well posedness for nonlinear schrodinger equations with boundary forces in low dimensions by strichartz estimates abstract in this paper we study the well posedness of solutions for nonlinear schrodinger equations on one and two dimensional domains with boundary where the boundary is disturbed by an external inhomogeneous type of dirichlet or neumann force we first prove the local existence of solutions at the energy level for quadratic and superquadratic sources using the strichartz estimates on domains secondly we obtain conditional uniqueness and local stability then we prove the boundedness of solutions in the energy space to pass from the local theory to the global theory regarding subquadratic sources we appeal to classical methods and trudinger s inequality to prove the uniqueness which combined with the existence of weak energy solutions mass and energy inequalities eventually implies the continuity of solutions in time

3 d sound source reconstruction and field reprediction using the helmholtz integral equation abstract a method for reconstruction of the boundary conditions of an arbitrarily shaped three dimensional 3 d sound source from a finite number of field pressures is presented and the associated field is repredicted by using the helmholtz integral equation to minimize the reconstruction error due to noise in the field pressures a filtering method is proposed and it is verified that numerically singular values which correspond to the higher order modes of the evanescent like components of the source velocities are eliminated for arbitrarily shaped sources with complex boundary conditions the reconstruction is performed accurately when the size of each element is less than a quarter wavelength and the field measurement positions are equally or randomly distributed on the surface of a sphere in the vicinity of the source our technique can be used in computer aided loudspeaker design that reconstructs loudspeaker diaphragm velocities from the finite number of field pressures and repredicts the variation of the sound field procuced by a redesigned loudspeaker cabinet on which the reconstructed diaphragm is mounted

singular value analysis of the jacobian matrix in microwave image reconstruction for non linear inverse scattering problems utilizing gauss newton methods the jacobian matrix encodes rich information concerning the system performance and algorithm efficiency in this paper we perform an analytical evaluation of a single iteration jacobian matrix based on a previously derived nodal adjoint representation concepts for studying linear ill posed problems such as the degree of ill posedness are used to assess the impact of important system parameters on the expected image quality analytical singular value decomposition svd of the jacobian matrix for a circular imaging domain is derived along with the numerical svd for optimizing imaging system configurations the results show significant reductions in the degree of ill posedness when signal frequency antenna array density and property parameter sampling are increased specifically the decay rate in the singular spectrum of the jacobian decreases monotonically with signal frequency being approximately 1 3 of its 0 1 ghz value at 3 ghz is improved with antenna array density up to about 35 equally spaced circumferentially positioned elements and drops significantly with increased property parameter sampling to more than twice the amount of measurement data these results should serve as useful guidelines in the development of design specifications for an optimized hardware installation

the generating hypothesis in the derived category of a ring we show that a strong form the fully faithful version of the generating hypothesis introduced by freyd in algebraic topology holds in the derived category of a ring r if and only if r is von neumann regular this extends results of the second author j pure appl algebra 208 2 2007 we also characterize rings for which the original form the faithful version of the generating hypothesis holds in the derived category of r these must be close to von neumann regular in a precise sense and given any of a number of finiteness hypotheses must be von neumann regular however we construct an example of such a ring that is not von neumann regular and therefore does not satisfy the strong form of the generating hypothesis

monotonicity method applied to the complex ginzburg landau and related equations abstract global existence of unique strong solutions is established for the complex ginzburg landau equation tu i u i u p 1u u 0 where 0 0 r p 1 and 1 2 p p 1 the key is a new inequality in monotonicity methods it is based on the sectorial estimates of in lp 1 and the nonlinear operator u u p 1u appearing in the equation the key inequality also yields the global existence of unique strong solutions of the nonlinear schrodinger type equation with monotone nonlinearity tu i u u p 1u 0 for all p 1

