Nonnegative solutions of a fourth-order nonlinear degenerate parabolic equation

Abstract

ELENA BERETTA, MICHIEL BERTSCH & ROBERTA DAL PASSO Communicated by M. GUaTIN O. Introduction In this paper we study nonnegative solutions of the fourth-order equation (0.1) ut + (u"uxxx)x = O, where n is a real positive constant. Equation (0.1) is a nonlinear parabolic partial differential equation which is degenerate at points at which u vanishes. It arises in several applications (see the references in [2, 4, 5, 6]) in which u(x, t) stands for a nonnegative quantity. For example, in the study of the dynamics of thin liquid films u denotes the thickness of the film or another relevant (nonnegative) quantity. The case n = 1 models the flow in a Hele-Shaw cell, while n = 3 corresponds to viscous flow on a solid surface without slip driven by surface tension (see also [6, 7, 8] for a related equation). It is well known that fourth-order equations do not satisfy a maximum principle. In particular, classical solutions of the linear equation ut + u .... ---- 0 may become negative in finite time. BERNIS 8~; FRIEDMAN [-2] have observed that if n is positive, equation (0.1) has (generalized) solutions which remain nonnegative for all times. In addition, they show that the qualitative positivity of these solutions strongly depends on the value of n. In more recent papers by KADANOFF et al., a combination of numerical calculations and matched asymptotics [4] and the analysis of travelling-wave solutions [-5] leads to results which suggest a very rich structure of the solutions of (0.1). It is the purpose of the present paper to clarify several aspects of the nonlinear equation (0.1) and its dependence on the parameter n. Our results lead to several conjectures and open problems. As was pointed out in [4], for sufficiently small values of n, solutions of (0.1) which are initially strictly positive may vanish at some point x 0 after a finite time to, but, to the best of our knowledge, there is no rigorous proof of this phenomenon in the literature. On the other hand, it is of particular interest for applications; for example, if u denotes the thickness of a liquid film, it implies that at time to the film breaks at xo.