optimal decay rates for solutions of a nonlinear wave equation with localized nonlinear dissipation of unrestricted growth and critical exponent source terms under mixed boundary conditions abstract this article investigates optimal decay rates for solutions to a semilinear hyperbolic equation with localized interior damping and a source term both dissipation and the source are fully nonlinear and the growth rate of the source map may include critical exponents for sobolev s embedding h 1 l 2 besides continuity and monotonicity no growth or regularity assumptions are imposed on the damping we analyze the system in the presence of neumann type boundary conditions including the mixed cases dirichlet neumann robin the damping affects a thin layer a collar near a portion of the boundary to cope with the lack of control on the remaining section we develop a special method that accounts for propagation of the energy estimates from the dissipative region onto the entire domain the neumann system does not satisfy the lopatinski condition in higher dimensions hence the study of energy propagation in the absence of damping near the neumann segment requires special geometric considerations

on a system of klein gordon type equations with acoustic boundary conditions we prove the existence uniqueness and uniform stabilization of global solutions for a generalized system of klein gordon type equations with acoustic boundary conditions on a portion of the boundary and the dirichlet boundary condition on the rest

on expressing a cubic form as a sum of cubes of linear forms abstract sufficient conditions for a cubic form in n variables to be representable as a sum of cubes of n m m n 2 2 linear forms are derived for m 0 1 the conditions are also necessary whether it is true for m 1 is an open question in each case the corresponding algorithm requires besides arithmetic operations over the coefficients of solving one algebraic equation of degree m n

a note on differential equations of coupled mode propagation in fluids there is a significant discrepancy between bathymetric contributions to ocean acoustic mode coupling matrix elements derived by various authors in this paper the discrepancy is shown to be due to incorrectness of the traditional approach to mode coupling theory originated by a d pierce j acoust soc am 37 19 27 1965 and d m milder j acoust soc am 46 1259 1263 1969 in the presence of a sloping interface or a range dependent rigid boundary physically their approach is consistent with neither energy conservation nor reciprocity mathematically the problem with the traditional approach consists of formal unjustified term by term differentiation of a series representing the acoustic pressure in terms of local normal modes the series is poorly convergent when a nonhorizontal interface of fluids with distinct densities is present a rigorous derivation of the acoustic mode coupling equations in media with piecewise continuous density and sound speed is given here on the basis of the reciprocity principle necessary corrections to the results presented recently by b e mcdonald j acoust soc am 100 219 224 1996 are discussed

a derivation of the differential equations of coupled mode propagation in this paper the differential equations of the coupled mode theory of acoustic propagation are derived the derivation properly accounts for the boundary conditions imposed at an interface and gives rise to terms not normally found in other coupled mode formulations the solutions of this couple mode system agree very well in numerical experiments with those obtained from a boundary integral equation method

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

on derivation of differential equations of coupled mode propagation from the reciprocity principle l in a letter to the editor mcdaniel j acoust soc am 114 3020 3023 2003 criticizes the derivation of equations of coupled mode propagation o a godin j acoust soc am 103 159 168 1998 based on the reciprocity principle and proposes another derivation a close examination of mcdaniel s arguments shows that her criticism is unwarranted the derivation presented by mcdaniel does not apply to acoustic waveguides with sloping boundaries and or interfaces which are the subject of analysis reported in the 1998 article two alternative derivations are discussed and shown to give the same mode coupling equations as originally obtained by godin

optimality of energy estimates for the wave equation with nonlinear boundary velocity feedbacks we consider the wave equation damped by a nonlinear boundary velocity feedback q ut first we consider the case where q has a linear growth at infinity we prove that the usual decay rate estimates proved by nakao differential integral equations 8 1995 pp 681 688 haraux and zuazua arch rational mech anal 100 1988 pp 191 206 conrad leblond and marmorat in proceedings of the fifth international federation of automatic control symposium on control of distributed parameter systems perpignan 1989 pp 101 116 zuazua siam j control optim 28 1990 pp 466 477 and komornik in control and estimation of distributed parameter systems nonlinear phenomena internat ser numer math 118 birkhauser basel 1994 pp 253 266 when q has a polynomial behavior at zero and by the second author esaim control optim calc var 4 1999 pp 419 444 in the general case are in fact optimal in one space dimension more generally we prove that the energy decays exactly like the solution of an explicit and simple ordinary differential equation next we study the problem when q is bounded at infinity we prove that strong solutions decay exponentially to zero and we exhibit a sequence of weak solutions for which the associated energy decays to zero at infinity as slowly as the iterated logarithms go to infinity at infinity

stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks in this paper we consider in a bounded and smooth domain the wave equation with a delay term in the boundary condition we also consider the wave equation with a delayed velocity term and mixed dirichlet neumann boundary condition in both cases under suitable assumptions we prove exponential stability of the solution these results are obtained by introducing suitable energies and by using some observability inequalities if one of the above assumptions is not satisfied some instability results are also given by constructing some sequences of delays for which the energy of some solutions does not tend to zero

nonlinear damped partial differential equations and their uniform discretizations abstract we establish sharp energy decay rates for a large class of nonlinearly first order damped systems and we design discretization schemes that inherit of the same energy decay rates uniformly with respect to the space and or time discretization parameters by adding appropriate numerical viscosity terms our main arguments use the optimal weight convexity method and uniform observability inequalities with respect to the discretization parameters we establish our results first in the continuous setting then for space semi discrete models and then for time semi discrete models the full discretization is then inferred from the previous results by adapting the ideas to deal with linear systems our results cover for instance the schrodinger equation with nonlinear damping the nonlinear wave equation the nonlinear plate equation the nonlinear transport equation as well as certain classes of equations with nonlocal terms

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

a unified approach via convexity for optimal energy decay rates of finite and infinite dimensional vibrating damped systems with applications to semi discretized vibrating damped systems the liapunov method is celebrated for its strength to establish strong decay of solutions of damped equations extensions to infinite dimensional settings have been studied by several authors see e g haraux 1991 11 and komornik and zuazua 1990 17 and references therein results on optimal energy decay rates under general conditions of the feedback is far from being complete the purpose of this paper is to show that general dissipative vibrating systems have structural properties due to dissipation we present a general approach based on convexity arguments to establish sharp optimal or quasi optimal upper energy decay rates for these systems and on comparison principles based on the dissipation property and interpolation inequalities in the infinite dimensional case for lower bounds of the energy we stress the fact that this method works for finite as well as infinite dimensional vibrating systems and as well as for applications to semi discretized nonlinear damped vibrating pde s a part of this approach has been introduced in alabau boussouira 2004 2005 1 2 in the present paper we identify a new simple and explicit criteria to select a class of nonlinear feedbacks for which we prove a simplified explicit energy decay formula comparatively to the more general but also more complex formula we give in alabau boussouira 2004 2005 1 2 moreover we prove optimality of the decay rates for this class in the finite dimensional case this class includes a wide range of feedbacks ranging from very weak nonlinear dissipation exponentially decaying in a neighborhood of zero to polynomial or polynomial logarithmic decaying feedbacks at the origin in the infinite dimensional case we establish a comparison principle on the energy of sufficiently smooth solutions through the dissipation relation this principle relies on suitable interpolation inequalities it allows us to give lower bounds for the energy of smooth initial data for the one dimensional wave equation with a distributed polynomial damping which improves haraux 1995 12 lower estimate of the energy for this case we also establish lower bounds in the multi dimensional case for sufficiently smooth solutions when such solutions exist we further mention applications of these various results to several classes of pde s namely the locally and boundary damped multi dimensional wave equation the locally damped plate equation and the globally damped coupled timoshenko beams system but it applies to several other examples furthermore we show that these optimal energy decay results apply to finite dimensional systems obtained from spatial discretization of infinite dimensional damped systems we illustrate these results on the one dimensional locally damped wave and plate equations discretized by finite differences and give the optimal energy decay rates for these two examples these optimal rates are not uniform with respect to the discretization parameter we also discuss and explain why optimality results have to be stated differently for feedbacks close to linear behavior at the origin

on the application of the reciprocity principle to coupled mode propagation l the reciprocity principle is applied to the derivation of coefficients describing the coupling of acoustic normal modes in waveguides characterized by range dependent density gradients and discontinuities the requirement of reciprocity alone is found insufficient to determine the mode coupling coefficients the stronger requirement of continuity of pressure and particle velocity at horizontal discontinuities of the waveguide is shown to provide coupling coefficients in agreement with those of fawcett j acoust soc am 92 290 295 1992

centers of higher degree forms let be a symmetric d linear form on a vector space v of dimension n over a field k its center cent is the analog of the space of symmetric matrices for a bilinear form if d 2 the center is a commutative subalgebra of end v it seems difficult to determine which subalgebras can be realized as cent for some some d linear form as a first step we conjecture that the center has dimension at most n the conjecture is proved for n 5

