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Very Singular Solutions for Thin Film Equations with Absorption
Author(s) -
Galaktionov V. A.
Publication year - 2010
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/j.1467-9590.2009.00461.x
Subject(s) - mathematical analysis , degenerate energy levels , mathematics , similarity solution , bounded function , zero (linguistics) , sign (mathematics) , similarity (geometry) , absorption (acoustics) , exponent , parameterized complexity , initial value problem , ode , boundary (topology) , boundary value problem , boundary layer , physics , image (mathematics) , combinatorics , optics , thermodynamics , quantum mechanics , linguistics , philosophy , artificial intelligence , computer science
The large‐time behavior of weak nonnegative and sign changing solutions of the fourth‐order thin film equation (TFE‐4) with absorptionwhere n ∈ (0, 3) and the absorption exponent p belongs to the subcritical rangeis studied. First, the standard free‐boundary problem with zero‐height, zero contact angle, and zero‐flux conditions at the interface and bounded compactly supported initial data is considered. Very singular similarity solutions (VSSs) have the formHere f solves the quasi‐linear degenerate elliptic equationthat becomes an ODE for N = 1 or in the radial setting in . By a combination of analytical, asymptotic, and numerical methods, existence of various branches of similarity profiles f parameterized by p is established. Secondly, changing sign VSSs of the Cauchy problem are described. This study is motivated by the detailed VSS results for the second‐order porous medium equation with absorption ( u ≥ 0)which have been known since the 1980s.