the survival probability for critical spread out oriented percolation above 4 1 dimensions i induction we consider critical spread out oriented percolation above 4 1 dimensions our main result is that the extinction probability at time n i e the probability for the origin to be connected to the hyperplane at time n but not to the hyperplane at time n 1 decays like 1 bn 2 as n to infty where b is a finite positive constant this in turn implies that the survival probability at time n i e the probability that the origin is connected to the hyperplane at time n decays like 1 bn as n to infty the latter has been shown in an earlier paper to have consequences for the geometry of large critical clusters and for the incipient infinite cluster the present paper is part i in a series of two papers in part ii we derive a lace expansion for the survival probability adapted so as to deal with point to plane connections this lace expansion leads to a nonlinear recursion relation for the survival probability in part i we use this recursion relation to deduce the asymptotics via induction

uniform stabilization of the damped cauchy ventcel problem with variable coefficients and dynamic boundary conditions in this paper the uniform stabilization of the cauchy ventcel problem with variable coefficients is considered and the uniform energy decay rate for the problem is established by riemannian geometry methods

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

on decay and blow up of the solution for a viscoelastic wave equation with boundary damping and source terms abstract in this paper we consider the decay and blow up properties of a viscoelastic wave equation with boundary damping and source terms we first extend the decay result for the case of linear damping obtained by lu et al on a viscoelastic equation with nonlinear boundary damping and source terms global existence and decay of the solution nonlinear analysis real world applications 12 1 2011 295 303 to the nonlinear damping case under weaker assumption on the relaxation function g t then we give an exponential decay result without the relation between g t and g t for the linear damping case provided that g l 1 0 is small enough finally we establish two blow up results one is for certain solutions with nonpositive initial energy as well as positive initial energy for both the linear and nonlinear damping cases the other is for certain solutions with arbitrarily positive initial energy for the linear damping case

on the control of a viscoelastic damped timoshenko type system abstract in this paper we consider the following timoshenko system tt x x 0 0 1 0 tt xx 0 t g t xx d x 0 0 1 0 with dirichlet boundary conditions where g is a positive nonincreasing function we establish an exponential and polynomial decay results with weaker conditions on g than those required in f ammar khodja a benabdallah j e munoz rivera r racke energy decay for timoshenko systems of memory type j differ equations 194 2003 82 115

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

on behaviour of solutions for a nonlinear viscoelastic equation with variable exponent nonlinearities abstract this paper deals with a viscoelastic wave equation with variable exponent nonlinearities subject to nonlinear boundary feedback under appropriate conditions a general decay result associated to solution energy is proved it is also shown that regarding arbitrary positive initial energy solutions blow up in a finite time

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

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

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

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

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

asymptotic stability and energy decay rates for solutions of the wave equation with memory in a star shaped domain we study the asymptotic stability and give the energy decay rates for solutions of the wave equation with boundary dissipation of the memory type

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

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

coarsening in the 1d ising model evolving with swendsen wang dynamics an unusual scaling we consider a simple model of domain growth the zero temperature 1d ising model evolving according to the swendsen wang dynamics we find that in the long time limit the pair correlation function scales with a characteristic length increasing as the square of the average domain size in that limit a few large domains occupy almost all the space with many small domains between them in contrast to the usual picture of coarsening the average domain size here is not a characteristic length of the growth problem instead one finds a power law distribution for the sizes of large domains with a cut off at a length which grows as the square of the average size of the domains

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

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

general stability of a viscoelastic variable coefficients plate equation with time varying delay in localized nonlinear internal feedback abstract this paper is concerned with the decay properties of a memory type euler bernoulli plate equation with variable coefficients and time varying delay in localized nonlinear internal feedback by using piecewise multiplier method and riemannian geometric analysis we establish the explicit and general decay rates of system whose nonlinear internal feedbacks act on a suitable subregion

stability of wave equation with locally viscoelastic damping and nonlinear boundary source abstract in this paper we consider a nonlinear wave equation with nonlinear boundary damping and source terms subject to a localized viscoelastic damping and localized frictional damping we obtain the global existence of solution via potential well method furthermore by introducing suitable lyapunov functionals using multiplier method and constructing convex differential inequality we compare the effect of locally viscoelastic term and frictional damping and establish an explicit and general decay rate result this stability result allows the localized viscoelastic damping and localized frictional damping acting simultaneously or complementarily in the domain which gives the competing relationship of the viscoelastic and frictional damping

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

stability of damped membranes and plates with distributed inputs abstract this paper proves the stability of boundary and distributed damped membranes and kirchhoff plates under distributed inputs distributed viscous or kelvin voigt damping ensures a weakly bounded response to a bounded transverse loading for pinned membranes and clamped plates damping on part of the boundary can also weakly stabilize the forced response provided the damped and undamped boundary normals satisfy certain conditions for example damping on half and one side of the boundary is sufficient for circular and rectangular domains respectively

upper semicontinuity of pullback attractors for a non autonomous viscoelastic equation this paper is concerned with upper semicontinuity of pullback attractors for a non autonomous viscoelastic equation with a past history with a nonautonomous perturbation in particular under some proper assumptions we prove that the pullback attractor a e t t r of eq 1 1 with e 0 and the global attractor a of eq 1 1 with e 0 satisfy lim e 0 dist x a e t a 0 for any email protected r

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

blow up of positive initial energy solutions of a nonlinear viscoelastic hyperbolic equation abstract in this paper we consider the nonlinear viscoelastic equation u t t u 0 t g t u d u t u t m 2 u u p 2 with initial conditions and dirichlet boundary conditions for nonincreasing positive functions g and for p m we prove that there are solutions with positive initial energy that blow up in finite time

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

well posedness and exponential stability for coupled lame system with viscoelastic term and strong damping abstract in this paper we consider a coupled lame system with a viscoelastic term and a strong damping we prove well posedness by using faedo galerkin method and establish an exponential decay result by introducing a suitable lyapunov functional

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

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

blow up in a nonlinear viscoelastic wave equation with strong damping abstract in this paper we consider the nonlinear viscoelastic equation u t t u 0 t g t u d u t u p 2 u in 0 t with initial conditions and dirichlet boundary conditions for nonincreasing positive functions g we show the finite time blow up of some solutions whose initial data have arbitrarily high initial energy

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

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

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

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

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

general decay of the solution for a viscoelastic wave equation with a time varying delay term in the internal feedback in this paper we consider a viscoelastic wave equation with a time varying delay term the coefficient of which is not necessarily positive by introducing suitable energy and lyapunov functionals under suitable assumptions we establish a general energy decay result from which the exponential and polynomial types of decay are only special cases