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The thin film equation with nonlinear diffusion. (English) Zbl 1001.35070
The paper deals with the fourth-order degenerate parabolic equation \[ u_t + \text{div}(u^n \bigtriangledown \Delta u - u^m \bigtriangledown u) = 0 \] where \(m \in \mathbb{R}\), \(n \in \mathbb{R}^+\). The unknown function \(u = u(x,t)\) (with \(t \in \mathbb{R}^+\), \(x\in \mathbb{R}^N\), \(N=1,2,3)\) is supposed to be nonnegative and describes the evolution of the height of a thin liquid film spreading on a solid surface. Two boundary value problems are considered: the Neumann problem (P) in a bounded domain with smooth boundary and the Cauchy problem (C) with compactly supported initial data. The paper contains theorems (given with detailed proofs) assuring the existence of a strong solution of (P), (C) and presents certain properties of this solution. Particularly it is proved that for \(m>0\), \(\frac 18 < n <2\) the solution propagates with finite speed. Some additional results concerning solutions of (C) with measure-valued data are given as well.

35K65 Degenerate parabolic equations
76A20 Thin fluid films
35K25 Higher-order parabolic equations
35K55 Nonlinear parabolic equations
76D45 Capillarity (surface tension) for incompressible viscous fluids
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