We study a generalized class of supersolutions, so-called supercaloric
functions to the porous medium equation in the fast diffusion case.
Supercaloric functions are defined as lower semicontinuous functions obeying a
parabolic comparison principle. We prove that bounded supercaloric functions
are weak supersolutions. In the supercritical range, we show that unbounded
supercaloric functions can be divided into two mutually exclusive classes
dictated by the Barenblatt solution and the infinite point-source solution, and
give several characterizations for these classes. Furthermore, we study the
pointwise behavior of supercaloric functions and obtain connections between
supercaloric functions and weak supersolutions.Comment: Corrected typographical errors and made minor notational adjustment