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

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