relating continuous and discrete pepa models of signalling pathways pepa and its semantics have recently been extended to model biological systems in order to cope with massive quantities of processes as is usually the case when considering biological reactions the model is interpreted in terms of a small set of coupled ordinary differential equations odes instead of a large state space continuous time markov chain ctmc so far the relationship between these two semantics of pepa had not been established this is the goal of the present paper after introducing a new extension of pepa denoted email protected that allows models to capture both mass action law and bounded capacity law cooperations the relationship between these two semantics is demonstrated the result relies on kurtz s theorem that expresses that a set of odes can be in some sense considered as the limit of pure jump markov processes

fluid computation of passage time distributions in large markov models recent developments in the analysis of large markov models facilitate the fast approximation of transient characteristics of the underlying stochastic process fluid analysis makes it possible to consider previously intractable models whose underlying discrete state space grows exponentially as model components are added in this work we show how fluid approximation techniques may be used to extract passage time measures from performance models we focus on two types of passage measure passage times involving individual components as well as passage times which capture the time taken for a population of components to evolve specifically we show that for models of sufficient scale global passage time distributions can be well approximated by a deterministic fluid derived passage time measure where models are not of sufficient scale we are able to generate upper and lower approximations for the entire cumulative distribution function of these passage time random variables using moment based techniques additionally we show that for passage time measures involving individual components the cumulative distribution function can be directly approximated by fluid techniques finally using the gpa tool we take advantage of the rapid fluid computation of passage times to show how a multi class client server system can be optimised to satisfy multiple service level agreements

abstract volterra integrodifferential equations with applications to parabolic models with memory in this paper we are concerned with local existence regularity and continuous dependence upon the initial data of epsilon regular mild solutions for the abstract integrodifferential equation 1 2 we also present a result on unique continuation and a blow up alternative for an epsilon regular mild solution of 1 2 finally we apply our results to three interesting models navier stokes equations with memory diffusion equations with memory and a strongly damped plate equation with memory

decay property for second order hyperbolic systems of viscoelastic materials we study a class of second order hyperbolic systems with dissipation which describes viscoelastic materials the considered dissipation is given by the sum of the memory term and the damping term when the dissipation is effective over the whole system we show that the solution decays in l 2 at the rate t n 4 as t provided that the corresponding initial data are in l 2 l 1 w heren is the space dimension the proof is based on the energy method in the fourier space also we discuss similar systems with weaker dissipation by introducing the operator 1 2 with 0 in front of the dissipation terms and

a fluid analysis framework for a markovian process algebra markovian process algebras such as pepa and stochastic p calculus bring a powerful compositional approach to the performance modelling of complex systems however the models generated by process algebras as with other interleaving formalisms are susceptible to the state space explosion problem models with only a modest number of process algebra terms can easily generate so many states that they are all but intractable to traditional solution techniques previous work aimed at addressing this problem has presented a fluid flow approximation allowing the analysis of systems which would otherwise be inaccessible to achieve this systems of ordinary differential equations describing the fluid flow of the stochastic process algebra model are generated informally in this paper we show formally that for a large class of models this fluid flow analysis can be directly derived from the stochastic process algebra model as an approximation to the mean number of component types within the model the nature of the fluid approximation is derived and characterised by direct comparison with the chapman kolmogorov equations underlying the markov model furthermore we compare the fluid approximation with the exact solution using stochastic simulation and we are able to demonstrate that it is a very accurate approximation in many cases for the first time we also show how to extend these techniques naturally to generate systems of differential equations approximating higher order moments of model component counts these are important performance characteristics for estimating for instance the variance of the component counts this is very necessary if we are to understand how precise the fluid flow calculation is in a given modelling situation

attractors for the nonclassical diffusion equations with fading memory abstract we discuss long time dynamical behavior of the nonclassical diffusion equation with fading memory when nonlinearity is critical the existence and regularity of global attractors in weak topological space and strong topological space are obtained while the forcing term only belongs to h 1 and l 2 respectively the results in this part are new and appear to be optimal corresponding to the forcing term

general decay of solutions of a viscoelastic equation abstract in this paper we consider the following viscoelastic equation u t t u 0 t g t u d 0 in a bounded domain and establish a general decay result which is not necessarily of exponential or polynomial type this work generalizes and improves earlier results in the literature

