strongly damped wave equation with exponential nonlinearities abstract in this paper we study the initial boundary value problem for two dimensional strongly damped wave equation with exponentially growing source and damping terms we first show the well posedness of this problem and then prove the existence of the global attractor in h 0 1 l l 2

an accelerated exponential time integrator for semi linear stochastic strongly damped wave equation with additive noise abstract this paper is concerned with the strong approximation of a semi linear stochastic wave equation with strong damping driven by additive noise based on a spatial discretization performed by a spectral galerkin method we introduce a kind of accelerated exponential time integrator involving linear functionals of the noise under appropriate assumptions we provide error bounds for the proposed full discrete scheme it is shown that the scheme achieves higher strong order in time direction than the order of temporal regularity of the underlying problem which allows for higher convergence rate than usual time stepping schemes particularly for the space time white noise case in two or three spatial dimensions the scheme still exhibits a good convergence performance another striking finding is that even for the velocity with low regularity the scheme always promises first order strong convergence in time numerical examples are finally reported to confirm our theoretical findings

on the strongly damped wave equation we prove the existence of the universal attractor for the strongly damped semilinear wave equation in the presence of a quite general nonlinearity of critical growth when the nonlinearity is subcritical we prove the existence of an exponential attractor of optimal regularity having a basin of attraction coinciding with the whole phase space as a byproduct the universal attractor is regular and of finite fractal dimension moreover we carry out a detailed analysis of the asymptotic behavior of the solutions in dependence of the damping coefficient

strongly damped wave problems bootstrapping and regularity of solutions abstract the aim of the article is to present a unified approach to the existence uniqueness and regularity of solutions to problems belonging to a class of second order in time semilinear partial differential equations in banach spaces our results are applied next to a number of examples appearing in literature which fall into the class of strongly damped semilinear wave equations the present work essentially extends the results on the existence and regularity of solutions to such problems previously these problems have been considered mostly within the hilbert space setting and with the main part operators being selfadjoint in this article we present a more general approach involving sectorial operators in reflexive banach spaces

an existence result for dissipative nonhomogeneous hyperbolic equations via a minimization approach abstract we discuss a purely variational approach to the study of a wide class of second order nonhomogeneous dissipative hyperbolic pdes precisely we focus on the wave like equations that present also a nonzero source term and a first order in time linear term the paper carries on the research program initiated in 14 and developed in 15 21 on the de giorgi approach to hyperbolic equations

strongly damped wave equation in uniform spaces abstract the cauchy problem for a semilinear strongly damped wave equation is considered in the whole of r n under suitable conditions imposed on nonlinear term which are much like for equations in bounded domains a dissipative semigroup t t of global solutions to this problem is constructed in a reach phase space being the product of the locally uniform spaces existence of an attractor for t t is then established

essential dimension of algebraic tori the essential dimension is a numerical invariant of an algebraic group g which may be thought of as a measure of complexity of g torsors over fields a recent theorem of n karpenko and a merkurjev gives a simple formula for the essential dimension of a finite p group we obtain similar formulas for the essential p dimension of a broad class of groups which includes all algebraic tori

finite dimensional attractors for the kirchhoff equation with a strong dissipation abstract the paper studies the existence of the finite dimensional global attractors and exponential attractors for the dynamical system associated with the kirchhoff type equation with a strong dissipation u t t m u 2 u u t h u t g u f x it proves that the above mentioned dynamical system possesses a global attractor which has finite fractal dimension and an exponential attractor for application the fact shows that for the concerned viscoelastic flow the permanent regime global attractor can be observed when the excitation starts from any bounded set in phase space and the dimension of the attractor that is the number of degree of freedom of the turbulent phenomenon and thus the level of complexity concerning the flow is finite

global regular solutions for the dynamic antiplane shear problem in nonlinear viscoelasticity here a is the two dimensional laplacian g is a scalar function and uo u f are given functions equation 1 occurs in the description of antiplane shear motions of certain viscoelastic solids for background material the reader is referred to 20 let y t denote the position of a particle with reference position at time t and consider an infinite column 2 r consisting of a homogeneous isotropic incompressible solid with the semilinear constitutive equation

stability of exponential attractors for a family of semilinear wave equations with gentle dissipation abstract the paper investigates the stability of exponential attractors for a family of semilinear wave equations with gentle dissipation u t t u u t f u g x with 0 1 2 i we propose a new criterion on the existence and stability of a family of exponential attractors depending on the perturbation parameters ii by applying this criterion to the equations we construct a family of exponential attractors a e x p and show their stability on the dissipative exponent

