a new phenomenon in the critical exponent for structurally damped semi linear evolution equations abstract in this paper we find the critical exponent for global small data solutions to the cauchy problem in r n for dissipative evolution equations with power nonlinearities u p or u t p u t t u t u u p u t p here n 0 with 2 we show that the critical exponent for each of the two nonlinearities is related to each of the two possible asymptotic profiles of the linear part of the equation which are described by the diffusion equations v t v 0 w t w 0 the nonexistence of global solutions in the critical and subcritical cases is proved by using the test function method under suitable sign assumptions on the initial data and lifespan estimates are obtained by assuming small initial data in sobolev spaces we prove the existence of global solutions in the supercritical case up to some maximum space dimension n and we derive l q estimates for the solution for q 1 for 2 the result holds in any space dimension n 1 the existence result also remains valid if and or are fractional

the construction of preconditioners for elliptic problems by substructuring i in earlier parts of this series of papers we constructed preconditioners for the discrete systems of equations arising from the numerical approximation of elliptic boundary value problems the resulting algorithms are well suited for implementation on computers with parallel architecture in this paper we will develop a technique which utilizes these earlier methods to derive even more efficient preconditioners the iterative algorithms using these new preconditioners converge to the solution of the discrete equations with a rate that is independent of the number of unknowns these preconditioners involve an incomplete chebyshev iteration for boundary interface conditions which results in a negligible increase in the amount of computational work theoretical estimates and the results of numerical experiments are given which demonstrate the effectiveness of the methods

asymptotic behavior for abstract evolution differential equations of second order abstract abstract evolution differential equations of second order in time are studied in order to get almost optimal decay estimates to the corresponding energy functional of the equations the framework is supported by a special energy method in the associated fourier space the constructed abstract theory can be applied to several concrete evolutionary partial differential equations as is illustrated in the last section of the paper

a finite element capacitance method for elliptic problems on regions partitioned into subregions a method is given for the solution of linear equations arising in the finite element method applied to a general elliptic problem this method reduces the original problem to several subproblems of the same form considered on subregions and an auxiliary problem very efficient iterative methods with the preconditioning operator and using fft are developed for the auxiliary problem

geometry of geometrically finite one dimensional maps we study the geometry of certain one dimensional maps as dynamical systems we prove the property of bounded and bounded nearby geometry of certainc 1 one dimensional maps with finitely many critical points this property enables us to give the quasisymmetric classification of geometrically finite one dimensional maps

markov partitions and feigenbaum like mappings we construct a markov partition for a feigenbaum like mapping we prove that this markov partition has bounded nearby geometry property similar to that for a geometrically finite one dimensional mappings 8 using this property we give a simple proof that any two feigenbaum like mappings are topologically conjugate and the conjugacy is quasisymmetric

on ulam von neumann transformations we define and study ulam von neumann transformations which are certain interval mappings and conjugate toq x 1 2x2 on 1 1 we use a singular metric on 1 1 to study a ulam von neumann transformation this singular metric is universal in the sense that it does not depend on any particular mapping but only on the exponent of this mapping at its unique critical point we give the smooth classification of ulam von neumann transformations by their eigenvalues at periodic points and exponents and asymmetries

geometry of cantor systems a cantor system is defined the geometry of a certain family of cantor systems is studied such a family arises in dynamical systems as hyperbolicity is created we prove that the bridge geometry of a cantor system in such a family is uniformly bounded and that the gap geometry is regulated by the size of the leading gap

smooth classification of geometrically finite one dimensional maps the scaling function of a one dimensional markov map is defined and studied we prove that the scaling function of a non critical geometrically finite one dimensional map is holder continuous while the scaling function of a critical geometrically finite one dimensional map is discontinuous we prove that scaling functions determine lipschitz conjugacy classes and moreover that the scaling function and the exponents and asymmetries of a geometrically finite one dimensional map are complete c1 invariants within a mixing topological conjugacy class