attractors for wave equations with degenerate memory abstract this paper is concerned with the long time dynamics of a semilinear wave equation with degenerate viscoelasticity u t t u 0 g s div a x u t s d s f u h x defined in a bounded domain of r 3 with dirichlet boundary condition and nonlinear forcing f u with critical growth the problem is degenerate in the sense that the function a x 0 in the memory term is allowed to vanish in a part of when a x does not degenerate and g decays exponentially it is well known that the corresponding dynamical system has a global attractor without any extra dissipation in the present work we consider the degenerate case by adding a complementary frictional damping b x u t which is in a certain sense arbitrarily small such that a b 0 in despite that the dissipation is given by two partial damping terms of different nature none of them necessarily satisfying a geometric control condition we establish the existence of a global attractor with finite fractal dimension

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

on a large class of kernels yielding exponential stability in viscoelasticity in this paper we prove exponential decay of solutions for a problem which appears in viscoelasticity the conditions on the admissible kernels are relaxed so as to allow for more kernels to be treated namely the smallness of the kernels is replaced by the smallness of the set where the kernel is flat this work extends previous works and improves in particular a recent result by pata 16 this is established by introducing two lemmas an idea due to pata and using the energy method

long time behavior of a quasilinear viscoelastic equation with past history abstract this paper is concerned with a class of quasilinear wave equations with memory u t u t t u u t t t t s u s d s u t f u h 0 which was considered by several authors with 0 since 2001 existing results are mainly devoted to global existence energy decay existence with small data and blow up of solutions however uniqueness seems to be an open problem and existence of attractors was no yet considered the objective of the present paper is to provide some results on the well posedness and long time behavior to this equation in a more general setting which includes past history that is by taking in the memory term

a general decay result for a memory type timoshenko thermoelasticity system with second sound abstract in this paper we consider a system of timoshenko thermoelasticity with second sound in the presence of an infinite history term we prove under suitable conditions on the coefficients and the relaxation function a general decay result from which the exponential and polynomial decay estimates are only special cases we also establish a lack of exponential decay result and discuss the optimality for some situation

upper semicontinuous attractors for a 2d mindlin timoshenko thermo viscoelastic model with memory abstract a nonlinear problem for a thermo viscoelastic mindlin timoshenko plate with hereditary heat conduction is considered here we prove the existence of a compact global attractor whose fractal dimension is finite the main aim of the work is to show the upper semicontinuity of the attractor as the relaxation kernels fade in a suitable sense

a variational formulation of the perturbed motion problem for a viscoelastic body abstract the analysis of perturbed motion is often very important for studying the progress of strain and stress in viscoelastic bodies the authors intend to provide a variational formulation of the problem as an alternative to the differential formulation used to date by solving the so called inverse problem of the calculus of variations this paper shows how the operator ruling the problem can be made symmetric by using a convolution bilinear form to obtain four functionals which are stationary at the solution of the differential problem in conclusion for example the two dimensional equations of the perturbed motion of a viscoelastic thin plate are derived from the stationary condition of the three dimensional functional

turning back time in markovian process algebra product form solutions in markovian process algebra mpa are constructed using properties of reversed processes the compositionality of mpas is directly exploited allowing a large class of hierarchically constructed systems to be solved for their state probabilities at equilibrium the paper contains new results on both reversed stationary markov processes as well as mpa itself and includes a mechanisable proof in mpa notation of jackson s theorem for product form queueing networks several examples are used to illustrate the approach

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 and blowup of solutions for a degenerate viscoelastic equation of kirchhoff type with source term abstract in this work we consider a nonlinear degenerate viscoelastic equation of kirchhoff type with source term in a bounded domain under suitable assumptions on the relaxation functions g and m we obtain global existence of solutions then we prove the general decay result finally we prove the finite time blow up result of solutions with negative initial energy this work generalizes and improves earlier results in the literature

on the decay of solutions in nonsimple elastic solids with memory the decay of solutions in nonsimple elasticity with memory is addressed analyzing how the decay rate is influenced by the different dissipation mechanisms appearing in the equations in particular a first order dissipation is shown to guarantee the asymptotic stability of the related solution semigroup but is not strong enough to entail exponential stability the latter occurs for a dissipation mechanism of the second order that is the same order as the one of the leading operator

long time behavior of a suspension bridge equations with past history in this paper we study a suspension bridge equation with memory effects for the suspension bridge equation without memory there are many classical results existing results mainly devoted to existence and uniqueness of a weak solution energy decay of solution and existence of global attractors however the existence of global attractors for the suspension bridge equation with memory was no yet considered the object of the present paper is to provide some results on the well posedness and long time behavior to the suspension bridge equation when the unique damping mechanism is given by the memory term

