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

Abstract

As a key example, the sixth-order doubly degenerate parabolic equation from thin film theory, u t = ( | u | m | u x x x x x | n u x x x x x ) x in R × 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 .