*(English)*Zbl 0856.35002

Fourth-order degenerate diffusion equations arise in a ‘lubrication approximation’ of a thin film or neck driven by surface tension. Numerical studies of the lubrication equation (LE) ${h}_{t}+{\left({h}^{n}{h}_{xxx}\right)}_{x}=0$ with various boundary conditions indicate that singularity formation in which $h\left(x\right(t),t)\to 0$ occurs for small enough $n$ with ‘anomalous’ or ‘second type’ scaling inconsistent with usual dimensional analysis.

This paper considers locally symmetric or even singularities in the (LE) and in the modified lubrication equation (MLE) ${h}_{t}+{h}^{n}{h}_{xxxx}=0$. Both equations have the property that entropy bounds forbid finite time singularities when $n$ is sufficiently large. Power series expansions for local symmetric similarity solutions are proposed for equation (LE) with $n<1$ and (MLE) for all $n\in \mathbb{R}$. In the latter case, special boundary conditions that force singularity formation are required to produce singularities when $n$ is large. Matching conditions at higher order terms in the expansion suggests a simple functional form for the time dependence of the solution.

Computer simulations presented here resolve the self similarity in the onset of the singularity of approximately 30 decades in ${min}_{x}\left(h(x,t)\right)$. Measurements of the similarity shape and time dependences show excellent agreement with the theoretical prediction.

One striking feature of the solution to (MLE) is a transition from a finite time singularity to an infinite time singularity $n=3/2$. Also both equations (LE) and (MLE) exhibit symmetric singularities for $n<0$ with derivatives of order $k$ vanishing at the singular point for all $2<k<4-2n$. They also exhibit a blow up in derivatives of order greater than $4-2n$.

##### MSC:

35A20 | Analytic methods, singularities (PDE) |

35C20 | Asymptotic expansions of solutions of PDE |

35K55 | Nonlinear parabolic equations |

76B45 | Capillarity (surface tension) |

76M20 | Finite difference methods (fluid mechanics) |