## Advances in Differential Equations

- Adv. Differential Equations
- Volume 3, Number 3 (1998), 417-440.

### The thin viscous flow equation in higher space dimensions

Michiel Bertsch, Roberta Dal Passo, Harald Garcke, and Günther Grün

#### Abstract

We prove local integral (entropy) estimates for nonnegative solutions of the fourth-order degenerate parabolic equation $$ u_t+ \div (u^n\nabla\Delta u)=0 $$ in space dimensions two and three. These estimates enable us to show that solutions have finite speed of propagation if $n\in(\frac 18,2)$ and that the support cannot shrink if the growth exponent $n$ is larger than $3/2$. In addition, we prove decay estimates for solutions of the Cauchy problem and a growth estimate for their support.

#### Article information

**Source**

Adv. Differential Equations, Volume 3, Number 3 (1998), 417-440.

**Dates**

First available in Project Euclid: 19 April 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1366399848

**Mathematical Reviews number (MathSciNet)**

MR1751951

**Zentralblatt MATH identifier**

0954.35035

**Subjects**

Primary: 35K55: Nonlinear parabolic equations

Secondary: 35K65: Degenerate parabolic equations 76D08: Lubrication theory

#### Citation

Bertsch, Michiel; Dal Passo, Roberta; Garcke, Harald; Grün, Günther. The thin viscous flow equation in higher space dimensions. Adv. Differential Equations 3 (1998), no. 3, 417--440. https://projecteuclid.org/euclid.ade/1366399848