Hocherman and Rosenau conjectured that long-wave unstable Cahn-Hilliard-type interface models develop finite-time singularities when the nonlinearity in the destabilizing term grows faster at large amplitudes than the nonlinearity in the stabilizing term (Phys. D 67, 1993, pp. 113-125). We consider this conjecture for a class of equations, often used to model thin films in a lubrication context, by showing that if the solutions are uniformly bounded above or below (e.g., are nonnegative), then the destabilizing term can be stronger than previously conjectured yet the solution still remains globally bounded.For example, they conjecture that for the long-wave unstable equationm > n leads to blowup. Using a conservation-of-volume constraint for the case m > n > 0, we conjecture a different critical exponent for possible singularities of nonnegative solutions. We prove that nonlinearities with exponents below the conjectured critical exponent yield globally bounded solutions. Specifically, for the above equation, solutions are bounded if m < n+ 2. The bound is proved using energy methods and is then used to prove the existence of nonnegative weak solutions in the sense of distributions. We present preliminary numerical evidence suggesting that m ≥ n + 2 can allow blowup.

We consider the fourth order degenerate diffusion equation

We study the linear stability of smooth steady states of the evolution equationunder both periodic and Neumann boundary conditions. If a = 0 we assume f ≡ 1. In particular we consider positive periodic steady states of thin film equations, where a = 0 and f, g might have degeneracies such as f (0) = 0 as well as singularities like g(0) = +∞.If a 0, we prove each periodic steady state is linearly unstable with respect to volume (area) preserving perturbations whose period is an integer multiple of the steady state's period. For area-preserving perturbations having the same period as the steady state, we prove linear instability for all a if the ratio g/f is a convex function. Analogous results hold for Neumann boundary conditions. The rest of the paper concerns the special case of a = 0 and power-law coefficients f (y) = y n and g(y) = By m . We characterize the linear stability of each positive periodic steady state under perturbations of the same period. For steady states that do not have a linearly unstable direction, we find all neutral directions. Surprisingly, our instability results imply a nonexistence result: there is a large range of exponents m and n for which there cannot be two positive periodic steady states with the same period and volume.

We consider nonnegative steady-state solutions of the evolution equationformula hereOur class of coefficients f, g allows degeneracies at h = 0, such as f(0) = 0, as well as divergences like g(0) = ±∞. We first construct steady states and study their regularity. For f, g > 0 we construct positive periodic steady states, and non-negative steady states with either zero or nonzero contact angles. For f > 0 and g < 0, we prove there are no non-constant positive periodic steady states or steady states with zero contact angle, but we do construct non-negative steady states with nonzero contact angle. In considering the volume, length (or period) and contact angle of the steady states, we find a rescaling identity that enables us to answer questions such as whether a steady state is uniquely determined by its volume and contact angle. Our tools include an improved monotonicity result for the period function of the nonlinear oscillator. We also relate the steady states and their scaling properties to a recent blow-up conjecture of Bertozzi and Pugh.

Abstract.We consider the effect of a second-order 'porous media' term on the evolution of weak solutions of the fourth-order degenerate diffnsion equation ht = -V. (h"VAh -Vhm) in one space dimension. The equation without the second-order term is derived from a 'lubrication approximation' and models surface tension dominated motion of thin viscous films and spreading droplets. Here h(x. I) is the thickness of the film, and the physical problem corresponds to n = 3.For simplicity, we consider periodic boundary conditions which has the physical interpretation of modelling a penodic "ray of droplets. In a previous work we studied the above equation without the second-order 'porous media' term. In particular we showed the existence of non-negative weak solutions with increasing support for 0 < n < 3 but the techniques failed for n > 3. This is consistent with the fact that, in this case, non-negative self-similar source-type solutions do not eltist for n 2 3.In this work, we discuss a physical justification for the 'porous media' term when n = 3 and 1 -= m < 2. We propose such behaviour as a cut off of the singular 'disjoining pressure' modelling long rangc van der Waals interactions.FQI all n > 0 and 1 c m c 2, we discuss possible behaviour at the edge of the support of the solution via leading order asymptotic analysis of travelling wave solutions. This analysis predicts a certain 'competition' beween the second-and fourth-order tem. We present rigorous weak existence theory for the above. equation for all n > 0 and 1 < m c 2. In particular, the presence of a second-order 'porous media' term in the above equation yields mn-nego~ive weak solutions that converge to their mean as t -+ m and that have additional regularity. Morwver, we show that there exists a time T' after which the weak solution is a posifive strong solution.For n =-312 we show that the regularity of the weak solutions is in exact agreement with Ulat pedicled by the asymptmics. Finally, we present several numerical computations of solutions. The simulations use a weighted implicit-explicit scheme on a dynamically adaptive mesh. The numerics suggest that the weak solution described by our existence thwry has compact support with a finite speed of propagation. The data confirms the local 'power law' behaviour at the edge of the support predicted by asymptotics.A M s classification scheme numbers: 35K55.35L15