In this thesis, we study the long-time behavior of nonnegative solutions of the degenerate parabolic equation u[subscript]t = (u[superscript]m)xx + K(u[superscript]n)x, 0 0, and u(x,0) given, where m ≥ 1, subject to one of two pairs of boundary conditions: (A) u(0,t) = 0, (u[superscript]m)[subscript]x(1,t) = au[superscript]p(1,t),t > 0, or (B) -(u[superscript]m)[subscript]x(0,t) = au[superscript]p(0,t), u(1,t) = 0. Here K ≡ [epsilon] /n, a, [epsilon] > 0, and p,n ≥ m ≥ 1. Solution diagrams for the steady states are given for all cases of n,m,p > 0, and the stability or instability of each branch is obtained in the cases p,n ≥ m ≥ 1. It is shown that some solutions can blow up in finite time. Generalizations replacing u[superscript]m by [phi](u), Ku[superscript]n by f(u), and au[superscript]p by g(u) are discussed;A second part of this thesis gives a development of existence, uniqueness, and comparison principles for more general equations u[subscript]t = [phi](u)[subscript]xx + f(u)[subscript]x under the same type of mixed boundary conditions as (A) and (B) above. These results are employed in the proofs of some of the results stated above;Such equations arise in the study of flow through a porous column, e.g., natural water filtration.