asymptotic profiles for wave equations with strong damping abstract we consider the cauchy problem in r n for strongly damped wave equations we derive asymptotic profiles of these solutions with weighted l 1 1 r n data by using a method introduced in 9 and or 10

diffusion phenomena for the wave equation with structural damping in the lp lq framework abstract in this paper we study diffusion phenomena for the wave equation with structural damping u t t u 2 a u t 0 u 0 x u 0 x u t 0 x u 1 x with a 0 and 0 1 2 we show that the solution u behaves like the solution v to v t 1 2 a 1 v 0 v 0 x v 0 x for suitable choice of initial data v 0 more precisely we derive l p l q decay estimates for the difference u v and its time and space derivatives where 1 p q possibly not on the conjugate line satisfying some additional condition related to in particular we show that under suitable assumptions on p q a double diffusion phenomenon appears that is the difference u v behaves like the solution to v t 2 a v 0 v 0 x v 0 x for a suitable choice of initial data v 0

the lp lq estimates of solutions to one dimensional damped wave equations and their application to the compressible flow through porous media we first obtain the lp lq estimates of solutions to the cauchy problem for one dimensional damped wave equation vtt vxx vt 0 v vt t 0 v0 v1 x x t r r corresponding to that for the parabolic equation t xx 0 t 0 v0 v1 x the estimates are shown by v t e t 2v0 t v0 t 2lp ct 121q 1p 1 v0 v1 lq t 1 etc for 1 q p to show the explicit formula of the damped wave equation will be used to apply the estimates to nonlinear problems is the second aim we will treat the system of a compressible flow through porous media the solution is expected to behave as the diffusion wave which is the solution to the porous media equation due to the darcy law when the initial data has the same constant state at a sharp lp convergence rate for p 2 has been recently obtained by nishihara proc roy soc edinburgh sect a 133a 2003 1 20 by choosing a suitably located diffusion wave we will show the l1 convergence applying

global existence for semi linear structurally damped σ evolution models abstract the main purpose of this paper is to study the global existence of small data solutions for semi linear structurally damped evolution models of the form u t t u u t f d a u u t u 0 x u 0 x u t 0 x u 1 x with 1 0 and 2 this is a family of structurally damped evolution models interpolating between models with exterior damping 0 and those with visco elastic type damping the function f d a u u t represents power non linearities d a u p for a 0 or u t p our goal is to propose a fujita type exponent diving the admissible range of powers p into those allowing global existence of small data solutions stability of zero solution and those producing a blow up behavior even for small data on the one hand we use new results from harmonic analysis for fractional gagliardo nirenberg inequality or for superposition operators see appendix a on the other hand our approach bases on l p l q estimates not necessarily on the conjugate line for solutions to the corresponding linear models assuming additional l 1 regularity for the data the linear models we have here in mind are v t t v v t 0 v 0 x v 0 x v t 0 x v 1 x with 1 0 and 0

an iterative method for elliptic problems on regions partitioned into substructures some new preconditioners for discretizations of elliptic boundary problems are studied with these preconditioners the domain under consideration is broken into subdomains and preconditioners are defined which only require the solution of matrix problems on the subdomains analytic estimates are given which guarantee that under appropriate hypotheses the preconditioned iterative procedure converges to the solution of the discrete equations with a rate per iteration that is independent of the number of unknowns numerical examples are presented which illustrate the theoretically predicted iterative convergence rates

uniform stabilizability of a full von karman system with nonlinear boundary feedback full von karman system accounting for in plane accelerations and describing the transient deformations of a thin elastic plate subject to edge loading is considered the energy dissipation is introduced via the nonlinear velocity feedback acting on a part of the edge of the plate it is known j puel and m tucsnak siam j control optim 33 1995 pp 255 273 that in the case of linear dissipation and star shaped domains boundary velocity feedback with the tangential derivatives of horizontal displacements leads to the exponential decay rates for the energy of the resulting closed loop system the main goal of the paper is to derive the uniform energy decay rates valid for the model without the above mentioned restrictions in particular it is shown that simple monotone nonlinear feedback without the tangential derivatives of the horizontal displacements provides the uniform decay rates for the energy in the absence of geometric hypotheses imposed on the controlled part of the boundary this is accomplished by establishing among other things sharp regularity results valid for the boundary traces of solutions corresponding to this nonlinear model and by employing a holmgren type uniqueness result proved recently in v isakov j differential equations 97 1997 pp 134 147 for the dynamical systems of elasticity which are overdetermined on the boundary

