generalized drazin invertibility of combinations of idempotents abstract the paper serves as a correction to j math anal appl 359 2009 731 738 dealing with the drazin invertibility of combinations of idempotents p q in a banach algebra as the arguments employed to do this work equally well for both the usual and generalized drazin inverse the latter is included in the discussion which covers the equivalence of the drazin invertibility of 1 pq p q p q and in general of p q with 0 as well as the equivalence of the drazin invertibility of the commutator pq qp and anticommutator pq qp of p q

optimal decay rates for solutions of a nonlinear wave equation with localized nonlinear dissipation of unrestricted growth and critical exponent source terms under mixed boundary conditions abstract this article investigates optimal decay rates for solutions to a semilinear hyperbolic equation with localized interior damping and a source term both dissipation and the source are fully nonlinear and the growth rate of the source map may include critical exponents for sobolev s embedding h 1 l 2 besides continuity and monotonicity no growth or regularity assumptions are imposed on the damping we analyze the system in the presence of neumann type boundary conditions including the mixed cases dirichlet neumann robin the damping affects a thin layer a collar near a portion of the boundary to cope with the lack of control on the remaining section we develop a special method that accounts for propagation of the energy estimates from the dissipative region onto the entire domain the neumann system does not satisfy the lopatinski condition in higher dimensions hence the study of energy propagation in the absence of damping near the neumann segment requires special geometric considerations

on the wave equation with hyperbolic dynamical boundary conditions interior and boundary damping and supercritical sources abstract the aim of this paper is to study the problem u t t u p x u t f x u in 0 u 0 on 0 0 u t t u u q x u t g x u on 0 1 u 0 x u 0 x u t 0 x u 1 x in where is a bounded open subset of r n with c 1 boundary n 2 1 is relatively open on denotes the laplace beltrami operator on is the outward normal to and the terms p and q represent nonlinear damping terms while f and g are nonlinear perturbations in the paper we establish local and global existence uniqueness and hadamard well posedness results when source terms can be supercritical or super supercritical

some results on the drazin invertibility and idempotents in this paper several equivalent conditions for the drazin invertibility of the sum difference and the product of idempotents in banach algebra are established

global attractors for the wave equation with nonlinear damping based on a new a priori estimate method so called asymptotic a priori estimate the existence of a global attractor is proved for the wave equation utt kg ut u f u 0 on a bounded domain r3 with dirichlet boundary conditions the nonlinear damping term g is supposed to satisfy the growth condition c1 s c2 g s c3 1 s p where 1 p 0 is arbitrary the nonlinear term f is supposed to satisfy the growth condition f s c4 1 s q where q 2 it is remarkable that when 2 p 5 we positively answer an open problem in chueshov and lasiecka i chueshov i lasiecka long time behavior of second evolution equations with nonlinear damping math scuola norm sup 2004 and improve the corresponding results in feireisl e feireisl global attractors for damped wave equations with supercritical exponent j differential equations 116 1995 431 447

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

global smooth solutions and stabilization of nonlinear elastodynamic systems with locally distributed dissipation we consider the existence of global solutions of the nonlinear elastodynamic system with a locally distributed damping in a bounded domain we assume that the energy function satisfies the strong ellipticity condition at the zero equilibrium the local dissipation is in the form a x u where the nonnegative function a x is only positive on a small portion of the domain we show the existence of global smooth solutions when initial data are small in particular we obtain the exponential decay of the energy which implies the exponential stabilization of the system by internal feedback

global existence and nonexistence for nonlinear wave equations with damping and source terms we consider an initial boundary value problem for a nonlinear wave equation in one space dimension the nonlinearity features the damping term u m 1 u t and a source term of the form u p 1 u with m p 1 we show that whenever m p then local weak solutions are global on the other hand we prove that whenever p m and the initial energy is negative then local weak solutions cannot be global regardless of the size of the initial data

existence of a solution of the wave equation with nonlinear damping and source terms abstract we study the nonlinear wave equation involving the nonlinear damping term u t u t m 1 and a source term of type u u p 1 for 1 p m we prove a global existence theorem with large initial data for 1 m p a blow up result is established for sufficiently large initial data

