A very singular solution for the slow diffusion equation with nonlinear convection (Q2473833)

We investigate an existence and uniqueness of the nontrivial, nonnegative solution of a nonlinear ordinary differential equation: \[ (f^m)''+\beta r+\alpha f'+\sigma(f^q)'=0 \] satisfying a specific decay rate: \(\lim_{r\to\infty}r^{\alpha /\beta}f(r)=0\) with \(\alpha =1/(2q-m-1)\) and \(\beta =(q-m)/(2q-m-1)\). Here \(m>1\), \(m<q<m+1\) and \(\sigma =1\) or \(-1\). Such a solution arises naturally when we study a very singular solution for a slow diffusion equation with nonlinear convection: \[ u_t=(u^m)_{xx}+(u^q)x \] defined either on the whole real line or on the half line \((-\infty,0]\).