Electron. J. Diff. Equ.,
Vol. 2015 (2015), No. 90, pp. 129.
Branching analysis of a countable family of global
similarity solutions of a fourthorder thin film equation
Pablo AlvarezCaudevilla, Victor A. Galaktionov
Abstract:
The main goal in this article is to justify that sourcetype and
other globalintime similarity solutions of the Cauchy problem
for the fourthorder 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 biharmonic equation
This approach leads to a countable family of various global similarity patterns
of the thin film equation, and describes their oscillatory signchanging
behav iour by using the known asymptotic properties of the fundamental
solution of biharmonic equation.
The branching from
for thin film equation requires Hermitian spectral
theory for a pair
of nonself 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; sourcetype similarity solutions;
finite interfaces; oscillatory signchanging behaviour;
Hermitian spectral theory; branching.
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Pablo AlvarezCaudevilla
Universidad Carlos III de Madrid
Av. Universidad 30, 28911Leganes, Spain
email: pacaudev@math.uc3m.es Phone: +34916249099


Victor A. Galaktionov
Department of Mathematical Sciences
University of Bath
Bath BA2 7AY, UK
email: vag@maths.bath.ac.uk Phone: +441225826988

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