interaction of diffusion and boundary conditions on considere le systeme d equations paraboliques u t d u f u x d u n e x u 0 x ou u r n r n n 3 est un ouvert borne de frontiere lisse d diag d 1 d n e diag e 1 e n chaque d j 0 est constant e j r est continu e j 0 j 1 2 n et e 0 est constant pour f e fixe on etudie comment le flot defini ci dessus depend des parametres d

structured condition numbers and small sample condition estimation of symmetric algebraic riccati equations abstract this paper is devoted to a structured perturbation analysis of the symmetric algebraic riccati equations by exploiting the symmetry structure based on the analysis the upper bounds for the structured normwise mixed and componentwise condition numbers are derived due to the exploitation of the symmetry structure our results are improvements of the previous work on the perturbation analysis and condition numbers of the symmetric algebraic riccati equations our preliminary numerical experiments demonstrate that our condition numbers provide accurate estimates for the change in the solution caused by the perturbations on the data moreover by applying the small sample condition estimation method we propose a statistical algorithm for practically estimating the condition numbers of the symmetric algebraic riccati equations

residual bounds of approximate solutions of the discrete time algebraic riccati equation let tilde x be a hermitian matrix which approximates the unique hermitian positive semi definite solution x to the discrete time algebraic riccati equation dare where f in cal c n times n c 2 in cal c m times m is hermitian positive definite g 1 in cal c n times m c in cal c r times n the pair f g 1 is stabilizable and the pair c f is detectable assume that i g tilde x is nonsingular and i g tilde x 1 f is stable let g g 1g 2 1 g 1 rm h h c rm h c and let be the residual of the dare with respect to tilde x define the linear operator vec l by the main result of this paper is if where denotes any unitarily invariant norm and then

delay nonlinear boundary conditions as limit of reactions concentrating in the boundary abstract in this work we are interested in the dynamic behavior of a parabolic problem with nonlinear boundary conditions and delay in the boundary we construct a reaction diffusion problem with delay in the interior where the reaction term is concentrated in a neighborhood of the boundary and this neighborhood shrinks to boundary as a parameter goes to zero we analyze the limit of the solutions of this concentrated problem and prove that these solutions converge in certain continuous function spaces to the unique solution of the parabolic problem with delay in the boundary this convergence result allows us to approximate the solution of equations with delay acting on the boundary by solutions of equations with delay acting in the interior and it may contribute to analyze the dynamic behavior of delay equations when the delay is at the boundary

parabolic evolution equations and nonlinear boundary conditions on etudie le probleme aux valeurs limites et initiales u 7b a t u f t u 7b b t u g t u 0 t t u 0 u 0 qui peut etre considere comme une contrepartie abstraite a des problemes paraboliques semilineaires

condition numbers of algebraic riccati equations in the frobenius norm consider the continuous time algebraic riccati equation care and the discrete time algebraic riccati equation dare which arise in linear control and system theory it is known that appropriate assumptions on the coefficient matrices guarantee the existence and uniqueness of hermitian positive semidefinite stabilizing solutions in this note we apply the theory of condition developed by rice to define condition numbers of the care and dare in the frobenius norm and derive explicit expressions of the condition numbers in a uniform manner both the complex case and real case are considered and connections to certain existing condition numbers of the care and dare are discussed

mixed and componentwise condition numbers of nonsymmetric algebraic riccati equation abstract in this paper the mixed and componentwise condition numbers of nonsymmetric algebraic riccati equation are considered the explicit expressions of two kinds of condition numbers are derived in the uniform manner and the upper bounds are proposed some simple numerical experiments show that the mixed and componentwise condition numbers can be sharp

scaling of the discrete time algebraic riccati equation to enhance stability of the schur solution method a simple scaling procedure for discrete time riccati equations is introduced this procedure eliminates instabilities which can be associated with the linear equation solution step of the generalized schur method without changing the condition of the underlying problem a computable bound for the relative error of the solution of the riccati equation is also derived

brief paper perturbation analysis and condition numbers of symmetric algebraic riccati equations this paper is devoted to the perturbation analysis of symmetric algebraic riccati equations based on our perturbation analysis the upper bounds for the normwise mixed and componentwise condition numbers are presented the results are demonstrated by our preliminary numerical experiments

bifurcation of positive solutions to scalar reaction diffusion equations with nonlinear boundary condition abstract the bifurcation of non trivial steady state solutions of a scalar reaction diffusion equation with nonlinear boundary conditions is considered using several new abstract bifurcation theorems the existence and stability of positive steady state solutions are proved using a unified approach the general results are applied to a laplace equation with nonlinear boundary condition and bistable nonlinearity and an elliptic equation with superlinear nonlinearity and sublinear boundary conditions