general stability result for viscoelastic wave equations in this paper we consider a viscoelastic equation and establish an explicit and general decay rate result without imposing restrictive assumptions on the behavior of the relaxation function at infinity our result allows a wider class of relaxation functions and improves earlier results in the literature

the longtime behavior of a nonlinear reissner mindlin plate with exponentially decreasing memory kernels abstract in this paper we investigate the longtime behavior of the mathematical model of a homogeneous viscoelastic plate based on reissner mindlin deformation shear assumptions according to the approximation procedure due to lagnese for the kirchhoff viscoelastic plate the resulting motion equations for the vertical displacement and the angle deflection of vertical fibers are derived in the framework of the theory of linear viscoelasticity assuming that in general both lame s functions and depend on time the coupling terms between the equations of displacement and deflection depend on hereditary contributions we associate to the model a nonlinear semigroup and show the behavior of the energy when time goes on in particular assuming that the kernels and decay exponentially and not too weakly with respect to the physical properties considered in the model then the energy decays uniformly with respect to the initial conditions i e we prove the existence of an absorbing set for the semigroup associated to the model

blow up of a hyperbolic equation of viscoelasticity with supercritical nonlinearities abstract we investigate a hyperbolic pde modeling wave propagation in viscoelastic media under the influence of a linear memory term of boltzmann type and a nonlinear damping modeling friction as well as an energy amplifying supercritical nonlinear source u t t k 0 u 0 k s u t s d s u t m 1 u t u p 1 u in 0 t u x t u 0 x t in 0 where is a bounded domain in r 3 with a dirichlet boundary condition the relaxation kernel k is monotone decreasing and k 1 we study blow up of solutions when the source is stronger than dissipations i e p max m k 0 under two different scenarios first the total energy is negative and the second the total energy is positive with sufficiently large quadratic energy this manuscript is a follow up work of the paper 30 in which hadamard well posedness of this equation has been established in the finite energy space the model under consideration features a supercritical source and a linear memory that accounts for the full past history as time goes to which is distinct from other relevant models studied in the literature which usually involve subcritical sources and a finite time memory

existence of random attractors for weakly dissipative plate equations with memory and additive white noise in this paper we prove the existence of random attractors for the continuous random dynamical systems generated by stochastic weakly dissipative plate equations with linear memory and additive white noise by defining the energy functionals and using the compactness translation theorem

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

carleman estimates for a class of degenerate parabolic operators given alpha in 0 2 and f in l 2 0 t times 0 1 we derive new carleman estimates for the degenerate parabolic problem w t x alpha w x x f where t x in 0 t times 0 1 associated to the boundary conditions w t 1 0 and w t 0 0 if 0 leq alpha 1 or x alpha w x t 0 0 if 1 leq alpha 2 the proof is based on the choice of suitable weighted functions and hardy type inequalities as a consequence for all 0 leq alpha 2 and omega subset subset 0 1 we deduce null controllability results for the degenerate one dimensional heat equation u t x alpha u x x h chi omega with the same boundary conditions as above

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

random attractors for damped non autonomous wave equations with memory and white noise abstract we first prove the existence of random attractors for the continuous random dynamical systems generated by stochastic damped non autonomous wave equations with linear memory and additive white noise when the nonlinearity has a critically growing exponent then we study the upper semicontinuity of random attractors when the coefficient of random term approaches zero

exact fluid lumpability in markovian process algebra quantitative analysis by means of discrete state stochastic processes is hindered by the well known phenomenon of state space explosion whereby the size of the state space may have an exponential growth with the number of agents of the system under scrutiny when the stochastic process underlies a markovian process algebra model this problem may be alleviated by suitable notions of behavioural equivalence that induce lumping at the underlying continuous time markov chain establishing an exact relation between a potentially much smaller aggregated chain and the original one for the analysis of massively parallel systems however lumping techniques may not be sufficient to yield a computationally tractable problem recently much work has been directed towards forms of fluid techniques that provide a set of ordinary differential equations odes approximating the expected path of the stochastic process unfortunately even fluid models of realistic systems may be too large for feasible analysis this paper studies a behavioural relation for process algebra with fluid semantics called projected label equivalence which is shown to yield an exactly fluid lumpable model i e an aggregated ode system which can be related to the original one without any loss of information project label equivalence relates sequential components of a process term in general for any two sequential components that are related in the fluid sense nothing can be said about their relationship from the stochastic viewpoint we define and study a notion of well posedness which allows us to relate fluid lumpability to the stochastic notion of semi isomorphism which is a weaker version of the common notion of isomorphism between the doubly labelled transition systems at the basis of the markovian interpretation

