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

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.

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

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