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,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.

The Cauchy problem for a tenth-order thin film equation II. Oscillatory source-type and fundamental similarity solutions

“…Equation (1.1) has been chosen as a typical difficult higher-order quasilinear degenerate parabolic model. Though the fourth-order version has been the most studied actually, since 1980s (and, later on, in the 2000s, various six-order ones), higher-order quasilinear degenerate equations are known to occur in several applications and, during the last ten-fifteen years, have began to steadily penetrate into modern nonlinear PDE theory; see a number of references/results in [12, § 1.1] and in [7,21,22]. Concerning the origin of the TFE-10 (1.1), as in [1], honestly, we have chosen this very difficult model in order to develop mathematical PDE techniques showing that such complicated quasilinear degenerate equations, anyway, admit a rather constructive study regardless their order.…”

On a fractional thin film equation

“…where K can be a local or nonlocal operator, even of higher order than 2. The case K = (−∆) m with m > 1 has been first studied to our knowledge in [12] and then in [34,32,33,24] and others. Work is mostly done in one space dimension.…”

The Cauchy Problem for a Tenth-Order Thin Film Equation I. Bifurcation of Oscillatory Fundamental Solutions

“…where b(x, t) is the fundamental solution of the operator D t − ∆ 5 . By the apparent connection between (1.1) and (1.7) (when n = 0), intuitively at least, this analysis provides us with a way to understand the CP for the TFE-10 by using the fact that the proper solution of the CP for (1.1), with the same initial data u 0 , is that one which converges to the corresponding unique solution of the CP for (3.1), as n → 0.…”

“…Equation (1.1) has been chosen as a typical higher-order quasilinear degenerate parabolic model, which is very difficult to study, and this is key for us; see below. Although the fourth-order version has been the most studied, the sixth-order TFE is known to occur in several applications and, during the last ten-fifteen years, has begun to steadily penetrate into modern nonlinear PDE theory; see references in [10, § 1.1] and more recently [5,19,20,21] and [17,3] where several applications of these problems are described, in particular image processing.…”