on mixed and componentwise condition numbers for moore penrose inverse and linear least squares problems classical condition numbers are normwise they measure the size of both input perturbations and output errors using some norms to take into account the relative of each data component and in particular a possible data sparseness componentwise condition numbers have been increasingly considered these are mostly of two kinds mixed and componentwise in this paper we give explicit expressions computable from the data for the mixed and componentwise condition numbers for the computation of the moore penrose inverse as well as for the computation of solutions and residues of linear least squares problems in both cases the data matrices have full column row rank

two kinds of condition numbers for the quadratic matrix equation in the situation that the quadratic matrix equation has the dominant or minimal solvents the mixed and componentwise condition numbers of the quadratic matrix equation are considered the explicit approximate expressions of two kinds of condition numbers are derived in the uniform manner and the upper bounds are proposed the theoretical results are illustrated by using some simple numerical examples

parabolic problems in h 1 with fast growing nonlinearities tdepartamento de matem itica do icmsc universidade de sao paulo 13 560 silo carlos s p brazil and institute of mathematics silesian university 40 007 katowice poland also institute of mathematics of the polish academy of sciences received 18 march 1996 received f or publication 1997 key words and phrases parabolic equations fast growing nonlinearities global solutions dissipative semigroups global attractors 1 introduction

a structured condition number for self adjoint polynomial matrix equations with applications in linear control abstract based on the classic definition of condition number a structured condition number is proposed for a class of self adjoint polynomial matrix equations the explicit formula of the structured condition number is derived with applying newly defined linear operators the structured condition number can be applied to some important polynomial matrix equations including the continuous time algebraic riccati equation care the discrete time algebraic lyapunov equation dale etc compared with the state of the art condition numbers for care and dale the newly proposed structured condition number can measure the sensitivity of the solution better which is validated by numerical examples

condition number and backward errors of nonsymmetric algebraic riccati equation abstract the normwise condition number and two kinds of backward errors are considered when nonsymmetric algebraic riccati equation nare has the minimal nonnegative solution based on the techniques for the symmetric case we apply the condition number theory developed by rice to define condition number for nare the explicit expression is derived in a uniform manner meanwhile two kinds of backward errors are defined and evaluated by the explicit formulas numerical experiments are listed to illustrate and compare the practical performance

residual bounds of approximate solutions of the algebraic riccati equation let tilde x geq 0 approximate the unique hermitian positive semi definite solution x to the algebraic riccati equation are where f g geq 0 a f is stabilizable and a g is detectable let be the residual of the are with respect to tilde x and define the linear operator vec l by by applying a new forward perturbation bound to the optimal backward perturbation corresponding to the approximate solution tilde x we obtained the following result if a f tilde x is stable and if 4 vec l 1 vec l 1 hat r f 1 for any unitarily invariant norm then

sensitivity analysis of the discrete time algebraic riccati equation abstract consider the discrete time algebraic riccati equation dare a t xa x a t xb s r b t xb 1 b t xa s t q 0 where a r n n b s r n m r rt r m m q qt r n n the available perturbation theory for the dare can only be applied to the case r 0 however in some control problems the matrix r can be singular in this paper we study perturbation properties of the dare without the restriction r 0 perturbation bounds and a relative condition number for the stabilizing solution of the dare are derived computable residual bounds for an approximate solution are also derived the theoretical results are illustrated by numerical examples

the sensitivity of the algebraic and differential riccati equations in this paper it is shown that the ideas developed by byers in proc summer research conference ams vol 47 contemporary math american mathematical society providence ri 1984 pp 35 49 on the sensitivity of the algebraic riccati equation can be sharpened and extended to norms other than the frobenius norm this extension is crucial from an interpretive point of view because use of the spectral norm allows an identification between the condition number of the algebraic riccati equation and the damping properties of the closed loop dynamical system moreover this approach has the pleasant feature that it carries over to a completely parallel theory for the sensitivity of the differential riccati equation an area that has not been considered previously

perturbation analysis of a quadratic matrix equation associated with an m matrix this paper is devoted to the perturbation analysis for the quadratic matrix equation x 2 e x f 0 where e is a diagonal matrix and f is an m matrix the quadratic matrix equation of this type arises in noisy wiener hopf problems for markov chains the solution of practical interest is a particular m matrix solution in this paper the perturbation bound of the m matrix solution is presented meanwhile the residual bound for an approximate solution to the m matrix solution is obtained the theoretical results are illustrated by using simple numerical examples

on mixed and componentwise condition numbers for indefinite least squares problem in this paper we mainly consider the mixed and componentwise condition numbers for the indefinite least squares problem without constraints or with least squares constraints the explicit expressions and the easily computable upper bounds for these condition numbers are presented two numerical examples are given to test these results and illustrate their superiority compared with the corresponding normwise condition numbers furthermore we also provide the mixed and componentwise condition numbers for the moore penrose inverse in indefinite inner product spaces

nonlinear balance for reaction diffusion equations under nonlinear boundary conditions dissipativity and blow up the authors consider a reaction diffusion equation in a bounded smooth domain r n with nonlinear flux terms on the boundary of they derive suitable conditions on the nonlinear terms of the problem which imply its dissipativity the authors explore different situations in which there exist solutions of the problem that blow up in finite time the main tools in the proof of the results are either energy estimates or sub and super solution techniques

