LONG-WAVE INSTABILITIES AND SATURATION IN THIN-FILM EQUATIONS

Hocherman and Rosenau conjectured that long-wave unstable Cahn-Hilliar d-type interface models develop finite-time singularities when the non linearity in the destabilizing term grows faster at large amplitudes t han the nonlinearity in the stabilizing term (Phys. D 67, 1993, pp. 11 3-125). We consider this conjecture for a class of equations, often us ed to model thin films in a lubrication context, by showing that if th e solutions are uniformly bounded above or below (e.g., are nonnegativ e), then the destabilizing term can be stronger than previously conjec tured yet the solution still remains globally bounded. For example, th ey conjecture that for the long-wave unstable equation h(t) = - (h(n)h (xxx))(x) - (h(m)h(x))(x), m > n leads to blowup. Using a conservation -of-volume constraint for the case m > n > 0, we conjecture a differen t critical exponent for possible singularities of nonnegative solution s. We prove that nonlinearities with exponents below the conjectured c ritical exponent yield globally bounded solutions. Specifically, for t he above equation, solutions are bounded if m < n + 2. The bound is pr oved using energy methods and is then used to prove the existence of n onnegative weak solutions in the sense of distributions. We present pr eliminary numerical evidence suggesting that m greater than or equal t o n + 2 can allow blowup. (C) 1998 John Wiley Br Sons, Inc.