## Advances in Differential Equations

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

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

#### 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

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1366896043

**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.