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Zbl 1173.35530
Evans, J.D.; Galaktionov, V.A.; King, J.R.
Unstable sixth-order thin film equation. II: Global similarity patterns.
(English)
[J] Nonlinearity 20, No. 8, 1843-1881 (2007). ISSN 0951-7715; ISSN 1361-6544/e

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)].
[Shigeru Sakaguchi (Hiroshima)]
MSC 2000:
*35K55 Nonlinear parabolic equations
35K65 Parabolic equations of degenerate type
35B40 Asymptotic behavior of solutions of PDE
35B33 Critical exponents

Keywords: free-boundary problem; Cauchy problem; self-similar solutions; interfaces

Citations: Zbl 1173.35562; Zbl 1221.35296; Zbl 1156.35387

Cited in: Zbl 1173.35562

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