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.