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

yetter drinfeld modules and projections of weak hopf algebras abstract in this paper we prove that if g b h is a morphism of weak hopf algebras which is split as an algebra coalgebra morphism then the subalgebra of coinvariants b h of b is a hopf algebra in the category of yetter drinfeld modules associated to h

cleft extensions of weak hopf algebras abstract in this paper we study the theory of cleft extensions for a weak bialgebra h among other results we determine when two unitary crossed products of an algebra a by h are equivalent and we prove that if h is a weak hopf algebra then the categories of h cleft extensions of an algebra a and of unitary crossed products of a by h are equivalent

weak c cleft extensions weak entwining structures and weak hopf algebras abstract we formulate the concept of weak cleft extension for a weak entwining structure in a braided monoidal category c with equalizers and coequalizers we prove that if a is a weak c cleft extension then there is an isomorphism of algebras between a and a subobject of the tensor product of a c and c where a c is a subalgebra of a also we prove the corresponding dual results and linking the information of this two parts we obtain a general property for a pair morphisms f c a and g a c of algebras and coalgebras satisfying certain conditions finally as particular instances we get the results of fernandez and rodriguez the theorems of radford majid and bespalov in the case of hopf algebras with projection and the ones obtained by alonso and gonzalez for weak hopf algebras living in a symmetric category with split idempotents for example the weak theorem of blattner cohen and montgomery for weak hopf algebras with coalgebra splitting is one of them

new trends towards lower energy estimates and optimality for nonlinearly damped vibrating systems abstract we present an approach based on comparison principles for energy and interpolation properties to derive lower energy estimates for nonlinearly either locally damped or boundary damped vibrating systems we show how the dissipation relation provides strong information on the asymptotic behavior of the energy of solutions the geometrical situations are either one dimensional or radial two dimensional or three dimensional for annulus domains we also consider the case of general domains but in this case for solutions with bounded velocities in time and space in all these cases the nonlinear damping function is assumed to have arbitrary strictly sublinear growth at the origin we give results for strong solutions and stronger lower estimates for smoother solutions the results are presented in two forms either on the side of energy comparison principles or through time pointwise lower estimates under additional geometric assumptions we give the resulting lower and upper estimates for four representative examples of damping functions we further give a weak lower estimate in the sense of a certain lim sup t and an upper estimate of the velocity for smoother solutions in case of general damping functions and for radial as well as multi dimensional domains we also discuss these estimates in the framework of optimality which is not proved here and indicate open problems raised by these results

optimality of energy estimates for the wave equation with nonlinear boundary velocity feedbacks we consider the wave equation damped by a nonlinear boundary velocity feedback q ut first we consider the case where q has a linear growth at infinity we prove that the usual decay rate estimates proved by nakao differential integral equations 8 1995 pp 681 688 haraux and zuazua arch rational mech anal 100 1988 pp 191 206 conrad leblond and marmorat in proceedings of the fifth international federation of automatic control symposium on control of distributed parameter systems perpignan 1989 pp 101 116 zuazua siam j control optim 28 1990 pp 466 477 and komornik in control and estimation of distributed parameter systems nonlinear phenomena internat ser numer math 118 birkhauser basel 1994 pp 253 266 when q has a polynomial behavior at zero and by the second author esaim control optim calc var 4 1999 pp 419 444 in the general case are in fact optimal in one space dimension more generally we prove that the energy decays exactly like the solution of an explicit and simple ordinary differential equation next we study the problem when q is bounded at infinity we prove that strong solutions decay exponentially to zero and we exhibit a sequence of weak solutions for which the associated energy decays to zero at infinity as slowly as the iterated logarithms go to infinity at infinity

stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks in this paper we consider in a bounded and smooth domain the wave equation with a delay term in the boundary condition we also consider the wave equation with a delayed velocity term and mixed dirichlet neumann boundary condition in both cases under suitable assumptions we prove exponential stability of the solution these results are obtained by introducing suitable energies and by using some observability inequalities if one of the above assumptions is not satisfied some instability results are also given by constructing some sequences of delays for which the energy of some solutions does not tend to zero

