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

A monograph containing significant new developments in the theory of reaction-diffusion systems, particularly those arising in chemistry and life sciences.

J. C. Meyer (Author), D. J. Needham (Author)

9781107477391, Cambridge University Press

Paperback, published 22 October 2015

173 pages
22.8 x 15.2 x 1 cm, 0.26 kg

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.

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.

Subject Areas: Mathematical modelling [PBWH], Integral calculus & equations [PBKL], Differential calculus & equations [PBKJ], Mathematics [PB]