alternating projections on manifolds we prove that if two smooth manifolds intersect transversally then the method of alternating projections converges locally at a linear rate we bound the speed of convergence in terms of the angle between the manifolds which in turn we relate to the modulus of metric regularity for the intersection problem a natural measure of conditioning we discuss a variety of problem classes where the projections are computationally tractable and we illustrate the method numerically on a problem of finding a low rank solution of a matrix equation

uniform stabilization of wave equation with localized damping and acoustic boundary condition abstract in this paper we deal with stability of the nonlinear wave equation with acoustic boundary conditions to non locally reacting boundary

boundary stabilization of linear elastodynamic systems it is proved that an elastic medium which occupies a bounded region in three dimensional euclidean space can be uniformly stabilized by means of traction forces applied on a portion of the boundary

global smooth solutions and stabilization of nonlinear elastodynamic systems with locally distributed dissipation we consider the existence of global solutions of the nonlinear elastodynamic system with a locally distributed damping in a bounded domain we assume that the energy function satisfies the strong ellipticity condition at the zero equilibrium the local dissipation is in the form a x u where the nonnegative function a x is only positive on a small portion of the domain we show the existence of global smooth solutions when initial data are small in particular we obtain the exponential decay of the energy which implies the exponential stabilization of the system by internal feedback

a generalized interpolation inequality and its application to the stabilization of damped equations in this paper we establish a generalized holder s or interpolation inequality for weighted spaces in which the weights are non necessarily homogeneous we apply it to the stabilization of some damped wave like evolution equations this allows obtaining explicit decay rates for smooth solutions for more general classes of damping operators in particular for 1 d models we can give an explicit decay estimate for pointwise damping mechanisms supported on any strategic point

new trends towards lower energy estimates and optimality for nonlinearly damped vibrating systems abstract we present an approach based on comparison principles for energy and interpolation properties to derive lower energy estimates for nonlinearly either locally damped or boundary damped vibrating systems we show how the dissipation relation provides strong information on the asymptotic behavior of the energy of solutions the geometrical situations are either one dimensional or radial two dimensional or three dimensional for annulus domains we also consider the case of general domains but in this case for solutions with bounded velocities in time and space in all these cases the nonlinear damping function is assumed to have arbitrary strictly sublinear growth at the origin we give results for strong solutions and stronger lower estimates for smoother solutions the results are presented in two forms either on the side of energy comparison principles or through time pointwise lower estimates under additional geometric assumptions we give the resulting lower and upper estimates for four representative examples of damping functions we further give a weak lower estimate in the sense of a certain lim sup t and an upper estimate of the velocity for smoother solutions in case of general damping functions and for radial as well as multi dimensional domains we also discuss these estimates in the framework of optimality which is not proved here and indicate open problems raised by these results

global existence to the initial boundary value problem for a system of semilinear wave equations abstract we prove the global existence of finite energy and h 2 solutions for a system of nonlinear wave equations in a bounded domain one equation of the system has a dissipative term while another equation has no dissipative mechanism to derive the delicate a priori estimates we employ a loan method

stabilization of local energy in an exterior domain for the wave equation with a localized dissipation abstract we derive an algebraic decay rate for the local energyeloc t of the solutions to the initial boundary value problem of the wave equation with a localized dissipation in an exterior domain the dissipative terma x utis assumed to be effective only on a neighbourhood of a part of the boundary x0 x x x0 x 0 for somex0 where x is unit outward normal atxwith respect to it should be noted that we make no geometrical condition on the boundary ifv rn 0 is star shaped with respect tox0 x0 is empty and hence we may assumea x 0 thus our result is a natural extension of the well known most classical result

decay of solutions of the wave equation with arbitrary localized nonlinear damping abstract we study the problem of decay rate for the solutions of the initial boundary value problem to the wave equation governed by localized nonlinear dissipation and without any assumption on the dynamics i e the control geometric condition is not satisfied we treat separately the autonomous and the non autonomous cases providing regular initial data without any assumption on an observation subdomain we prove that the energy decays at last as fast as the logarithm of time our result is a generalization of lebeau in a boutet de monvel v marchenko eds algebraic and geometric methods in mathematical physics kluwer academic publishers dordrecht the netherlands 1996 pp 73 result in the autonomous case and nakao adv math sci appl 7 1 1997 317 work in the non autonomous case in order to prove that result we use a new method based on the fourier bross iaglintzer fbi transform