multilevel preconditioners for elliptic problems by substructuring we consider the construction of preconditioners for reduced schur problems for solving finite element elliptic equations which are obtained by block gaussian elimination of the unknowns corresponding to a number of subregions the preconditioners are based on approximations of the considered elliptic problem on a sequence of meshes and are derived in a purely algebraic manner it is proved that the proposed algebraic multilevel preconditioner is of nearly optimal order that is its relative condition number with respect to the corresponding reduced schur complement is proportional to l 2 where l is the number of discretization levels used this condition number is bounded uniformly with respect to the number of subregions the way the region is partitioned for example into strips or into boxes and the discontinuity of the coefficients of the elliptic operator if they are constants on the elements from the initial triangulation the data transport except on the coarsest level is of local type and is proportional to the number of nodes on the interface boundary

structural instability of cookie cutter sets abstract it is proved that the cookie cutter set in r is structurally instable in c 1 topology that means for the invariant set e of the ifs f i i we can always perturb f i i arbitrarily small in c 1 topology to provide an ifs g i i with its invariant set f such that dim h e dim h f and e f are not lipschitz equivalent

an asymptotic universal focal decomposition for non isochronous potentials galileo in the xvii century observed that the small os cillations of a pendulum seem to have constant period in fact the taylor expansion of the period map of the pendulum is constant up to second order in the initial angular velocity around the stable equi librium it is well known that for small oscillations of the pendulum and small intervals of time the dynamics of the pendulum can be ap proximated by the dynamics of the harmonic oscillator we study the dynamics of a family of mechanical systems that includes the pendulum at small neighbourhoods of the equilibrium but after long intervals of time so that the second order term of the period map can no longer be neglected we analyze such dynamical behaviour through a renor malization scheme acting on the dynamics of this family of mechanical systems the main theorem states that the asymptotic limit of this renormalization scheme is universal it is the same for all the elements in the considered class of mechanical systems as a consequence we obtain an universal asymptotic focal decomposition for this family of mechanical systems this paper is intended to be the first of a series of articles aiming at a semiclassical quantization of systems of the pendu lum type as a natural application of the focal decomposition associated to the two point boundary value problem

algebraic multigrid for linear systems obtained by explicit element reduction we consider sparse linear systems where a set of interior degrees of freedom have been eliminated in order to reduce the problem size this elimination is assumed to be local so the interior principal submatrix is block diagonal and the resulting schur complement is still sparse for it to be beneficial the elimination process should lead to reduced memory requirements but equally importantly it should also result in an algebraic problem that can be solved efficiently in this paper we propose a general element reduction approach and show how the elimination process can exploit a particular subzonal discretization to maintain the sparsity of the schur complement we also investigate algebraic multigrid amg solution algorithms applied to the reduced problem and we discuss the influence of the local elimination on solver related properties of the matrix such as near nullspace preservation and the availability of stable subspace decompositions we focus on boomeramg a parallel variant of classical ruge stuben amg applied to scalar diffusion problems v henson and u yang appl numer math 41 2002 pp 155 177 and the auxiliary space maxwell solver ams for electromagnetic diffusion applications t kolev and p vassilevski j comput math 27 2009 pp 604 623 in the electromagnetic case we establish algebraically a reduced version of the hiptmair xu decomposition from r hiptmair and j xu siam j numer anal 45 2007 pp 2483 2509 and consider a modification of the reduction process that targets the singular problems arising in simulations with pure void zero conductivity regions for scalar diffusion problems our particular stencil analysis shows that the reduction has a positive effect on meshes with stretched elements we present a number of two dimensional three dimensional and axisymmetric numerical experiments which demonstrate that the combination of an appropriately chosen local elimination with the use of the boomeramg and ams solvers can lead to significant improvements in the overall solution time

