Quasi-linear parabolic equations: some properties of physical significance

Abstract

A nonlinear transport equation with a source term, describing e.g., the temperature evolution in a hot plasma, is analyzed extensively, assuming power nonlinearities in the diffusion coefficient as well as in the source term. Various analytic approaches to the problem are developed and compared. A method of Painlevé is used to test the equation with regard to the possibilities of obtaining exact formal solutions for arbitrary initial conditions. It is found that such solutions can in general not be expressed in terms of known transcendental functions. A global understanding of the behaviour of the evolution is, however, reached for a great variety of possibilities, characterized by different values of the powers of the nonlinearities in the diffusion and source terms. Part of the analysis is based on a method of central expansion which, combined with suitable computational aid, provides a new tool for describing one, two and three dimensional situations for radially symmetric cases.