Advances in Differential Equations

The thin viscous flow equation in higher space dimensions

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

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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

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

First available in Project Euclid: 19 April 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35K55: Nonlinear parabolic equations
Secondary: 35K65: Degenerate parabolic equations 76D08: Lubrication theory


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.

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