derivatives of spectral functions a spectral function of a hermitian matrix x is a function which depends only on the eigenvalues of x 1x 2x nx and hence may be written f 1x 2x nx for some symmetric function f such functions appear in a wide variety of matrix optimization problems we give a simple proof that this spectral function is differentiable at x if and only if the function f is differentiable at the vector x and we give a concise formula for the derivative we then apply this formula to deduce an analogous expression for the clarke generalized gradient of the spectral function a similar result holds for real symmetric matrices

global existence and asymptotic behavior for a coupled hyperbolic system with localized damping abstract in this paper we investigate the global existence and energy decay rate for the solution of a coupled hyperbolic system the semi explicit energy decay rate is established by using piecewise multiplier techniques and weighted integral inequality we extend the energy decay result in alabau boussouira f alabau boussouira convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems appl math optim 51 2005 61 105 for a single equation to the coupled hyperbolic system

the lack of exponential stability in certain transmission problems with localized kelvin voigt dissipation in this paper we consider the transmission problem of a material composed of three components one of them is a kelvin voigt viscoelastic material the second is an elastic material no dissipation and the third is an elastic material inserted with a frictional damping mechanism the main result of this paper is that the rate of decay will depend on the position of each component when the viscoelastic component is not in the middle of the material then there exists exponential stability of the solution instead when the viscoelastic part is in the middle of the material there is not exponential stability in this case we show that the corresponding solution decays polynomially as 1 t 2 moreover we show that the rate of decay is optimal over the domain of the infinitesimal generator finally using a second order scheme that ensures the decay of energy newmark beta method we give some numerical examples which demonstrate this asymptotic behavior

weighted l2 estimates for dissipative wave equations with variable coefficients abstract we establish weighted l 2 estimates for the wave equation with variable damping u t t u a u t 0 in r n where a x a 0 1 x with a 0 0 and 0 1 in particular we show that the energy of solutions decays at a polynomial rate t n 2 1 if a x a 0 x for large x we derive these results by strengthening significantly the multiplier method this approach can be adapted to other hyperbolic equations with damping

sharp energy decay estimates for the wave equation with a local degenerate dissipation we derive a fast decay estimate for the wave equation with a local degenerate dissipation of the type a x u t in a bounded domain w the dissipative coefficient a x is a nonnegative function only on a neighborhood of some part of the boundary w and may vanish somewhere in w the results obtained extend and improve on earlier results of nakao as well as those of tcheugoue tebou the method of proof is based on multipliers technique and some modified interpolation and difference inequalities

on a navier stokes allen cahn model with inertial effects abstract a mathematical model describing the flow of two phase fluids in a bounded container is considered under the assumption that the phase transition is influenced by inertial effects the model couples a variant of the navier stokes system for the velocity u with an allen cahn type equation for the order parameter relaxed in time in order to introduce inertia at the microscopic level the resulting model is characterized by second order material derivatives which constitute the main difficulty in the mathematical analysis in order to obtain a tractable problem a viscous relaxation term is then included in the phase equation the mathematical results consist in existence of weak solutions in 3d and under additional assumptions existence and uniqueness of strong solutions in 2d a partial characterization of the long time behavior of solutions is also given and in particular some issues related to dissipation of energy are discussed

existence and energy decay rates of solutions to the variable coefficient euler bernoulli plate with a delay in localized nonlinear internal feedback abstract this paper is concerned with the existence and energy decay estimates of solutions to the variable coefficient euler bernoulli plate with a delay in localized nonlinear internal feedback the existence of solutions to the nonlinear system is obtained by using galerkin method combined with some energy estimates furthermore applying riemannian geometry method we establish uniform decay rates of the plate equation with variable coefficients and a nonlinear delay term

well posedness and stability of a hinged plate equation with a localized nonlinear structural damping abstract we consider an n dimensional plate equation in a bounded domain with a locally distributed nonlinear dissipation involving the laplacian the dissipation is effective in a neighborhood of a suitable portion of the boundary when the space dimension equals two the associated linear equation corresponds to the plate equation with a localized viscoelastic or structural damping first we prove existence uniqueness and smoothness results then using an appropriate perturbed energy coupled with multiplier technique we directly prove exponential and polynomial decay estimates for the underlying energy to the author s best knowledge the perturbed energy approach is new in the framework of stabilization of second order evolution equations with locally distributed damping

