Two generalisations of the thin film equation

This paper is concerned with two generalisations of the widely studied thin film equation u(t) = -(u(n)u(xxx))(x). Both are degenerate fourth-order parabolic equations in conservation form; the first is u(t) = - (u(n)u(xxx) + alphau(n-1)u(x)u(xx) + betau(n-2)u(x)(3))(x), which shares the scaling properties of the thin film equation (with special cases arising in applications and in earlier analyses), while the second is a doubly nonlinear equation u(t) = - (u(n)\u(xxx)\ (m-1)u(xxx))(x), which is relevant to capillary driven flows of thin films of power-law fluids, We focus on giving a characterisation of nonnegative mass preserving compactly supported solutions, exploiting local analyses about the edge of the support and special closed form solutions; however, other properties are also noted and open questions raised. (C) 2001 Elsevier Science Ltd. All rights reserved.