on contracting interval iteration for nonlinear problems in rn part ii applications governs classes of evolution type transport processes in the theories of diffusion conduction of heat chemical kinetics etc explicit solutions of 1 1 with boundary conditions are known only in a few exceptional cases e g 4 therefore finite dimensional approximations are of considerable practical interest according to the equivalence theorem e g 7 the convergence of a sequence of such approximations is ensured if the conditions of consistency and stability are satisfied gershgorin s method is a general approach for establishing the stability property in the case of nonlinear bvp boundary value problems this method transfers the property of inverse monotonicity from the bvp to the finite dimensional approximation where it has to be shown that this property holds uniformly as a grid parameter m e n tends to infinity if a is an inverse monotone operator then the operator inequality au d aw making use of a partial ordering of the space u on which a operates implies that u d w for every pair of elements u w e u such that au aw the uniformity of the property of inverse monotonicity can be shown by use of available sufficient conditions if the classes of admissible functions c k and c in 1 1 are rather severely restricted in addition for fixed m the qualitative property of convergence does not yield a quantitative estimate of the relationship between the local and the global discretization error 13 p 131 where local error refers to the order of consistency of the difference scheme and the global error for any fured m is the distance between the restriction to the grid of the solution of the bvp and the solution of the difference scheme rather than verifying the qualitative properties of stability and convergence for fmed m here a quantitative error estimate is derived which relates the local to the global discretization error this estimate depends on both the grid parameter m and the number of the iteration cycle of the numerical method to be employed for the deduction of this relationship it is sufficient that the finite system is individually i e not necessarily uniformly inverse monotone for each value of the grid parameter m e n

finite element approximation of multi scale elliptic problems using patches of elements in this paper we present a method for the numerical solution of elliptic problems with multi scale data using multiple levels of not necessarily nested grids the method consists in calculating successive corrections to the solution in patches whose discretizations are not necessarily conforming this paper provides proofs of the results published earlier see c r acad sci paris ser i 337 2003 679 684 gives a generalization of the latter to more than two domains and contains extensive numerical illustrations new results including the spectral analysis of the iteration operator and a numerical method to evaluate the constant of the strengthened cauchy buniakowski schwarz inequality are presented

multi specialization and multi asymptotic expansions abstract in this paper we extend the notion of specialization functor to the case of several closed submanifolds satisfying some suitable conditions applying this functor to the sheaf of whitney holomorphic functions we construct different kinds of sheaves of multi asymptotically developable functions whose definitions are natural extensions of the definition of strongly asymptotically developable functions introduced by majima

global solvability of nonlinear diffusion equations with forcing at the boundary abstract under the influence of a sufficiently weak nonlinear source term it is by now well known that a degenerate diffusion equation is globally solvable a similar result is known when the nonlinear source is present as a forcing term at the boundary such results are usually established via comparison with solutions of a related differential equation or some other form of the maximum principle however these techniques do not appear to apply in general situations when forcing occurs on only part of the boundary and convection is present in this work we obtain the global existence of solutions for such problems by deriving a differential inequality involving thelr 1norms of solutions the result is similar to that described above for equations with a nonlinear source term and furthermore establishes the global existence of small solutions even for strong forcings the technique also applies to a more general class of problems involving nonlinear reaction diffusion and convection

local enrichment of nurbs patches using a non intrusive coupling strategy geometric details local refinement inclusion fracture in this work we apply a non intrusive global local coupling strategy for the modelling of local phenomena in a nurbs patch the idea is to consider the nurbs patch to be enriched as the global model this results in a simple flexible strategy first the global nurbs patch remains unchanged which completely eliminates the need for costly re parametrization procedures even if the local domain is expected to evolve then easy merging of a linear nurbs code with any other existing robust codes suitable for the modelling of complex local behaviour is possible the price to pay is the number of iterations of the non intrusive solver but we show that this can be strongly reduced by means of acceleration techniques the main development for nurbs is to be able to handle non conforming geometries only slight changes in the implementation process including the setting up of suitable quadrature rules for the evaluation of the interface reaction forces are made in response to this issue a range of numerical examples in two dimensional linear elasticity are given to demonstrate the performance of the proposed methodology and its significant potential to treat any case of local enrichment in a nurbs patch simply

