On a Fourth Order Degenerate Parabolic Equation: Global Entropy Estimates, Existence and Qualitative Behaviour of Solutions

By means of energy and entropy estimates, we prove existence and positivity results in higher space dimensions for degenerate parabolic equations of fourth order with nonnegative initial values. We discuss their asymptotic behaviour for t ! 1 and give a counterexample to uniqueness. 0. Introduction In this paper we will present new results on existence, (non)uniqueness, positivity and asymptotic behaviour in higher space dimensions of solutions to degenerate parabolic equations of fourth order of the form: u t + div ? m(u)ru =0 in (0; T); @u @ = @ @ u =0 on @ 0; T]; (1) u(0;) =u 0 () in : We assume that the nonnegative diiusion coeecient m vanishes at zero and has at most polynomial growth. We denote by n its growth exponent near zero. Equation (1) can be seen as the archetype of a class of parabolic equations of higher order which appear in material sciences and uid dynamics. For instance, in lubrication theory (cf. 3], 8] and the references therein), u describes the height of a viscous droplet spreading on a plain, solid surface; in the Cahn-Hilliard model of phase separation for binary mixtures, u plays the role of the concentration of one component (cf. 10]), and in a plasticity model (cf. 13] and the references therein) u stands for the density of disloca-tions. Crucial for these applications is the fact, that it is possible to construct solutions of (1) which preserve nonnegativity as has been proved for space dimension N = 1 by Bernis and Friedman 6] and for higher space dimensions in the papers by Gr un 13] and by Elliott, Garcke 10]. This behaviour is in strong contrast to that of solutions to nondegenerate parabolic equations of fourth order which in general become negative even in the case of strictly positive initial values. Moreover, the publications of Beretta, Bertsch, Dal Passo 2] and of Bertozzi, Pugh 8] who study this equation in space dimension N = 1 reveal a rich structure of qualitative behaviour of solutions depending on the diiusion growth exponent n. To put it concisely, the larger n is, the stronger is the tendency of solutions to stay positive and the weaker is the regularity at the boundary of the set where u vanishes.