exponential decay of solutions of maxwell s equations coupled with a first order ordinary differential equation for the polarization this paper is concerned with maxwell s equations involving a first order ordinary differential equation for the dielectric polarization the main goal is the exponential convergence of the electromagnetic field to an asymptotic state which is determined by the initial data this is achieved by the construction of a suitable energy functional involving a vector potential which provides an additional dissipative term for the magnetic field

strong decay for wave equations with nonlinear nonmonotone damping thus the energy is nonincreasing and the term p represents damping here and always the ymbols i ii denote the norm on the sobolev space wk p o 1 the classical example p d u d a positive constant shows that under reasonable coercivity hypotheses concerning f any solution u u decays to zero exponentially with respect to the trong topology on the energy space hd o 1 x l o 1 this property is preserved even if tl d x with d 0 on a certain nonvoid subinterval i e 0 l see l 4 moreover the same is true if p d g u d as above and g is globally lipschitz and satisfies the growth condition g w w 2 const w2 wer 1 2 cf 3 in the case f 0 3 d x g ut g monotone the strong decay to zero follows from a variant of the lasalle invariance principle see 5 91 in some particular cases the speed of convergence is obtained lo

energy decay and periodic solution for the wave equation in an exterior domain with half linear and nonlinear boundary dissipations abstract we first consider the wave equation in an exterior domain in r n with two separated boundary parts 0 1 on 0 the dirichlet condition u 0 0 is imposed while on 1 neumann type nonlinear boundary dissipation u g u t is assumed further a half linear localized dissipation is attached on for such a situation we derive a precise rate of decay of the energy e t for solutions of the initial boundary value problem we impose no geometrical condition on the shape of the boundary 0 1 secondly when a t periodic forcing term works we prove the existence of a t periodic solution on r under an additional growth assumption on x v and g v

frictional versus viscoelastic damping in a semilinear wave equation in this article we show exponential and polynomial decay rates for the partially viscoelastic nonlinear wave equation subject to a nonlinear and localized frictional damping the equation that models this problem is given by begin eqnarray u tt kappa 0 delta u int 0 t mbox div a x g t s nabla u s ds f u b x h u t 0 mbox in omega times bbb r quad end eqnarray where a b are nonnegative functions a in c 1 overline omega b in l infty omega satisfying the assumption begin eqnarray a x b x geq delta 0 quad forall x in omega end eqnarray and f and h are power like functions we observe that the assumption 0 2 gives us a wide assortment of possibilities from which to choose the functions a x and b x and the most interesting case occurs when one has simultaneous and complementary damping mechanisms taking this point of view into account a distinctive feature of our paper is exactly to consider different and localized damping mechanisms acting in the domain but not necessarily strategically localized dissipations as considered in the prior literature

oblique projection local feedback stabilization of nonautonomous semilinear damped wave like equations abstract the stabilization of a class of nonlinear weakly damped wave equations by means of a finite dimensional feedback control is investigated the stabilizing control is constructed based on an appropriate oblique projection and it enters as a time dependent linear combination of a finite number of suitable indicator functions supported in small regions firstly it is shown that an oblique projection feedback is able to globally exponentially stabilize linear nonautonomous weakly damped wave equations then relying on this result the local stabilization for semilinear equations is proven for a suitable class of nonlinearities finally numerical experiments are given which validate the theoretical results

total energy decay for the wave equation in exterior domains with a dissipation near infinity we show that the energy of solutions to the initial boundary value problem for the wave equation in exterior domains with a dissipation which is localized only near infinity tends to zero as the time goes to infinity we do not make any geometrical condition like star shapedness on the boundary

on the energy decay for dissipative nonlinear wave equations in one space dimension abstract the energy decay problem is studied for the nonlinear dissipative wave equation in one space dimension it is shown by mochizuki and motai 6 that the decay rate is at least logarithmic when the exponent of the nonlinearity is greater than one and less than three in this paper an improvement is found which implies a polynomial decay rate for the same range of exponents

on a first order hyperbolic system including bean s model for superconductors with displacement current abstract this paper deals with a class of first order systems with generally multivalued damping terms including bean s model for superconductors with displacement current the main goal is to prove the convergence of solutions to stationary states as t

