Branching analysis of a countable family of global similarity solutions of a fourth-order thin film equation

Alvarez-Caudevilla, Pablo ; Galaktionov, Victor A.

Electronic journal of differential equations, Tome 2015 (2015)

Résumé

The main goal in this article is to justify that source-type and other global-in-time similarity solutions of the Cauchy problem for the fourth-order thin film equation

$ u_t=-\nabla \cdot (|u|^n \nabla \Delta u) \quad {in }\mathbb{R}^N \times \mathbb{R}_ + {where }n>0,\; N \ge 1 $

can be obtained by a continuous deformation (a homotopy path) as

$ u_t = - \Delta^2 u \quad{in }\mathbb{R}^N \times \mathbb{R}_ +, { where } \mathbf{B}=-\Delta^2 +\frac 14 y \cdot \nabla+ \frac N4 I. $

This approach leads to a countable family of various global similarity patterns of the thin film equation, and describes their oscillatory sign-changing behav iour by using the known asymptotic properties of the fundamental solution of bi-harmonic equation. The branching from $n=0^+$ for thin film equation requires Hermitian spectral theory for a pair $\{\mathbf{B}, \mathbf{B}^*\}$ of non-self adjoint operators and leads to a number of difficult mathematical problems. These include, as a key part, the problem of multiplicity of solutions, which is under particular scrutiny.

Détail

Classification : 35K55, 35B32, 35G20, 35K41, 35K65
Keywords: thin film equation, Cauchy problem, source-type similarity solutions, finite interfaces, oscillatory sign-changing behaviour, Hermitian spectral theory, branching