estimation of acoustic source strength by inverse methods part i conditioning of the inverse problem this paper deals with the discrete inverse problem in acoustics it is assumed that a number of acoustic sources are located at known spatial positions and that the acoustic pressure is measured at a number of spatial positions in the radiated field the transfer functions relating the strengths of the acoustic sources to the radiated pressures are also assumed known in principle the strengths of the acoustic sources can be deduced from the measured acoustic pressures by inversion of this matrix of transfer functions the accuracy of source strength reconstruction in the presence of noise which contaminates the measured pressures is crucially dependent on the conditioning of the matrix to be inverted this paper examines the conditioning of this inverse problem particularly with regard to the geometry and number of sources and measurement positions and the non dimensional frequency a preliminary investigation is also presented of methods such as tikhonov regularization and singular value discarding which can improve the accuracy of source strength reconstruction in poorly conditioned cases results are also presented which enable the solution of the inverse problem when the time histories of the acoustic sources are time stationary random processes and the spectra and cross spectra are measured at a number of positions in the radiated field the paper illustrates the possibilities and limitations of the use of inverse methods in the deduction of acoustic source strength from radiated field measurements

spherical harmonics singular value decomposition and the head related transfer function abstract the singular value decomposition svd can be used to analyze the matrix of green functions relating the acoustic pressure at a number of field points to the strengths of a number of point sources on the surface of a body which radiates or scatters sound the left and right singular vectors of the resulting decomposition yield at a given frequency two sets of orthogonal basis functions describing field mode shapes and source mode shapes respectively this paper attempts to make the connection between this decomposition and the basis functions of classical acoustics in particular it is found that for a spherical co ordinate system when the source points and field points are chosen in order to sample the source and field appropriately then the matrices of left and right singular vectors are related to the sampled spherical harmonics by a unitary transformation preliminary investigations are presented of the sensitivity of this relationship to the manner in which the source and field are sampled in addition the great utility of the method is illustrated with some new results of numerical analysis of the scattering of sound by the outer ear pinna resonances and their associated mode shapes are identified at certain frequencies where high values of the dominant singular values indicate a strong coupling between source and field

local smoothing properties of dispersive equations is it possible for time evolution partial differential equations which are reversible and conservative to smooth locally the initial data for the linear wave equation for instance the answer is no however in 10 t kato found a local smoothing property of the korteweg de vries equation the solution of the initial value problem is locally one derivative smoother than the initial datum kato s proof uses in a curcial way the algebraic properties of the symbol for the korteweg de vries equation and the fact that the underlying spatial dimension is one actually judging from the way several integrations by parts and cancellations conspire to reveal a smoothing effect one would be inclined to believe this was a special property of the k dv equation this is not however the case in this paper we attempt to describe a general local smoothing effect for dispersive equations and systems the smoothing effect is due to the dispersive nature of the linear part of the equation all the physically significant dispersive equations and systems known to us have linear parts displaying this local smoothing property to mention only a few the k dv benjamin ono intermediate long wave various boussinesq and schrodinger equations are included we study thus equations and systems of the form

asymptotic stability of the wave equation on compact surfaces and locally distributed damping a sharp result this paper is concerned with the study of the wave equation on compact surfaces and locally distributed damping described by u tt m u x g u t 0 on m 0 where m c r 3 is a smooth oriented embedded compact surface without boundary denoting by g the riemannian metric induced on m by r 3 we prove that for each 0 there exist an open subset v c m and a smooth function f m r such that meas v meas m e hessf g on v and inf f x 0 x v in addition we prove that if x a o 0 on an open subset m c m which contains m v and if g is a monotonic increasing function such that k s g s k s for all s 1 then uniform and optimal decay rates of the energy hold

strichartz inequalities and the nonlinear schrodinger equation on compact manifolds kklwe prove strichartz estimates with fractional loss of derivatives for the schrodinger equation on any riemannian compact manifold as a consequence we infer low regularity local well posedness results in any dimension as well as global existence results for the cauchy problem of nonlinear schrodinger equations on surfaces in the case of defocusing polynomial nonlinearities and on three manifolds in the case of cubic defocusing nonlinearities we also discuss the optimality of these strichartz estimates on spheres

