uniform stability of semilinear wave equations with arbitrary local memory effects versus frictional dampings abstract this paper is concerned with the mixed initial boundary value problem for semilinear wave equations with complementary frictional dampings and memory effects we successfully establish uniform exponential and polynomial decay rates for the solutions to this initial boundary value problem under much weak conditions concerning memory effects more specifically we obtain the exponential and polynomial decay rates after removing the fundamental condition that the memory effect region includes a part of the system boundary while the condition is a necessity in the previous literature moreover for the polynomial decay rates we only assume minimal conditions on the memory kernel function g without the usual assumption of g controlled by g

asymptotic stability of abstract dissipative systems with infinite memory we consider in this paper the problem of asymptotic behavior of solutions to an abstract linear dissipative integrodifferential equation with infinite memory past history modeling linear viscoelasticity we show that the stability of the system holds for a much larger class of the convolution kernels than the one considered in the literature and we provide a relation between the decay rate of the solutions and the growth of the kernel at infinity some applications are also given

time dependent periodic navier stokes flows on a two dimensional torus the dynamical behaviour of an incompressible viscous fluid flow on a two dimensional torus externally excited by a spatially periodic force is investigated the flow field described by navier stokes equations is found to possess a sequence of time periodic solutions which bifurcate from a single steady state solution i e hopf bifurcations this result is based on a combination of analysis and computations and each provides corroborative evidence to the findings of the other

stabilization and asymptotic behavior of dispersive medium models abstract we analyze the stability of different vectorial and scalar dispersive medium models set in a bounded domain of r d d 1 they correspond to the coupling between the maxwell system or the wave equation and some first order differential equations of parabolic type we show that some models have an exponential decay rate while the other ones are only polynomially stable in that last case we show that the obtained polynomial decay is optimal

mildly dissipative nonlinear timoshenko systems global existence and exponential stability we consider nonlinear systems of timoshenko type in a one dimensional bounded domain the system has a dissipative mechanism being present only in the equation for the rotation angle it is a damping effect through heat conduction the global existence of small smooth solutions as well as the exponential stability are investigated

fem ids for a second order strongly damped wave equation with memory abstract a linear triangular finite element method implicit difference scheme fem ids for the solution of second order strongly damped wave equation sdwe with memory on domain with interfaces is proposed sufficient conditions that guarantee the existence of a unique solution are given the finite element discretization is such that the arbitrary but smooth interface is first approximated by a polygon with a boundary whose vertices all lie on the interface with the interface being at distance from the approximate interface linear interpolation operator incorporating the influence of isproved this together with ritz volterra operator and some auxiliary error estimates in the neighborhood of the interface are used to obtain the convergence estimate of the proposed scheme the scheme is applied to some test examples and the numerical results confirm the computational efficiency of the method

moore gibson thompson equation with memory part ii general decay of energy abstract we study a temporally third order moore gibson thompson equation with a memory term previously it was known that in non critical regime the global solutions exist and the energy functionals decay to zero more precisely it is known that the energy has exponential decay if the memory kernel decays exponentially the current work is a generalization of the previous one part i in that it allows the memory kernel to be more general and shows that the energy decays the same way as the memory kernel does exponentially or not

applying projective geometry to transformations on rank one idempotents abstract we present a short proof of molnar s characterization of bijective transformations on the set of all rank one idempotent operators on a banach space which preserve zero products in both directions an improvement in the finite dimensional case is given we apply these results to describe automorphisms of standard operator semigroups and to improve uhlhorn s version of wigner s theorem

an explicit example of hopf bifurcation in fluid mechanics it is observed that a complete and explicit example of hopf bifurcation appears not to be known in fluid mechanics such an example is presented for the rotating benard problem with free boundary conditions on the upper and lower faces and horizontally periodic solutions normal modes are found for the linearization and the veronis computation of the wave numbers is modified to take into account the imposed horizontal periodicity an invariant subspace of the phase space is found in which the hypotheses of the joseph sattinger theorem are verified thus demonstrating the hopf bifurcation the criticality calculations are carried through to demonstrate rigorously that the bifurcation is subcritical for certain cases and to demonstrate numerically that it is subcritical for all the cases in the paper