general decay rate estimates and numerical analysis for a transmission problem with locally distributed nonlinear damping in this paper we obtain very general decay rate estimates associated to a wavewave transmission problem subject to a nonlinear damping locally distributed employing arguments firstly introduced in lasiecka and tataru 1993 and we shall present explicit decay rate estimates as considered in alabau boussouira 2005 and cavalcanti etal 2007 in addition we implement a precise and efficient code to study the behavior of the transmission problem when k1k2 and when one has a nonlinear frictional dissipation g ut more precisely we aim to numerically check the general decay rate estimates of the energy associated to the problem established in first part of the paper

dispersive estimates for principally normal pseudodifferential operators the aim of these notes is to describe some recent re sults concerning dispersive estimates for principally normal pseu dodifferential operators the main motivation for this comes from unique continuation problems such estimates can be used to prove l q carleman inequalities which in turn yield unique continuation results for various partial differential operators with rough poten tials dispersive estimates are l q estimates for nonelliptic partial differ ential operators which are a consequence of the decay properties of their fundamental solutions these decay properties follow from spa tial spreading of the singularities of the solutions since solutions prop agate in directions conormal to the characteristic set of the operator this spreading can be related to nonzero curvatures of the characteristic set dispersive estimates for constant coefficient operators are closely related to the restriction theorem in harmonic analysis various types of dispersive estimates are known to be true for op erators such as the wave operator the schrodinger operator and the linear kdv see ginibre velo 4 keel tao 11 they have proved to be useful in the study of nonlinear problems as well as of problems with unbounded potentials more recently similar estimates have been obtained for wave op erators with variable coefficients beginning with the smooth case in kapitanskii 10 mockenhaupt seeger and sogge 14 up to operators with c 2 coefficients in smith 15 and tataru 21 23 similar results were obtained for the schrodinger equation in staffilani tataru 19 c 2 coefficients and in burq gerard tzvetkov 1 smooth coeffic ients in the variable coefficient elliptic case one should also mention sogge s l q

a potential well theory for the wave equation with a nonlinear boundary condition soit d un sous ensemble connexe ouvert borne de r n a frontiere de lipschitz d soit d l union de deux sous varietes disjointes a n 1 dimensions et et leur confluence de lipschitz on considere le probleme aux valeurs limites et initiales 2 u t 2 u dans d o t u x o u x u x o t v x dans d u x t o sur o t u n f u x t sur o t

a note on the boundary stabilization of the wave equation we study the energy decay rates of the wave equation in a domain where boundary damping is present we generalize the geometrical conditions obtained earlier in j math purer appl 58 1979 pp 249 273 by using some more general multipliers of strauss comm pure appl math 28 1975 pp 265 278 the interaction between distributed damping and boundary damping is discussed a regulator problem is also formally discussed by the synthesis method

regularity of higher energies of wave equation with nonlinear localized damping and a nonlinear source abstract wave equation driven by a nonlinear dissipative source and subjected to a nonlinear damping that is localized on a small region near the boundary is considered while finite energy h 1 l 2 solutions to this problem are bounded uniformly for all times t 0 this property generally fails for higher energy norms h 2 h 1 this is due to a generation of higher energy by the source and the ultimate loss of dissipativity however the presence of the damping may turn things around by counteracting the sources also at the higher energy levels the benefits of this counteraction depend on the nonlinear characteristics of dissipation any deviation from linearity be it origin or infinity causes degradation of the damping hence of decay rates of finite energy this in turn is shown to have an adverse effect on the stability of higher energies the main aim of this paper is to provide a quantitative analysis of the interaction between nonlinearity of the damping and nonlinearity of the source we show that under some correlation between growth rates of the damping and the source the norms of topological order above the finite energy level remain globally bounded for all times

well posedness and optimal decay rates for the wave equation with nonlinear boundary damping source interaction we establish subject to some natural additional assumptions imposed on the relation between the source and the damping both well posedness and effective optimal decay rates for the solutions of a semilinear model of the wave equation the theory presented allows to consider both superlinear and sublinear behaviours of the dissipation in the presence of unstructured sources

asymptotic stability for a strongly coupled klein gordon system in an inhomogeneous medium with locally distributed damping abstract we consider a strongly coupled klein gordon system posed in an inhomogeneous medium with smooth boundary subject to a local damping distributed around a neighborhood of the boundary according to the geometric control condition we show that the energy of the system goes uniformly and exponentially to zero for all initial data of finite energy taken in bounded sets of finite energy phase space for this purpose refined microlocal analysis arguments are considered by exploiting ideas due to burq and gerard 16 by using sharp l p l q carleman estimates we prove a unique continuation property for coupled systems we also consider a strongly coupled system of wave equations with localized nonlinear damping acting only on one equation considering a compact riemannian manifold we show that the energy of the coupled system goes uniformly to zero

