A DISCRETIZATION SCHEME FOR AN ONE-DIMENSIONAL REACTION-DIFFUSION EQUATION WITH DELAY AND ITS DYNAMICS

The goal of this paper is to present an approximation scheme for a reaction-diffusion equation with finite delay, which has been used as a model to study the evolution of a population with density distribution u, in such a way that the resulting finite dimensional ordinary differential system contains the same asymptotic dynamics as the reaction-diffusion equation.

Reaction-diffusion systems coupled at the boundary and the Morse-Smale property

We study an one-dimensional nonlinear reaction-diffusion system coupled on the boundary. Such system comes from modeling problems of temperature distribution on two bars of same length, jointed together, with different diffusion coefficients. We prove the transversality property of unstable and stable manifolds assuming all equilibrium points are hyperbolic. To this end, we write the system as an equation with noncontinuous diffusion coefficient. We then study the nonincreasing property of the number of zeros of a linearized nonautonomous equation as well as the Sturm-Liouville properties of the solutions of a linear elliptic problem. (C) 2008 Elsevier Inc. All rights reserved.

Dynamics in dumbbell domains III. Continuity of attractors

In this paper we conclude the analysis started in [J.M. Arrieta, AN Carvalho, G. Lozada-Cruz, Dynamics in dumbbell domains I. Continuity of the set of equilibria, J. Differential Equations 231 (2006) 551-597] and continued in [J.M. Arrieta, AN Carvalho, G. Lozada-Cruz, Dynamics in dumbbell domains II. The limiting problem, J. Differential Equations 247 (1) (2009) 174-202 (this issue)] concerning the behavior of the asymptotic dynamics of a dissipative reaction-diffusion equation in a dumbbell domain as the channel shrinks to a line segment. In [J.M. Arrieta, AN Carvalho. G. Lozada-Cruz, Dynamics in dumbbell domains I. Continuity of the set of equilibria, J. Differential Equations 231 (2006) 551-597], we have established an appropriate functional analytic framework to address this problem and we have shown the continuity of the set of equilibria. In [J.M. Arrieta, AN Carvalho, G. Lozada-Cruz. Dynamics in dumbbell domains II. The limiting problem, J. Differential Equations 247 (1) (2009) 174-202 (this issue)], we have analyzed the behavior of the limiting problem. In this paper we show that the attractors are Upper semicontinuous and, moreover, if all equilibria of the limiting problem are hyperbolic, then they are lower semicontinuous and therefore, continuous. The continuity is obtained in L(p) and H(1) norms. (C) 2008 Elsevier Inc. All rights reserved.

Cascades of Hopf bifurcations from boundary delay

We consider a 1-dimensional reaction-diffusion equation with nonlinear boundary conditions of logistic type with delay. We deal with non-negative solutions and analyze the stability behavior of its unique positive equilibrium solution, which is given by the constant function u equivalent to 1. We show that if the delay is small, this equilibrium solution is asymptotically stable, similar as in the case without delay. We also show that, as the delay goes to infinity, this equilibrium becomes unstable and undergoes a cascade of Hopf bifurcations. The structure of this cascade will depend on the parameters appearing in the equation. This equation shows some dynamical behavior that differs from the case where the nonlinearity with delay is in the interior of the domain. (C) 2009 Elsevier Inc. All rights reserved.

ON THE SUPERCRITICALITY OF THE FIRST HOPF BIFURCATION IN A DELAY BOUNDARY PROBLEM

The goal of this paper is to analyze the character of the first Hopf bifurcation (subcritical versus supercritical) that appears in a one-dimensional reaction-diffusion equation with nonlinear boundary conditions of logistic type with delay. We showed in the previous work [Arrieta et al., 2010] that if the delay is small, the unique non-negative equilibrium solution is asymptotically stable. We also showed that, as the delay increases and crosses certain critical value, this equilibrium becomes unstable and undergoes a Hopf bifurcation. This bifurcation is the first one of a cascade occurring as the delay goes to infinity. The structure of this cascade will depend on the parameters appearing in the equation. In this paper, we show that the first bifurcation that occurs is supercritical, that is, when the parameter is bigger than the delay bifurcation value, stable periodic orbits branch off from the constant equilibrium.

Dynamics in dumbbell domains II. The limiting problem

In this work we continue the analysis of the asymptotic dynamics of reaction-diffusion problems in a dumbbell domain started in [J.M. Arrieta, AN Carvalho, G. Lozada-Cruz, Dynamics in dumbbell domains I. Continuity of the set of equilibria, J. Differential Equations 231 (2) (2006) 551-597]. Here we study the limiting problem, that is, an evolution problem in a ""domain"" which consists of an open, bounded and smooth set Omega subset of R(N) with a curve R(0) attached to it. The evolution in both parts of the domain is governed by a parabolic equation. In Omega the evolution is independent of the evolution in R(0) whereas in R(0) the evolution depends on the evolution in Omega through the continuity condition of the solution at the junction points. We analyze in detail the linear elliptic and parabolic problem, the generation of linear and nonlinear semigroups, the existence and structure of attractors. (C) 2009 Elsevier Inc. All rights reserved.

