a mackey imprimitivity theory for algebraic groups let g be an affine algebraic group over an algebraically closed field k and let h be a closed subgroup of g if v is a rational h module a comodule for the coordinate ring of h there is a now well known notion of an induced module vi g for g defined as the space morph g v of all h equivariant morphisms from g to a finite dimensional subspace of v with obvious g action the question arises given a rational g module m how can one recognize m as an induced module vig for a finite group g the answer is the mackey imprimitivity theorem the module m is induced if and only if it is a direct sum of subspaces permuted transitively by g with h the stabilizer of one of these subspaces called v one uses this result for example in proving the famous mackey decomposition theorem which describes the restriction of any induced module to a second subgroup l as a direct sum of suitable induced modules given the imprimitivity theorem the proof is just a matter of grouping the summands permuted by g into their orbits under l in the case of algebraic groups the situation is quite different for some subgroups h all g modules are induced this occurs for example if g is connected and k g h k e g if h is parabolic also if g is a connected unipotent group then a rational g module m is induced from some proper subgroup if and only if its endomorphism ring contains a two dimensional submodule e for the conjugation action of g with e k 1 and such that k l is precisely the subspace annihilated by the action of the lie algebra of g on e 27 cf also 5 5 below the latter is an application of a general criterion in case g h is affine a rational g module m is induced if and only if there is an action of the coordinate ring a of g h compatible with the action of g on both a and m note that this generalizes the imprimitivity theorem in the case of finite groups since then a has a k basis of g h orthogonal idempotents permuted transitively by g

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

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

asymptotic behavior of a nonlinear kirchhoff type equation with spring boundary conditions in this paper we consider the initial boundary value problem for a nonlinear kirchhoff type equation with the damping term and spring boundary conditions we establish the global existence and uniqueness of solutions to this problem in time and give an example and simulation to illustrate our results for the proof we use the faedo galerkin approximate method finally we study the asymptotic behavior of solutions and some of its simulation results results of this paper are able to apply industrial parts such as a typical model widely used to represent threads wires magnetic tapes belts band saws and so on

on the measure of the set of factorizable polynomial bispectra in a recent work 1989 the authors have shown that factorization of bispectrum is not always possible in the present work they show that the subset of factorizable bispectra has lebesgue measure zero in the set of polynomial bispectra i e those that are obtained from finite support bicumulants hence a polynomial bispectrum cannot almost always be exactly realized as that of the output of a linear model driven by a third order white input this result can be generalized to multidimensional polynomial bispectra although it follows that a linear model driven by a third order white input cannot almost always realize a given bispectrum or a bicumulant sequence the use of a linear model as an approximation in certain applications can be justified if the computed bispectrum has an index value close to unity

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

local and global wellposedness of weak solutions for the wave equation with nonlinear boundary and interior sources of supercritical exponents and damping abstract we consider the wave equation with interior and boundary nonlinear sources and damping and we are interested in local and global wellposedness of finite energy solutions the main difficulty is represented by the fact that the lopatinski condition fails to hold unless the dim 1 and thus the analysis of the boundary nonlinearities becomes a subtle issue

on the wave equation with semilinear porous acoustic boundary conditions the goal of this work is to study a model of the wave equation with semilinear porous acoustic boundary conditions with nonlinear boundary interior sources and a nonlinear boundary interior damping first applying the nonlinear semigroup theory we show the existence and uniqueness of local in time solutions the main difficulty in proving the local existence result is that the neumann boundary conditions experience loss of regularity due to boundary sources using an approximation method involving truncated sources and adapting the ideas in lasiecka and tataru 1993 28 we show that the existence of solutions can still be obtained second we prove that under some restrictions on the source terms then the local solution can be extended to be global in time in addition it has been shown that the decay rates of the solution are given implicitly as solutions to a first order ode and depends on the behavior of the damping terms in several situations the obtained ode can be easily solved and the decay rates can be given explicitly third we show that under some restrictions on the initial data and if the interior source dominates the interior damping term and if the boundary source dominates the boundary damping then the solution ceases to exists and blows up in finite time moreover in either the absence of the interior source or the boundary source then we prove that the solution is unbounded and grows as an exponential function

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

global existence and uniform stabilization of a generalized dissipative klein gordon equation type with boundary damping in this paper we prove the existence uniqueness and uniform stability of strong and weak solutions of the nonlinear generalized klein gordon equation 1 1 1 see sec i in bounded domains with nonlinear damped boundary conditions given by 1 1 3 see sec i with some restrictions on function f u h u g ut and b x we prove the existence and uniqueness by means of nonlinear semigroup method and obtain the uniform stabilization by using the multiplier technique

