The Cauchy Problem for Non-Lipschitz Semi-Linear Parabolic Partial Differential Equations
London Mathematical Society Lecture Note Series

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.

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 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.