quasilinear parabolic and elliptic equations with nonlinear boundary conditions abstract this paper is concerned with a class of quasilinear parabolic and elliptic equations in a bounded domain with both dirichlet and nonlinear neumann boundary conditions the equation under consideration may be degenerate or singular depending on the property of the diffusion coefficient the consideration of the class of equations is motivated by some heat transfer problems where the heat capacity and thermal conductivity are both temperature dependent the aim of the paper is to show the existence and uniqueness of a global time dependent solution of the parabolic problem existence of maximal and minimal steady state solutions of the elliptic problem including conditions for the uniqueness of a solution and the asymptotic behavior of the time dependent solution in relation to the steady state solutions applications are given to some heat transfer problems and an extended logistic reaction diffusion equation

quasilinear parabolic and elliptic systems with mixed quasimonotone functions abstract this paper deals with a class of quasilinear parabolic and elliptic systems with mixed quasimonotone reaction functions the boundary condition in the system may be dirichlet nonlinear or a combination of these two types the elliptic operators in the system are allowed to be degenerate the aim is to show the existence and uniqueness of a classical solution to the parabolic system the existence of maximal and minimal solutions or quasisolutions of the elliptic system and the asymptotic behavior of the solution of the parabolic system this consideration leads to a global attractor of the parabolic system as well as an one sided stability of the maximal and minimal solutions applications of these results are given to three models arising from biology and ecology where diffusion coefficients are density dependent and are degenerate these applications exhibit quite distinct dynamical behavior of the population species between degenerate density dependent diffusion and constant diffusion

integral transforms with exponential kernels and laplace transform let x z y be a correspondence of complex manifolds f g we study integral transforms associated to kernels exp o with p meromor phic on z acting on formal or moderate cohomologies our main application is the laplace transform in this case x is the projective compactification of the vector space v c y is its dual space z x x y and o z w z w we obtain the isomorphisms f 0 ov f n f ov thom f v thom far n ov this content downloaded from 207 46 13 59 on mon 13 jun 2016 05 54 06 utc all use subject to http about jstor org terms 972 masaki kashiwara and pierre schapira where f is a conic and rll constructible sheaf on v and fa is its fourier sato transform some applications are discussed rims kyoto university kyoto 606 01 japan institut de mathematiques universit9 paris vi case 82 4 pl jussieu 75252 paris france e mail address schapirafmath jussieu fr this content downloaded from 207 46 13 59 on mon 13 jun 2016 05 54 06 utc all use subject to http about jstor org terms

existence and dynamics of quasilinear parabolic systems with time delays abstract this paper is concerned with a coupled system of quasilinear parabolic equations where the effect of time delays is taken into consideration in the reaction functions of the system the partial differential operators in the system may be degenerate and the reaction functions possess some mixed quasimonotone property including quasimonotone nondecreasing functions the aim of the paper is to show the existence and uniqueness of a global solution to the parabolic system the existence of positive quasisolutions or maximal minimal solutions of the corresponding elliptic system and the asymptotic behavior of the solution of the parabolic system in relation to the quasisolutions or maximal minimal solutions of the elliptic system applications are given to three reaction diffusion models arising from mathematical biology and ecology where the diffusion coefficients are density dependent and are degenerate this degenerate density dependent diffusion leads to some interesting distinct asymptotic behavior of the time dependent solution when compared with density independent diffusion

traveling wave profiles for nonlinear equations abstract exact traveling wave profiles are here obtained for a general reaction diffusion convection equation a two phase flow equation a generalized harry dym equation a generalized korteweg de vries kdv equation a modified mkdv equation a branching network model equation a nagumo equation and a third order equation of fluid mechanics

the role of critical exponents in blow up theorems the sequel abstract in 27 fujita showed that for positive solutions the initial value problem in r n for u t u u p with p 1 exhibited the following behavior if p p c 1 2 n then the initial value problem does not have any nontrivial non negative solution existing on r n 0 a global solution whereas if p p c there exist global small data positive solutions as well as solutions which are non global we call such a result a blow up theorem of fujita type in 50 levine discussed the various theorems of this type that had appeared in the literature prior to 1990 in this paper we revisit the literature since 1990

