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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.

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