Comparison Principle, Regularity, and Uniform Estimates

In this chapter, the authors mainly study comparison principle, regularity and uniform estimates, and uniform bounds from bounded L^p norms. Comparison principle based on the maximum principle plays an important role in estimating upper and/or lower bounds of solutions and judging the sign of solution, as well as establishing the upper and lower solutions method. Regularities and uniform estimates based on the L^p theory and Schauder theory are fundamental problems in the study of parabolic equations. The regularity theory is used to investigate smoothness of solutions. In order to prove global existence of classical or mild solutions of parabolic equations, some priori bounds in the uniform norm are needed. Some interesting examples are given to show that one can use the structure of reaction terms to obtain uniform bounds from bounded L^p norms.