invariants for smooth conjugacy of hyperbolic dynamical systems ii we show that the eigenvalues of the derivatives at periodic points form a complete set of invariants for smooth local conjugacy of anosov diffeomorphisms oft2

an application of lp lq decay estimates to the semi linear wave equation with parabolic like structural damping abstract in this paper we study the global existence of small data solutions to u t t u 2 a u t u p u 0 x u 0 x u t 0 x u 1 x where a 0 0 1 2 and p 1 assuming small data in some sobolev spaces we obtain the global existence for p 1 2 n 2 in space dimension n n where n n as 1 2 in particular our result holds in any space dimension n 2 if 1 2

l1 l1 estimates for the strongly damped plate equation abstract in this paper we derive l 1 l 1 long time estimates for the strongly damped plate equation u t t 2 u 2 u t 0 x r n t r u 0 x u 0 x u t 0 x u 1 x in particular we prove that u t l 1 c 1 t n 4 u 0 l 1 1 t 1 2 u 1 l 1 for any t 0 in space dimension n 5

global existence of small solutions to a class of nonlinear evolution equations on etudie le comportement global des solutions d une classe d equations d evolution non lineaires on considere le probleme aux valeurs initiales pour des equations quasilineaires qui sont des perturbations d equations lineaires dissipatives

asymptotic differentiable structure on cantor set we study hyperbolic maps depending on a parameter e each of them has an invariant cantor set as e tends to zero the map approaches the boundary of hyperbolicity we analyze the asymptotics of scaling function of the invariant cantor set as e goes to zero we show that there is a limiting scaling function of the limiting map and this scaling function has dense jump discontinuities because the limiting map is not expanding removing these discontinuities by continuous extension we show that we obtain the scaling function of the limiting map with respect to a ulam von neumann type metric

a capacitance matrix method for dirichlet problem on polygon region an efficient algorithm for the solution of linear equations arising in a finite element method for the dirichlet problem is given the cost of the algorithm is proportional ton 2log2 n n 1 h where the cost of solving the capacitance matrix equations isnlog2 n on regular grids andn 3 2log2 n on irregular ones

renormalization and conjugacy of piecewise linear lorenz maps abstract we investigate the uniform piecewise linearizing question for a family of lorenz maps let f be a piecewise linear lorenz map with different slopes and positive topological entropy we show that f is conjugate to a linear mod one transformation and the conjugacy admits a dichotomy it is either bi lipschitz or singular depending on whether f is renormalizable or not f is renormalizable if and only if its rotation interval degenerates to be a rational point furthermore if the endpoints are periodic points with the same rotation number then the conjugacy is quasisymmetric

lp lq estimates of solutions to the damped wave equation in 3 dimensional space and their application it has been asserted that the damped wave equation has the diffusive structure as t in this paper we consider the cauchy problem in 3 dimensional space for the linear damped wave equation and the corresponding parabolic equation and obtain the lp lq estimates of the difference of each solution which represent the assertion precisely explicit formulas of the solutions are analyzed for the proof the second aim is to apply the lp lq estimates to the semilinear damped wave equation with power nonlinearity if the power is larger than the fujita exponent then the time global existence of small weak solution is proved and its optimal decay order is obtained