an upper bound on the essential dimension of a central simple algebra abstract we prove a new upper bound on the essential p dimension of the projective linear group pgl n

long time behavior of strongly damped nonlinear wave equations abstract we consider the long time behavior of a strongly damped nonlinear wave equation we will show that the initial boundary value problem has a global solution and that there exists a compact global attractor with finite dimension

continuation and asymptotics of solutions to semilinear parabolic equations with critical nonlinearities abstract in this paper we discuss continuation properties and asymptotic behavior of regular solutions to abstract semilinear parabolic problems in case when the nonlinear term satisfies critical growth conditions a necessary and sufficient condition for global in time existence of regular solutions is given we also formulate sufficient conditions to construct a piecewise regular solutions continuation beyond maximal time of existence for regular solutions applications to strongly damped wave equations and to higher order semilinear parabolic equations are finally discussed in particular global solvability and the existence of a global attractor for u t t d 1 2 u t d u f u in h 0 1 l 2 is achieved in case when a nonlinear term f satisfies a critical growth condition and a dissipativeness condition similar result is obtained for a 2 m th order semilinear parabolic initial boundary value problem in a hilbert space h 2 b j m

essential p dimension of pgln abstract an improved upper bound for the essential p dimension of pgl n is computed using algebraic tori it follows that ed pgl 8 2 17 if char k 2

large time behavior via the method of l trajectories abstract the method of l trajectories is presented in a general setting as an alternative approach to the study of the large time behavior of nonlinear evolutionary systems it can be successfully applied to the problems where solutions suffer from lack of regularity or when the leading elliptic operator is nonlinear here we concentrate on systems of a parabolic type and apply the method to an abstract nonlinear dissipative equation of the first order and to a class of equations pertinent to nonlinear fluid mechanics in both cases we prove the existence of a finite dimensional global attractor and the existence of an exponential attractor

global attractors for a third order in time nonlinear dynamics abstract long time behavior of a third order in time nonlinear pde equation is considered this type of equations arises in the context of nonlinear acoustics 12 20 22 24 where modeling accounts for a finite speed of propagation paradox the latter results in hyperbolic nature of the dynamics it will be proved that the underlying pde generates a well posed dynamical system which admits a global and finite dimensional attractor the main difficulty associated with the problem studied is the lack of lyapunov function along with the lack of compactness of trajectories which fact prevents applicability of standard tools in the area of dynamical systems

gradient estimates for a class of parabolic systems we establish local calderon zygmund type estimates for a class of parabolic problems whose model is the nonhomogeneous degenerate singular parabolic p laplacian system ut div du p 2du div f p 2f proving that f lqloc du lqloc q p we also treat systems with discontinuous coefficients of vanishing mean oscillation vmo type

global existence and exponential stability of small solutions to nonlinear viscoelasticity the global existence of smooth solutions to the equations of nonlinear hyperbolic system of 2nd order with third order viscosity is shown for small and smooth initial data in a bounded domain ofn dimensional euclidean space with smooth boundary dirichlet boundary condition is studied and the asymptotic behaviour of exponential decay type of solutions ast tending to is described time periodic solutions are also studied as an application of our main theorem nonlinear viscoelasticity strongly damped nonlinear wave equation and acoustic wave equation in viscous conducting fluid are treated

regularity of solutions on the global attractor for a semilinear damped wave equation we consider attractors a 0 1 corresponding to a singularly perturbed damped wave equation utt 2 a 1 2 ut aut au f u

comparison of some field invariants in all this paper p is a prime number and f is a field of characteristic 6 p the invariants we are interested in are the p cohomological dimension cdp f 16 the diophantine dimension dd f inf i f is ci 5 for p 2 the quadratic diophantine dimension ddq f inf i f is c i where c q i is the condition introduced by pfister 13 for p 2 the u invariant u f 8 ch 11 the p invariant 7 for an element c pbr f h f p p c inf n c is a sum of n classes of algebras of degree p p f sup p c c pbr f the p invariant 7 for c as above p c logp ind c where ind c is the schur index of any central simple algebra representing c p f sup p c c pbr f to muddy water a little more we shall also consider the stable p and p invariants p f sup p e e f finite separable e f p 1 p f sup p e e f finite separable e f p 1

