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 $$ u_\gamma(x,t) = t^{-\alpha_\gamma} f_\gamma(y),\,\,\mbox{with the rescaled variable}\,\,\, y= x/{t^{\beta_\gamma}}, \,\, \beta_\gamma= \frac {1-n \alpha_\gamma}{10}, $$ where $\alpha_\gamma>0$ are real <em> nonlinear eigenvalues</em> ($\gamma$ is a multiindex in $\mathbb{R}^N$) of the tenth-order thin film equation (TFE-10) \begin{eqnarray*} \label{i1a} u_{t} = \nabla \cdot (|u|^{n} \nabla \Delta^4 u) \quad in \quad \mathbb{R}^N \times \mathbb{R}_+ \,, \quad n>0, (0.1) \end{eqnarray*} are studied. The present paper continues the study began in [1]. Thus, the following questions are also under scrutiny: <br> (<strong>I</strong>) Further study of the limit $n \to 0$, where the behaviour of finite interfaces and solutions as $y \to \infty$ are described. In particular, for $N=1$, the interfaces are shown to diverge as follows: $$ |x_0(t)| \sim 10 ( \frac{1}{n}\sec ( \frac{4\pi}{9} ) )^{\frac 9{10}} t^{\frac 1{10}} \to \infty as n \to 0^+. $$ <br> (<strong>II</strong>) For a fixed $n \in (0, \frac 98)$, oscillatory structures of solutions near interfaces. <br> (<strong>III</strong>) Again, for a fixed $n \in (0, \frac 98)$, global structures of some nonlinear eigenfunctions $\{f_\gamma\}_{|\gamma| \ge 0}$ by a combination of numerical and analytical methods.