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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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