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, ut=(|u|m|uxxxxx|nuxxxxx)xinR×R+, with two parameters, n≥0 and m∈(−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≈1.