# Interfaces and Free Boundaries

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**Volume 12, Issue 4, 2010, pp. 463–496**

**DOI: 10.4171/IFB/242**

The thin film equation with backwards second order diffusion

Amy Novick-Cohen^{[1]}and Andrey Shishkov

^{[2]}(1) Department of Mathematics, Technion - Israel Institute of Technology, Technion City, 32000, HAIFA, ISRAEL

(2) Institute of Applied Mathematics and Mechanics, Academy of Sciences of Ukraine, R. Luxemburg St. 74, 83114, DONETSK, UKRAINE

We focus on the thin film equation with lower order “backwards” diffusion which can describe, for example, the evolution of thin viscous films in the presence of gravity and thermo-capillary effects, or the thin film equation with a “porous media cutoff” of van der Waals forces. We treat in detail the equation

*u _{t}* + {

*u*(

^{n}*u*+ν

_{xxx}*u*–

^{m–n}u_{x}*Au*)}

^{M–n}u_{x}*= 0,*

_{x}where ν = +/– 1,

*n*> 0,

*M*>

*m*, and

*A*≥ 0. Global existence of weak nonnegative solutions is proven when

*m–n*> –2 and

*A*> 0 or ν = –1, and when –2 <

*m–n*< 2,

*A*= 0, ν = 1. From the weak solutions, we get strong entropy solutions under the additional constraint that

*m–n*> –3/2 if ν = 1. A local energy estimate is obtained when 2 ≤ n < 3 under some additional restrictions. Finite speed of propagation is proven when

*m*>

*n*/2, for the case of “strong slippage”, 0 <

*n*< 2, when ν = 1 based on local entropy estimates, and for the case of “weak slippage”, 2 ≤

*n*< 3, when ν = +/–1 based on local entropy and energy estimates.

*Keywords: *Thin film equation; backwards diffusion; higher order parabolic equations; degenerate parabolic equations; finite speed of propagation

Novick-Cohen A, Shishkov A. The thin film equation with backwards second order diffusion. *Interfaces Free Bound.* 12 (2010), 463-496. doi: 10.4171/IFB/242