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1996 Finite speed of propagation and continuity of the interface for thin viscous flows
Francisco Bernis
Adv. Differential Equations 1(3): 337-368 (1996).


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.


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Francisco Bernis. "Finite speed of propagation and continuity of the interface for thin viscous flows." Adv. Differential Equations 1 (3) 337 - 368, 1996.


Published: 1996
First available in Project Euclid: 25 April 2013

zbMATH: 0846.35058
MathSciNet: MR1401398

Primary: 35K65
Secondary: 35K55 , 35R35

Rights: Copyright © 1996 Khayyam Publishing, Inc.


Vol.1 • No. 3 • 1996
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