The thin viscous flow equation in higher space dimensions

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.