asymptotics of viscoelastic materials with nonlinear density and memory effects abstract this paper is concerned with the nonlinear viscoelastic equation t u t t u t t u u 0 s u t s d s f u h suitable to modeling extensional vibrations of thin rods with nonlinear material density t u t u and presence of memory effects this class of equations was studied by many authors but well posedness in the whole admissible range 0 4 and for f growing up to the critical exponent were established only recently the existence of global attractors was proved in presence of an additional viscous or frictional damping in the present work we show that the sole weak dissipation given by the memory term is enough to ensure existence and optimal regularity of the global attractor a for 4 and critical nonlinearity f

global strong solution of fourth order nonlinear wave equation abstract in this paper we consider a class of fourth order nonlinear wave equation in a bounded domain with smooth boundary and discuss the existence and uniqueness of the global strong solutions by gradually smoothing method

characterizations of the commutators and the anticommutator involving idempotents in this paper we characterize the idempotency the invertibility or generalized invertibility the positivity and the range relations of the commutator c pq qp and the anticommutator d pq qp involving idempotents particular attention is paid to the idempotent operators p and q and some further characteristics when p and q are positive are derived as well

sharp norm estimations for moore penrose inverses of stable perturbations of hilbert c module operators it is well known that norm upper bound vert bar t dag vert leq frac vert t dag vert 1 vert t dag vert cdot vert delta t vert plays a fundamental role in stable resp rank preserving perturbation analysis for moore penrose inverses of hilbert space operators resp matrices in this paper in the general setting of hilbert c module operators we provide a new approach to the study of norm upper bounds of bar t dag and obtain a sharp estimation for vert bar t dag vert this obtained estimation is applied to the study of the linear least squares problem and of perturbation analysis for the schur complement of a positive semidefinite operator matrix some new norm upper bounds for schur complements of positive semidefinite operator matrices are derived

systems of nonlinear wave equations with damping and supercritical boundary and interior sources utt u g1 ut f1 u v vtt v g2 vt f2 u v in a bounded domain r with robin and dirichlet boundary conditions on u and v respectively the nonlinearities f1 u v and f2 u v are with supercritical exponents representing strong sources while g1 ut and g2 vt act as damping in addition the boundary condition also contains a nonlinear source and a damping term by employing nonlinear semigroups and the theory of monotone operators we obtain several results on the existence of local and global weak solutions and uniqueness of weak solutions moreover we prove that such unique solutions depend continuously on the initial data

the existence of global attractors for the norm to weak continuous semigroup and application to the nonlinear reaction diffusion equations abstract in this paper first we introduce a new concept called the norm to weak continuous semigroup in a banach space and give a technical theorem to verify this notion of continuity then we establish a general method which is necessary and sufficient to obtain the existence of the global attractor for this kind of semigroup as an application we obtain the existence of the global attractor for a nonlinear reaction diffusion equation with a polynomial growth nonlinearity of arbitrary order and with some weak derivatives in the inhomogeneous term the global attractors are obtained in l p h 0 1 and h 2 h 0 1 respectively a new a priori estimate method called asymptotic a priori estimate has been introduced since the solutions of the equation has no higher regularity and the semigroup associated the solutions is not continuous in l p h 0 1 and h 2 h 0 1 the results in this part are new and appear to be optimal

global existence to the initial boundary value problem for a system of semilinear wave equations abstract we prove the global existence of finite energy and h 2 solutions for a system of nonlinear wave equations in a bounded domain one equation of the system has a dissipative term while another equation has no dissipative mechanism to derive the delicate a priori estimates we employ a loan method

mixed problem for some semi linear wave equation with a nonhomogeneous condition on considere le probleme aux valeurs limites et initiales suivant u tt u xx u t sign u t 0 x t 0 1 0 t u x 0 t g t u 1 t 0 u x 0 u 0 x u t x 0 u 1 x avec 0 1 avec u 0 h 1 0 1 u 1 dans l 2 0 1 et g t dans l 2 0 t un theoreme d existence globale et d unicite est demontre

regularity theory of hyperbolic equations with non homogeneous neumann boundary conditions ii general boundary data abstract this paper studies the regularity of solutions of general mixed second order time dependent hyperbolic problems of neumann type in a previous paper i lasiecka and r triggiani ann mat pura appl iv clvii 1990 285 367 using pseudo differential calculus we have provided sharp regularity results of the solutions and their traces when the non homogeneous data are in l 2 now we complement this study by providing a regularity when the non homogeneous data are more regular than as well as less regular than l 2 in contrast with our previous paper a functional analytic approach based on the l 2 results of our previous paper is used throughout

