The Cauchy Problem for Non-Lipschitz Semi-Linear Parabolic Partial Differential Equations

J. C. Meyer, D. J. Needham

• Publisher: Cambridge University Press
• Year: 2015
• ISBN:
  9781316311097
• Topic: Mathematics
• Subtopic: Mathematics General
• Index: Mathematics

Reaction-diffusion theory is a topic which has developed rapidly over the last thirty years, particularly with regards to applications in chemistry and life sciences. Of particular importance is the analysis of semi-linear parabolic PDEs. This monograph provides a general approach to the study of semi-linear parabolic equations when the nonlinearity, while failing to be Lipschitz continuous, is Hölder and/or upper Lipschitz continuous, a scenario that is not well studied, despite occurring often in models. The text presents new existence, uniqueness and continuous dependence results, leading to global and uniformly global well-posedness results (in the sense of Hadamard). Extensions of classical maximum/minimum principles, comparison theorems and derivative (Schauder-type) estimates are developed and employed. Detailed specific applications are presented in the later stages of the monograph. Requiring only a solid background in real analysis, this book is suitable for researchers in all areas of study involving semi-linear parabolic PDEs.

Table of contents

• Cover
• Series page
• Title page
• Copyright page
• Contents
• List of Notations
• 1 Introduction
• 2 The Bounded Reaction-Diffusion Cauchy Problem
• 3 Maximum Principles
• 4 Diffusion Theory
• 5 Convolution Functions, Function Spaces, Integral Equations and Equivalence Lemmas
• 6 The Bounded Reaction-Diffusion Cauchy Problem with f ∈ L
• 7 The Bounded Reaction-Diffusion Cauchy Problem with f ∈ L[sub(u)]
• 8 The Bounded Reaction-Diffusion Cauchy Problem with f ∈ H[sub(α)]
• 9 Application to Specific Problems
• 10 Extensions and Concluding Remarks
• References
• Index