Abstract
The large-time behavior of weak nonnegative and sign changing solutions of the fourth-order thin film equation (TFE-4) with absorption where n ∈ (0, 3) and the absorption exponent p belongs to the subcritical range is 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 form Here f solves the quasi-linear degenerate elliptic equation that 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.
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