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

exponential stability for the wave model with localized memory in a past history framework abstract in this paper we discuss the asymptotic stability as well as the well posedness of the damped wave equation posed on a bounded domain of r n n 2 x u t t u 0 g s div a x u t s d s b x u t 0 subject to a locally distributed viscoelastic effect driven by a nonnegative function a x and supplemented with a frictional damping b x 0 acting on a region a of where a 0 in a assuming that x is constant considering that the well known geometric control condition t 0 holds and supposing that the relaxation function g is bounded by a function that decays exponentially to zero we prove that the solutions to the corresponding partial viscoelastic model decay exponentially to zero even in the absence of the frictional dissipative effect in addition in some suitable cases where the material density x is not constant it is also possible to remove the frictional damping term b x u t that is the localized viscoelastic damping is strong enough to assure that the system is exponentially stable the semi linear case is also considered

note on intrinsic decay rates for abstract wave equations with memory in this paper we consider a viscoelastic abstract wave equation with memory kernel satisfying the inequality g h g 0 s 0 where h s is a given continuous positive increasing and convex function such that h 0 0 we shall develop an intrinsic method based on the main idea introduced by lasiecka and tataru uniform boundary stabilization of semilinear wave equation with nonlinear boundary dissipation differential and integral equations 6 507 533 1993 for determining decay rates of the energy given in terms of the function h s this will be accomplished by expressing the decay rates as a solution to a given nonlinear dissipative ode we shall show that the obtained result while generalizing previous results obtained in the literature is also capable of proving optimal decay rates for polynomially decaying memory kernels h s sp and for the full range of admissible parameters p 1 2 while such result has been known for certain restrictive ranges of the parameters p 1 3 2

decay estimates for second order evolution equations with memory abstract this paper develops a unified method to derive decay estimates for general second order integro differential evolution equations with semilinear source terms depending on the properties of convolution kernels at infinity we show that the energy of a mild solution decays exponentially or polynomially as t our approach is based on integral inequalities and multiplier techniques these decay results can be applied to various partial differential equations we discuss three examples a semilinear viscoelastic wave equation a linear anisotropic elasticity model and a petrovsky type system

asymptotic behavior for coupled systems of second order abstract evolution equations with one infinite memory abstract we are concerned with the asymptotic behavior for a coupled system of two second order abstract evolution equations with one infinite memory and the damping effect given by only one equation which describes the dynamics of many viscoelastic systems and covers the well known timoshenko system we present an optimal energy decay estimate under very weak conditions more specifically we prove that the energy of the system decays to zero at least at the rate of t 1 only under the condition that the related kernel functions are non increasing and integrable moreover we apply our abstract results to three classes of coupled systems of second order partial differential equations arising in viscoelasticity

recovery of a space dependent vector source in anisotropic thermoelastic systems abstract we investigate the theoretical and numerical determination of a space dependent vector source load in an anisotropic thermoelastic system of type iii from the knowledge of an additional final time measurement the uniqueness of a solution to this inverse source problem is proved for various assumptions made on the convolution kernel a convergent and stable iterative algorithm is proposed for the recovery of the unknown vector source in the linear case and at the same time a stopping criterion is also given three numerical experiments are considered to validate the properties of the proposed iterative procedure and the regularizing stabilizing character of the corresponding stopping criterion the numerical experiments carried out showed that it exists a certain limitation of the method with respect to the recovery of non symmetric sources

the cauchy problem in one dimensional nonlinear viscoelasticity abstract the initial value problem for a nonlinear hyperbolic volterra equation which models the motion of an unbounded viscoelastic bar is studied under physically motivated assumptions the existence of a unique globally defined classical solution is established provided the initial data are sufficiently smooth and small boundedness and asymptotic behavior are also discussed this analysis is based on energy estimates in conjunction with properties of strongly positive definite kernels

global solutions to the equation of viscoelasticity with fading memory abstract we consider the initial history value problem for the one dimensional equation of thermoelasticity with fading memory it is proved that if the data are smooth and small then a unique smooth solution exists globally in time and converges to the constant equilibrium state as time goes to infinity our proof is based on a technical energy method which makes use of the strict convexity of the entropy function and the properties of strongly positive definite kernels

