ABSTRACT

This chapter deals with a discussion on some connections between weak convergence in L1 and almost everywhere convergence, and first recall Vitali’s theorem. It provides a variant of the Arzela–Ascoli theorem for time-dependent functions with values in a Banach space X, but which are only continuous with respect to the weak topology of X. An important issue in the study of C-F equations is the existence of either stationary or self-similar solutions. Investigating this question on the associated stationary problems proves to be rather intractable, but, fortunately, progress can be made by utilising dynamical systems theory. The chapter examines an elementary proof of a version of Gronwall’s inequality that is due to Henry for alternative proofs and for several variants.