weak stability for coupled wave and or petrovsky systems with complementary frictional damping and infinite memory abstract in this paper we consider coupled wave wave petrovsky petrovsky and wave petrovsky systems in n dimensional open bounded domain with complementary frictional damping and infinite memory acting on the first equation we prove that these systems are well posed in the sense of semigroups theory and provide a weak stability estimate of solutions where the decay rate is given in terms of the general growth of the convolution kernel at infinity and the arbitrary regularity of the initial data we finish our paper by considering the uncoupled wave and petrovsky equations with complementary frictional damping and infinite memory and showing a strong stability estimate depending only on the general growth of the convolution kernel at infinity

uniform decay in a timoshenko type system with past history abstract fernandez sare and rivera h d fernandez sare j e munoz rivera stability of timoshenko systems with past history j math anal appl 339 1 2008 482 502 considered the following timoshenko type system 1 t t k x x 0 2 t t b x x 0 g t x x t s d s k x 0 where g is a positive differentiable exponentially decaying function they established an exponential decay result in the case of equal wave speed propagation and a polynomial decay result in the case of nonequal wave speed propagation in this paper we study the same system for g decaying polynomially and prove polynomial stability results for the equal and nonequal wave speed propagation our results are established under conditions on the relaxation function weaker than those in h d fernandez sare j e munoz rivera stability of timoshenko systems with past history j math anal appl 339 1 2008 482 502

decay estimate and asymptotic profile for a plate equation with memory abstract in this paper the initial value problem for a plate equation with a memory is considered under weaker assumptions on the memory function we obtain the decay estimates of solutions which is of the regularity loss type by applying the energy method in fourier space moreover if the memory kernel is an exponential function we derive the large time behavior and asymptotic profile of solutions in the one dimensional space by virtue of the technique of spectral analysis

spectral analysis and exponential stability of one dimensional wave equation with viscoelastic damping abstract this paper presents the exponential stability of a one dimensional wave equation with viscoelastic damping using the asymptotic analysis technique we prove that the spectrum of the system operator consists of two parts the point and continuous spectrum the continuous spectrum is a set of n points which are the limits of the eigenvalues of the system and the point spectrum is a set of three classes of eigenvalues one is a subset of n isolated simple points the second is approaching to a vertical line which parallels to the imagine axis and the third class is distributed around the continuous spectrum moreover the riesz basis property of the generalized eigenfunctions of the system is verified consequently the spectrum determined growth condition holds true and the exponential stability of the system is then established

the concept of a minimal state in viscoelasticity new free energies and applications to pdes we show here the impact on the initial boundary value problem and on the evolution of viscoelastic systems of the use of a new definition of state based on the stress response see e g 48 16 41 comparisons are made between this new approach and the traditional one which is based on the identification of histories and states we shall refer to a stress response definition of state as the minimal state 29 materials with memory and with relaxation are discussed the energetics of linear viscoelastic materials is revisited and new free energies expressed in terms of the minimal state descriptor are derived together with the related dissipations furthermore both the minimum and the maximum free energy are recast in terms of the minimal state variable and the current strain the initial boundary value problem governing the motion of a linear viscoelastic body is re stated in terms of the minimal state and the velocity field through the principle of virtual power the advantages are i the elimination of the need to know the past strain history at each point of the body and ii the fact that initial and boundary data can now be prescribed on a broader space than resulting from the classical approach based on histories these advantages are shown to lead to natural results about well posedness and stability of the motion finally we show how the evolution of a linear viscoelastic system can be described through a strongly continuous semigroup of linear contraction operators on an appropriate hilbert space the family of all solutions of the evolutionary system obtained by varying the initial data in such a space is shown to have exponentially decaying energy

on the energy decay of the linear thermoelastic plate with memory abstract we analyze the longterm properties of a c 0 semigroup describing the solutions to a linear evolution system that models a thermoelastic thin plate with memory effects in the heat flux law under reasonably general assumptions on the memory kernel all single trajectories are shown to decay to zero in spite of that the semigroup is not exponentially stable

