Stability of translating solutions to mean curvature flow

We prove stability of rotationally symmetric translating solutions to mean curvature flow. For initial data that converge spatially at infinity to such a soliton, we obtain convergence for large times to that soliton without imposing any decay rates. The authors are members of SFB 647/B3 “Raum – Zeit – Materie: Singularity Structure, Long-time Behaviour and Dynamics of Solutions of Non-linear Evolution Equations”.     

Transesterification of an RNA model in buffer solutions in H2O and D2O

Transesterification of a phosphodiester bond of RNA models has been studied in various buffer solutions, under neutral and slightly alkaline conditions in H₂O and D₂O. The results show that imidazole is the only buffer system where a clear buffer catalysis on the cleavage of a phosphodiester bond is observed. The rate enhancement in sulphonic acid buffers is smaller, and a sulphonate base, particularly, is inactive as a catalyst. The rate‐enhancing effect of imidazole is, however, catalytic, and the catalytic inactivity of sulphonate buffers can be attributed to their structure and/or charge. The catalysis by imidazole is a complex system which, in addition to first‐order reactions, involves a process that shows a second‐order dependence in imidazole concentration. The latter reaction becomes significant in acidic imidazole buffers (pH < pK_a), as the buffer concentration increases. The kinetic solvent deuterium isotope effect k_H/k_D, referring to first‐order catalysis by imidazole base, is 2.3 ± 0.3. That referring to second‐order catalysis is most probably much larger, but an accurate value could not be obtained. Copyright © 2007 John Wiley & Sons, Ltd.     

Entire radial solutions of elliptic systems and inequalities of the mean curvature type

In this paper we study first nonexistence of radial entire solutions of elliptic systems of the mean curvature type with a singular or degenerate diffusion depending on the solution u. In particular we extend a previous result given in [R. Filippucci, Nonexistence of radial entire solutions of elliptic systems, J. Differential Equations 188 (2003) 353-389]. Moreover, in the scalar case we obtain nonexistence of all entire solutions, radial or not, of differential inequalities involving again operators of the mean curvature type and a diffusion term. We prove that in the scalar case, nonexistence of entire solutions is due to the explosion of the derivative of every nonglobal radial solution in the right extremum of the maximal interval of existence, while in that point the solution is bounded. This behavior is qualitatively different with respect to what happens for the m-Laplacian operator, studied in [R. Filippucci, Nonexistence of radial entire solutions of elliptic systems, J. Differential Equations 188 (2003) 353-389], where nonexistence of entire solutions is due, even in the vectorial case, to the explosion in norm of the solution at a finite point. Our nonexistence theorems for inequalities extend previous results given by Naito and Usami in [Y. Naito, H. Usami, Entire solutions of the inequality div(A(|Du|)Du)?f(u), Math. Z. 225 (1997) 167-175] and Ghergu and Radulescu in [M. Ghergu, V. Radulescu, Existence and nonexistence of entire solutions to the logistic differential equation, Abstr. Appl. Anal. 17 (2003) 995-1003].     

Asymptotic behavior of solutions for a class of predator-prey reaction-diffusion systems with time delays

The aim of this paper is to investigate the asymptotic behavior of solutions for a class of three-species predator-prey reaction-diffusion systems with time delays under homogeneous Neumann boundary condition. Some simple and easily verifiable conditions are given to the rate constants of the reaction functions to ensure the convergence of the time-dependent solution to a constant steady-state solution. The conditions for the convergence are independent of diffusion coefficients and time delays, and the conclusions are directly applicable to the corresponding parabolic-ordinary differential system and to the corresponding system without time delays.     

Continuity of the Solution Map in Parametric Affine Variational Inequalities

A systematic study of the upper semicontinuity and the lower semicontinuity of the solution map in parametric affine variational inequalities is given in this paper. Several examples are constructed to analyze the results. This work was supported by Korea Research Foundation Grant (KRF 2001-015-DP0049), the APEC Postdoctoral Fellowships Program, and the KOSEF Brain Pool Program.     

The solution existence of general variational inclusion problems

We propose general variational inclusion problems which are slightly different from corresponding problems considered in several recent papers in the literature and show that they are advantageous. Sufficient conditions for the solution existence are established. As applications we derive consequences for several special cases of variational inclusion problems, quasioptimization problems, equilibrium problems and implicit variational inequalities and show that they improve the results of some recent existing papers.     

Monotonic solutions of a quadratic integral equation of fractional order

In the paper we indicate an error made in the proof of the main result of the paper [M.A. Darwish, On quadratic integral equation of fractional orders, J. Math. Anal. Appl. 311 (2005) 112-119]. Moreover, we provide correct proof of a slightly modified version of the mentioned result. The main tool used in our proof is the technique associated with the Hausdorff measure of noncompactness.     

GLOBAL WEAK SOLUTIONS OF THE RELATIVISTIC VLASOV-KLEIN-GORDON SYSTEM IN TWO DIMENSIONS

In this paper we consider the system of classical particles coupled with a Klein-Gordon field in two dimensions.We establish a-priori-bounds on the solutions of this system with initial data satisfying a size restriction derived from conservation of energy.This result,together with the smoothing of"velocity averaging",yields the existence of global weak solutions to the corresponding restricted initial value problem.The size restriction is necessary since energy of the system is indefinite.Finally,we show that the weak solutions preserve the total mass.     

Global smooth solutions to the multi-dimensional hydrodynamic model for two-carrier plasmas

The existence of global smooth solutions to the multi-dimensional hydrodynamic model for plasmas of electrons and positively charged ions is shown under the assumption that the initial densities are close to a constant. The model consists of the conservation laws for the particle densities and the current densities, coupled to the Poisson equation for the electrostatic potential. Furthermore, it is proved that the particle densities converge exponentially fast to the (constant) steady state. The proof uses a higher-order energy method inspired from extended thermodynamics.     

Idempotency of Extensions via the Bicompletion

Let Top ₀ be the category of topological T ₀-spaces, QU ₀ the category of quasi-uniform T ₀-spaces, T : QU ₀ → Top ₀ the usual forgetful functor and K : QU ₀ → QU ₀ the bicompletion reflector with unit k : 1 → K. Any T-section F : Top ₀ → QU ₀ is called K-true if KF = FTKF, and upper (lower) K-true if KF is finer (coarser) than FTKF. The literature considers important T-sections F that enjoy all three, or just one, or none of these properties. It is known that T(K,k)F is well-pointed if and only if F is upper K-true. We prove the surprising fact that T(K,k)F is the reflection to Fix(TkF) whenever it is idempotent. We also prove a new characterization of upper K-trueness. We construct examples to set apart some natural cases. In particular we present an upper K-true F for which T(K,k)F is not idempotent, and a K-true F for which the coarsest associated T-preserving coreflector in QU ₀ is not stable under K. We dedicate this paper to the memory of Sérgio de Ornelas Salbany (1941–2005).