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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 =·(|u| n Δ 2 u)-Δ(|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=u=Δu=𝐧·{|u| n Δ 2 u-(|u| p-1 u)}=0

at the singularity interface which is the lateral boundary of supp u with the unit outward normal 𝐧. It is shown by the authors that, for the first critical exponent p=n+1+4 N for n(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 -N nN+6 f(y),y=xt -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)].


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