Pablo Alvarez-Caudevilla, Victor A. Galaktionov
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
can be obtained by a continuous deformation (a homotopy path) as . This is done by reducing to similarity solutions (given by eigenfunctions of a rescaled linear operator ) of the classic bi-harmonic equation
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 for thin film equation requires Hermitian spectral theory for a pair 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.
Submitted April 11, 2014. Published April 10, 2015.
Math Subject Classifications: 35K55, 35B32, 35G20, 35K41, 35K65.
Key Words: Thin film equation; Cauchy problem; source-type similarity solutions; finite interfaces; oscillatory sign-changing behaviour; Hermitian spectral theory; branching.
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| Pablo Alvarez-Caudevilla |
Universidad Carlos III de Madrid
Av. Universidad 30, 28911-Leganes, Spain
email: email@example.com Phone: +34-916249099
| Victor A. Galaktionov |
Department of Mathematical Sciences
University of Bath
Bath BA2 7AY, UK
email: firstname.lastname@example.org Phone: +44-1225826988
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