hydrogenlike atom fractional bohr atom and of a fractional oscillator in the semiclassical approximation an equation for the fractional probability current density is developed and discussed we also discuss the relationships between the fractional and standard schrodinger equations

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

asymptotics for large time of solutions to the nonlinear schrodinger and hartree equations we study the asymptotic behavior in time of solutions to the cauchy problems for the nonlinear schrodinger equation with a critical power nonlinearity and the hartree equation we prove the existence of modified scattering states and the sharp time decay estimate in the uniform norm of solutions to the cauchy problem with small initial data this estimate is very important for the proof of the existence of modified scattering states to the nonlinear schrodinger equations with a critical nonlinearity and the hartree equation in order to derive the desired estimates we introduce a certain phase function since the previous methods based solely on a priori estimates of the operator x it acting on the solution without specifying any phase function do not work for the critical case under consideration the well known nonexistence of the usual l 2 scattering states shows that our result is sharp

long range scattering for non linear schrodinger and hartree equations in space dimension n 2 we consider the scattering problem for the non linear schrodinger nls equation with a power interaction with critical powerp 1 2 n in space dimensionsn 2 and 3 and for the hartree equation with potential x 1 in space dimensionn 2 we prove the existence of modified wave operators in thel2 sense on a dense set of small and sufficiently regular asymptotic states

cm theory of damped wave equations with stabilisation abstract the aim of this note is to extend the energy decay estimates from j wirth wave equations with time dependent dissipation i non effective dissipation j differential equations 222 2006 487 514 to a broader class of time dependent dissipation including very fast oscillations this is achieved using stabilisation conditions on the coefficient in the spirit of f hirosawa on the asymptotic behavior of the energy for wave equations with time depending coefficients math ann 339 4 2007 819 839

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

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

fuzzy bilinear equations abstract when a 1 and a 2 are m n fuzzy matrices x is an n 1 fuzzy matrix and is the max min composition the equations a 1 x a 2 x are called fuzzy bilinear equations our first main result of this paper is the determination of the greatest solution and result of the equation then we obtain an upper bounded number of steps for finding the greatest solution and result of the equation finally we present some radimentary properties of solutions and results

sharp asymptotic behavior of solutions for cubic nonlinear schrodinger equations with a potential we consider the cubic nonlinear schrodinger equation with a potential we obtain a sharp time decay for the global in time solution and we get the large time asymptotic profile of small solutions to the cauchy problem

nonlinear schrodinger equations with exceptional potentials abstract we consider the cubic nonlinear schrodinger equation with an exceptional potential we obtain a sharp time decay for the global in time solution and we get the large time asymptotic profile of small solutions we prove the existence of modified scattering for this model that is linear scattering modulated by a phase our approach is based on the spectral theorem for the perturbed linear schrodinger operator and a factorization technique that allows us to control the resonant nonlinear term we make some parity assumptions in order to control the small energy behavior of the scattering coefficients and of the wave functions

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

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

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