global existence and open loop exponential stabilization of weak solutions for nonlinear schrodinger equations with localized external neumann manipulation abstract in this paper we consider the nonlinear schrodinger equations nls with focusing defocusing interior source and possibly nonlinear damping on a bounded regular domain in the euclidean space moreover it is assumed that the solutions are subject to external neumann boundary manipulation on one portion of the boundary our aim is to obtain global existence of the weak solutions under various assumptions on the sign of the source and powers of the nonlinearities in the case of a linear damping we also prove that solutions decay exponentially under the assumption that the neumann type control at the boundary decays in a similar manner

well posedness and energy decay estimates in the cauchy problem for the damped defocusing schrodinger equation abstract in this paper we study the existence at the h 1 level as well as the stability for the damped defocusing schrodinger equation in r d the considered damping coefficient is time dependent and may vanish at infinity to prove the existence we employ the method devised by ozsari kalantarov and lasiecka 27 which is based on monotone operators theory in particular when d 1 or d 2 we obtain the uniqueness decay estimates for the l 2 level and h 1 l p 2 level energies are established with the help of direct multipliers method coupled with refined energy estimates and a lower semi continuity argument

the radiation modes of baffled finite plates the modal style approach for representing the exterior radiation characteristics of structures generally seeks to find a set of orthogonal functions or radiation modes that diagonalize a discretized radiation operator in the exterior domain of the structure the choice of basis functions for the modal representation is arbitrary though use of the structural modes of vibration tends to provide some physical insight the radiation modes are found through an eigenanalysis or singular value decomposition analysis of the radiation operator the eigenvalue or singular value associated with a given radiation mode is directly proportional to the radiation efficiency of that radiation mode in this paper the dependency of the radiation efficiencies and radiation mode shapes on the number of degrees of freedom permitted in the derivation of the radiation operator is investigated for a baffled finite rectangular plate the accuracy of the acoustic modal representation depends on the number of degrees of freedom in the radiation operator with the least efficient radiation modes converging slowest further the rate of convergence is dependent on the particular basis function selection the convergence behavior has significant impact on those applications of the exterior acoustic modal approach that seek to exploit the least efficient radiation modes

on smooth solutions to the initial boundary value problem for the nonlinear schro dinger equation in two space dimensions on considere le probleme aux valeurs limites et initiales iu t u f u 2 u 0 x t 0 u x 0 u 0 x x u 0 t 0 ou est un domaine borne de r 2 a frontiere lisse

inverse aeroacoustic problem for a streamlined body ii accuracy of solutions previously established methods for performing the aeroacoustic inversion for a fiat plate airfoil are tested to determine their sensitivity to errors in both the far field input data and the mean flow input parameters the results of the sensitivity analysis for the input data show that the magnification of input error due to ill posedness can be controlled by optimal choices of the regularization parameter and the input data locations a similar sensitivity analysis of the effect of mean flow parameters mach number and reduced frequency shows strong dependence of the inverse process on these parameters in this case improvements in the reconstructions can be obtained by using input data from the midfield

the relationship of singular value decomposition to wave vector filtering in sound radiation problems singular value decomposition svd has recently been suggested as a filtering tool to analyze acoustic radiation the relationship between this filtering technique and the more familiar wave vector filtering is examined for a geometry with a preferred axis the source modes are found to consist approximately of a single wave number and the accompanying radiation modes are thus beams some simple examples are worked out to illustrate the general behavior of the filtering and some of the properties of the beams are given

weakly damped focusing nonlinear schrodinger equations with dirichlet control abstract in this article we consider the weakly damped focusing nonlinear schrodinger equations on bounded domains at the natural h 1 energy level with dirichlet control acting on a portion of the boundary we introduce the dynamic extension method for homogenizing the inhomogeneous boundary input then we construct approximate solutions using monotone operator theory a hidden trace regularity is proved to control the norm of the solutions in a global sense this allows the use of compactness techniques by which we prove the existence of weak solutions finally using multiplier techniques we prove the exponential decay of solutions under the assumption that the boundary control also decays in a similar fashion