nonlinear second order equations with applications to partial differential equations where a and b are usually unbounded linear operators in a banach space these problems arise often in the study of partial differential equations as is usual we control a nonlinear perturbation possibly involving spatial derivatives by the linear terms which contain higher order spatial derivatives but contrary to the usual method of reducing the problem to a first order system in some energy norm space as in 9 5 we use the factoring method of lo this method allows the equation to be written as an integral equation containing a double integral involving the nonlinearity reflecting the fact that the equation is second order there are several advantages to this approach which will be illustrated fully in the examples first we show that the equation is locally well posed if loosely the nonlinearity satisfies the local lipschitz condition

finite dimensional attractors for the quasi linear strongly damped wave equation abstract we present a new method of investigating the so called quasi linear strongly damped wave equations t 2 u t x u x u f u x x u g in bounded 3d domains this method allows us to establish the existence and uniqueness of energy solutions in the case where the growth exponent of the non linearity is less than 6 and f may have arbitrary polynomial growth rate moreover the existence of a finite dimensional global and exponential attractors for the solution semigroup associated with that equation and their additional regularity are also established in a particular case 0 which corresponds to the so called semi linear strongly damped wave equation our result allows to remove the long standing growth restriction f u c 1 u 5

global attractor for damped abstract nonlinear hyperbolic systems this work is concerned with the long time dynamics of a class of abstract nonlinear second order in time systems with damping this class of systems describes nonlinear dissipative elastic models with the nonlinear term produced by neo hookean type stress strain relationships in the present work we use an analysis based in part on the results of h t banks d s gilliam and v i shubov on the existence and uniqueness of the weak solutions to show that these systems generate a strong dynamical system also more importantly we are able to prove the existence of a compact strong global attractor finally we make several comments concerning the regularity of this attractor and present some examples

essential dimension of central simple algebras let p be a prime integer 1 s r be integers and f be a field of characteristic different from p we find upper and lower bounds for the essential p dimension ed p al g p r p s of the class al g p r p s of central simple algebras of degree p r and exponent dividing p s in particular we show that ed alg 8 2 ed2 alg 8 2 8 and ed p al g p 2 p p 2 p for p odd

cauchy problem for quasi linear wave equations with nonlinear damping and source terms abstract the paper studies the existence and non existence of global weak solutions to the cauchy problem for a class of quasi linear wave equations with nonlinear damping and source terms it proves that when max m p where m 1 1 and p 1 are respectively the growth orders of the nonlinear strain terms the nonlinear damping term and the source term under rather mild conditions on initial data the cauchy problem admits a global weak solution especially in the case of space dimension n 1 the weak solutions are regularized and so generalized and classical solution both prove to be unique on the other hand if 0 1 and the initial energy is negative then under certain opposite conditions any weak solution of the cauchy problem blows up in finite time and an example is shown

dynamics of strongly damped wave equations in locally uniform spaces attractors and asymptotic regularity this paper is dedicated to analyzing the dynamical behavior of strongly damped wave equations with critical nonlinearity in locally uniform spaces after proving the global well posedness we first establish the asymptotic regularity of the solutions which appears to be optimal and the existence of a bounded in h 2 lu r n x h 1 lu r n subset which attracts exponentially every initial h 1 lu r n x l 2 lu r n bounded set with respect to the h 1 lu r n x l 2 lu r n norm then we show there is a h 1 lu r n x l 2 lu r n h 1 r n x h 1 r n global attractor which reflects the strongly damped property of u t to some extent

quasilinear equations for viscoelasticity of strain rate type we consider the quasilinear m m system of partial differential equations that governs the motion of a viscoelastic material of strain rate type on a bounded domain in mathbb r n the dependence of the stress on both the strain and strain rate is nonlinear and our hypotheses allow for a potential energy which is a nonconvex function of the strain the critical hypothesis is that the dependence of the stress function on the strain rate is uniformly strictly monotone in the sense of minty and browder we prove the existence and uniqueness of weak solutions to a natural initial boundary value problem for a large class of constitutive functions we then treat the question of h2 regularity of solutions and show that while regularity in the initial data is preserved solutions do not in general become more regular than their initial data this generalizes a result for the semilinear case due to rybka

tori and essential dimension abstract the present paper deals with algebraic tori and essential dimension in three unrelated contexts after some preliminaries on essential dimension versal torsors and tori we explicitly construct a versal torsor for pgl n n 5 odd defined over a field of transcendence degree 1 2 n 1 n 2 over the base field this recovers a result of lorenz reichstein rowen and saltman we also discuss the so called tori method which gives a geometric proof of a result of ledet on the essential dimension of a cyclic p group in the last section we compute the essential dimension of the functor k h 1 k gl n z the latter set being in bijection with the isomorphism classes of n dimensional k tori