the higher order derivatives of spectral functions in this paper we are interested in the higher order derivatives of functions of the eigenvalues of symmetric matrices with respect to the matrix argument we describe the formula for the k th derivative of such functions in two general cases the flrst case concerns the derivatives of the composition of an arbitrary not necessarily symmetric k times difierentiable function with the eigenvalues of symmetric matrices at a symmetric matrix with distinct eigenvalues the second case describes the derivatives of the composition of a k times difierentiable separable symmetric function with the eigenvalues of symmetric matrices at an arbitrary symmetric matrix we show that the formula signiflcantly simplifles when the separable symmetric function is k times continuously difierentiable as an application of the developed techniques we re derive the formula for the hessian of a general spectral function at an arbitrary symmetric matrix the new tools lead to a shorter cleaner derivation than the original one in 16 to make the exposition as self contained as possible we have included the necessary background results and deflnitions the proofs of the intermediate technical results are collected in the appendices

the rate at which energy decays in a viscously damped hinged euler bernoulli beam we study the best decay rate of the solutions of a damped euler bernoulli beam equation with a homogeneous dirichlet boundary conditions we show that the fastest decay rate is given by the supremum of the real part of the spectrum of the infinitesimal generator of the underlying semigroup if the damping coefficient is in l infty 0 1

energy decay of a microbeam model with a locally distributed nonlinear feedback control abstract in this paper we address the problem of internal stabilization of the deflection of a microbeam which is modeled by a sixth order hyperbolic equation employing multiplier techniques and an integral inequality we prove that a locally distributed nonlinear feedback control forces the energy associated to the deflection to decay exponentially or polynomially to zero as a consequence of this the deflection goes to the rest position as the time goes to infinity

exponential stabilization of conservation systems with interior disturbance abstract in this paper we consider the stabilization problem of the conservation systems with interior disturbance employing the idea of sliding mode control we design a nonlinear distributed feedback controller we prove the solvability of the resulted closed loop system by the maximal monotone operator theory further we prove the exponential stability of the closed loop system in particular we prove that the requirement of a classical solution in the lyapunov approach is unnecessary

exact controllability stabilization and perturbations for distributed systems exact controllability is studied for distributed systems of hyperbolic type or for petrowsky systems like plate equations the control is a boundary control or a local distributed control exact controllability consists in trying to drive the system to rest in a given finite time the solution of the problems depends on the function spaces where the initial data are taken and also depends on the function space where the control can be chosen a systematic method named hum for hilbert uniqueness method is introduced as the terminology indicates it is based on uniqueness results classical or new and on hilbert spaces constructed in infinitely many ways by using uniqueness a number of applications are indicated having a general method for exact controllability implies having a general method for stabilization this leads to new and complicated nonlinear riccati s type pdes to be compared with direct methods when available in the last part of the paper we consider how all this behaves fo

stability analysis of dissipative systems subject to nonlinear damping via lyapunov techniques in this paper we provide a general strategy based on lyapunov functionals to analyze global asymptotic stability of linear infinite dimensional systems subject to nonlinear dampings under the assumption that the origin of the system is globally asymptotically stable with a linear damping to do so we use the fact that for any linear infinite dimensional system that is globally exponentially stable there exists a lyapunov functional then we derive a lyapunov functional for the nonlinear system which is the sum of a lyapunov functional coming from the linear system and another term that compensates the nonlinearity our results are then applied to the linearized korteweg de vries equation and some wave equations

exponential decay of energy of evolution equations with locally distributed damping consider an evolution equation with energy dissipation frac partial partial t y x t ay x t by x t on a bounded x domain omega where by signifies a dissipative perturbation to an otherwise energy conserving system this dissipation may be due to medium impurities viscous effects or artificially imposed dampers and stabilizers it is distributed over only part of the domain omega the question of when the dissipation is effective enough to cause uniform exponential decay of energy is examined because of the locally distributed nature of energy dissipation the problem lacks coercivity and is not directly solvable by energy identities thus to get conditions sufficient for uniform exponential decay a different approach needs to be taken provided here is a set of tight sufficient conditions in terms of the influence of the dissipative operatorbon the separated eigenmodes or clustered eigenmodes ofa the main theorems are general enough to treat the wave and bea

rate of decay of solutions of the wave equation with arbitrary localized nonlinear damping abstract we study the rate of decay of solutions of the wave equation with localized nonlinear damping without any growth restriction and without any assumption on the dynamics providing regular initial data the asymptotic decay rates of the energy functional are obtained by solving nonlinear ode moreover we give explicit uniform decay rates of the energy more precisely we find that the energy decays uniformly at last as fast as 1 ln t 2 2 0 when the damping has a polynomial growth or sublinear and for an exponential damping at the origin the energy decays at last as fast as 1 ln ln t e 2 2 0