inverse aeroacoustic problem for a streamlined body i basic formulation the inverse aeroacoustic problem associated with a flat plate airfoil in unsteady compressible flow is formulated in terms of a fredholm integral equation of the first kind the feasibility of determining the unsteady pressure along the airfoil surface from the radiated acoustic signal is demonstrated it is shown that the hadamard conditions of existence and uniqueness of the inverse solution can be satisfied the third hadamard condition for the continuous dependence of the solution on the input acoustic data shows the problem to be ill posed the singular value decomposition method with regularization is used to treat the associated ill conditioned algebraic problem discretization and collocation techniques are used to represent the unsteady pressure on the airfoil both methods give very accurate reconstructions when perfect input data are used the collocation method requires only a small amount of input data either from the far field or the near field thus making it more suitable for applications

qualitative properties of solutions for nonlinear schrodinger equations with nonlinear boundary conditions on the half line in this paper we study the interaction between a nonlinear focusing robin type boundary source a nonlinear defocusing interior source and a weak damping term for nonlinear schrodinger equations posed on the infinite half line we construct solutions with negative initial energy satisfying a certain set of conditions which blow up in finite time in the h1 sense we obtain a sufficient condition relating the powers of nonlinearities present in the model which allows construction of blow up solutions in addition to the blow up property we also discuss the stabilization property and the critical exponent for this model

the generating hypothesis for the stable module category of a p group freyd s generating hypothesis interpreted in the stable module category of a finite p group g is the statement that a map between finite dimensional kg modules factors through a projective if the induced map on tate cohomology is trivial we show that freyd s generating hypothesis holds for a non trivial finite p group g if and only if g is either c2 or c3 we also give various conditions which are equivalent to the generating hypothesis

phantom maps in the stable module category abstract we study phantom maps in the stable module category stmod kg wherekis a field andgis a finite group in this article we almost exclusively deal with maps out of countably generated modules we show that the space ph kg m n of stabilized phantom maps m n has an expression as a lim 1 space which allows some control on its vanishing then we present a situation where all maps are phantom and need not be trivial we provide explicit details for such a particular situation finally we construct a universal phantom map we use it to show that the composite of two phantom maps is trivial and to characterize the modules with no nontrivial outbound phantom maps

qualitative aspects for the cubic nonlinear schrodinger equations with localized damping exponential and polynomial stabilization abstract results of exponential polynomial decay rates of the energy in l 2 level related to the cubic nonlinear schrodinger equation with localized damping posed on the whole real line will be established in this paper we use kato s theory and a priori estimates to obtain the result of global well posedness and we determine exponential polynomial stabilization combining the ideas of unique continuation due to zhang semigroup property and komornik s approach

transfinite adams representability abstract we consider the following problems in a well generated triangulated category t let be a regular cardinal and t t the full subcategory of compact objects is every functor h t op ab that preserves products of h t x t for some x in t is every natural transformation t x t t y t of the form t f t for some f x y in t if the answer to both questions is positive we say that t satisfies adams representability a classical result going back to brown and adams shows that the stable homotopy category satisfies 0 adams representability the case 0 is well understood thanks to the work of christensen keller and neeman in this paper we develop an obstruction theory to decide whether t satisfies adams representability we derive necessary and sufficient conditions of homological nature and we compute several examples in particular we show that there are rings satisfying adams representability for all 0 and rings which do not satisfy adams representability for any 0 moreover we exhibit rings for which the answer to both questions is no for all 2

numerical modelling of the spatial acoustic response of the human pinna abstract this paper is concerned with the spatial acoustic response of the human pinna modelled with the boundary element method bem accurate geometric models of five pinnae are captured by using state of the art 3 d laser scanners and digitisers these computer models are converted to valid bem models and analysed acoustically up to 20 khz using baffled pinnae models we first repeat and validate numerically the normal concha modes as measured in the classical experiments made by e a g shaw in the 1970s we then study the spatial mode shapes of a baffled cylinder as a simplified model of the concha and accurate pinnae by applying the singular value decomposition svd technique the method is used to analyse at discrete frequencies a matrix of green functions relating the acoustic pressure at field points and source points in space we demonstrate how the singular vectors which appear as spatial mode shapes couple to the singular vectors of the sound field the relationship between these basis functions found with the svd and the normal concha mode shapes is demonstrated the method is also used in producing reduced order transfer functions by taking into account only the most dominant features of the singular vectors for the cases of a baffled cylinder and accurate baffled pinnae