fractional order theory of thermoelasticity in this work a new theory of thermoelasticity is derived using the methodology of fractional calculus the theories of coupled thermoelasticity and of generalized thermoelasticity with one relaxation time follow as limit cases a uniqueness theorem for this model is proved a variational principle and a reciprocity theorem are derived

asymptotic behavior and attractors for reaction diffusion equations in unbounded domains abstract in this paper we give general and flexible conditions for a reaction diffusion equation to be dissipative in an unbounded domain the functional setting is based on standard lebesgue and sobolev lebesgue spaces we show how the reaction and diffusion mechanisms have to work together to obtain the asymptotic compactness of solutions and therefore the existence of the compact attractor in particular cases our results allow us to improve some previous known results

blow up results and localization of blow up points for the heat equation with a nonlinear boundary condition abstract in this paper we study the large time behavior of positive solutions of the heat equation under the nonlinear boundary condition u f u where is the outward normal and f is nondecreasing with f u 0 for u 0 we show that if b r and 1 f is integrable at infinity there is finite time blow up for any initial datum in the two dimensional case we show that this is true for any smooth simply connected domain in the radially symmetric case if f c2 is convex and satisfies the properties above we show that blow up occurs only at the boundary

continuity of attractors for a reaction diffusion problem with nonlinear boundary conditions with respect to variations of the domain in this work we show for a class of dissipative semilinear parabolic problems that the global compact attractor varies continuously with respect to parameters in the equations applications to a parabolic problem with nonlinear boundary conditions are also obtained

patterns in parabolic problems with nonlinear boundary conditions we obtain existence of asymptotically stable nonconstant equilibrium solutions for semilinear parabolic equations with nonlinear boundary conditions on small domains connected by thin channels we prove the convergence of eigenvalues and eigenfunctions of the laplace operator in such domains this information is used to show that the asymptotic dynamics of the heat equation in this domain is equivalent to the asymptotic dynamics of a system of two ordinary differential equations diffusively weakly coupled the main tools employed are the invariant manifold theory and a uniform trace theorem

regularity and asymptotic behavior for the second order parabolic equation with nonlinear boundary conditions in lp abstract the equation lu x u on b 0 b bounded smooth domain in r n with nonlinear boundary conditions u v g x u on b 0 is studied l being the uniformly parabolic operator with time independent coefficients under suitable conditions on the nonlinearities that do not involve monotonicity global existence uniqueness compactness of the orbits and certain regularizing effects of the semigroup are established in the case that l is in divergence form it is shown that under generic conditions orbits tend as t to some equilibrium and that the stable equilibria attract essentially baire category the whole space l2 b

the dynamics of a one dimensional parabolic problem versus the dynamics of its discretization abstract in this paper we prove that the spatial discretization of a one dimensional system of parabolic equations with suitably small step size contains exactly the same asymptotic dynamics as the continuous problem

delay partial differential equations with some large diffusion it is often the case in infinite dimensional dynamical systems that a certain unbounded operator depends upon a parameter v its eigenvalues and eigenfunctions will also depend upon v in many cases see for example l 7 there is a finite number of eigenvalues of the unbounded operator that stay bounded as the parameter varies the remaining eigenvalues blow up at a certain rate in this case one expects the dynamics to be dictated by a system of ordinary differential equations this is indeed the case in several applications and its proof strongly uses the invariant manifold theory in the cases considered in this paper we study the dynamics of infinite dimensional dynamical systems for which a countable number of eigenvalues of the unbounded operator converges to the eigenvalues of an unbounded operator in a lower dimensional domain whereas the remaining blow up this is the case for example in parabolic equations in thin domains studied by hale and raugel s we will not be able to address the general case of thin domains but only the case of a thin straight channel however the property that allows this theory to work is present in the general case that is there is a sequence of eigenvalues that stay bounded and a sequence of eigenvalues that blow up see also arrieta 9 the theory of invariant manifolds will no longer work in this case since it is centered around the existence of backward solutions which cannot be expected in this situation the main tools in this work are a converse theorem on existence of compact attractors and an eigenvalue problem a physical interpretation of a parabolic problem in a thin straight channel is that there is a very high diffusion in one of the directions from which our intuition would say that in that direction spatial changes will not be observed the same interpretation can be given to a damped wave problem in a thin straight channel the wave speed can be interpreted as being very large in one direction this would intuitively imply that there would be no spatial changes for the amplitude of the wave in that direction we also consider the problem called shadow systems where the diffusion is very large for some of the substances in the reaction and not so large for the remaining substances in this

large diffusivity and asymptotic behavior in parabolic systems abstract for systems of reaction diffusion equations with neumann boundary conditions it is shown that the solutions are asymptotic to the solutions of an ordinary differential equation if the diffusivity is large the methods apply also to reaction diffusion systems with time delays