The Cauchy problem for a tenth-order thin film equation II. Oscillatory source-type and fundamental similarity solutions

Abstract

Fundamental global similarity solutions of the standard form uγ (x, t) = t −αγ fγ (y), with the rescaled variable y = x/t βγ , βγ = 1−nαγ 10, where αγ > 0 are real nonlinear eigenvalues (γ is a multiindex in R N ) of the tenth-order thin film equation (TFE-10)are studied. The present paper continues the study began in [1]. Thus, the following questions are also under scrutiny:(I) Further study of the limit n → 0, where the behaviour of finite interfaces and solutions as y → ∞ are described. In particular, for N = 1, the interfaces are shown to diverge as follows:(II) For a fixed n ∈ (0, 9 8 ), oscillatory structures of solutions near interfaces. (III) Again, for a fixed n ∈ (0, 9 8 ), global structures of some nonlinear eigenfunctions {fγ } |γ|≥0 by a combination of numerical and analytical methods.

The Cauchy Problem for a Tenth-Order Thin Film Equation I. Bifurcation of Oscillatory Fundamental Solutions

“…2 ), we can assure that the number of solutions for (4.23) is exactly two. This is the dimension of the kernel of the operator B + 1 10 I (as we expected in our more general conjecture). The above particular example shows how difficult the questions on existence and multiplicity of solutions for such non-variational branching problems are.…”

“…The eigenvalue-eigenfunction pairs where the eigenvalues are not explicitly known, but have to be solved for, requires the solution of a 12th-order system. This will be discussed in [1].…”

“…Indeed, this ODE creates a 10-dimensional phase space and a construction of suitable homotopic connections of equilibria, which are admissible for necessary nonlinear eigenfunctions, is not easy at all. In the forthcoming paper [1], we study the first eigenfunction of (1.9), i.e., the fundamental source-type profile f 0 (y) by using a variety of other analytical, asymptotic, and numerical methods.…”