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Unstable sixth-order thin film equation. I: Blow-up similarity solutions. (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 $$u_t = \nabla\cdot (|u|^n\nabla \Delta^2 u)- \Delta(|u|^{p-1}u),$$ 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: $$u = \nabla u = \Delta u = {\bold n}\cdot\{|u|^n\nabla\Delta^2u-\nabla(|u|^{p-1}u)\} = 0$$ at the singularity interface which is the lateral boundary of supp $u$ with the unit outward normal $\bold n$. By a formal matched expansion technique, the authors show that, for the first critical exponent $p= n+1+\frac 4N$ 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 $$u_k(x,t) = (T-t)^{-\frac N{nN+6}} f_k(y),\ y= x(T-t)^{-\frac 1{nN+6}},\ k= 1,2, \dots,$$ 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
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