local exact controllability and stabilizability of the nonlinear schrodinger equation on a bounded interval this paper studies the exact controllability and the stabilization of the cubic schrodinger equation posed on a bounded interval both internal and boundary controls are considered and the results are given first in a periodic setting and next with dirichlet resp neumann boundary conditions it is shown that the systems with either an internal control or a boundary control are locally exactly controllable in the classical sobolev space h s for any s geq0 it is also shown that the systems with an internal stabilization are locally exponentially stabilizable in h s for any s geq0

orlov spectra bounds and gaps the orlov spectrum is a new invariant of a triangulated category it was introduced by d orlov building on work of a bondal m van den bergh and r rouquier the supremum of the orlov spectrum of a triangulated category is called the ultimate dimension in this work we study orlov spectra of triangulated categories arising in mirror symmetry we introduce the notion of gaps and outline their geometric significance we provide the first large class of examples where the ultimate dimension is finite categories of singularities associated to isolated hypersurface singularities similarly given any nonzero object in the bounded derived category of coherent sheaves on a smooth calabi yau hypersurface we produce a generator by closing the object under a certain monodromy action and uniformly bound this generator s generation time in addition we provide new upper bounds on the generation times of exceptional collections and connect generation time to braid group actions to provide a lower bound on the ultimate dimension of the derived fukaya category of a symplectic surface of genus greater than one

an inhomogeneous boundary value problem for nonlinear schrodinger equations abstract we consider a nonlinear schrodinger equation in a domain r n with the inhomegeneous dirichlet boundary condition u q where q is a given smooth function the nonlinear term contributes a positive term to the energy we prove the existence of global solutions of finite energy

stabilization and control for the nonlinear schrodinger equation on a compact surface in this paper we study the stabilization property and the exact controllability for the nonlinear schrodinger equation on a two dimensional compact riemannian manifold without boundary we use a pseudo differential dissipation the proofs are based on a result of propagation of singularities and on recent dispersion estimates strichartz type inequalities due to n burq p gerard and n tzvetkov

ideals in triangulated categories phantoms ghosts and skeleta we begin by showing that in a triangulated category specifying a projective class is equivalent to specifying an ideal i of morphisms with certain properties and that if i has these properties then so does each of its powers we show how a projective class leads to an adams spectral sequence and give some results on the convergence and collapsing of this spectral sequence we use this to study various ideals in the stable homotopy category we examine phantom maps skeletal phantom maps superphantom maps and ghosts a ghost is a map which induces the zero map of homotopy groups we show that ghosts lead to a stable analogue of the lusternik schnirelmann category of a space and we calculate this stable analogue for low dimensional real projective spaces we also give a relation between ghosts and the hopf and kervaire invariant problems in the case ofa modules over ana ring spectrum the ghost spectral sequence is a universal coefficient spectral sequence from the phantom projective class we derive a generalized milnor sequence for filtered diagrams of finite spectra and from this it follows that the group of phantom maps fromxtoycan always be described as alim 1group the last two sections focus on algebraic examples in the derived category of an abelian category we study the ideal of maps inducing the zero map of homology groups and find a natural setting for a result of kelly on the vanishing of composites of such maps we also explain how pure exact sequences relate to phantom maps in the derived category of a ring and give an example showing that phantoms can compose non trivially