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

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

global existence for elastic waves with memory we treat the cauchy problem for nonlinear systems of viscoelasticity with a memory term we study the existence and the time decay of the solution to this nonlinear problem the kernel of the memory term includes integrable singularity at zero and polynomial decay at infinity we prove the existence of a global solution for space dimensions n 3 and arbitrary quadratic nonlinearities

remarks on the time dependent periodic navier stokes flows on a two dimensional torus this note is a continuation of the earlier paper chen z m price w g time dependent periodic navier stokes flows on a two dimensional torus commun math phys bf 179 577 597 1996 where the evidences of the occurrence for the time dependent periodic navier stokes flows were found based on a combination of analysis and computation this investigation is now confirmed via rigorous analysis the existence of the time dependent periodic navier stokes flows on a two dimensional torus is proved

energy decay rates via convexity for some second order evolution equation with memory and nonlinear time dependent dissipation abstract the stabilization of the following abstract integro differential equation u t a u t 0 t g t s a u s d s q t u t f u t is investigated we establish the general decay rate of the solution energy in terms of the behavior of the nonlinear feedback and the relaxation function g without imposing any restrictive growth assumption on the damping at the origin and strongly weakening the usual assumption of the relaxation function g our approach is based on the multiplier method and make use of some properties of the convex functions these decay results can be applied to various concrete models we shall study some examples to illustrate our result

stochastic nonlinear wave equation with memory driven by compensated poisson random measures in this paper we study a class of stochastic nonlinear wave equation with memory driven by levy noise we first show the existence and uniqueness of global mild solutions using a suitable energy function second under some additional assumptions we prove the exponential stability of the solutions

integro differential equations of hyperbolic type with positive definite kernels abstract the main purpose of this work is to study the damping effect of memory terms associated with singular convolution kernels on the asymptotic behavior of the solutions of second order evolution equations in hilbert spaces for kernels that decay exponentially at infinity and possess strongly positive definite primitives the exponential stability of weak solutions is obtained in the energy norm it is also shown that this theory applies to several examples of kernels with possibly variable sign and to a problem in nonlinear viscoelasticity

uniform decay of solutions for a quasilinear system of viscoelastic equations abstract in this paper we consider a system of two coupled quasilinear viscoelastic equations in canonical form with dirichlet boundary condition we use the perturbed energy method to show that the dissipations given by the viscoelastic terms are strong enough to ensure uniform decay with exponential and polynomial rates of the solutions energy which extends some existing results for a single equation to the case of a coupled system

a non local in time telegraph equation abstract in this work we derive a non local in time telegraph equation our model includes as particular cases the classical telegraph equation and the fractional in time telegraph equation among others further we define the fundamental solution of the problem and we prove that it can be interpreted as a probability density function finally using versions of the karamata feller tauberian theorem we study the temporal behavior of the variance of the distribution process associated with the solution of the equation in large times as well as in short times

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

general decay of solutions in one dimensional porous elastic system with memory abstract in this work we consider a one dimensional porous elastic system with memory effects it is well known that porous elastic system with a single dissipation mechanism lacks exponential decay in contrary we prove that the unique dissipation given by the memory term is strong enough to exponentially stabilize the system depending on the kernel of the memory term and the wave speeds of the system in fact we prove a general decay result for which exponential and polynomial decay results are special cases our result is new and improves previous results in the literature

maps leaving functional values of operator products invariant abstract let a be a standard operator algebra on a complex hilbert space h of dimension greater than 2 by invariants of certain functional values of operator products we characterize some surjective maps on a furthermore several kinds of general preserver problems on standard operator algebras are solved when we take respectively the functional as for example k numerical radius k 1 operator norm ky fan k norm schatten p norm 1 p and so on