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

global existence and asymptotic decay of solutions to the nonlinear timoshenko system with memory abstract in this paper we consider the initial value problem for the nonlinear timoshenko system with a memory term due to the regularity loss property and weak dissipation we have to assume stronger nonlinearity than usual by virtue of the semi group arguments we obtain the global existence and optimal decay of solutions to the nonlinear problem under smallness and enough regularity assumptions on the initial data where we employ a time weighted l 2 energy method combined with the optimal l 2 decay of lower order derivatives of solutions

hadamard well posedness for a hyperbolic equation of viscoelasticity with supercritical sources and damping abstract presented here is a study of a viscoelastic wave equation with supercritical source and damping terms we employ the theory of monotone operators and nonlinear semigroups combined with energy methods to establish the existence of a unique local weak solution in addition it is shown that the solution depends continuously on the initial data and is global provided the damping dominates the source in an appropriate sense

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

on singular semilinear elliptic equations abstract for the semilinear elliptic equation u p x u 0 x r n n 3 0 we show via the barrier method the existence of a positive entire solution behaving like x 2 n near

asymptotic behaviour of imperfect viscoelastic beams abstract in this paper the problem of the lateral stability of imperfect viscoelastic beams is analysed the problem is examined by means of the quasi static approach and leads to a system of integro differential equations which can be resolved by a series expansions and the laplace transforms the viscous critical load is defined according to the asymptotic behaviour of the beam and can be evaluated by introducing the assumption of weak fading memory solid material and thermodynamic compatibility the critical load does not depend either on the type or the entity of the imperfections furthermore some characteristic aspects of the problem like the faster progress of the torque moment with respect to the progress of the lateral bending moment are underlined for a three element model it is possible to reach a closed form solution in the case of the poisson ratio constant in time and three element model an interesting analogy with the asymptotic behaviour of the imperfect column is observed

a new approach to the stability of an abstract system in the presence of infinite history abstract in this paper we consider the following problem u t t t a u t 0 g s a u t s d s 0 t 0 u t u 0 t t 0 u t 0 u 1 where a is a self adjoint positive definite operator and g is a positive nonincreasing function we adopt the method introduced in 19 for finite history with some modifications imposed by the nature of our problem to establish a general decay result which depends only on the behavior of the relaxation function our result extends the decay result obtained for problems with finite history to those with infinite history in addition it improves in some cases some decay results obtained earlier in 15

long time dynamics for a class of kirchhoff models with memory this paper is concerned with a class of kirchhoff models with memory effects utt 2u div u p 2 u 0 s 2u t s ds ut f u h defined in a bounded domain of rn this non autonomous equation corresponds to a viscoelastic version of kirchhoff models arising in dynamics of elastoplastic flows and plate vibrations under assumptions that the exponent p and the growth of f u are up to the critical range it turns out that the model corresponds to an autonomous dynamical system in a larger phase space by adding a component which describes the relative displacement history then the existence of a global attractor is granted furthermore in the subcritical case this global attractor has finite hausdorff and fractal dimensions

on the spectrum of euler bernoulli beam equation with kelvin voigt damping abstract the spectral property of an euler bernoulli beam equation with clamped boundary conditions and internal kelvin voigt damping is considered the essential spectrum of the system operator is rigorously identified to be an interval on the left real axis under some assumptions on the coefficients it is shown that the essential spectrum contains continuous spectrum only and the point spectrum consists of isolated eigenvalues of finite algebraic multiplicity the asymptotic behavior of eigenvalues is presented

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

general decay of the solution energy in a viscoelastic equation with a nonlinear source abstract in a bounded domain we consider u t t u 0 t g t u d u u where 0 and g is a nonnegative and decaying function we prove that for certain class of relaxation functions and certain initial data the rate of decay of energy is similar to that of g this result improves earlier ones in the literature in which only the exponential and polynomial decay rates are considered