Lower semicontinuity of attractors for non-autonomous dynamical systems

This paper is concerned with the lower semicontinuity of attractors for semilinear non-autonomous differential equations in Banach spaces. We require the unperturbed attractor to be given as the union of unstable manifolds of time-dependent hyperbolic solutions, generalizing previous results valid only for gradient-like systems in which the hyperbolic solutions are equilibria. The tools employed are a study of the continuity of the local unstable manifolds of the hyperbolic solutions and results on the continuity of the exponential dichotomy of the linearization around each of these solutions.

Semilinear parabolic problems in thin domains with a highly oscillatory boundary

In this paper, we study the behavior of the solutions of nonlinear parabolic problems posed in a domain that degenerates into a line segment (thin domain) which has an oscillating boundary. We combine methods from linear homogenization theory for reticulated structures and from the theory on nonlinear dynamics of dissipative systems to obtain the limit problem for the elliptic and parabolic problems and analyze the convergence properties of the solutions and attractors of the evolutionary equations. (C) 2011 Elsevier Ltd. All rights reserved.

A discontinuous-Galerkin-based immersed boundary method

A numerical method to approximate partial differential equations on meshes that do not conform to the domain boundaries is introduced. The proposed method is conceptually simple and free of user-defined parameters. Starting with a conforming finite element mesh, the key ingredient is to switch those elements intersected by the Dirichlet boundary to a discontinuous-Galerkin approximation and impose the Dirichlet boundary conditions strongly. By virtue of relaxing the continuity constraint at those elements. boundary locking is avoided and optimal-order convergence is achieved. This is shown through numerical experiments in reaction-diffusion problems. Copyright (c) 2008 John Wiley & Sons, Ltd.

Reaction microtextures of monazite: Correlation between chemical and age domains in the Nazare Paulista migmatite, SE Brazil

Back-scattered imaging, X-ray element mapping and electron microprobe analyzer (EMPA) chemical dating reveal complex compositional and age zoning in monazite crystals from different layers and textural positions in a garnet-bearing migmatite in SE Brazil. Y-rich (variable Y(2)O(3), averaging 2.5 wt.%) relict cores are preserved in mesosome and melanosome monazite, and correspond to 793 +/- 6 Ma inherited crystals possibly generated in a previous metamorphic event. These cores are overgrown and widely replaced by two generations of monazite, which are present in all migmatite layers. The first, also Y-rich (average 2.5 wt.% Y(2)O(3)), was produced at similar to 635 Ma during prograde metamorphism under subsolidus conditions, while the second has an Y-poor (<1.5 wt.% Y(2)O(3)), low Th/U signature, and precipitated from low Y and HREE anatectic melts produced by reactions in which garnet was inert. Quartz-rich trondhjemitic leucosome represents lower temperature melt (bearing some subsolidus quartz and garnet with included monazite) formed at temperatures below muscovite breakdown; its Y-poor monazite indicates an age of 617 +/- 6 Ma. Granitic leucosomes formed close to peak metamorphic conditions (T>750 degrees C) above muscovite breakdown have their slightly younger character confirmed by a 609 +/- 7 Ma low-Y monazite age. A similar 606 +/- 5 Ma age was obtained for low-Y monazite rims and domains in mesosome and melanosome, and reflects the time of monazite saturation in interstitial granitic melt that was trapped in these layers. Our results confirm that inherited monazite crystals can be preserved during partial melting at temperatures above muscovite breakdown. Moreover, careful textural control aided by X-ray chemical mapping may allow monazite generated at different stages in a similar to 25 Myr prograde metamorphic path to be identified and dated using an electron microprobe. (C) 2008 Elsevier B.V. All rights reserved.

Radical acetylation of 2`-deoxyguanosine and L-histidine coupled to the reaction of diacetyl with peroxynitrite in aerated medium

