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Finite speed of propagation and continuity of the interface for thin viscous flows. (English) Zbl 0846.35058
Summary: 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.

35K55 Nonlinear parabolic equations
35K65 Degenerate parabolic equations
35R35 Free boundary problems for PDEs
76D08 Lubrication theory