Theoretical aspects of the Cauchy problem for non-Lipschitz semi-linear parabolic partial differential equations

• Main Author: Meyer, John Christopher
• Published: University of Birmingham 2013
• Subjects: 510; QA Mathematics
• Online Access: https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.573521

The aim of this thesis is to provide a generic approach to the study of semi-linear parabolic partial differential equations when the nonlinearity fails to be Lipschitz continuous, but is in the class of Hӧlder continuous functions or the class of upper Lipschitz continuous functions. New results are obtained concerning the well-posedness (in the sense of Hadamard) of the initial value problem, namely, uniqueness and conditional continuous dependence results for upper Lipschitz continuous nonlinearities, and an existence result for Hӧlder continuous nonlinearities. To obtain these results, two new maximum principles have been obtained, for which examples have been provided to exhibit their applications and limitations. Additionally, new derivative estimates of Schauder-type have been obtained. Once the general theory has been established, specific problems are studied in detail. These show how one can apply the general theory, as well as problem specific approaches, to obtain well-posedness results.