Strong Maximum and Comparison Principles for Degenerate and Singular Parabolic Equations

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Analytic investigations of the validity of the strong maximum and comparison principles for degenerate and singular quasi-linear parabolic partial differential equations.This research project is concerned with the open problem of the validity of the strong maximum and comparison principles for degenerate and singular quasi-linear parabolic partial differential equations.This question has been partially answered in the monograph by Pucci and Serrin for the case of degenerate and singular quasi-linear elliptic partial differential equations. However, this problem is almost entirely open for the parabolic case. Further closely related problems to this question arethe compact support principle, Harnack's inequalities, and the uniqueness and the regularity of weak solutions. The research will be focused primarily on the Hopf comparison principle for degenerate and singular quasi-linear parabolic partial differential equations. The methods of investigation will be taken from the theories of Partial Differential Equations, Calculus of Variations, and Nonlinear Functional Analysis.