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

radford s biproducts and yetter drinfeld modules for monoidal hom hopf algebras let h be a monoidal hom bialgebra and b be a left h hom module algebra and also a left h hom comodule coalgebra then in this paper we first introduce the notion of a hom smash coproduct which is a monoidal hom coalgebra second we find sufficient and necessary conditions for the hom smash product algebra structure and the hom smash coproduct coalgebra structure on b h to afford b h a monoidal hom bialgebra structure generalizing the well known radford s biproduct where the conditions are equivalent to that b is a bialgebra in the category of hom yetter drinfeld modules hydhh finally we illustrate the category of hom yetter drinfeld modules hydhh and prove that the category hydhh is a braided monoidal category

a unified approach via convexity for optimal energy decay rates of finite and infinite dimensional vibrating damped systems with applications to semi discretized vibrating damped systems the liapunov method is celebrated for its strength to establish strong decay of solutions of damped equations extensions to infinite dimensional settings have been studied by several authors see e g haraux 1991 11 and komornik and zuazua 1990 17 and references therein results on optimal energy decay rates under general conditions of the feedback is far from being complete the purpose of this paper is to show that general dissipative vibrating systems have structural properties due to dissipation we present a general approach based on convexity arguments to establish sharp optimal or quasi optimal upper energy decay rates for these systems and on comparison principles based on the dissipation property and interpolation inequalities in the infinite dimensional case for lower bounds of the energy we stress the fact that this method works for finite as well as infinite dimensional vibrating systems and as well as for applications to semi discretized nonlinear damped vibrating pde s a part of this approach has been introduced in alabau boussouira 2004 2005 1 2 in the present paper we identify a new simple and explicit criteria to select a class of nonlinear feedbacks for which we prove a simplified explicit energy decay formula comparatively to the more general but also more complex formula we give in alabau boussouira 2004 2005 1 2 moreover we prove optimality of the decay rates for this class in the finite dimensional case this class includes a wide range of feedbacks ranging from very weak nonlinear dissipation exponentially decaying in a neighborhood of zero to polynomial or polynomial logarithmic decaying feedbacks at the origin in the infinite dimensional case we establish a comparison principle on the energy of sufficiently smooth solutions through the dissipation relation this principle relies on suitable interpolation inequalities it allows us to give lower bounds for the energy of smooth initial data for the one dimensional wave equation with a distributed polynomial damping which improves haraux 1995 12 lower estimate of the energy for this case we also establish lower bounds in the multi dimensional case for sufficiently smooth solutions when such solutions exist we further mention applications of these various results to several classes of pde s namely the locally and boundary damped multi dimensional wave equation the locally damped plate equation and the globally damped coupled timoshenko beams system but it applies to several other examples furthermore we show that these optimal energy decay results apply to finite dimensional systems obtained from spatial discretization of infinite dimensional damped systems we illustrate these results on the one dimensional locally damped wave and plate equations discretized by finite differences and give the optimal energy decay rates for these two examples these optimal rates are not uniform with respect to the discretization parameter we also discuss and explain why optimality results have to be stated differently for feedbacks close to linear behavior at the origin

weak hopf bialgebras in this chapter the r bialgebroids and hopf algebroids of the previous chapter are investigated further in the particular case when the base algebra r carries a so called separable frobenius structure separable frobenius structures on some algebra r are shown to correspond to separable frobenius structures on the forgetful functor from the category of r bimodules to the category of vector spaces based on that the bijection of chap 5 is refined to bijections between three structures for any algebra a first monoidal structures on the category of a modules together with separable frobenius structures on the forgetful functor to the category of vector spaces second bialgebroid structures on a over some base algebra r together with separable frobenius structures on r finally weak bialgebra structures on a the r bialgebroid a is a hopf algebroid if and only if the corresponding weak bialgebra a is a weak hopf algebra

local points of twisted mumford quotients and shimura curves we determine over which fields twisted mumford quotients have rational points using the p adic uniformization we apply these results to shimura curves and show some new cases for which the jacobians are even in the sense of ps

general decay rate estimates and numerical analysis for a transmission problem with locally distributed nonlinear damping in this paper we obtain very general decay rate estimates associated to a wavewave transmission problem subject to a nonlinear damping locally distributed employing arguments firstly introduced in lasiecka and tataru 1993 and we shall present explicit decay rate estimates as considered in alabau boussouira 2005 and cavalcanti etal 2007 in addition we implement a precise and efficient code to study the behavior of the transmission problem when k1k2 and when one has a nonlinear frictional dissipation g ut more precisely we aim to numerically check the general decay rate estimates of the energy associated to the problem established in first part of the paper

weak hopf algebras and some new solutions of the quantum yang baxter equation abstract in this paper the concepts of a weak hopf algebra and a quasi braided almost bialgebra are introduced it is shown that the quantum quasi doubles of some weak hopf algebras are quasi braided almost bialgebras this fact implies that some new solutions of the quantum yang baxter equation can be constructed from some weak hopf algebras in particular when the weak hopf algebra is a finite clifford monoid algebra

