Asymptotic behaviour of solutions of a nonlinear transport equation

We investigate the asymptotic behaviour of solutions of the convection- diffusion equation $$ b(u)_t + divleft( u q - n u right) = 0 qquad hbox{for r = |x| > e quadhbox{andquad t>0, $$ where $q=l/r, er $, $l>0$. The asymptotic limits that we consider are $ttoinfty$ and $e downto0$. We prove that self-similar solutions of this equation exist, and that are attractors in a class of solutions of the initial-boundary value problem with a flux boundary condition at $r=e$. We give estimates of the rate of convergence in an integral sense and in the $L^infty$-norm.