This publication considers parabolic problems based on the Liapunov method within the qualitative theory of ordinary differential equations. Here lie, mainly, the problems which have been investigated most thoroughly, the construction of Liapunov functionalis which naturally generalize Liapunov functions for non-linear parabolic equations of the second order with one spatial variable. The authors establish stabilizing solutions theorems, and the necessary and sufficient conditions of general and asymptotic stability of stationary solutions, including the so-called critical case. Attraction domains for stable solutions of mixed problems for these equations are described. Furthermore, estimates for the number of stationary solutions are obtained. The book should be of interest to researchers in the field of mathematical physics.