accurate bloch analysis of 1 d periodic lines through the simulation of truncated structures a common approach to perform dispersive analyses of waveguides periodic along one direction is based on the electromagnetic simulation of a single spatial period of the structure however the resulting equivalent two port network representation of the single cell may lead to inaccurate modal results since mutual coupling between cells has been neglected when a finite number of adjacent cells are simulated with the aim of improving the accuracy of the analysis spurious solutions are introduced they are shown here to be related to the nonuniqueness of root extraction operations in the complex plane a simple automatic method is proposed to recover the correct solution and to test the convergence of the analysis as the number of simulated cells is increased

limiting behavior for strongly damped nonlinear wave equations in this paper we investigate both the existence and the limiting behavior for the equation utt aut au t u ut where a is a sectorial operator is periodic in t and satisfies certain regularity and growth assumptions in most results on limiting behavior we will assume a has compact resolvent we consider the equation as an abstract ode defined on a paired space x x 0 1 with regard to the limiting behavior one of our principal results will be to show that if there is a bounded set in one of the spaces considered for which all points or trajectories enter into and remain then there is a set j consisting of very smooth functions defined on all of the spaces considered which is the maximum compact invariant set uniformly asymptotically stable connected and having very strong attractivity properties in all these spaces we will often show it attracts all points in a bounded set uniformly we will give a few sharper results for the case where a the work is motivated by recent papers of webb and fitzgibbon and applies techniques found in recent papers by the author

global existence for a quasi linear evolution equation with a non convex energy we establish the existence of global in time weak solutions to the initial boundary value problem related to the dynamics of coherent solid solid phase transitions in viscoelasticity the class of the stored energy functionals includes the double well potential and a general convolution damping term is considered

finite dimensional attractors for the kirchhoff models with critical exponents the paper studies the existence of the finite dimensional global attractor and exponential attractor for the dynamical system associated with the kirchhoff models utt u m 1 u ut 2u h ut g u f x it proves that for the subcritical and critical cases 1 m n 2 n 2 with n 6 the above mentioned dynamical system possesses in phase spaces x1 a finite dimensional global attractor and a weak exponential attractor respectively

cauchy problem for quasi linear wave equations with viscous damping abstract the paper studies the global existence and asymptotic behavior of weak solutions to the cauchy problem for quasi linear wave equations with viscous damping it proves that when p max m where m 1 1 and p 1 are respectively the growth orders of the nonlinear strain terms the nonlinear damping term and the source term the cauchy problem admits a global weak solution which decays to zero according to the rate of polynomial as t as long as the initial data are taken in a certain potential well and the initial energy satisfies a bounded condition especially in the case of space dimension n 1 the solutions are regularized and so generalized and classical solution both prove to be unique comparison of the results with previous ones shows that there exist clear boundaries similar to thresholds among the growth orders of the nonlinear terms the states of the initial energy and the existence asymptotic behavior and nonexistence of global solutions of the cauchy problem

well posedness and attractor on the 2d kirchhoff boussinesq models abstract the paper studies the well posedness and the existence of attractors for a class of 2d kirchhoff boussinesq models u t t k u t 2 u d i v u 1 u 2 g u with 0 0 0 we show that i the ibvp of the equations is well posed in natural energy space x 2 and strong solution space x 4 respectively provided that g s c 1 s 2 ii the related solution semigroup has a global and an generalized exponential attractor in x 2 provided that the damping parameter k is suitably large and g s c iii in particular when 0 the corresponding boussinesq model has a subclass j of limit solutions and the subclass j has a weak global attractor in energy space x 1 without any upper bound restriction for the growth exponent of g u iv in the cases that either 0 or 0 the corresponding model has a global attractor in x 4 provided that g s c 1 s and without any restriction for the damping parameter k 0 especially when 0 the corresponding results extend those in grassell et al 2009

asymptotic behaviour and changes of phase in one dimensional nonlinear viscoelasticity in this paper we study the asymptotic behaviour as t co of solutions u x t to the nonlinear partial differential equation ut m4 a o o 1 1 with initial conditions u 0 q x u x u x o 0 1 3a

longtime dynamics of the kirchhoff equation with strong damping and critical nonlinearity on rn abstract the paper studies the longtime dynamics of the kirchhoff equation with strong damping and critical nonlinearity on r n u t t u t m u 2 u u t g x u f x we establish the well posedness the existence of the global and exponential attractors in natural energy space h h 1 r n l 2 r n in critical nonlinearity case these results improve not only the recent ones achieved by yang 2007 32 but also ones achieved by conti pata and squassina 2005 9 to some extent