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

decay rates of solutions of an anisotropic inhomogeneous n dimensional viscoelastic equation with polynomially decaying kernels we consider the anisotropic and inhomogeneous viscoelastic equation and we prove that the first and second order energy decay polynomially as time goes to infinity when the relaxation function also decays polynomially to zero that is if the kernelg ijkl satisfies dot g ijkl leqq c 0 g ijkl 1 frac 1 p and g ijkl g ijkl 1 frac 1 p in l 1 mathbb r for p 2such that 2 m 1 p then the first and second order energy decay as frac 1 1 t q withq 2 m 1

convergence to equilibrium of solutions to a nonautonomous semilinear viscoelastic equation with finite or infinite memory abstract in this paper we consider the nonautonomous semilinear viscoelastic equation u t t u 0 k s u t s d s f x u g t in r with dirichlet boundary conditions and finite t or infinite memory here is a bounded domain in r n with smooth boundary and the nonlinearity f r r is analytic in the second variable uniformly with respect to the first one for this equation we derive an appropriate lyapunov function and we use the ojasiewicz simon inequality to show that the dissipation given by the memory term is strong enough to prove the convergence to a steady state for any global bounded solution in addition we discuss the rate of convergence to equilibrium which is polynomial or exponential depending on the ojasiewicz exponent and the decay of the time dependent right hand side g

stability for semilinear hyperbolic coupled system with frictional and viscoelastic localized damping abstract in this paper we prove stability results for a semilinear hyperbolic coupled system subject to a viscoelastic localized damping acting in the first equation and a frictional localized one acting in the second equation of the system we divide the proof into two parts in the first part the equations are posed in a homogeneous medium with the damping acting in a boundary neighbourhood in the second part the equations posed in an inhomogeneous medium with the damping acting in a collar of the boundary and in an appropriate mesh in the interior due to the assumptions on the nonlinear function involving the frictional damping general decay rates are obtained to prove the results we used the tools of the microlocal analysis theory

sharp energy estimates for nonlinearly locally damped pdes via observability for the associated undamped system abstract we consider the problem of sharp energy decay rates for nonlinearly damped abstract infinite dimensional systems direct methods for nonlinear stabilization generally rely on multiplier techniques and thus are valid under restrictive geometric conditions compared to the optimal geometric optics condition of bardos et al 1992 10 we prove sharp simple and quasi optimal energy decay rates through an indirect method namely an observability estimate for the corresponding undamped system one of the main advantage of these results is that they allow to combine optimal geometric conditions as for instance that of bardos et al 1992 10 and the optimal weight convexity method of the first author alabau boussouira 2010 6 alabau boussouira 2005 2 to deduce very simple and quasi optimal energy decay rates for nonlinearly locally damped systems we also show that using arguments based on russell s principle russell 1978 24 one can deduce sharp energy decay rates from the exponential stabilization of the linearly damped system our results extend to nonlinearly damped systems those of haraux 1989 14 and ammari and tucsnak 2001 9 which concern linearly damped systems

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

asymptotic behavior for second order stochastic evolution equations with memory abstract in this paper a general second order integro differential evolution equation with memory driven by multiplicative noise is considered we prove the existence of global mild solution and asymptotic stability of the zero solution using lyapunov function techniques moreover we discuss three examples to show that the asymptotic stability results can be applied to various partial differential equations

coupled second order evolution equations with fading memory optimal energy decay rate abstract of concern is the cauchy problem for coupled second order semilinear evolution equations in a hilbert space with indirect memory damping we find an approach to obtain successfully an optimal rate of uniform decay for the system energy only under basic conditions on the memory kernels simultaneously the same rate is also obtained with less difficulty for the corresponding single memory dissipative second order evolution equations as can be seen our results essentially improve the previously related ones in the literature the abstract results are then applied to several concrete problems in the real world