global existence and open loop exponential stabilization of weak solutions for nonlinear schrodinger equations with localized external neumann manipulation abstract in this paper we consider the nonlinear schrodinger equations nls with focusing defocusing interior source and possibly nonlinear damping on a bounded regular domain in the euclidean space moreover it is assumed that the solutions are subject to external neumann boundary manipulation on one portion of the boundary our aim is to obtain global existence of the weak solutions under various assumptions on the sign of the source and powers of the nonlinearities in the case of a linear damping we also prove that solutions decay exponentially under the assumption that the neumann type control at the boundary decays in a similar manner

asymptotic behavior for a semilinear second order evolution equation this paper is devoted to the qualitative analysis for a second order evolution equation u tt delta u delta u t epsilon delta u tt f u g x epsilon is an element of 0 1 with critical nonlinearity some uniformly w r t epsilon is an element of 0 1 asymptotic regularity about the solutions has been established for both g x is an element of l 2 omega and g x is an element of h 1 which shows that the solutions are exponentially approaching a more regular fixed subset uniformly w r t epsilon is an element of 0 1 as an application of this regularity result a family epsilon epsilon epsilon is an element of 0 1 of finite dimensional exponential attractors has been constructed moreover to characterize the relation with a strongly damped wave equation epsilon 0 the upper semicontinuity at epsilon 0 of the global attractors has been proved

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

a note on a strongly damped wave equation with fast growing nonlinearities a strongly damped wave equation including the displacement depending nonlinear damping term and nonlinear interaction function is considered the main aim of the note is to show that under the standard dissipativity restrictions on the nonlinearities involved the initial boundary value problem for the considered equation is globally well posed in the class of sufficiently regular solutions and the semigroup generated by the problem possesses a global attractor in the corresponding phase space these results are obtained for the nonlinearities of an arbitrary polynomial growth and without the assumption that the considered problem has a global lyapunov function

global well posedness of solutions for fourth order dispersive wave equation with nonlinear weak damping linear strong damping and logarithmic nonlinearity abstract to understand the characteristics of dynamical behavior especially the kinetic evolution for logarithmic nonlinearity we aim to study the global well posedness of nonlinear fourth order wave equations with logarithmic source term where the dispersive the nonlinear weak damping and linear strong damping are taken into account based on the potential well method the main ingredient of this paper is to construct several conditions for initial data leading to the solution global existence or infinite time blow up with subcritical initial energy moreover we extend the results to the critical initial energy which is realized by introducing scaling technique finally surrounding the blow up at arbitrarily high initial energy we compare and discuss the research program stemming from two different strategies in the first one it is assumed that the blow up result is bound to the original logarithmic source by weakening the dispersive dissipative structure while the second one is based on the nonlinear wave equation with complete dispersive dissipative structure but the logarithmic source is replaced by an enhanced version through these two strategies we explain the mechanism of these two structures respectively and establish two kinds of sufficient conditions for initial data leading to high energy blow up

a gentle guide to the basics of two projections theory abstract this paper is a survey of the basics of the theory of two projections it contains in particular the theorem by halmos on two orthogonal projections and roch silbermann gohberg and krupnik s theorem on two idempotents in banach algebras these two theorems which deliver the desired results usually very quickly and comfortably are missing or wrongly cited in many recent publications on the topic the paper is intended as a gentle guide to the field the basic theorems are precisely stated some of them are accompanied by full proofs others not but precise references are given in each case and many examples illustrate how to work with the theorems

local well posedness asymptotic behavior and asymptotic bootstrapping for a class of semilinear evolution equations of the second order in time a class of semilinear evolution equations of the second order in time of the form u tt au au t au tt f u is considered where a is the dirichlet laplacian is a smooth bounded domain in r n and f c 1 r r a local well posedness result is proved in the banach spaces w 1 p 0 x w 1 p 0 when f satisfies appropriate critical growth conditions in the hilbert setting if f satisfies an additional dissipativeness condition the nonlinear semigroup of global solutions is shown to possess a gradient like attractor existence and regularity of the global attractor are also investigated following the unified semigroup approach bootstrapping and the interpolation extrapolation techniques