Diacetyl, like other alpha-dicarbonyl compounds, is reportedly cytotoxic and genotoxic. A food and cigarette contaminant, it is related with alcohol hepatotoxicity and lung disease. Peroxynitrite is a potent oxidant formed in vivo by the diffusion-controlled reaction of the superoxide radical anion with nitric oxide, which is able to form adducts with carbon dioxide and carbonyl compounds. Here, we investigate the nucleophilic addition of peroxynitrite to diacetyl forming acetyl radicals, whose reaction with molecular oxygen leads to acetate. Peroxynitrite is shown to react with diacetyl in phosphate buffer (bell-shaped pH profile with maximum at 7.2) at a very high rate constant (k(2) = 1.0 X 10(4) M-1 s(-1)) when compared with monocarbonyl substrates (k(2) < 10(3) M-1 s(-1)). Phosphate ions (100-500 MM) do not affect the rate of spontaneous peroxynitrite decay, but the H2PO4- anion catalyzes the nucleophilic addition of the peroxynitrite anion to diacetyl. The intermediacy of acetyl radicals is suggested by a three-line spectrum (a(N) = a(H) = 0.83 mT) obtained by EPR spin trapping of the reaction mixture with 2-methyl-2-nitrosopropane. The peroxynitrite reaction is accompanied by concentration-dependent oxygen uptake. Stoichiometric amounts of acetate from millimolar amounts of peroxynitrite and diacetyl were obtained under nonlimiting conditions of dissolved oxygen. In the presence of either L-histidine or 2`-deoxyguanosine, the peroxynitrite/diacetyl system afforded the corresponding acetylated molecules identified by HPLC-MS"". These studies provide evidence that the peroxynitrite/diacetyl reaction yields acetyl radicals and raise the hypothesis that protein and DNA nonenzymatic acetylation may occur in cells and be implicated in aging and metabolic disorders in which oxygen and nitrogen reactive species are putatively involved.

Acetylation of cellulose in LiCl-N,N-dimethylacetamide: first report on the correlation between the reaction efficiency and the aggregation number of dissolved cellulose

The acylation of three cellulose samples by acetic anhydride, Ac(2)O, in the solvent system LiCl/N,N-dimethylacetamide, DMAc (4 h, 110 A degrees C), has been revisited in order to investigate the dependence of the reaction efficiency on the structural characteristics of cellulose, and its aggregation in solution. The cellulose samples employed included microcrystalline, MCC; mercerized cotton linters, M-cotton, and mercerized sisal, M-sisal. The reaction efficiency expresses the relationship between the degree of substitution, DS, of the ester obtained, and the molar ratio Ac(2)O/AGU (anhydroglucose unit of the biopolymer); 100% efficiency means obtaining DS = 3 at Ac(2)O/AGU = 3. For all celluloses, the dependence of DS on Ac(2)O/AGU is described by an exponential decay equation: DS = DS(o) - Ae(-[(Ac2O/AGU)/B]); (A) and (B) are regression coefficients, and DS(o) is the calculated maximum degree of substitution, achieved under the conditions of each experiment. Values of (B) are clearly dependent on the cellulose employed: B((M-cotton)) > B((M-sisal)) > B((MCC)); they correlate qualitatively with the degree of polymerization of cellulose, and linearly with the aggregation number, N(agg), of the dissolved biopolymer, as calculated from static light scattering measurements: (B) = 1.709 + 0.034 N(agg). To our knowledge, this is the first report on the latter correlation; it shows the importance of the physical state of dissolved cellulose, and serves to explain, in part, the need to use distinct reaction conditions for MCC and fibrous celluloses, in particular Ac(2)O/AGU, time, temperature.

Ethanol oxidation reaction on PtCeO(2)/C electrocatalysts prepared by the polymeric precursor method

This paper presents a study of the electrocatalysis of ethanol oxidation reactions in an acidic medium on Pt-CeO(2)/C (20 wt.% of Pt-CeO(2) on carbon XC-72R), prepared in different mass ratios by the polymeric precursor method. The mass ratios between Pt and CeO(2) (3:1, 2:1, 1:1, 1:2, 1:3) were confirmed by Energy Dispersive X-ray Analysis (EDAX). X-ray diffraction (XRD) structural characterization data shows that the Pt-CeO(2)/C catalysts are composed of nanosized polycrystalline non-alloyed deposits, from which reflections corresponding to the fcc (Pt) and fluorite (CeO(2)) structures were clearly observed. The mean crystallite sizes calculated from XRD data revealed that, independent of the mass ratio, a value close to 3 nm was obtained for the CeO(2) particles. For Pt, the mean crystallite sizes were dependent on the ratio of this metal in the catalysts. Low platinum ratios resulted in small crystallites. and high Pt proportions resulted in larger crystallites. The size distributions of the catalysts particles, determined by XRD, were confirmed by Transmission Electron Microscope (TEM) imaging. Cyclic voltammetry and chronoamperometic experiments were used to evaluate the electrocatalytic performance of the different materials. In all cases, except Pt-CeO(2)/C 1:1, the Pt-Ceo(2)/C catalysts exhibited improved performance when compared with Pt/C. The best result was obtained for the Pt-CeO(2)/C 1:3 catalyst, which gave better results than the Pt-Ru/C (Etek) catalyst. (C) 2009 Elsevier B.V. All rights reserved.