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

mappings that preserve pairs of operators with zero triple jordan product abstract let f be a field and n 3 suppose s 1 s 2 m n f contain all rank one idempotents the structure of surjections s 1 s 2 satisfying aba 0 a b a 0 is determined similar results are also obtained for a subsets of bounded operators acting on a complex or real banach space x b the space of hermitian matrices acting on n dimensional vectors over a skew field d c subsets of self adjoint bounded linear operators acting on an infinite dimensional complex hilbert space it is then illustrated that the results can be applied to characterize mappings on matrices or operators such that f aba f a b a for all a b for functions f such as the spectral norm schatten p norm numerical radius and numerical range etc

energy decay for timoshenko systems of memory type abstract linear systems of timoshenko type equations for beams including a memory term are studied the exponential decay is proved for exponential kernels while polynomial kernels are shown to lead to a polynomial decay the optimality of the results is also investigated

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

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

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

boundary stabilisation of the wave equation in the presence of singularities we study the boundary stabilisation of the wave equation by a nonlinear feedback active on a part of the boundary in geometric situations for which the solutions have singularities these singulari

a generalization of gudder nagy s theorem with numerical ranges of operators in this paper we prove that for a b c which are bounded linear operators on a separable complex hilbert space h ax x bx x cx x for all x h with x 1 if and only if there exists a complex number c c such that a ci and c cb or b ci and c ca

energy decay rate for a coupled hyperbolic system with nonlinear damping abstract in this paper we investigate the energy decay rate for the solution of a coupled hyperbolic system the explicit energy decay rate is established by using multiplier techniques and constructing a suitable energy functional

existence results for some damped second order volterra integro differential equations abstract in this paper we make a subtle use of operator theory techniques and the well known schauder fixed point principle to establish the existence of pseudo almost automorphic solutions to some second order damped integro differential equations with pseudo almost automorphic coefficients in order to illustrate our main results we will study the existence of pseudo almost automorphic solutions to a structurally damped plate like boundary value problem

blow up of solution for a nonlinear petrovsky type equation with memory in this paper we consider the nonlinear petrovsky type equation u t t 2 u 0 t g t s 2 u t s d s u t m 2 u t u p 2 u with initial conditions and dirichlet boundary conditions under suitable conditions of the initial data and the relaxation function we prove that the solution with upper bounded initial energy blows up in finite time moreover for the linear damping case we show that the solution blows up in finite time by different method for nonpositive initial energy

on the stabilization of timoshenko systems with memory and different speeds of wave propagation in this work we consider a one dimensional timoshenko system with different speeds of wave propagation and with only one control given by a viscoelastic term on the angular rotation equation for a wide class of relaxation functions and for sufficiently regular initial data we establish a general decay result for the energy of solution unlike the past history and internal feedback cases the second energy is not necessarily decreasing to overcome this difficulty a precise estimate of the second energy in terms of the initial data and the relaxation function is proved

global existence and asymptotic behavior for a fractional differential equation this paper is concerned with the global existence and asymptotic behavior of solutions to an initial boundary value problem of hyperbolic type we investigate the interaction between a polynomial source and a dissipation of fractional order this fractional dissipation is defined by a temporal nonlocal term

general decay result for nonlinear viscoelastic equations abstract in this paper we consider a nonlinear viscoelastic equation with minimal conditions on the l 1 0 relaxation function g namely g t t h g t where h is an increasing and convex function near the origin and is a nonincreasing function with only these very general assumptions on the behavior of g at infinity we establish optimal explicit and general energy decay results from which we can recover the optimal exponential and polynomial rates when h s s p and p covers the full admissible range 1 2 we get the best decay rates expected under this level of generality and our new results substantially improve several earlier related results in the literature

coupled second order semilinear evolution equations indirectly damped via memory effects abstract in this paper we study a system of coupled second order semilinear evolution equations in a hilbert space which is partially damped through memory effects a global existence and uniqueness theorem regarding the solutions to its cauchy problem is given following this we analyze stability of the system energy and obtain various decay rates corresponding to the integral kernel some of them being optimal moreover we apply our abstract theory to concrete systems including that of timoshenko type for which we essentially improve the previous related results