weak solutions and blow up for wave equations of p laplacian type with supercritical sources this paper investigates a quasilinear wave equation with kelvin voigt damping utt pu ut f u in a bounded domain 3 and subject to dirichlet boundary conditions the operator p 2 p 3 denotes the classical p laplacian the nonlinear term f u is a source feedback that is allowed to have a supercritical exponent in the sense that the associated nemytskii operator is not locally lipschitz from w01 p into l2 under suitable assumptions on the parameters we prove existence of local weak solutions which can be extended globally provided the damping term dominates the source in an appropriate sense moreover a blow up result is proved for solutions with negative initial total energy

range inclusion and operator equations abstract letaanddbe positive operators on a complex hilbert spaceh in this work we show that the operator equationldx x a a l 0 has a unique solutionclsatisfying certain conditions if there exists a positive operatorbsuch thatab d b a 0 this implies several previously known results in special cases on this class of operator equations also along with some definitions a counterexample to a previous conjecture will be given

nonsingularity of linear combinationsof idempotent matrices abstract gros and trenkler siam j matrix anal appl 21 1999 390 pointed out that if a difference of idempotent matrices p 1 and p 2 is nonsingular then so is their sum and koliha et al linear algebra appl in press expressed explicitly a condition which combined with the nonsingularity of p 1 p 2 ensures the nonsingularity of p 1 p 2 in the present note these results are strengthened by showing that the nonsingularity of p 1 p 2 is in fact equivalent to the nonsingularity of any linear combination c 1 p 1 c 2 p 2 wherein c1 c2 0 some other nonsingularity type relationships referring to linear combinations of p 1 and p 2 are also established

generalized browder s and weyl s theorems for banach space operators abstract we find necessary and sufficient conditions for a banach space operator t to satisfy the generalized browder s theorem we also prove that the spectral mapping theorem holds for the drazin spectrum and for analytic functions on an open neighborhood of t as applications we show that if t is algebraically m hyponormal or if t is algebraically paranormal then the generalized weyl s theorem holds for f t where f h t the space of functions analytic on an open neighborhood of t we also show that if t is reduced by each of its eigenspaces then the generalized browder s theorem holds for f t for each f h t

global existence and blow up of solutions to systems of nonlinear wave equations with degenerate damping and source terms abstract we focus on the global well posedness of the system of nonlinear wave equations u t t u d u k e v l u t m 1 u t f 1 u v v t t v d v e u v t r 1 v t f 2 u v in a bounded domain r n n 1 2 3 with dirichlet boundary conditions the nonlinearities f 1 u v and f 2 u v act as a strong source in the system under some restriction on the parameters in the system we obtain several results on the existence of local solutions global solutions and uniqueness in addition we prove that weak solutions to the system blow up in finite time whenever the initial energy is negative and the exponent of the source term is more dominant than the exponents of both damping terms

superclose and superconvergence of finite element discretizations for the stokes equations with damping a class of conforming mixed finite element methods fems are applied to the stokes equations with damping with the help of the prior estimates for the exact and approximated solutions and the appropriate choice of the parameters a n and r appeared in the problem the superclose and superconvergence results for the velocity in h 1 norm and the pressure in l 2 norm are derived to our best knowledge this is the first superconvergence analysis of conforming mixed fems for this kind nonlinear problem

the lifespan for 3d quasilinear wave equations with nonlinear damping terms abstract in this paper we study the initial boundary value problem for a system of nonlinear wave equations involving nonlinear damping terms in a bounded domain the nonexistence of global solutions is discussed under some conditions on the given parameters estimates on the lifespan of solutions are also given our results extend and generalize the recent results in k agre m a rammaha system of nonlinear wave equations with damping and source terms differential integral equations 19 2006 1235 1270 especially the blow up of weak solutions in the case of non negative energy

controllability for one dimensional nonlinear wave equations with degenerate damping abstract internal exact controllability for the nonlinear wave equation in one space dimension y t t y x x g y y t h is studied where g is a nonnegative function for the case lim sup s g s ln s we obtain the global exact controllability for the equation with dirichlet boundary condition the proof is based on the combination of fixed point arguments and explicit observability estimates for the linearized wave equation with a potential that depends on both x and t for the case g s s we only get a local exact controllability by means of banach fixed point theorem