liapunov razumikhin pairs and the location of positive limit sets for precompact functional differential equations with infinite delay in a recent paper l haddock and terjeki addressed the problem of locating positive limit sets for autonomous functional differential equations with infinite delay the authors obtained invariance principles and other results concerning asymptotic behavior of solutions by employing liapunov razumikhin techniques the autonomous nature of the equations they considered was central to their work the purpose of this paper is to extend their results to a class of nonautonomous equations such a task was undertaken for finite delay equations in 2 here as in 2 the assumption of autonomy is replaced by a continuity condition and our invariance principles involve limiting equations of our original equation we feel that the setting for our theory is quite natural and that our methods illustrate the role of limiting equations in extending results from the autonomous theory to a significantly larger class of equations several examples are given to illustrate our techniques

asymptotic behavior of electromagnetic waves in nonlinear dielectric media this paper is concerned with maxwell s equations including a nonlinear dielectric polarization and a generally nonlinear law for the electric current the large time behavior of the electromagnetic field is investigated in the case of a bounded spatial domain as well as in the exterior domain case

exact controllability for a thermodiffusion system with locally distributed controls abstract in this work we establish a exact controllability result for a thermodiffusion system modeled by cattaneo s law posed in a one dimensional domain in the present model the control mechanisms are effective in a small subinterval of the domain to obtain the desired results we prove an observability inequality for the adjoint system which together with the multiplier methods and the hilbert uniqueness method hum developed by j l lions gives the controllability

uniform exponential long time decay for the space semi discretization of a locally damped wave equation via an artificial numerical viscosity we consider the finite difference space semi discretization of a locally damped wave equation the damping being supported in a suitable subset of the domain under consideration so that the energy of solutions of the damped wave equation decays exponentially to zero as time goes to infinity the decay rate of the semi discrete systems turns out to depend on the mesh size h of the discretization and tends to zero as h goes to zero we prove that adding a suitable vanishing numerical viscosity term leads to a uniform with respect to the mesh size exponential decay of the energy of solutions this numerical viscosity term damps out the high frequency numerical spurious oscillations while the convergence of the scheme towards the original damped wave equation is kept we discuss this problem in 1d and 2d in the interval and the square respectively our method of proof relies on discrete multiplier techniques

sharp sufficient conditions for the observation control and stabilization of waves from the boundary for the observation or control of solutions of second order hyperbolic equation in mathbb r t times omega ralston s construction of localized states comm pure appl math 22 1969 pp

on the decay of solutions of some nonlinear dissipative wave equations in higher dimensions on etudie la propriete de decroissance des solutions des equations d onde dissipatives non lineaires de la forme 2 u t 2 u g u t f x t x t r avec les conditions u x o u 0 x u t x o u 1 x x et u x t 0 sur r ou est un domaine borne de r n avec une frontiere lisse et g v est une fonction du genre k v v k o o

energy decay rates for the semilinear wave equation with nonlinear localized damping and source terms abstract in this paper we develop an intrinsic approach to derivation of energy decay rates for the semilinear wave equation with localized interior nonlinear monotone damping g u t and a source term f u the proposed approach allows to consider in an unified way much more general classes of hyperbolic problems than addressed before in the literature these generalizations refer to both geometric and topological aspects of the problem the method leads to optimal decay rates for solutions of semilinear hyperbolic equations driven by a source of critical exponent and subjected to a nonlinear damping localized in a small region adjacent to a portion of the boundary the distinct features of the model include i neumann boundary conditions are assumed and ii no growth conditions are imposed on the damping g s it is well known that neumann boundary does not satisfy lopatinski condition and therefore the analysis of propagation of energy in the absence of the damping on the neumann part of the boundary requires special geometric considerations in addition the sole conditions assumed on g s are monotonicity continuity and g 0 0 in particular no differentiability and no growth conditions are imposed on the damping both at the origin and at the infinity the asymptotic decay rates for the energy function are obtained from an intrinsic algorithm driven by solutions of simple ode several examples illustrate the theory by exhibiting various decay rates exponential algebraic rational logarithmic etc for the energy functional an important corollary of our energy decay theorem is a stability result which shows that under certain conditions when dissipation is sublinear at infinity the solution of the system remains uniformly bounded for all time in the norms above the finite energy level even in the presence of a nonlinear source term