hopf bifurcation of the three dimensional navier stokes equations this paper is concerned with the three dimensional navier stokes flows excited by a unidirectional external force and gives an explicit example of hopf bifurcation phenomenon occurring in the navier stokes problem complete rigorous analysis on the existence of this instability behavior is provided

decay rates for the three dimensional linear system of thermoelasticity we consider the two and three dimensional system of linear thermoelasticity in a bounded smooth domain with dirichlet boundary conditions we analyze whether the energy of solutions decays exponentially uniformly to zero as t to infty first of all by a decoupling method we reduce the problem to an observability inequality for the lame system in linear elasticity and more precisely to whether the total energy of the solutions can be estimated in terms of the energy concentrated on its longitudinal component we show that when the domain is convex the decay rate is never uniform in fact the lack of uniform decay holds in a more general class of domains in which there exist rays of geometric optics of arbitrarily large length that are always reflected perpendicularly or almost tangentially on the boundary we also show that in three space dimensions the lack of uniform decay may also be due to a critical polarization of the energy on the transversal component of the displacement in two space dimensions we prove a sufficient and almost necessary condition for the uniform decay to hold in terms of the propagation of the transversal characteristic rays under the further assumption that the boundary of the domain does not have contacts of infinite order with its tangents we also give an example due to d hulin in which these geometric properties hold in three space dimensions we indicate without proof how a careful analysis of the polarization of singularities may lead to sharp sufficient conditions for the uniform decay to hold in two space dimensions we prove that smooth solutions decay polynomially in the energy space to a finite dimensional subspace of solutions except when the domain is a ball or an annulus finally we discuss some closely related controllability and spectral issues

boundary control of the timoshenko beam it is shown that the timoshenko beam can be uniformly stabilized by means of a boundary control a numerical study on the spectrum is also presented

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

general stability result for a viscoelastic plate equation with past history and general kernel abstract in this paper we consider the following plate problem u t t u t t 2 u 0 k s 2 u t s d s 0 and we show that the stability of this problem holds for a much larger class of kernels more precisely we consider the kernel k 0 0 satisfies k t t k t t 0 where and are functions satisfying some specific properties under this very general assumption on the behavior of k at infinity we establish a relation between the decay rate of the solution and the growth of k at infinity this work generalizes and improves earlier results in the literature moreover we drop the boundedness assumption on the history data usually made in the literature

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

existence and decay of solutions for a viscoelastic wave equation with acoustic boundary conditions abstract in this paper we consider a multi dimensional damped semilinear viscoelastic wave equation with variable coefficient and acoustic boundary conditions first we prove a local existence theorem by using the faedo galerkin approximations combined with a contraction mapping theorem second we show that under suitable conditions on the initial data and the relaxation function the solution exists globally in time and we prove the uniform decay rate of the energy

blow up at infinity of solutions for integro differential equation in this paper we consider the blow up at infinity of solutions for the integro differential equation with strong damping term as well as a time dependent nonlinear dissipative function q and a driving force term f under the homogeneous dirichlet boundary conditions some interesting applications are given in particular subcases of q and f this improves earlier results in the literatures

propagation observation and control of waves approximated by finite difference methods this paper surveys several topics related to the observation and control of wave propagation phenomena modeled by finite difference methods the main focus is on the property of observability corresponding to the question of whether the total energy of solutions can be estimated from partial measurements on a subregion of the domain or boundary the mathematically equivalent property of controllability corresponds to the question of whether wave propagation behavior can be controlled using forcing terms on that subregion as is often desired in engineering applications observability controllability of the continuous wave equation is well understood for the scalar linear constant coefficient case that is the focus of this paper however when the wave equation is discretized by finite difference methods the control for the discretized model does not necessarily yield a good approximation to the control for the original continuous problem in other words the classical convergence consistency stability property of a numerical scheme does not suffice to guarantee its suitability for providing good approximations to the controls that might be needed in applications observability controllability may be lost under numerical discretization as the mesh size tends to zero due to the existence of high frequency spurious solutions for which the group velocity vanishes this phenomenon is analyzed and several remedies are suggested including filtering tychonoff regularization multigrid methods and mixed finite element methods we also briefly discuss these issues for the heat beam and schrodinger equations to illustrate that diffusive and dispersive effects may help to retain the observability controllability properties at the discrete level we conclude with a list of open problems and future subjects for research

