Evolution of a stable profile for a class of nonlinear diffusion equations with fixed boundaries

Abstract

A class of quasilinear parabolic equations with fixed boundaries arising in studies of cross-field diffusion in toroidal multipole plasmas is presented. It is well known that these equations have separable solutions which decay in time. Surprisingly, both octupole and numerical experiments show, in particular cases, that the separable solution evolves from an arbitrary initial distribution of particles. The evolution and stability properties of these solutions are demonstrated in this paper. When the coefficients of the equations are independent of the spatial variable, infinitesimal perturbations decay as the fourth power (or higher) of the separable solution time dependence; the separable solution is therefore stable. When the initial particle distribution has no nulls except at the boundaries, an approximate analysis shows that large perturbations decay exponentially causing the rapid evolution of the separable solution. The analysis allows the asymptotic behavior of the system to be predicted approximately from knowledge of the initial particle distribution.