# The Cauchy Problem for NonLipschitz SemiLinear Parabolic Partial Differential Equations: 419 (London Mathematical Society Lecture Note Series, Series Number 419)

Cambridge University Press, 10/22/2015

EAN 9781107477391, ISBN10: 1107477395

Paperback, 176 pages, 22.9 x 15.2 x 1 cm

Language: English

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 HÃƒÂ¶lder 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.