representations of hopf algebras arising fromlie algebras of cartan type c w x 1 1 this is a continuation of 13 in an effort to use algebraic group actions to study the representations of restricted lie algebras of cartan type in this paper we study the representation theory of certain finite diw x mensional hopf algebras that arise from the construction in 13 the representation theory of these hopf algebras closely resembles the representation theory of frobenius kernels of reductive algebraic groups in positive characteristics let g be a connected reductive algebraic group scheme over an algebraically closed field k of characteristic p 0 it is well known that if f g a g is the r th iteration of the frobenius map and g s ker f r r there is a chain of normal subgroup schemes g eg e eg in earlier 1 2 terminology this is equivalent to the existence of a chain of finite dimen sional cocommutative hopf algebras dist g dist g 1 2 dist g where dist g is known as the r th hyperalgebra of the distribur tion algebra dist g of g let g be the lie algebra of g and u g be its restricted universal enveloping algebra there exists an isomorphism of

local hadamard well posedness for nonlinear wave equations with supercritical sources and damping abstract we consider the wave equation with supercritical interior and boundary sources and damping terms the main result of the paper is local hadamard well posedness of finite energy weak solutions the results obtained 1 extend the existence results previously obtained in the literature by allowing more singular sources 2 show that the corresponding solutions satisfy hadamard well posedness conditions during the time of existence this result provides a positive answer to an open question in the area and it allows for the construction of a strongly continuous semigroup representing the dynamics governed by the wave equation with supercritical sources and damping

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

algebraic group actions in the cohomology theory of lie algebras of cartan type abstract given a lie algebra g of cartan type we construct an infinite dimensional cocommutative hopf algebrad g u g which is the analog of the distribution algebra of a connected reductive algebraic group we show that the simpleu g modules lift to ad g u g structure this additional structure is used to formulate and prove relative projectivity theorems for lie algebras of cartan type the support varieties of certaind g u g modules are computed by using these algebraic group techniques

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

principal indecomposable representations for restricted lie algebras of cartan type abstract let l be any one of w n 1 s n 1 h n 1 and k n 1 over an algebraically closed held f of characteristic p 3 in this paper we extend the results concerning modular representations of classical lie algebras and semisimple groups to the case of l and obtain some properties of principal indecomposable modules of u l which parallel closely those of classical lie algebras

global existence for the wave equation with nonlinear boundary damping and source terms abstract the paper deals with local and global existence for the solutions of the wave equation in bounded domains with nonlinear boundary damping and source terms the typical problem studied is u tt u 0 in 0 u 0 on 0 0 u u t m 2 u t u p 2 u on 0 1 u 0 x u 0 x u t 0 x u 1 x on where r n n 1 is a regular and bounded domain 0 1 m 1 2 p r where r 2 n 1 n 2 when n 3 r when n 1 2 and the initial data are in the energy space we prove local existence of the solutions in the energy space when m r r 1 p or n 1 2 and global existence when p m or the initial data are inside the potential well associated to the stationary problem

brauer type reciprocity for a class of graded associative algebras classical brauer reciprocity can be stated roughly as follows let g be a finite group and let k be an algebraically closed field of characteristic p 0 if s is a simple kg module and p s is its projective cover then the multiplicity of a simple module l as a composition factor of a characteristic zero lift of p s is the same as the multiplicity of s as a composition factor of a modulo p reduction of l all modules in this paper are assumed to be finite dimensional one thinks of the module l as playing an intermediate role between p s and s reciprocities similar to brauer reciprocity here called brauer type reciprocities have subsequently been found to occur in many other settings for instance if g is a classical modular lie algebra definition given in section 2 then there exists a set 9 of g modules with the following property given a simple g module s the projective cover p s of s has a filtration with each successive quotient isomorphic to a module in 5 and for each such filtration the number of times ze 9 occurs is the same as the multiplicity of s as a composition factor of z this was proved for g of type a i by pollack lo and for arbitrary g by humphreys 3 the reciprocity in this setting is often called humphreys reciprocity inspired by humphreys result bernstein gelfand and gelfand sought and found a brauer type reciprocity bgg reciprocity in a certain truncated category their category 8 of modules for a complex semisimple lie algebra see 11 later this was generalized by mirollo and

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

trace regularity of the solutions of the wave equation with homogeneous neumann boundary conditions and data supported away from the boundary let rn be a smooth domain with boundary let u be the solution of a second order hyperbolic scalar equation with homogeneous neumann boundary conditions defined on due to initial conditions u0 u1 and to a forcing term the present paper studies the regularity of the map u0 u1 u where u0 u1 l2 q h1 l2 q 0 t and u is the dirichlet trace of the solution in particular the case of compactly supported data u0 u1 is considered and contrasted with corresponding results where the data do not have compact support in this latter case an optimal regularity result is given for the wave equation on the half space of dimension 2 it states contrary to expectations from elliptic or parabolic theory that u h34 but u h34 e e 0 where 0 t

complexes and vanishing of cohomology for group schemes abstract completely reducible homology complexes were originally introduced in the context of finite groups and used to study the question of vanishing of cohomology in this paper we study these complexes and the vanishing of cohomology for arbitrary finite dimensional cocommutative hopf algebras applications are later provided for infinitesimal group schemes of reductive and solvable algebraic groups

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