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

Abstract

Abstract. Countable families of global-in-time and blow-up similarity signchanging patterns of the Cauchy problem for the fourth-order thin film equation (TFE-4)are studied. The similarity solutions are of standard "forward" and "backward" formsand α ∈ 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 {fγ, αγ}, a bifurcation-branching analysis is performed by using a homotopy path n → 0 + in (0.1), where B ± 0 (α, f ) become associated with a pair {B, B * } of linear non-self-adjoint operatorswhich are known to possess a discrete real spectrum,|γ|≥0 (γ is a multiindex in R N ). These operators occur after corresponding global and blow-up scaling of the classic bi-harmonic equation ut = −Δ 2 u. 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

Towards optimal regularity for the fourth-order thin film equation in RN: Graveleau-type focusing self-similarity

“…For the one-dimensional case, similar results were obtained in [7], establishing the Hölder continuity in C 0, 1 2 with respect to the variable x and in C 0, 1 8 with respect to t. However, in the N -dimensional case, the optimal regularity has been unsolved besides the interest of the specialised mathematical community; see [8,10,15,16]. In many of those works it is also assumed that the solutions are non-negative for the Cauchy problem (CP).…”

“…The second expansion cannot be interpreted pointwise for oscillatory changing sign solutions f (y), though now these functions are assumed to have finite number of zero surfaces (as the generalised Hermite polynomials for n = 0 do). Indeed, as discussed in [1] for (2.3) this is true if the zeros are transversal. Furthermore, in order to apply the Lyapunov-Schmidt branching analysis we suppose the expansion Moreover, we write…”

“…We can guarantee that the first profile is unique but for the rest there could be more than branch of solutions emanating at n = 0. The number of branches will depend on the dimension of the eigenspace so that the dimension is somehow involved; see [1] for any further comments and a detailed analysis of a similar branching analysis.…”

“…Due to our analysis is possible to establish a connection between the radial solutions of (1.1) and the corresponding to (4.1) via the self-similar associated equations when n → 0 + . To this end we apply the Lyapunov-Schmidt method to ascertain qualitative properties of the self-similar equation (2.3) − following a similar analysis as the one carried out in [1].…”

The p-Laplace Equation in Domains with Multiple Crack Section via Pencil Operators

“…We now show, using a branching approach at n = 0 (following a similar philosophy to [2]), how a polynomial behaviour (3.7) of nonlinear eigenfunctions is connected with that for analytic harmonic polynomials. Namely, looking at the equation (3.10) as a perturbed linear ODE (2.7) for small n > 0, with perturbations of order O(n).…”