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Source-type solutions to thin-film equations in higher dimensions. (English) Zbl 0894.76019
Summary: We prove that the thin film equation \(h_t+ \text{div} (h^n \text{grad} (\Delta h))=0\) in dimension \(d\geq 2\) has a unique \(C^1\) source-type radial self-similar non-negative solution if \(0<n <3\) and has no solution of this type if \(n\geq 3\). When \(0<n <3\), the solution \(h\) has finite speed of propagation, and we obtain the first order asymptotic behaviour of \(h\) at the interface or free boundary separating the regions where \(h>0\) and \(h=0\). (The case \(d=1\) was already known).

MSC:
76D08 Lubrication theory
35Q35 PDEs in connection with fluid mechanics
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