Local bifurcation analysis of global and "blow-up" patterns for a fourth-order thin film equation Articles uri icon

publication date

  • October 2011

start page

  • 483

end page

  • 537

issue

  • 5

volume

  • 18

international standard serial number (ISSN)

  • 1021-9722

electronic international standard serial number (EISSN)

  • 1420-9004

abstract

  • Countable families of global-in-time and blow-up similarity sign-changing patterns of the Cauchy problem for the fourth-order thin film equation (TFE-4) u t =−∇⋅(|u| n ∇Deltau)inR N ×R + wheren>0, are studied. The similarity solutions are of standard "forward" and "backward" forms u ± (x,t)=(±t) −alfa f(y),y=x/(±t) beta , beta=1−alfan 4 ,±t>0,wherefsolve B ± n (alfa,f)≡−∇⋅(|f| n ∇Deltaf)±betay⋅∇f±alfaf=0inR N ,(0.1) and alfa∈R is a parameter (a "nonlinear eigenvalue"). The sign " + ", i.e., t > 0, corresponds to global asymptotics as t → + ∞, while "−" (t < 0) yields blow-up limits t → 0− describing possible "micro-scale" (multiple zero) structures of solutions of the PDE. To get a countable set of nonlinear pairs {fgamma, alfagamma}, a bifurcation-branching analysis is performed by using a homotopy path n → 0+ in (0.1), where B ± 0 (alfa,f) become associated with a pair {B, B*} of linear non-self-adjoint operators B=−Delta 2 +1 4 y⋅∇+N 4 IandB ∗ =−Delta 2 −1 4 y⋅∇(so(B) ∗ L 2 =B ∗ ), which are known to possess a discrete real spectrum, sigma(B)=sigma(B ∗ )={lambda gamma =−∣ ∣ gamma∣ ∣ 4 } ∣ ∣ gamma∣ ∣ ≥0 (gammaisamultiindexinR N ) . These operators occur after corresponding global and blow-up scaling of the classic bi-harmonic equation ut = − Delta2u. This allows us to trace out the origin of a countable family of n-branches of nonlinear eigenfunctions by using simple or semisimple eigenvalues of the linear operators {B, B*} leading to important properties of oscillatory sign-changing nonlinear patterns of the TFE, at least, for small n > 0.

keywords

  • thin film equation; local bifurcation analysis; source-type and blow-up similarity solutions; the cauchy problem; finite interfaces; oscillatory sign-changing behaviour