the moore penrose inverses of m n block matrices and their applications abstract this article gives the expressions for the moore penrose inverses of m n block matrices when they satisfy rank additivity conditions and presents some of their special cases and applications

global existence and global nonexistence of solutions of the cauchy problem for a nonlinearly damped wave equation abstract in this paper we consider the long time behavior of solutions of the initial value problem for semilinear wave equations of the form formula

blow up for nonlinear dissipative wave equations in rn abstract we consider the cauchy problem u t t u u t m 1 u t u p 1 u t x 0 r n u 0 x u 0 x u t 0 x v 0 x for 1 m p p n n 2 for n 3 we prove that for any given numbers 0 0 there exist infinitely many data u 0 v 0 in the energy space such that the initial energy e 0 the gradient norm u 0 2 and the solution of the above cauchy problem blows up in finite time

global nonexistence for nonlinear kirchhoff systems in this paper we consider the problem of non continuation of solutions of dissipative nonlinear kirchhoff systems involving the p x laplacian operator and governed by nonlinear driving forces f f t x u as well as nonlinear external damping terms q q t x u u t both of which could significantly dependent on the time t the theorems are obtained through the study of the natural energy eu associated to the solutions u of the systems thanks to a new approach of the classical potential well and concavity methods we show the nonexistence of global solutions when the initial energy is controlled above by a critical value that is when the initial data belong to a specific region in the phase plane several consequences interesting in applications are given in particular subcases the results are original also for the scalar standard wave equation when p 2 and even for problems linearly damped

on nonlinear wave equations with degenerate damping and source terms in this article we focus on the global well posedness of the differential equation u tt u u k j u t u p 1 u in x 0 t where j is a sub differential of a continuous convex function j under some conditions on j and the parameters in the equations we obtain several results on the existence of global solutions uniqueness nonexistence and propagation of regularity under nominal assumptions on the parameters we establish the existence of global generalized solutions with further restrictions on the parameters we prove the existence and uniqueness of a global weak solution in addition we obtain a result on the nonexistence of global weak solutions to the equation whenever the exponent p is greater than the critical value k m and the initial energy is negative we also address the issue of propagation of regularity specifically under some restriction on the parameters we prove that solutions that correspond to any regular initial data such that u 0 h 2 h 1 0 u 1 h 1 0 are indeed strong solutions

energy decay rates for the semilinear wave equation with nonlinear localized damping and source terms abstract in this paper we develop an intrinsic approach to derivation of energy decay rates for the semilinear wave equation with localized interior nonlinear monotone damping g u t and a source term f u the proposed approach allows to consider in an unified way much more general classes of hyperbolic problems than addressed before in the literature these generalizations refer to both geometric and topological aspects of the problem the method leads to optimal decay rates for solutions of semilinear hyperbolic equations driven by a source of critical exponent and subjected to a nonlinear damping localized in a small region adjacent to a portion of the boundary the distinct features of the model include i neumann boundary conditions are assumed and ii no growth conditions are imposed on the damping g s it is well known that neumann boundary does not satisfy lopatinski condition and therefore the analysis of propagation of energy in the absence of the damping on the neumann part of the boundary requires special geometric considerations in addition the sole conditions assumed on g s are monotonicity continuity and g 0 0 in particular no differentiability and no growth conditions are imposed on the damping both at the origin and at the infinity the asymptotic decay rates for the energy function are obtained from an intrinsic algorithm driven by solutions of simple ode several examples illustrate the theory by exhibiting various decay rates exponential algebraic rational logarithmic etc for the energy functional an important corollary of our energy decay theorem is a stability result which shows that under certain conditions when dissipation is sublinear at infinity the solution of the system remains uniformly bounded for all time in the norms above the finite energy level even in the presence of a nonlinear source term

