*(English)*Zbl 1173.35562

This paper studies blow-up singularity formation phenomena for the sixth-order quasilinear parabolic thin film equation (TFE) with unstable (backward parabolic) second-order homogeneous term

where $n>0$ and $p>1$. This equation is degenerate at the singularity set $\{u=0\}$, and for this equation the authors consider the free-boundary problem with zero-height, zero-contact-angle, zero-moment, and conservation of mass conditions:

at the singularity interface which is the lateral boundary of supp $u$ with the unit outward normal $\mathbf{n}$. By a formal matched expansion technique, the authors show that, for the first critical exponent $p=n+1+\frac{4}{N}$ for $n\in (0,5/4)$, where $N$ is the space dimension, this free-boundary problem admits a countable set of continuous branches of radially symmetric self-similar blow-up solutions

where $T>0$ is the blow-up time. In the Cauchy problem, one needs more regular connections (the necessary maximal regularity) with the singularity level $\{{f}_{k}=0\}$ that make it possible to extend the solution by ${f}_{k}=0$ beyond the support. The authors show that the Cauchy problem admits a countable set of self-similar blow-up solutions of maximal regularity, which are oscillatory near the interfaces.

The fourth-order TFE has been studied by the authors in two papers [Eur. J. Appl. Math. 18, No. 2, 195–231 (2007; Zbl 1221.35296) and ibid. 18, No. 3, 273–321 (2007; Zbl 1156.35387)].

[For part II see ibid., 1843–1881 (2007; Zbl 1173.35530).]

##### MSC:

35K65 | Parabolic equations of degenerate type |

35K55 | Nonlinear parabolic equations |

35B40 | Asymptotic behavior of solutions of PDE |