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Unstable sixth-order thin film equation. II: Global similarity patterns. (English) Zbl 1173.35530
The authors continue the study in [Nonlinearity 20, No. 8, 1799--1841 (2007; Zbl 1173.35562)] of the asymptotic behavior of solutions of 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 $p > 1$. The previous paper concerns blow-up similarity solutions, and the present paper concerns global similarity solutions. 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 $\bold n$. It is shown by the authors 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 solutions defined for all $t > 0$ of the form $$ u(x,t) = t^{-\frac N{nN+6}} f(y),\qquad y= xt^{-\frac 1{nN+6}}. $$ In the Cauchy problem, one needs more regular connections (the necessary maximal regularity) with the singularity level $\{ f=0\}$ that make it possible to extend the solution by $f=0$ beyond the support. The authors show that the Cauchy problem admits a countable set of self-similar global 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)].

35K55Nonlinear parabolic equations
35K65Parabolic equations of degenerate type
35B40Asymptotic behavior of solutions of PDE
35B33Critical exponents (PDE)
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