Countable branching of similarity solutions of higher-order porous medium type equations

Abstract

A countable set of self-similar solutions of the fourth-order porous medium equation (the PME$-4$), \begin{equation} u_t = - (|u|^n u)_{xxxx} \quad \mbox{in} \,\,\, {{\bf R}} \times {{\bf R}}_+ \quad (n>0), \tag*{(0.1)} \end{equation} is described. The similarity solutions under consideration have the standard form $$ u_l(x,t) = t^{-{\alpha}_l} f_l(y), \quad y= x/{t^{{\beta}_l}}, \quad {\beta}_l = \tfrac {1- {\alpha}_l n}2, \quad l=0,1,2,... \, , $$ where ${\alpha}_l$ and $f_l(y)$ are eigenvalues and eigenfunctions of a nonlinear eigenvalue problem (actually, for a linear pencil of ordinary differential operators) written as $$ {\bf B}_n(f) \equiv -(|f|^n f)^{(4)} + \tfrac {1-{\alpha} n}4 \, y \, f' + {\alpha} f=0 \quad \mbox{in} \quad {{\bf R}}, \quad f \not =0, \,\,\, f \in C_0({{\bf R}}). $$ First four nonlinear eigenfunctions were obtained by Bernis and McLeod in the 1980s. In order to identify the full countable set of eigenfunctions $\{f_l\}$, we check their appearance at the branching point $n=0$ from the eigenfunctions $\{\psi_l\}$ of the non self-adjoint operator $ {\bf B}_0 = - D_y^4 + \frac 14 \, y D_y + \frac 14 \, I, \quad \mbox{with the point spectrum} \,\,\, {\sigma}({\bf B}_0)= \{{\lambda}_l=- \frac l4, \, l \ge 0\}, $ where $I$ is the identity. These eigenfunctions give the solutions $$ u_l(x,t)={t}^{- \frac N4 + {\lambda}_l} \psi_l(y), \quad y=x/t^{\frac 14}, \quad l=0,1,2,... \, , $$ of the linear bi-harmonic equation $ u_t = -u_{xxxx}, $ which is (0.1) for $n=0$. The results extend to the PME$-4$ posed in ${{\bf R}^N}$. A similar classification is performed for the PME$-6$ \begin{equation} \tag*{(0.2)} u_t = (|u|^n u)_{xxxxxx} \quad \mbox{in} \,\,\, {{\bf R}} \times {{\bf R}}_+ \quad (u_t=u_{xxxxxx} \,\,\, \mbox{for} \,\,\, n=0). \end{equation} The general methodology of the study of (0.1) and (0.2) is associated with that developed for the classic second-order PME $ u_t = (|u|^n u)_{xx}$ in $ {{\bf R}} \times {{\bf R}}_+, $ which systematic study was began by the discovery of the first ZKB source-type solution by Zel'dovich, Kompaneetz, and Barenblatt in 1950-52.