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 u_\g(x,t)=t^{-\a_\g} f_\g(y), with the rescaled variable y= x/{t^{\b_\g}}, \b_\g= \frac {1-n \a_\g}{10}, where \a_\g>0 are real nonlinear eigenvalues (\g is a multiindex in R^N) of the tenth-order thin film equation (TFE-10) u_{t} = \nabla \cdot(|u|^{n} \n \D^4 u) in R^N \times R_+, n>0, are studied. The present paper continues the study began by the authors in the previous paper P. Alvarez-Caudevilla, J.D.Evans, and V.A. Galaktionov, The Cauchy problem for a tenth-order thin film equation I. Bifurcation of self-similar oscillatory fundamental solutions, Mediterranean Journal of Mathematics, No. 4, Vol. 10 (2013), 1759-1790. Thus, the following questions are also under scrutiny: (I) Further study of the limit n \to 0, where the behaviour of finite interfaces and solutions as y \to infinity are described. In particular, for N=1, the interfaces are shown to diverge as follows: |x_0(t)| \sim 10 \left( \frac{1}{n}\sec\left( \frac{4\pi}{9} \right) \right)^{\frac 9{10}} t^{\frac 1{10}} \to \infty as n \to 0^+. (II) For a fixed n \in (0, \frac 98), oscillatory structures of solutions near interfaces. (III) Again, for a fixed n \in (0, \frac 98), global structures of some nonlinear eigenfunctions \{f_\g\}_{|\g| \ge 0} by a combination of numerical and analytical methods.