global attractors and convergence to equilibrium for degenerate ginzburg landau and parabolic equations abstract we study the dynamics of a degenerate complex ginzburg landau cgl and a real parabolic equation with a variable generally nonsmooth diffusion coefficient which may vanish at some points or be unbounded for the cgl equation there exists a global attractor in l 2 for the parabolic equation we show the existence of a global branch of nonnegative stationary states the global bifurcation result is used in order to establish in conjunction with the definition of a gradient dynamical system in the natural phase space that any solution with nonnegative initial data tends to the trivial or the nonnegative equilibrium

a block equation solver for large unsymmetric matrices arising in the boundary integral equation method an equation solver for the large unsymmetric systems of linear equations arising in the application of the boundary integral equation method to problems of linear elasticity in homogeneous or piecewise homogeneous solids is presented the solver uses gaussian elimination the advantages of the solver over many existing eliminational or iterational methods are that it ignores many of the large groups of zeros that occur in piecewise homogeneous work and also restrains growth of the number of non zero matrix entries memory requirements and computation times therefore are reduced for many piecewise homogeneous problems

the zero set of a solution of a parabolic equation on etudie l ensemble nul d une solution u t x de l equation u t a x t u xx b x t u x c x t u sous des hypotheses tres generales sur les coefficients a b et c

convergence for degenerate parabolic equations abstract we prove that any bounded global solution to a degenerate parabolic problem in one spatial dimension converges to a unique stationary state

positivity versus localization in degenerate diffusion equations on considere le probleme de cauchy u t u m f u 0 dans s r n o u x o u 0 x x r n ou m 1 n 1 et les fonctions f et u 0 satisfont les hypotheses 1 f c 1 o f o 0 2 u 0 est une fonction continue non negative definie sur r n a support borne

generic domain decomposition and iterative solvers for 3d bem problems summary in the past two decades considerable improvements concerning integration algorithms and solvers involved in boundary element formulations have been obtained first a great deal of efficient techniques for evaluating singular and quasi singular boundary element integrals have been definitely established and second iterative krylov solvers have proven to be advantageous when compared to direct ones also including non hermitian matrices the former fact has implied in cpu time reduction during the assembling of the system of equations and the latter fact in its faster solution in this paper a triangle polar co ordinate transformation and the telles co ordinate transformation applied in previous works independently for evaluating singular and quasi singular integrals are combined to increase the efficiency of the integration algorithms and so to improve the performance of the matrixassembly routines in addition the jacobi preconditioned biconjugate gradient j bicg solver is used to develop a generic substructuring boundary element algorithm in this way it is not only the system solution accelerated but also the computer memory optimized discontinuous boundary elements are implemented to simplify the coupling algorithm for a generic number of subregions several numerical experiments are carried out to show the performance of the computer code with regard to matrix assembly and the system solving in the discussion of results expressed in terms of accuracy and cpu time advantages and potential applications of the be code developed are highlighted copyright 2006 john wiley sons ltd

stabilization for degenerate diffusion with absorption abstract the purpose of this paper is to study the limit in l 1 of solutions of general initial boundary value problems of the form ut w g x u and u w in a bounded domain with general boundary conditions of the form w w 0 where and are maximal monotone graphs and g r r is a nonincreasing continuous function in r r we prove that a solution stabilizes in l 1 as t to a function u l 1 which satisfies u x 1 c g x 1 0 a e x with c 1 0 so if for instance 1 0 1 0 g x 1 0 0 then a solution stabilizes by converging to 0 in l 1 as t

solution of bem systems of equations via iterative techniques the present paper is concerned with the application of iterative techniques to solve boundary element method bem systems of equations initially a brief description of some algorithms which have been employed for symmetric positive definite matrices is given finite element method matrices for instance subsequently non symmetric versions of these algorithms are described and their performance is checked for bem matrices through various numerical experiments preconditioned algorithms were found to work quite well

nonlinear heat conduction with absorption space localization and extinction in finite time additional lower order terms in the nonlinear heat conduction equation can lead to essentially new behavior of the solutions the statements answer the questions when the support of the solution is bounded in space and when the solution dies in finite time

a poincare bendixson theorem for scalar reaction diffusion equations for scalar equations u t u xx f x u u x with x e s 1 and f e c 2 we show that the classical theorem of poincare and bendixson holds the limit set of any bounded solution satisfies exactly one of the following alternatives this is surprising because the system is genuinely infinite dimensional

