### Finite speed of propagation and continuity of the interface for thin viscous flows

Francisco Bernis

#### Abstract

We consider the fourth-order nonlinear degenerate parabolic equation $$u_t+(|u|^n u_{xxx})_x=0$$ which arises in lubrication models for thin viscous films and spreading droplets as well as in the flow of a thin neck of fluid in a Hele-Shaw cell. We prove that if $0<n<2$ this equation has finite speed of propagation for nonnegative strong" solutions and hence there exists an interface or free boundary separating the regions where $u>0$ and $u=0$. Then we prove that the interface is Hölder continuous if $1/2<n<2$ and right-continuous if $0<n\leq 1/2$. Finally we study the Cauchy problem and obtain optimal asymptotic rates as $t\to\infty$ for the solution and for the interface when $0<n<2$; these rates exactly match those of the source-type solutions. If $0<n<1$ the property of finite speed of propagation is also proved for changing sign solutions.

#### Article information

Source
Adv. Differential Equations Volume 1, Number 3 (1996), 337-368.

Dates
First available in Project Euclid: 25 April 2013

Mathematical Reviews number (MathSciNet)
MR1401398

Zentralblatt MATH identifier
0846.35058

Subjects
Primary: 35K65: Degenerate parabolic equations
Secondary: 35K55: Nonlinear parabolic equations 35R35: Free boundary problems

#### Citation

Bernis, Francisco. Finite speed of propagation and continuity of the interface for thin viscous flows. Adv. Differential Equations 1 (1996), no. 3, 337--368. https://projecteuclid.org/euclid.ade/1366896043.