algebraic multilevel preconditioning methods i a recursive way of constructing preconditioning matrices for the stiffness matrix in the discretization of selfadjoint second order elliptic boundary value problems is proposed it is based on a sequence of nested finite element spaces with the usual nodal basis functions using a nodeordering corresponding to the nested meshes the finite element stiffness matrix is recursively split up into two level block structures and is factored approximately in such a way that any successive schur complement is replaced approximated by a matrix defined recursively and thereform only implicitely given to solve a system with this matrix we need to perform a fixed number v of iterations on the preceding level using as an iteration matrix the preconditioning matrix already defined on that level it is shown that by a proper choice of iteration parameters it suffices to use v left 1 gamma 2 right tfrac 1 2 iterations for the so constructedv foldv cycle wherev 2 corresponds to aw cycle preconditioning matrices to be spectrally equivalent to the stiffness matrix the conditions involve only the constant in the strengthened c b s inequality for the corresponding two level hierarchical basis function spaces and are therefore independent of the regularity of the solution for instance if we use successive uniform refinements of the meshes the method is of optimal order of computational complexity if gamma 2 tfrac 8 9 under reasonable assumptions of the finite element mesh the condition numbers turn out to be so small that there are in practice few reasons to use an accelerated iterative method like the conjugate gradient method for instance

preconditioning and two level multigrid methods of arbitrary degree of approximation let h be a mesh parameter corresponding to a finite element mesh for an elliptic problem we describe preconditioning methods for two level meshes which for most prob lems solved in practice behave as methods of optimal order in both storage and computa tional complexity namely per mesh point these numbers are bounded above by relatively small constants for all h ho where ho is small enough to cover all but excessively fine meshes we note that in practice multigrid methods are actually solved on a finite often even a fixed number of grid levels in which case also these methods are not asymptotically optimal as h 0 numerical tests indicate that the new methods are about as fast as the best implementations of multigrid methods applied on general problems variable coefficients general domains and boundary conditions for all but excessively fine meshes furthermore most of the latter methods have been implemented only for difference schemes of second order of accuracy whereas our methods are applicable to higher order approximations we claim that our scheme could be added fairly easily to many existing finite element codes 1 introduction consider the numerical solution of elliptic boundary value prob lems discretized by finite element methods we assume that the boundary is polygonal or consists of planes we note that in practical problems one often has a fine enough grid already after the definition of the boundary and the minimal number of vertices needed for a first coarse triangulation anyhow if not so in most cases one makes only a few steps of mesh refinement hence the power of multigrid methods their optimal order of computational complexity is most often not achieved fully because optimality requires a large number of recursively defined meshes for details see e g 4 and for further references see 7 hence one might as well consider other methods perhaps simpler and more effective on a fixed mesh but which are not asymptotically optimal here we shall describe a method which uses only a fixed mesh but for which one nevertheless achieves a low order of computational complexity and of seemingly optimal order except for from a practical viewpoint excessively small meshes to be more precise the computational cost per mesh point is bounded by clog n for n no where n is the number of mesh points no is large enough to cover most applications and c is small enough that the method is competitive with multigrid methods as is well known the latter need recursion and the usual smoothing followed by corrections of the solutions on the different mesh levels we claim that the new method is more suitable for implementation in existing finite element packages in fact most packages for the multigrid methods are only for second order difference methods

algebraic multilevel preconditioning methods ii a recursive way of constructing preconditioning matrices for the stiffness matrix in the discretization of selfadjoint second order elliptic boundary value problems is proposed it is based on a sequence of nested finite element spaces with the usual nodal basis functions using a nodeordering corresponding to the nested meshes the finite element stiffness matrix is recursively split up into two level block structures and is factored approximately in such a way that any successive schur complement is replaced approximated by a matrix defined recursively and thereform only implicitely given to solve a system with this matrix we need to perform a fixed number v of iterations on the preceding level using as an iteration matrix the preconditioning matrix already defined on that level it is shown that by a proper choice of iteration parameters it suffices to use v left 1 gamma 2 right tfrac 1 2 iterations for the so constructedv foldv cycle wherev 2 corresponds to aw cycle preconditioning matrices to be spectrally equivalent to the stiffness matrix the conditions involve only the constant in the strengthened c b s inequality for the corresponding two level hierarchical basis function spaces and are therefore independent of the regularity of the solution for instance if we use successive uniform refinements of the meshes the method is of optimal order of computational complexity if gamma 2 tfrac 8 9 under reasonable assumptions of the finite element mesh the condition numbers turn out to be so small that there are in practice few reasons to use an accelerated iterative method like the conjugate gradient method for instance