Variational Convergence and Stochastic Homogenization of Nonlinear Reaction-Diffusion Problems

Variational Convergence and Stochastic Homogenization of Nonlinear Reaction-Diffusion Problems

Omar Anza Hafsa, Jean-Philippe Mandallena, Gérard Michaille

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Description

A substantial number of problems in physics, chemical physics, and biology, are modeled through reaction-diffusion equations to describe temperature distribution or chemical substance concentration. For problems arising from ecology, sociology, or population dynamics, they describe the density of some populations or species. In this book the state variable is a concentration, or a density according to the cases. The reaction function may be complex and include time delays terms that model various situations involving maturation periods, resource regeneration times, or incubation periods. The dynamics may occur in heterogeneous media and may depend upon a small or large parameter, as well as the reaction term. From a purely formal perspective, these parameters are indexed by n. Therefore, reaction-diffusion equations give rise to sequences of Cauchy problems.

The first part of the book is devoted to the convergence of these sequences in a sense made precise in the book. The second part is dedicated to the specific case when the reaction-diffusion problems depend on a small parameter ∊ₙ intended to tend towards 0. This parameter accounts for the size of small spatial and randomly distributed heterogeneities. The convergence results obtained in the first part, with additionally some probabilistic tools, are applied to this specific situation. The limit problems are illustrated through biological invasion, food-limited or prey-predator models where the interplay between environment heterogeneities in the individual evolution of propagation species plays an essential role. They provide a description in terms of deterministic and homogeneous reaction-diffusion equations, for which numerical schemes are possible.

Contents:

  • Preface
  • Introduction
  • Sequences of Reaction-Diffusion Problems: Convergence:
    • Variational Convergence of Nonlinear Reaction-Diffusion Equations
    • Variational Convergence of Nonlinear Distributed Time Delays Reaction-Diffusion Equations
    • Variational Convergence of Two Components Nonlinear Reaction-Diffusion Systems
    • Variational Convergence of Integrodifferential Reaction-Diffusion Equations
    • Variational Convergence of a Class of Functionals Indexed by Young Measures
  • Sequences of Reaction-Diffusion Problems: Stochastic Homogenization:
    • Stochastic Homogenization of Nonlinear Reaction-Diffusion Equations
    • Stochastic Homogenization of Nonlinear Distributed Time Delays Reaction-Diffusion Equations
    • Stochastic Homogenization of Two Components Nonlinear Reaction-Diffusion Systems
    • Stochastic Homogenization of Integrodifferential Reaction-Diffusion Equations
    • Stochastic Homogenization of Non Diffusive Reaction Equations and Memory Effect
  • Appendices:
    • Grönwall Type Inequalities
    • Basic Notions on Variational Convergences
    • Ergodic Theory and Subadditive Processes
    • Large Deviations Principle
    • Measure Theory
    • Inf-Convolution and Parallel Sum
  • Bibliography
  • Notation
  • Index

Readership: Graduate students and researchers.

Key Features:

  • The convergence of reaction-diffusion problems or systems with possibly time delays terms, integrodifferential operators, and random coefficients, is not approached in a unified way in the mathematical literature. This book covers this lack
  • The choice retained for the drafting, which move gradually towards an abstract framework, must allow students and researchers to familiarize themselves with the variational convergences adapted to the framework of PDEs which may include reaction terms
  • Attention is paid to applications; in particular to the convergence of reaction-diffusion equations with random coefficients resulting from the modeling of heterogeneous systems steming from physics, biology or population dynamics
  • Thanks to numerous examples and the help of the appendix, this book may be of interest to students and researchers in a first approach to ecosystems or physical modellings


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Omar Anza Hafsa:
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