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- W2065295443 abstract "We consider the fourth-order thin film equation, with a stable second-order diffusion term. For the first critical exponent, where N ≥ 1 is the space dimension, the Cauchy problem is shown to admit countable continuous branches of source-type self-similar very singular solutions of the form These solutions are inherently oscillatory in nature and will be shown in Part II to be the limit of appropriate free-boundary problem solutions. For p ≠ p 0 , the set of very singular solutions is shown to be finite and to be consisting of a countable family of branches (in the parameter p ) of similarity profiles that originate at a sequence of critical exponents { p l , l ≥ 0}. At p = p l , these branches appear via a non-linear bifurcation mechanism from a countable set of similarity solutions of the second kind of the pure thin film equation Such solutions are detected by the ‘Hermitian spectral theory’, which allows an analytical n -branching approach. As such, a continuous path as n → 0 + can be constructed from the eigenfunctions of the linear rescaled operator for n = 0, i.e. for the bi-harmonic equation u t = −Δ 2 u . Numerics are used, wherever appropriate, to support the analysis." @default.
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- W2065295443 date "2011-02-21" @default.
- W2065295443 modified "2023-09-26" @default.
- W2065295443 title "On continuous branches of very singular similarity solutions of a stable thin film equation. I – The Cauchy problem" @default.
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- W2065295443 doi "https://doi.org/10.1017/s0956792511000039" @default.
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