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

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

general decay rate estimate for a viscoelastic equation with weakly nonlinear time dependent dissipation and source terms a viscoelastic wave equation in canonical form with weakly nonlinear time dependent dissipation and source terms is investigated in this paper for a wider class of relaxation functions and without imposing any restrictive growth assumption on the damping term at the origin we establish an explicit and general energy decay rate result

asymptotic stability of semigroups associated with linear weak dissipative systems with memory we study the asymptotic behavior of the solutions of a class of linear dissipative integral differential equations we show in the abstract setting a necessary and sufficient condition to get an exponential decay of the solution in the case of the lack of exponential decay we find the polynomial rate of decay of the solution some examples are given

exponential and polynomial decay for a quasilinear viscoelastic equation abstract we consider the following nonlinear viscoelastic equation u t u t t u u t t 0 t g t s u s d s 0 in a bounded domain for 0 we show that the dissipation induced by the integral term is strong enough to stabilize the solution this result improves an earlier one given by cavalcanti et al in m m cavalcanti v n d cavalcanti j ferreira existence and uniform decay for nonlinear viscoelastic equation with strong damping math methods appl sci 24 2001 1043 1053

global attractors for a semilinear hyperbolic equation in viscoelasticity a semilinear partial differential equation of hyperbolic type with a convolution term describing simple viscoelastic materials with fading memory is considered regarding the past history memory of the displacement as a new variable the equation is transformed into a dynamical system in a suitable hilbert space the dissipation is extremely weak and it is all contained in the memory term longtime behavior of solutions is analyzed in particular in the autonomous case the existence of a global attractor for solutions is achieved

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

some well posedness and general stability results in timoshenko systems with infinite memory and distributed time delay in this paper we consider a timoshenko system in one dimensional bounded domain with infinite memory and distributed time delay both acting on the equation of the rotation angle without any restriction on the speeds of wave propagation and under appropriate assumptions on the infinite memory and distributed time delay convolution kernels we prove first the well posedness and second the stability of the system where we present some decay estimates depending on the equal speed propagation case and the opposite one the obtained decay rates depend on the growths of the memory and delay kernels at infinity in the nonequal speed case the decay rate depends also on the regularity of initial data our stability results show that the only dissipation resulting from the infinite memory guarantees the asymptotic stability of the system regardless to the speeds of wave propagation and in spite of the presence of a distributed time delay applications of our approach to specific coupled timoshenko heat and timoshenko wave systems as well as the discrete time delay case are also presented

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

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

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

asymptotic behavior of the energy for a class of weakly dissipative second order systems with memory a class of second order abstract systems with memory and dirichlet boundary conditions is investigated by suitable liapunov functionals existence of solutions as well as asymptotic behavior are determined in particular when the memory kernel decays exponentially the polynomially decay of the solutions is proved

existence of global attractors for the coupled system of suspension bridge equations a mathematical model of the suspension bridge describes the vibration of the road bed in the vertical plain and that of the main cable we show the existence of an absorbing set for the solution of the problem furthermore the global attractors of the coupled system of suspension bridge are studied by a new semigroup approach

existence of strong global attractors for hyperbolic equation with linear memory we prove that the existence of global attractors of strong solutions for the hyperbolic equations with linear memory using the semigroup approach

stability of timoshenko systems with past history we consider vibrating systems of timoshenko type with past history acting only in one equation we show that the dissipation given by the history term is strong enough to produce exponential stability if and only if the equations have the same wave speeds otherwise the corresponding system does not decay exponentially as time goes to infinity in the case that the wave speeds of the equations are different which is more realistic from the physical point of view we show that the solution decays polynomially to zero with rates that can be improved depending on the regularity of the initial data

a general decay result of a nonlinear system of wave equations with infinite memories in this paper we consider a system of two wave equations with nonlinear damping and source terms acting in both equations in the presence of infinite memory terms and prove an explicit and general decay result our approach allows a wide class of kernels among which those of exponential decay type are only special cases

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

model checking single agent behaviours by fluid approximation in this paper we investigate a potential use of fluid approximation techniques in the context of stochastic model checking of csl formulae we focus on properties describing the behaviour of a single agent in a large population of agents exploiting a limit result known also as fast simulation in particular we will approximate the behaviour of a single agent with a time inhomogeneous ctmc which depends on the environment and on the other agents only through the solution of the fluid differential equation and model check this process we will prove the asymptotic correctness of our approach in terms of satisfiability of csl formulae we will also present a procedure to model check time inhomogeneous ctmc against csl formulae

boundary controllability and observability of a viscoelastic string in this paper we consider an integrodifferential system which governs the vibration of a viscoelastic one dimensional object we assume that we can act on the system at the boundary and we prove that it is possible to control both the position and the velocity at every point of the body and at a certain time t large enough we shall prove this result using moment theory and we shall prove that the solution of this problem leads to identification of a riesz sequence which solves controllability and observability the results presented here are constructive and can lead to simple numerical algorithms