This chapter examines the Cauchy problem, or pure initial value problem, for the PME and the GPME in d-dimensional space, d = 1. It concentrates the main effort on the PME with zero forcing term. It considers solutions which are integrable with respect to the space variables, so-called solutions with finite mass, and develops the corresponding L ¹ theory. It establishes well-posedness for the Cauchy problem in this setting, which is the one most often found in relevant literature and applications.
Keywords: Cauchy problem, pure initial value problem, PME, GPME, solutions with finite mass, L1