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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
##### Keywords:
lubrication models; finite speed of propagation