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

nonexistence of global solutions to semilinear wave equations in high dimensions on considere le probleme de cauchy pour l equation d onde semi lineaire 2 t 2 u x t u x t p p 1 u x o f x u t x o g x avec f g c o r n

weak hopf algebras i integral theory and c structure we give an introduction to the theory of weak hopf algebras proposed as a coassociative alternative of weak quasi hopf algebras we follow an axiomatic approach keeping as close as possible to the classical theory of hopf algebras the emphasis is put on the new structure related to the presence of canonical subalgebras al and ar in any weak hopf algebra a that play the role of non commutative numbers in many respects a theory of integrals is developed in which we show how the algebraic properties of a such as the frobenius property or semisimplicity or innerness of the square of the antipode are related to the existence of non degenerate normalized or haar integrals in case of c weak hopf algebras we prove the existence of a unique haar measure h a and of a canonical grouplike element g a implementing the square of the antipode and factorizing into left and right elements g glg 1r gl al gr ar further discussion of the c case will be presented in part ii

weak hopf algebras with projection and weak smash bialgebra structures abstract in this paper we study weak hopf algebras with projection if f h b g b h are morphisms of weak hopf algebras such that g f idh we prove that it is possible to find an object bh in the new category of weak yetter drinfeld modules that verifies similar conditions to the ones include in the definition of weak hopf algebra finally we define weak smash bialgebra structures and prove that under central and cocentral conditions bh and h determine an example of them

smooth p adic analytic spaces are locally contractible 0 introduction 293 1 piecewise rs linear spaces 298 2 r colored polysimplicial sets 308 3 r colored polysimplicial sets of length l 313 4 the skeleton of a nondegenerate pluri stable formal scheme 327 5 a colored polysimplicial set associated with a nondegenerate poly stable fibration 336 6 p adic analytic and piecewise linear spaces 346 7 strong local contractibility of smooth analytic spaces 355 8 cohomology with coefficients in the sheaf of constant functions 362

stabilization of a transmission wave plate equation abstract we consider a stabilization problem for a model arising in the control of noise coupling the damped wave equation with a damped kirchhoff plate equation we prove an exponential stability result under some geometric condition our method is based on an identity with multipliers that allows to show an appropriate energy estimate

weak c cleft extensions and weak galois extensions abstract for a weak entwining structure a c we formulate the notion of weak c galois extension with normal basis and we show that these galois extensions are equivalent to the weak c cleft extensions introduced in j n alonso alvarez j m fernandez vilaboa r gonzalez rodriguez a b rodriguez raposo weak c cleft extensions weak entwining structures and weak hopf algebras j algebra 284 2005 679 704

weak hopf algebras and weak yang baxter operators abstract in this paper we introduce the notions of weak yang baxter operator and weak braided hopf algebra we prove that it is possible to obtain examples of these notions working with yetter drinfeld modules associated to a weak hopf algebra h with invertible antipode finally we complete the study of the structure of weak hopf algebras with a projection obtaining a categorical equivalence between the category of weak hopf algebra projections associated to h and the category of hopf algebras in the non strict braided monoidal category of left left yetter drinfeld modules over h

a note on the boundary stabilization of the wave equation we study the energy decay rates of the wave equation in a domain where boundary damping is present we generalize the geometrical conditions obtained earlier in j math purer appl 58 1979 pp 249 273 by using some more general multipliers of strauss comm pure appl math 28 1975 pp 265 278 the interaction between distributed damping and boundary damping is discussed a regulator problem is also formally discussed by the synthesis method

regularity of higher energies of wave equation with nonlinear localized damping and a nonlinear source abstract wave equation driven by a nonlinear dissipative source and subjected to a nonlinear damping that is localized on a small region near the boundary is considered while finite energy h 1 l 2 solutions to this problem are bounded uniformly for all times t 0 this property generally fails for higher energy norms h 2 h 1 this is due to a generation of higher energy by the source and the ultimate loss of dissipativity however the presence of the damping may turn things around by counteracting the sources also at the higher energy levels the benefits of this counteraction depend on the nonlinear characteristics of dissipation any deviation from linearity be it origin or infinity causes degradation of the damping hence of decay rates of finite energy this in turn is shown to have an adverse effect on the stability of higher energies the main aim of this paper is to provide a quantitative analysis of the interaction between nonlinearity of the damping and nonlinearity of the source we show that under some correlation between growth rates of the damping and the source the norms of topological order above the finite energy level remain globally bounded for all times

