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 and . This equation is degenerate at the singularity set , 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 with the unit outward normal . By a formal matched expansion technique, the authors show that, for the first critical exponent for , where is the space dimension, this free-boundary problem admits a countable set of continuous branches of radially symmetric self-similar blow-up solutions
where is the blow-up time. In the Cauchy problem, one needs more regular connections (the necessary maximal regularity) with the singularity level that make it possible to extend the solution by 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).]