On source-type solutions and the Cauchy problem for a doubly degenerate sixth-order thin film equation, I: Local oscillatory properties

As a key example, the sixth-order doubly degenerate parabolic equation from thin film theory,

u(t) = (vertical bar u vertical bar(m)vertical bar u(xxxxx)vertical bar(n)u(xxxxx))(x) in R x R+,

with two parameters, n >= 0 and m is an element of (-n, n + 2), is considered. In this first part of the research, various local properties of its particular travelling wave and source-type solutions are studied. Most complete analytic results on oscillatory structures of these solutions of changing sign are obtained for m = 1 by an algebraic-geometric approach, with extension by continuity for m approximate to 1.