non linear numerical radius isometries on atomic nest algebras and diagonal algebras abstract a nonlinear map between operator algebras is said to be a numerical radius isometry if w t s w t s for all t s in its domain algebra where w t stands for the numerical radius of t let n and m be two atomic nests on complex hilbert spaces h and k respectively denote alg n the nest algebra associated with n and d n alg n alg n the diagonal algebra we give a thorough classification of weakly continuous numerical radius isometries from alg n onto alg m and a thorough classification of numerical radius isometries from d n onto d m

existence and decay of solutions of a viscoelastic equation with a nonlinear source abstract in a bounded domain we consider u tt u 0 t g t u d u u where 0 and g is a nonnegative and decaying function we prove a local existence theorem and show for certain initial data and suitable conditions on g and that this solution is global with energy which decays exponentially or polynomially depending on the rate of the decay of the relaxation function g

general decay rate estimates for viscoelastic dissipative systems abstract the linear viscoelastic equation is considered we prove uniform decay rates of the energy by assuming a nonlinear feedback acting on the boundary without imposing any restrictive growth assumption on the damping term and strongly weakening the usual assumptions on the relaxation function our estimate depends both on the behavior of the damping term near zero and on the behavior of the relaxation function at infinity the proofs are based on the multiplier method and on a general lemma about convergent and divergent series for obtaining the uniform decay rates

the existence and the asymptotic behavior of a plate equation with frictional damping and a logarithmic source term abstract in this paper we consider a plate equation with a logarithmic nonlinearity in the presence of a frictional damping we prove for a suitably chosen initial data a global existence as well as a decay result

polynomial stability of the timoshenko system by one boundary damping abstract in this paper we study the indirect boundary stabilization of the timoshenko system with only one dissipation law this system which models the dynamics of a beam is a hyperbolic system with two wave speeds assuming that the wave speeds are equal we prove non uniform stability and an optimal polynomial energy decay rate is obtained otherwise if the ratio of the wave speeds is a rational number we show that the decay rate is of polynomial type

exponential stability for second order evolutionary problems abstract we study the exponential stability of evolutionary equations the focus is laid on second order problems and we provide a way to rewrite them as a suitable first order evolutionary equation for which the stability can be proved by using frequency domain methods the problem class under consideration is broad enough to cover integro differential equations delay equations and classical evolution equations within a unified framework

asymptotic stability in linear viscoelasticity with supplies abstract we present some results about the asymptotic behavior of a linear viscoelastic system making use of the approach based on the concept of minimal state this approach allows to obtain results in a larger class of solutions and data with respect to the classical one based on the histories of the deformation gradient recently a lot of attention has been paid to find unified approaches which permit to study the asymptotic behavior with memory kernels presenting a temporal decay of which the exponential and polynomial decays are only special cases here we extend this unified approach to the dynamic problem in presence of supplies by using the minimal state and compare our results with those present in literature

quadratic optimization for infinite dimensional linear differential difference type systems syntheses via the fredholm equation a closed loop solution for quadratic optimization is presented for control systems whose model is given by infinite dimensional linear differential difference equations frac dx t dt ax t sum i 1 n a i x left t h i right bu t sum j 1 m b j u left t r j right a nonsemigroup evolution formula for the fundamental solution is given first the system is then reduced to a volterra integral system by the semicausality approach a new optimality principle is established and the closed loop optimal control is given by real time state feedback plus retarded state integral feedback the feedback operator is determined by solving a linear fredholm integral equation the two crucial methods semicausal dynamical optimization for volterra integral systems and nonsemigroup evolution property are brought together to generalize the syntheses results to infinite dimensional cases