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

two grid mfeas for the incompressible stokes type variational inequality with damping abstract in this paper we study two grid mixed finite element algorithms mfeas for the stationary incompressible stokes equations with damping and nonlinear slip boundary conditions we set up the weak formulation and obtain the well posedness then employing the mixed finite element method some optimal error estimates between the exact solution and the approximation solution are derived two kinds of two grid mfeas based on the stokes correction and newton type correction are established some error estimates for two grid mfeas are obtained several numerical results are presented to confirm our theoretical findings

on the initial and terminal value problem for a class of semilinear strongly material damped plate equations abstract we study the semilinear strongly damped plate equation by considering its two different problems for initial value problem we prove the local well posedness and blow up results of solution for the problem with polynomial nonlinear source terms for terminal value problem given the ill posedness in the sense of hadamard we propose a regularization method for the problem with logarithmic nonlinearities under the a priori assumption on the exact solution belonging to a gevrey space we propose the fourier truncation method to stabilize the ill posed problem also by establishing some stability estimates of logarithmic type in l q space

global solutions and finite time blow up for fourth order nonlinear damped wave equation in this paper we study the initial boundary value problem and global well posedness for a class of fourth order wave equations with a nonlinear damping term and a nonlinear source term which was introduced to describe the dynamics of a suspension bridge the global existence decay estimate and blow up of solution at both subcritical e 0 0 in this paper we study the initial boundary value problem and global well posedness for a class of fourth order wave equations with a nonlinear damping term and a nonlinear source term which was introduced to describe the dynamics of a suspension bridge the global existence decay estimate and blow up of solution at both subcritical e 0 0

estimating tails of probability distributions we study the asymptotic properties of estimators of the tail of a distribution based on the excesses over a threshold a key idea is the use of pickands generalised pareto distribution and its fitting in most cases by the method of maximum likelihood the results cover all three limiting types of extreme value theory we propose a new estimator for an index of regular variation and show that it often performs better than hill s estimator we give new results for estimating the endpoint of a distribution extending earlier work by hall and by smith and weissman finally we give detailed results for the domain of attraction of exp e x and show that in most cases our proposed estimator is more efficient than two others one based on the exponential distribution and the other due to davis and resnick we also touch briefly on the problem of large deviations from a statistical point of view the results make extensive use of existing work on rates of convergence

the existence of global attractors for a class nonlinear evolution equation abstract in this paper using a new method or framework we establish the existence of global attractors for a class nonlinear evolution equation in h 0 1 h 0 1 where the nonlinear term f satisfies a critical exponential growth condition

a note on generalized pareto distributions and thek upper extremes denote byd n k the maximum distance between the distribution function of thekth largest order statistic in aniid sample of sizen equally standardized ink and of its corresponding limiting extreme value distribution

global nonexistence theorems for a class of evolution equations with dissipation we study abstract evolution equations with nonlinear damping terms and source terms including as a particular case a nonlinear wave equation of the type ba cl u tt delta u b u t m 2 u t c u p 2 u t x in 0 t times omega 6pt u t x 0 t x in 0 t times partial omega 6pt u 0 cdot u 0 in h 0 1 omega quad u t 0 cdot v 0 in l 2 omega es ea where 0 p 2 m 1 we prove a global nonexistence theorem for positive initial value of the energy when 1 1

the difference and sum of projectors abstract the aim of this paper is to present new results on the invertibility of the sum of projectors and new relations between the nonsingularity of the difference and the sum of projectors we also give simple proofs without referring to rank theory of some results on the nonsingularity of the difference of projectors obtained by gros and trenkler siam j matrix anal appl 21 1999 390

blow up for semilinear wave equation with boundary damping and source terms abstract in this paper we consider the semilinear wave equation with boundary damping and source terms this work is devoted to prove a finite time blow up result under suitable condition on the initial data and positive initial energy

a mixed fem for navier stokes type variational inequality with nonlinear damping term abstract in this work we consider a mixed finite element method fem for incompressible navier stokes equations with nonlinear damping term and friction boundary conditions after establishing the variational formulation we show the well posedness of this problem subsequently we focus our attention on the mixed fem and analyze the galerkin approximation to this system then some optimal error estimates are deduced in the end some iterative algorithms are presented and numerical results are given to verify the theoretical analysis

global existence and blow up of solutions for a system of nonlinear viscoelastic wave equations with damping and source abstract in this paper we investigate the global existence and finite time blow up of solutions to the system of nonlinear viscoelastic wave equations u t t u 0 t g 1 t u d u t m 1 u t f 1 u v v t t v 0 t g 2 t v d v t r 1 v t f 2 u v in 0 t with initial and dirichlet boundary conditions where is a bounded domain in r n n 1 2 3 under suitable assumptions on the functions g i f i i 1 2 the initial data and the parameters in the equations we establish several results concerning local existence global existence uniqueness and finite time blow up property