Local bifurcation-branching 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) ut= -del . (|u|(n)del Delta u) in R(N) x R(+), where n > 0, are studied. The similarity solutions are of standard "forward" and "backward" forms u(+/-)(x, t) = (+/- t)(-alpha)f(y), y = x/(+/- t)(beta), beta = 1-alpha n/4, +/- t>0, where f solve Bn+(alpha, f)equivalent to-del.(|f|n del Delta f)+/-beta y.del f +/-alpha f = 0 in RN, (0.1) and alpha is an element of R is a parameter (a "nonlinear eigenvalue"). The sign "+", i.e.,t > 0, corresponds to global asymptotics as t -> +infinity, 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 {f(gamma), alpha(gamma)}, a bifurcation-branching analysis is performed by using a homotopy path n -> 0(+) in (0.1), where B(0)(+/-) (alpha, f) become associated with a pair {B, B*} of linear non-self-adjoint operators B=-Delta(2) + 1/4 y.del+N/4 I and B*=-Delta(2) - 1/4y.( so (B)*(L2) = B*), which are known to possess a discrete real spectrum, sigma(B) = sigma(B*) = {lambda(gamma) = - |gamma/4|}(|gamma|>0) (gamma is a multiindex in R(N)). These operators occur after corresponding global and blow-up scaling of the classic bi-harmonic equation u(t) = -Delta(2)u. This allows us to trace out the origin of a countable family of n-branches of nonlinear eigenfunctions by using simple or semi-simple 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.