Large Time Solution of an Inhomogeneous Nonlinear Diffusion Equation

The paper considers the large time solution of the nonlinear diffusion equation ρ(x)partial u/partial t = partial/partial xbig\{C(x) u^βpartial u/partial x big\}, β > 0, for initial data with compact support. The asymptotic behaviour of ρ(x) and C(x) as |x| ->∞ defines a parameter space, and the nature of the solution at large time depends crucially on the location therein. The paper examines the nature and uniformity of the asymptotic solutions for xin (- ∞,∞) in a restricted region of the entire parameter space, and in certain cases gives the leading error terms. A new integral invariant of the solution is also given.