on the rate of decay in interacting continua with memory abstract this paper is concerning the linear theory of isothermal interacting continua with memory we consider anti plane shear deformations in a mixture of two elastic solids where the dissipation mechanisms can be the viscosity in one of the components and the viscosity with respect to the relative displacement we have seen that when the only dissipation mechanism applies on the relative displacement we cannot expect the exponential decay for the solutions we have also analyzed the case when the viscosity mechanism applies on a constituent we have seen that generically the decay is of exponential type however if the coupling constitutive parameter vanishes the decay is slow

on the general decay of a nonlinear viscoelastic plate equation with a strong damping and p x t laplacian abstract in this paper we consider a nonlinear viscoelastic plate equation with a lower order perturbation of a p x t laplacian operator of the form u t t 2 u p x t u 0 t g t s u s d s u t f u 0 x t q t 0 t associated with initial and dirichlet neumann boundary conditions under suitable conditions on g f and the variable exponent of the p x t laplacian operator we prove a general decay result in the presence of a strong damping u t 0 acting in the domain this equation corresponds to a viscoelastic version arising in dynamics of elastoplastic flows and plate vibrations

frictional versus viscoelastic damping in timoshenko systems with different speeds of wave propagation abstract we consider the decay rate of timoshenko type equations for beams with frictional damping and memory effect in an interval when the speeds of wave propagation are different we show that the solution energy of timoshenko systems decays at the rate t 1 under only basic conditions on the memory kernels without any decaying conditions the proof is based on some appropriate functionals and it improves the previously related results in the literature

maps preserving numerical radius and cross norms of operator products let h be a complex hilbert space with dim h 3 and x1d4b2 x1d4b1 h be subsets that contain all rank one idempotents let f stand for numerical radius w or a cross operator norm n all the general surjective maps x1d4b2 x1d4b1 satisfying f ab f a b resp f aba f a b a for all a b x1d4b2 are characterized

stabilizability of systems of one dimensional wave equations by one internal or boundary control force we study the internal and boundary stabilizability of a system of wave equations by one control force we prove that the classical internal damping applied to only one of the equations never gives exponential stability if the wave speeds are different and if the wave speeds are the same we give explicit necessary and sufficient conditions for the stability to occur we also study the simultaneous boundary stabilization of the same system

locally distributed control and damping for the conservative systems in this paper we note the equivalence between exact controllability and exponential stabilizability for an abstract conservative system with bounded control this enables us to establish a frequency domain characterization for the exact controllability uniform exponential decay property of second order elastic systems such as the wave equation and the petrovsky equation with locally distributed control damping a piecewise multiplier method for frequency domain is introduced for several classes of pdes on regions which are not necessarily smooth we obtain a sufficient condition for the subregion on which the application of control damping will yield the exact controllability uniform exponential decay property this result provides useful information for designing the location of controllers dampers for distributed systems with a law of conservation

on the spectrum of evolution operators generated by hyperbolic systems abstract we prove that the asymptotic behavior of semigroups generated by mixed initial value problems for hyperbolic systems in one space variable is determined by the zeroes of the characteristic equation the proof is based on the following reduction principle we construct a reduced system which is much simpler than the given one such that the difference of the semigroups is a compact operator the construction of the reduced system is presented in the non autonomous case because it has implifications in the study of systems which are time periodic

general decay and blow up of solution for a quasilinear viscoelastic problem with nonlinear source abstract in this paper we consider a quasilinear viscoelastic wave equation in canonical form with the homogeneous dirichlet boundary condition we prove that for certain class of relaxation functions and certain initial data in the stable set the decay rate of the solution energy is similar to that of the relaxation function this result improves earlier ones obtained by messaoudi and tatar s a messaoudi n e tatar global existence and uniform stability of solutions for a quasilinear viscoelastic problem math methods appl sci 30 2007 665 680 in which only the exponential and polynomial decay rates are considered conversely for certain initial data in the unstable set there are solutions that blow up in finite time the last result is new since it allows a larger class of initial energy which may take positive values