# The thin film equation with backwards second order diffusion - Mathematics > Analysis of PDEs

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Abstract: In this paper, we focus on the thin film equation with lower order-backwards- diffusion which can describe, for example, the evolution of thinviscous films in the presence of gravity and thermo-capillary effects, or thethin film equation with a -porous media cutoff- of van der Waals forces. Wetreat in detail the equation $$u t + \{u^nu {xxx} + u u^{m-n}u x -A u^{M-n}u x\} x=0,$$ where $u=\pm 1,$ $n>0,$ $M>m,$ and $A \ge 0.$ Global existenceof weak nonnegative solutions is proven when $m-n> -2$ and $A>0$ or $u=-1,$and when $-2< m-n<2,$ $A=0,$ $u=1.$ From the weak solutions, we get strongentropy solutions under the additional constraint that $m-n> -{3}-{2}$ if$u=1.$ A local energy estimate is obtained when $2 \le n<3$ under someadditional restrictions. Finite speed of propagation is proven when $m>n-2,$for the case of -strong slippage,- \$0

Autor: Amy Novick-Cohen, Andrey Shishkov

Fuente: https://arxiv.org/