stabilization of solutions of a degenerate nonlinear diffusion problem i 4 u f u in l l x r u tl t o in r u x 0 uo x in l l where m 1 is a parameter f is locally lipschitz continuous f 0 0 and u is bounded problems of this form arise in a number of areas of science for instance in models for gas or fluid flow in porous media 2 and for the spread of certain biological populations 13 161 this paper is divided into two parts in part i we consider what may be called the motivating example problem i which consists of problem i with the special choice f u u 1 u u a 1 for suitably restricted parameters a we begin by describing in detail the set l of nonnegative equilibrium solutions of problem i clearly 8 l contains the trivial solution u 0 for all l 0 write 8 l e l o in the description of z l there are two critical parameter values lo and l1 with 0 lo l1 j we show that i g l 4 for 0 l lo

regularizing effects for ut aϑ u 0 in l1 abstract various initial boundary value problems and cauchy problems can be written in the form du dt a u 0 where r r is nondecreasing and a is the linear generator of strongly continuous nonexpansive semigroup e ta in an l1 space for example if a subject perhaps to suitable boundary conditions we obtain equations arising in flow in a porous medium or plasma physics depending on the choice of while if a x acting in l1 r we have a scalar conservation law in this paper we show that if m m 0 and m 2 m 2 where 1 1 then roughly speaking the norm of t du dt may be estimated in terms of the initial data u0 in l1 such estimates give information about the regularity of solutions asymptotic behaviour etc in applications side issues such as the introduction of sufficiently regular approximate problems on which estimates can be made and the assignment of a precise meaning to the operator a are also dealt with these considerations are of independent interest

out of core solver for large multi zone boundary element matrices in this paper an out of core solver is developed for a three dimensional elastostatic boundary element program this program includes multi regions which enables a large model to be divided into a number of homogeneous sub models each with their own material properties the system matrix produced by multi regions is sparse blocked and unsymmetric in character and so reduces disc space and solution time the solver presented in this paper utilizes efficiently the structure of this matrix by holding only non zero parts of this matrix in a sequential unformatted direct access file reading in as much of the file as possible into the working space performing gauss elimination then writing part of the matrix back to file the maximum utilization of working space is particularly important on vector machines as vector activity is maximized whilst record read write is minimized the effect of multi regions on cpu is demonstrated on both the convex and cray machines

efficient analytical integration of symmetric galerkin boundary integrals over curved elements thermal conduction formulation abstract substantial improvements are reported in the computational efficiency of galerkin boundary element analysis bea employing curved continuous boundary elements a direct analytical treatment of the singular double integrations involved in galerkin bea adapting a limit to the boundary concept used successfully in collocation bea is used to obviate significant computation in the determination of the galerkin coefficient matrices symbolic manipulation has been strategically employed to aid in the analytical evaluation of the singular contributions to these double integrals the analytical regularization procedure separates the potentially singular galerkin integrands into an essentially singular but simple part plus a regular remainder that can be integrated numerically the finite contribution from the simplified singular term is then computed analytically it is shown that the key to containing the explosive growth in the length of the formulae associated with such a hybrid analytical numerical integration scheme is the strategic timing of when to take the limit to the boundary this regularization also isolates the contribution from the curvature of the boundary element thus facilitating enhanced computational efficiency in problems with many straight elements example problems are presented to quantify the performance of this approach it is concluded that with these techniques galerkin symmetric bea can be more efficient than its collocation based counterpart

convergence asymptotic periodicity and finite point blow up in one dimensional semilinear heat equations abstract we consider the initial value problem for the semilinear heat equation u t u xx f u t 0 x l t 0 under the dirichlet the neumann or the periodic boundary conditions we show that each solution whether it exists globally for t 0 or blows up in a finite time possesses an asymptotic profile in a certain sense and tends to this profile as time increases in the special case where f u t t f u t for some t 0 among other things the above statement is interpreted as saying that any bounded global solution converges as t to a time t periodic solution having some specific spatial structure in the case where the solution blows up in a finite time say at t t 0 assuming simply that f is a smooth function satisfying some growth conditions and that the initial data is a nonconstant bounded function we prove that the blow up set is a finite set and that lim t t 0 u x t x exists with being a smooth function having at most finitely many singular points

symmetry and convergence properties for non negative solutions of nonautonomous reaction diffusion problems nonautonomous parabolic equations of the form u t u f u t on a symmetric domain are considered using the moving hyperplane method it is proved that any bounded nonnegative solution symmetrises as t this is then used to show that for nonlinearities periodic in t any non negative bounded solution approaches a periodic solution

stabilization of solutions of nonlinear and degenerate evolution equations on etudie le comportement asymptotique des solutions bornees des equations de la forme u t u f x u dans r u x o u 0 x dans u x t u 1 x t dans r ou est un domaine borne de r n avec de classe c 2 0 1