ABSTRACT

This chapter reviews some basic theories and results such as Lp theory, Schauder theory and maximum principle for linear parabolic equations, which are fundamental tools in the study of nonlinear parabolic equations. The chapter starts with the L p theory and Schauder theory for the first initial-boundary value problems, second initial-boundary value problems, and mixed initial-boundary value problems. It then moves on to topics such as the Hopf boundary lemma, maximum principle of classical solutions, and strong and weak solutions for scalar equations and systems including problems with non-classical boundary conditions and non-local terms. For further discussions on the existence, uniqueness, regularity, estimation, and maximum principle of weak solutions, readers can refer to monographs.