well posedness and optimal decay rates for the wave equation with nonlinear boundary damping source interaction we establish subject to some natural additional assumptions imposed on the relation between the source and the damping both well posedness and effective optimal decay rates for the solutions of a semilinear model of the wave equation the theory presented allows to consider both superlinear and sublinear behaviours of the dissipation in the presence of unstructured sources

cross products by braided groups and bosonization let h be a braided cocommutative hopf algebra in a braided monoidal category c and b a hopf algebra in c on which h acts we construct a cross product hopf algebra b formula h in c as an application we show that every b in a certain class can be converted to an equivalent ordinary hopf algebra by a process of bosonization the class includes for example all super hopf algebras the constructions respect any quasitriangular structures on b as a corollary we show that if h r is an ordinary quasitriangular hopf algebra then the smash product by the adjoint action of h on itself had formula h can be given the structure of a hopf algebra we prove a similar result for any hopf algebra to which a quasitriangular hopf algebra maps

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

strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation the aim of this article is twofold first we consider the wave equation in the hyperbolic space hi and obtain the counterparts of the strichartz type estimates in this context next we examine the relationship between semilinear hyperbolic equations in the minkowski space and in the hyperbolic space this leads to a simple proof of the recent result of georgiev lindblad and sogge on global existence for solutions to semilinear hyperbolic problems with small data shifting the space time strichartz estimates from the hyperbolic space to the minkowski space yields weighted strichartz estimates in rn x 1r which extend the ones of georgiev lindblad and sogge

completed cohomology of shimura curves and a p adic jacquet langlands correspondence we study indefinite quaternion algebras over totally real fields f and give an example of a cohomological construction of p adic jacquet langlands functoriality using completed cohomology we also study the tame levels of p adic automorphic forms on these quaternion algebras and give an analogue of mazur s level lowering principle

the structure of hopf algebras with a projection the smash product algebra and the smash coproduct coalgebra are well known in the context of hopf algebras 1 5 and these notions can be viewed as being motivated by the semidirect product construction in the theory of groups and in the theory of afiine group schemes respectively let

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

crossed products for weak hopf algebras with coalgebra splitting in this paper we prove that if g b h is a morphism of weak hopf algebras which is split as a coalgebra morphism then there exists a subalgebra bh of b morphisms bh h bh bh bh h h bh and an isomorphism of algebras and right h comodules bb b bh h being bh h a subobject of bh h with its algebra structure defined by a crossed product also we obtain the dual results and as a consequence we prove the radford s theorem for weak hopf algebras

uniform decay rates for the wave equation with nonlinear damping locally distributed in unbounded domains with finite measure this paper is concerned with the study of the uniform decay rates of the energy associated with the wave equation with nonlinear damping locally distributed u tt delta u a x g u t 0 hbox in omega times 0 infty subject to dirichlet boundary conditions where omega subset mathbb r n n geq 2 is an unbounded open set with finite measure and unbounded smooth boundary partial omega gamma the function a x responsible for the localized effect of the dissipative mechanism is assumed to be nonnegative and essentially bounded and in addition a x geq a 0 0 hbox a e in omega where omega omega cup x in omega x r r 0 and omega is a neighborhood in omega of the closure of partial omega cap b r where b r x in omega x r

weak hopf algebras ii representation theory dimensions and the markov trace abstract if a is a weak c hopf algebra then the category of finite dimensional unitary representations of a is a monoidal c category with its monoidal unit being the gns representation d e associated to the counit e this category has isomorphic left dual and right dual objects which leads as usual to the notion of a dimension function however if e is not pure the dimension function is matrix valued with rows and columns labeled by the irreducibles contained in d e this happens precisely when the inclusions a l a and a r a are not connected still there exists a trace on a which is the markov trace for both inclusions we derive two numerical invariants for each c wha of trivial hypercenter these are the common indices i and of the haar respectively markov conditional expectations of either one of the inclusions a l r a or a l r a in generic cases i in the special case of weak kac algebras we reproduce d nikshych s result 2000 j operator theory to appear by showing that i and is always an integer