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On the Benilov-Vynnycky blow-up problem

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  • We study an initial-boundary value problem for a fourth-order parabolic partial differential equation with an unknown velocity. The equation originated from the linearization of a two-dimensional Couette flow model, that was recently proposed by Benilov and Vynnycky. In the case of a $180^{\circ}$-- contact angle between liquid and a moving plate Benilov and Vynnycky conjectured that the speed of the contact line blows up to infinity in finite time. In this paper we present numerical simulations and qualitative analysis of the model. We show that depending on the initial data and parameter values different long time behaviors of velocity can be observed. The speed of the contact line may blow up to infinity or converge to a constant.
    Mathematics Subject Classification: Primary: 76D08, 35K30; Secondary: 65M06.


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