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Droplet spreading under weak slippage: The optimal asymptotic propagation rate in the multi-dimensional case. (English) Zbl 1056.35072
Summary: We prove optimal estimates on the growth rate of the support of solutions to the thin-film equation $$u_t+\text{div}(|u|^n \nabla\Delta u)=0$$ in space dimensions $$N=2$$ and $$N=3$$ for parameters $$n\in[2,3)$$ which correspond to Navier’s slip condition $$(n=2)$$ or certain variants modeling weaker slippage effects. Our approach relies on a new class of weighted energy estiates. It is inspired by the one-dimensional technique of J. Hulshof and A. E. Shishkov, [Adv. Differ. Equ. 3, No. 5, 625–642 (1998; Zbl 0953.35072)], and it simplifies their method, mainly with respect to basic integral estimates to be used.

##### MSC:
 35K15 Initial value problems for second-order parabolic equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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