powers of the phantom ideal the mono epi me exact structure on the morphisms of an exact category a e is introduced and used to prove ideal versions of salce s lemma christensen s ghost lemma and wakamatsu s lemma for an exact category salce s lemma establishes a bijective correspondence i i between the class of special precovering ideals of a e and that of its special preenveloping ideals me extensions of morphisms are used to define an extension i j of ideals christensen s lemma asserts that the class of special precovering respectively special preenveloping ideals is closed under products and extensions and that the bijective correspondence of salce s lemma satisfies ij j i and i j j i wakamatsu s lemma asserts that if a covering ideal i is closed under me extensions then it is a special precovering ideal as an application it is proved that if g is a finite group and is the ideal of phantom morphisms in the category k g mod then n 1 is the object ideal generated by projective modules where n is the nilpotency index of the jacobson radical j if r is a semiprimary ring with jn 0 then n is generated by projective modules for a right coherent ring r over which every cotorsion left r module has a coresolution of length n by pure injective modules n 1 is generated by flat modules

bilinear eigenfunction estimates and the nonlinear schrodinger equation on surfaces we study the cubic non linear schrodinger equation nls on compact surfaces on the sphere mathbb s 2 and more generally on zoll surfaces we prove that for s 1 4 nls is uniformly well posed in hs which is sharp on the sphere the main ingredient in our proof is a sharp bilinear estimate for laplace spectral projectors on compact surfaces

escape function conditions for the observation control and stabilization of the wave equation for the linear wave equation with time invariant coefficients on a connected compact riemannian manifold omega g with c 3 boundary the geodesics condition of bardos lebeau and rauch siam j control optim 30 1992 pp 1024 1065 is characterized in terms of escape functions which are some lyapunov functions on the phase space s bar omega the unit sphere cotangent bundle differentiable escape functions yield a sufficient condition which is slightly less sharp but does not refer to geodesics the escape function condition yields a straightforward geometric proof that the geodesics condition holds in the situations where first order differential multiplier methods apply using microlocal control results it allows us to generalize some control results that were obtained by multiplier methods to variable coefficients and lower order terms it also allows us to prove in some class of simple situations e g in mathbb r 2 with constant coefficients that no first order differential multiplier method can reach the optimal control time or control regions

exact boundary controllability of the nonlinear schrodinger equation abstract this paper studies the exact boundary controllability of the semi linear schrodinger equation posed on a bounded domain r n with either the dirichlet boundary conditions or the neumann boundary conditions it is shown that if s n 2 or 0 s n 2 with 1 n 2 2 s or s 0 1 with n 2 then the systems are locally exactly controllable in the classical sobolev space h s around any smooth solution of the cubic schrodinger equation

estimation of acoustic source strength by inverse methods part ii experimental investigation of methods for choosing regularization parameters two regularization methods tikhonov regularization and singular value discarding are used to improve the accuracy of reconstruction of acoustic source strength by inverse techniques in this paper some methods are investigated for choosing the tikhonov regularization parameter and the singular values to be discarded of these we concentrate on the use of ordinary cross validation and generalized cross validation these methods can provide an appropriate regularization parameter without prior knowledge of either the acoustic source strength or the contaminating measurement noise some experimental results obtained using a randomly vibrating simply supported plate mounted in a baffle are presented to illustrate the performance of the methods for choosing the regularization parameters

uniform decay rate estimates for schrodinger and plate equations with nonlinear locally distributed damping on a compact n dimensional riemannian manifold m g we establish uniform decay rate estimates for the linear schrodinger and plate equations subject to an internal nonlinear damping locally distributed on the manifold our approach can be also employed for other equations provided that inverse inequality for the linear model occurs in the particular case of the wave equation where the well known geometric control condition gcc is equivalent to the observability inequality our method generalizes the results due to cavalcanti et al 2010 2009 9 10 regarding the optimal choice of dissipative regions

ghosts in modular representation theory a ghost over a finite p group g is a map between modular representations of g which is invisible in tate cohomology motivated by the failure of the generating hypothesis the statement that ghosts between finite dimensional g representations factor through a projective we define the ghost number of kg to be the smallest integer l such that the composite of any l ghosts between finite dimensional g representations factors through a projective in this paper we study ghosts and the ghost numbers of p groups we begin by showing that a weaker version of the generating hypothesis where the target of the ghost is fixed to be the trivial representation k holds for all p groups we then compute the ghost numbers of all cyclic p groups and all abelian 2 groups with c2 as a summand we obtain bounds on the ghost numbers for abelian p groups and for all 2 groups which have a cyclic subgroup of index 2 using these bounds we determine the finite abelian groups which have ghost number at most 2 our methods involve techniques from group theory representation theory triangulated category theory and constructions motivated from homotopy theory

