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

Part of London Mathematical Society Lecture Note Series

  • Date Published: January 2016
  • availability: This ISBN is for an eBook version which is distributed on our behalf by a third party.
  • format: Adobe eBook Reader
  • isbn: 9781316311097

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  • 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.

    • A novel new approach to the study of semi-linear parabolic PDEs, of interest to those working in reaction-diffusion theory and its applications
    • Presents a number of specific applications in combustion, autocatalysis, biochemical reactions, epidemiology and population dynamics
    • Requires only a solid appreciation of real analysis, making it suitable for a wide range of researchers in applied mathematics and the theoretical aspects of physical, chemical and biological sciences
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    Product details

    • Date Published: January 2016
    • format: Adobe eBook Reader
    • isbn: 9781316311097
    • availability: This ISBN is for an eBook version which is distributed on our behalf by a third party.
  • Table of Contents

    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 e L
    7. The bounded reaction-diffusion Cauchy problem with f e Lu
    8. The bounded reaction-diffusion Cauchy problem with f e La
    9. Application to specific problems
    10. Concluding remarks.

  • Authors

    J. C. Meyer, University of Birmingham
    J. C. Meyer is University Fellow in the School of Mathematics at the University of Birmingham, UK. His research interests are in reaction-diffusion theory.

    D. J. Needham, University of Birmingham
    D. J. Needham is Professor of Applied Mathematics at the University of Birmingham, UK. His research areas are applied analysis, reaction-diffusion theory and nonlinear waves in fluids. He has published over 100 papers in high-ranking journals of applied mathematics, receiving over 2000 citations.

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