global nonexistence for abstract evolution equations with positive initial energy abstract in this paper we consider the problem of noncontinuation of solutions of the initial value problem for abstract evolution equations of the form pu tt q t u t a t u f t u t j 0 wherepandqare linear self adjoint operators anda t u andf t u are respectively a linear operator inu typically of differential type and a nonlinear driving force our principal concern is the noncontinuation or blow up of solutions when the initial energy is positive but appropriately bounded

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

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

energy decay rates for solutions of the wave equations with nonlinear damping in exterior domain abstract in this paper we study the behaviors of the energy of solutions of the wave equations with localized nonlinear damping in exterior domains

on waring s problem for cubic forms let w be a cubic form in n variables over the complex field and r be the smallest integer such that w can be expressed as a sum of cubes of r linear forms some theorems are proved that connect r with the ranks of certain matrices associated with w for arbitrary n and for n 5 based on these theorems an algorithm to express a cubic form in five variables as a sum of cubes of linear forms is proposed the paper is concluded with two examples of indecomposable cubic forms for the first form it is shown that r 7 and that there exists a one parameter set of representations of the form as a sum of cubes of seven linear forms while for the second form it is shown that r 8

consistent coupled mode theory of sound propagation for a class of nonseparable problems this article examines the effects of boundary condition approximations that arise whenever the coupled mode theories of a d pierce j acoust soc am 37 19 27 1965 and d m milder j acoust soc am 46 1259 1263 1969 are applied to propagation problems involving range dependent boundaries this boundary condition approximation requires that the depth derivative rather than the normal derivative of the field be continuous across a sloping interface the approximation is necessary in order to carry out the partial separation of depth and range variables effected in the mathematical formulation of the theory this article will show that a consequence of this approximation is that conventional coupled mode theory applied to dissipationless media with nonhorizontal boundaries does not conserve energy it is shown that a correction to coupled mode theory can be derived such that the proper boundary conditions are satisfied and energy is conserved to first order in the local bottom slope moreover the corrections are not prohibitive in terms of added computational complexity numerical examples are presented which illustrate the nonconservation of energy effect and the corrections to the theory

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

on partial polynomial interpolation thealexander hirschowitztheoremsaysthatageneralcollectionof k double points in p n imposes independent conditions on homogeneous polynomials of degree d with a well known list of exceptions we generalize this theorem to arbitrary zero dimensional schemes contained in a general union of double points we work in the polynomial interpolation setting in this framework our main result says that the affine space of polynomials of degree d in n variables with assigned values of any number of general linear combinations of first partial derivatives has the expected dimension if d 2w ith only five exceptional cases if d 2 the exceptional cases are fully described

global nonexistence theorems for a class of evolution equations with dissipation we study abstract evolution equations with nonlinear damping terms and source terms including as a particular case a nonlinear wave equation of the type ba cl u tt delta u b u t m 2 u t c u p 2 u t x in 0 t times omega 6pt u t x 0 t x in 0 t times partial omega 6pt u 0 cdot u 0 in h 0 1 omega quad u t 0 cdot v 0 in l 2 omega es ea where 0 p 2 m 1 we prove a global nonexistence theorem for positive initial value of the energy when 1 1

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

numerical analysis for the wave equation with locally nonlinear distributed damping in this paper we present spectral methods in order to solve wave equation subject to a locally distributed nonlinear damping thanks to the efficiency and the accuracy of spectral method we can check that discrete energy decreases to zero as time goes to infinity uniformly with respect to the mesh size when the damping is supported in a suitable subset of the domain of consideration we prove the convergence of the full fourier galerkin discretization thus we apply our schemes to illustrate the uniform discrete energy decay rates of the solution for a wide range of damping functions

on symmetric operators of higher degree and their applications abstract the concept of a symmetric operator relative to a quadratic form is extended to a k form acting in n space ln for any k 2 it is shown how to embed l2 into lr with the smallest possible r and expand to a k form g4 acting in lr such that there exists an operator lr lr that is symmetric with respect to g4 and whose eigenvalues are all distinct as an application it is shown how to express a polynomial in one variable as a sum of r kth powers of the linear binomials with the smallest possible r as a particular case a formula that generalizes the cardan formula for solving cubic equations is derived

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