Asymptotic properties of nonlinear diffusion, nonlinear drift-diffusion, and nonlinear reaction-diffusion equations

We review a Fokker-Planck approach to nonlinear evolution equations such as nonlinear diffusion equations and nonlinear drift-diffusion equations and extend this approach to nonlinear reaction-diffusion equations. Using this Fokker-Planck approach along with appropriately defined entropy and free energy measures, we show that transient solutions converge to stationary ones in the long time limit. Implications for bifurcation theory are also addressed. (C) 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.