exact controllability for the schrodinger equation the exact controllability of schrodinger equation in bounded domains with dirichlet boundary condition is studied both the boundary controllability and the internal controllability problems are considered concerning the boundary controllability the paper proves the exact controllability in h 1 omega with l 2 boundary control on the other hand the exact controllability in l 2 omega is proved with l 2 controls supported in a neighborhood of the boundary both results hold for an arbitrarily small time the method of proof combines both the hum hilbert uniqueness method and multiplier techniques

existence of solutions for schrodinger evolution equations we study the existence uniqueness and regularity of the solution of the initial value problem for the time dependent schrodinger equationi u t 1 2 u v t x u u 0 u0 we provide sufficient conditions onv t x such that the equation generates a unique unitary propagatoru t s and such thatu t s u0ec1 l2 c0 h2 n foru0eh2 n the conditions are general enough to accommodate moving singularities of type x 2 n 4 or x n 2 n 3

on global solutions to the initial boundary value problem for the damped nonlinear schrodinger equations where a 0 and q is a bounded domain in rn with smooth boundary x when a 0 undamped case the global existence of smooth solutions to the initial boundary value problem l 3 has been proved only when n 2 and f lu12 w lulp with p 2 brezis and gallouet cl with 2 i p i3 m tsutsumi 9 higher dimensional cases are very difficult pribl s and remain unproved as far as the author knows if we add a linear damping term iau a 0 then the mass ilull iztr1 and the energy ivu x t l dx j f lu x t l dx decay exponentially as t cc as is seen below which may assert the global existence of small smooth solutions even if the nonlinear term f lul u does not have a proper sign it is our purpose to establish the global existence and decay estimates of smooth solutions of the problem l 3 with small data there are a number of works on the nonlinear evolution equations see 3 5 71 and their references in the following spirit energy and decay estimates give global existence of solutions if the damping term is not added there is no hope that the decay estimates of solutions in some f q p 2 2 can be obtained since

a simple approach to the cauchy problem for complex ginzburg landau equations by compactness methods abstract this paper is concerned with the cauchy problem cgl in l 2 r n for complex ginzburg landau equations with laplacian and nonlinear term u q 2 u multiplied by the complex coefficients i and i respectively q 2 0 0 r the global existence of strong solutions to cgl is established without any upper restriction on q 2 but with some restriction on and the result corresponds to ginibre and velo 1996 3 proposition 5 1 which is technically proved by combining convolution regularizing methods with compactness localizing methods while our proof here is fairly simplified the key to our proof is the cauchy problem cgl r which is cgl with replaced with v r where v r x x r 2 x r v r x 0 x r the solvability of cgl r is a direct consequence of okazawa and yokota 2002 16 theorem 4 1 taking the limit of global strong solutions to cgl r as r yields a global strong solution to cgl the result gives also an unbounded version of okazawa and yokota 2002 16 theorem 1 1 with p 2 for the initial boundary value problem on bounded domains

uniform decay rates for the energy of weakly damped defocusing semilinear schrodinger equations with inhomogeneous dirichlet boundary control abstract in this paper we study the open loop stabilization as well as the existence and regularity of solutions of the weakly damped defocusing semilinear schrodinger equation with an inhomogeneous dirichlet boundary control first of all we prove the global existence of weak solutions at the h 1 energy level together with the stabilization in the same sense it is then deduced that the decay rate of the boundary data controls the decay rate of the solutions up to an exponential rate secondly we prove some regularity and stabilization results for the strong solutions in h 2 sense the proof uses the direct multiplier method combined with monotonicity and compactness techniques the result for weak solutions is strong in the sense that it is independent of the dimension of the domain the power of the nonlinearity and the smallness of the initial data however the regularity and stabilization of strong solutions are obtained only in low dimensions with small initial and boundary data