weak asymptotic decay via a relaxed invariance principle for a wave equation with nonlinear non monotone damping this paper considers the problem of asymptotic decay as t of solutions of the wave equation u tt u a x u t u t x a bounded open connected set in n n 1 with smooth boundary u 0 on the nonlinear function is not assumed to be globally lipschitz continuous 0 y 2 y n 1 0 y 1 y 1 y n 1 0 for all y n 1 is not assumed to be monotone in y 1 under additional restrictions on the kernel of conditions are given which imply that u u t converges to 0 0 weakly in h h 1 0 l 2 as t the work generalises earlier results of dafermos and haraux where strong decay in h as t was obtained in the case y 1 y n 1 q y 1 q monotone on

on the location of positive limit sets for autonomous functional differential equations with infinite delay in 1983 the authors published a paper lo on an invariance principle for finite delay functional differential equations fdes in 9 t krisztin and the authors extended many of the ideas of lo to include fdes with infinite delay the basic results in both 9 and lo centered around the use of liapunov razumikhin techniques and the location of positive limit sets of precompact orbits the main purpose of this paper is to extend the results of 9 and to provide several examples to illustrate the main theorems of particular interest will be to establish asymptotic constancy of solutions of equations for which each constant function is itself a solution this idea was one of the main themes in lo but was not examined in 9 likewise we will see how a family of spaces as opposed to a single phase space can be useful in examining certain equations cf example 3 4 in considering the use of limit sets in conjunction with razumikhin techniques and invariance principles for infinite delay equations there are several difficulties that are encountered these include i choice of space s ii precompactness of orbits and iii appropriate comparison principles these topics are discussed individually in 11 7 and 8

stabilization of trajectories for some weakly damped hyperbolic equations in this paper we study the phenomenon of stabilization of trajectories for a wave equation in a bounded open domain endowed with a weak dissipative mechanism by stabilization we mean that the self oscillations induced by the wave equation are damped out asymptotically and so we are left when time tends to infinity either with an equilibrium if the system is autonomous or with a forced oscillation if the system was submitted to an exterior periodic or almost periodic excitation the common point between the two situations is the extinction of selfoscillations in the presence of a weak damping and it is extremely satisfactory to discover that the difficulty is overcome in both cases by the use of one simple and elegant idea due to c m dafermos and which appeared in s precisely in the study of some partially damped hyperbolic systems the paper is divided in four sections in section 1 we recall the idea of s which will be used in sections 2 and 3 in section 2 we generalize the result of s proposition 2 31 in a framework which allows the existence of nontrivial equilibria a simple example from continuum mechanics is given in section 3 we apply the same device to establish total damping in the sense of s for a wave equation with single valued damping term and almost periodic forcing the result which we obtain is not covered by the general theorems of 7 or 8 a simple example is studied in section 4 we give two counterexamples which show that the result of section 3 cannot be extended to the framework of weak solutions or to other boundary conditions in each specific case a careful analysis will be necessary the notation which is used throughout the text is the standard one with 145 0022 0396185 3 00

approximating functions on stratified sets we investigate smooth approximations of functions with prescribed gradient behavior on a distinguished stratified subset of the domain as an application we outline how our results yield important consequences for a recently introduced class of stochastic processes called the matrix valued bessel processes

singular internal stabilization of the wave equation abstract we consider an initial and boundary value problem for the one and two dimensional wave equation with nonlinear damping concentrated on an interior point and respectively on an interior curve in the two dimensional case our main result asserts that generically i e for almost all interior curves the solutions decay to zero in the energy space when the domain is strictly convex we show that whatever the interior curve is the decay is not uniform we generalize in this way results known in one space dimension our main improvement of existing one dimensional results consists in giving sharp decay rates provided that the initial data are regular and the damping term is linear a crucial intermediate step is the proof of a generalization of ingham s inequality on nonharmonic fourier series

smooth approximation of lipschitz functions on riemannian manifolds abstract we show that for every lipschitz function f defined on a separable riemannian manifold m possibly of infinite dimension for every continuous e m 0 and for every positive number r 0 there exists a c smooth lipschitz function g m r such that f p g p e p for every p m and lip g lip f r consequently every separable riemannian manifold is uniformly bumpable we also present some applications of this result such as a general version for separable riemannian manifolds of deville godefroy zizler s smooth variational principle

