Travelling wavefronts of reaction-diffusion equations in cylindrical domains

Abstract

Function spaces with asymptotics is a usual tool in the analysis on manifolds with singularities. The asymptotics are singular ingredients of the kernels of pseudodifferential operators in the calculus. They correspond to potentials supported by the singularities of the manifold, and in this form asymptotics can be treated already on smooth configurations. This article is aimed at describing refined asymtotics in the Dirichlet problem in a ball. The beauty of explicit formulas actually highlights the structure of asymptotic expansions in the calculi on singular varieties.     

Optimal control of self-organized dynamics in cellular signal transduction

We demonstrate how model-based optimal control can be exploited in biological and biochemical modelling applications in several ways. In the first part, we apply optimal control to a detailed kinetic model of a glycolysis oscillator, which plays a central role in immune cells, in order to analyse potential regulatory mechanisms in the dynamics of associated signalling pathways. We demonstrate that the formulation of inverse problems with the aim to determine specific time-dependent input stimuli can provide important insight into dynamic regulations of self-organized cellular signal transduction. In the second part, we present an optimal control study aimed at target-oriented manipulation of a biological rhythm, an internal clock mechanism related to the circadian oscillator. This oscillator is responsible for the approximate endogenous 24 h (latin: circa dies) day-night rhythm in many organisms. On the basis of a kinetic model for the fruit fly Drosophila, we compute switching light stimuli via mixed-integer optimal control that annihilate the oscillations for a fixed time interval. Insight gained from such model-based specific manipulation may be promising in biomedical applications.     

Asymptotic expansions for ornstein-uhlenbeck semigroups perturbed by potentials over banach spaces

An asymptotic expansion of Schilder-type integrals with general phase function on abstract Wiener spaces is given and good control on remainders is obtained. For Ornstein –Uhlenbeck semigroups perturbed by potentials on Banach spaces the asymptotic expansion is given in terms of explicitly discussed “classical orbits”, in the case of finitely many non-degenerate maxima of the phase function. A representation of the leading term by a solution of an infinite dimensional Sturm-Liouville problem is also provided     

Probabilistic gradient approximation for a viscous scalar conservation law in space dimension d≥2

In this paper, we are interested in the solution of a viscous scalar conservation law. We remark that its first order spatial derivatives solve a system of partial differential equations presenting a nonlocal nonlinearity. We associate a nonlinear martingale problem with this system. After proving existence and uniqueness for the martingale problem, we obtain a propagation of chaos result for a system of interacting diffusion processes. We deduce that it is possible to approximate the solution of the viscous scalar conservation law thanks to the interacting diffusions     

Markov processes generated by linear stochastic evolution equations †

A class of Hilbert space-valued Markov processes which can be expressed as the mild solution of a linear abstract evolution equation is studied. Sufficient conditions for the generator of the Markov process to be well-defined are given and Kolmogorov's equation and an equation for the characteristic function of the process are derived. The theory is illustrated by examples of parabolic, hyperbolic and delay stochastic differential equations.     

A central limit theorem in nonlinear filtering

It is shown that the probability law of a diffusion process conditioned on weakly corrupted observations is asymptotically Gaussian when properly scaled. The method of proof involves Fisher information matrices and a Cramér-Rao inequality.     

On the estimation of nonlinear time series models

In the present paper, a framework for parametric estimation in nonlinear time series is developed. Strong consistency and asymptotic normality of minimum Hellinger distance estimates for a determined class of nonlinear models are investigated. The main Interest for these estimates is motivated by their robustness under perturbations as it has been emphazized in Beran [2]. The first part of the paper is devoted to the study of some probabilistic properties which ensure the existence and the optimal properties of the estimates     

A BARRIER BOUNDARY VALUE PROBLEM FOR PARABOLIC AND ELLIPTIC EQUATIONS

Barrier type boundary conditions are modeled for describing the substance diffusion in a medium when obstacles in the medium are considered. The existence of solutions of parabolic and elliptic differential equations with barrier boundary condition is presented in this paper.     

Nonlinear Wave Equations in Infinite Waveguides

Abstract

We present sharp decay rates as time tends to infinity for solutions to linear Kleinc-Gordon and wave equations in domains with infinite boundaries like infinite waveguides, as well as the global well-posedness and the asymptotics for small data for the solutions to the associated nonlinear initial-boundary value problems.