Asymptotic Behaviour of the Thin Film Equation in Bounded Domains

We investigate the extinction behaviour of a fourth order degenerate diffusion equation in a bounded domain, the model representing the flow of a viscous fluid over edges at which zero contact angle conditions hold. The extinction time may be finite or infinite and we distinguish between the two cases by identification of appropriate similarity solutions. In certain cases an unphysical mass increase may occur for early time and the solution may become negative; an appropriate remedy for this is noted. Numerical simulations supporting the analysis are included. 1 Introduction The role of the separable solutions u t \Gamma 1 n f(x) as t ! +1;n ? 0; (1a) and u (t c \Gamma t) \Gamma 1 n f(x) as t ! t \Gamma c ; \Gamma1 ! n ! 0; (1b) in describing the extinction behaviour of the porous medium equation @u @t = @ @x ` u n @u @x ' ; (2) subject to u = 0 at x = \Sigma1 and with suitable initial conditions, is well established (see, for example, Berryman & Holland (1980), Ar..