stabilization of the wave equation with localized nonlinear damping abstract we prove some decay estimates of the energy of the wave equation in a bounded domain the damping is nonlinear and is effective only in a neighborhood of a suitable subset of the boundary the method of proof is direct and is based on the multipliers technique on some integral inequalities due to haraux and komornik and on a judicious idea of conrad and rao

stabilization of the elastodynamic equations with a degenerate locally distributed dissipation we consider the dynamic elasticity equations with a locally distributed damping in a bounded domain the local dissipation of the form a x yt involves coefficients a that vanish on a negligible portion of the subset where the damping is effective using multiplier techniques interpolation inequalities and a judicious application of holder inequality we prove sharp energy decay estimates the results obtained generalize and improve on earlier works by nakao and the author in the framework of the ordinary wave equation

nonsmooth analysis of eigenvalues the eigenvalues of a symmetric matrix depend on the matrix nonsmoothly this paper describes the nonsmooth analysis of these eigenvalues in particular i present a simple formula for the approximate limiting frechet subdifferential of an arbitrary function of the eigenvalues subsuming earlier results on convex and clarke subgradients as an example i compute the subdifferential of the k th largest eigenvalue

on a uniqueness theorem of l amerio and g prouse we give two generalizations of a theorem of l amerio and g prouse concerning uniqueness of the almost periodic solution of the wave equation with a local multivalued damping term and almost periodic forcing counterexamples are given which show that these results may fail if some hypotheses of the theorems are dropped they also show that the two generalizations are relatively independent

asymptotic stability of the wave equation on compact manifolds and locally distributed damping a sharp result let m g be a n dimensional n geqq 2 compact riemannian manifold with boundary where g denotes a riemannian metric of class c this paper is concerned with the study of the wave equation on m g with locally distributed damping described by left begin array l u tt delta bf g u a x g u t 0 quad hbox on thinspace m times left 0 infty right u 0 hbox on partial m times left 0 infty right end array right where m represents the boundary of m and a x g ut is the damping term the main goal of the present manuscript is to generalize our previous result in cavalcanti et al trans ams 361 9 4561 4580 2009 treating the conjecture in a more general setting and extending the result for n dimensional compact riemannian manifolds m g with boundary in two ways i by reducing arbitrarily the region m ast subset m where the dissipative effect lies this gives us a totally sharp result with respect to the boundary measure and interior measure where the damping is effective ii by controlling the existence of subsets on the manifold that can be left without any dissipative mechanism namely a precise part of radially symmetric subsets an analogous result holds for compact riemannian manifolds without boundary

stabilization of the euler bernoulli plate with variable coefficients by nonlinear internal feedback we consider the energy decay for solutions of the euler bernoulli plate equation with variable coefficients where a nonlinear internal feedback acts in a suitable subregion of the domain the riemannian geometric method is used to deal with variable coefficient problems when the feedback region has a structure similar to that for the wave equation with constant coefficients we establish the stabilization of the system in the case of fixed boundary conditions several energy decay rates are established according to various growth restrictions on the nonlinear feedback near the origin and at infinity we further show that unlike for the case of constant coefficients choices of such feedback regions depend not only on the type of boundary conditions but also on the curvature of a riemannian metric based on the coefficients of the system

uniform decay rates for the wave equation with nonlinear damping locally distributed in unbounded domains with finite measure this paper is concerned with the study of the uniform decay rates of the energy associated with the wave equation with nonlinear damping locally distributed u tt delta u a x g u t 0 hbox in omega times 0 infty subject to dirichlet boundary conditions where omega subset mathbb r n n geq 2 is an unbounded open set with finite measure and unbounded smooth boundary partial omega gamma the function a x responsible for the localized effect of the dissipative mechanism is assumed to be nonnegative and essentially bounded and in addition a x geq a 0 0 hbox a e in omega where omega omega cup x in omega x r r 0 and omega is a neighborhood in omega of the closure of partial omega cap b r where b r x in omega x r

on solutions of the wave equation with a sublinear dissipative term the existence of periodic solution and a certain decay property for the nonlinear dissipative wave equation utt u ut f in r or r with the boundary condition u 0 are proved where is a bounded domain in rn and v is a sublinear function say v k0 v rv k0 0 0 r 1