singular reaction diffusion equations of porous medium type abstract a class of reaction diffusions equations with nonlinear boundary conditions and porous medium type of diffusion is investigated by the method of upper and lower solutions the diffusion term in the equation includes the so called slow diffusion m 1 fast diffusion 0 m 1 and super diffusion m 0 and the reaction term may involve a singular function the aim of this paper is to show the existence of a global time dependent solution the existence of positive minimal and maximal steady state solutions including the uniqueness of the positive solution and the asymptotic behavior of the time dependent solution in relation to the positive steady state solutions of particular concern is the convergence of the time dependent solution to a unique positive steady state solution and the global attraction property of the reaction diffusion problem applications are given to several specific problems which are of considerable interest in the current literature

global existence for nonlinear diffusion equations solutions of nonlinear possibly degenerate reaction diffusion models are known to exist for all time if the asymptotic growth of the reaction term is not greater than that of the diffusion term via concavity methods it is also known that negating such a condition results in solutions which blow up in finite time on the other hand two recent studies regarding the case of linear diffusion have reported on the ability for convective terms in a model to create global existence of solutions where no such result is true in their absence in this paper we develop a theory of global existence for a general class of reaction diffusion convection models which only requires a similar balance of diffusion and reaction our result extends previous work to models including convection but it does not reveal the exact role of convection in yielding global solutions further analysis of a slightly simplified model establishes the global existence of all solutions if the reaction term grows asymptotically at a rate less than that of either diffusion or convection

positive solutions of quasilinear parabolic systems with nonlinear boundary conditions abstract the aim of this paper is to investigate the existence uniqueness and asymptotic behavior of solutions for a coupled system of quasilinear parabolic equations under nonlinear boundary conditions including a system of quasilinear parabolic and ordinary differential equations also investigated is the existence of positive maximal and minimal solutions of the corresponding quasilinear elliptic system as well as the uniqueness of a positive steady state solution the elliptic operators in both systems are allowed to be degenerate in the sense that the density dependent diffusion coefficients d i u i may have the property d i 0 0 for some or all i our approach to the problem is by the method of upper and lower solutions and its associated monotone iterations it is shown that the time dependent solution converges to the maximal solution for one class of initial functions and it converges to the minimal solution for another class of initial functions and if the maximal and minimal solutions coincide then the steady state solution is unique and the time dependent solution converges to the unique solution applications of these results are given to three model problems including a porous medium type of problem a heat transfer problem and a two component competition model in ecology these applications illustrate some very interesting distinctive behavior of the time dependent solutions between density independent and density dependent diffusions

on multigrid for overlapping grids the solution of elliptic partial differential equations on composite overlapping grids using multigrid is discussed an approach is described that provides a fast and memory efficient scheme for the solution of boundary value problems in complex geometries the key aspects of the new scheme are an automatic coarse grid generation algorithm an adaptive smoothing technique for adjusting residuals on different component grids and the use of local smoothing near interpolation boundaries other important features include optimizations for cartesian component grids the use of over relaxed red black smoothers and the generation of coarse grid operators through galerkin averaging numerical results in two and three dimensions show that very good multigrid convergence rates can be obtained for dirichlet neumann and mixed boundary conditions a comparison to krylov based solvers shows that the multigrid solver can be much faster and require significantly less memory

determination of an unknown coefficient in a nonlinear heat equation abstract the problem of determining an unknown term k u in the equation k u u t k u u x x is considered in this paper applying tikhonov s regularization approach we develop a procedure to find an approximate stable solution to the unknown coefficient from the overspecified data

leray s quantization of projective duality 2 review on correspondences for sheaves and d modules 4 2 1 notations 4 2 1 1 geometry 4 2 1 2 sheaves 4 2 1 3 d modules 5 2 1 4 e modules 6 2 2 correspondences for sheaves and d modules 6

numerical zoom for multiscale problems with an application to nuclear waste disposal we analyse here a computational technique and error estimates for the numerical solution of some problems with multiple scales when the small scale is confined to geometrically small regions such as jumps of coefficients on curves and surfaces or complex variations of coefficients in small regions where numerical zooms can be made the method is an adaptation of the hilbert subspace decomposition method studied by the second author in a different context so the method is restated with all known results combined with the layer decomposition of s delpino o pironneau asymptotic analysis and layer decomposition for the couplex exercise in alain bourgeat michel kern eds computational geosciences vol 8 no 2 kluwer academics publishers 2004 pp 149 162 the method is applied to the numerical assessment of a nuclear waste repository site