Stability and Diffusive Dynamics on Extended Domains

We consider dissipative systems on the real axis in situations when the evolution is dominated by a dynamics similar to the one of a linear diffusion equation. It is surprising that such a diffusive behavior occurs in relatively complicated systems. After a discussion of the linear and nonlinear diffusion equation, we give a brief introduction into the methods which are available to describe diffusive behavior in nonlinear systems. These are L1–L1 estimates, Lyapunov functions and discrete and continuous renormalization groups. In the second part of the paper we show examples, where such a diffusive dynamics can be seen. For the Ginzburg–Landau equation we consider the nonlinear stability of Eckhaus– stable equilibria and the diffusive mixing of two different Eckhaus–stable equilibria. Diffusive dynamics also occurs in pattern forming systems as the Swift–Hohenberg equation or hydrodynamical stability problems as Bénard’s problem. In such cases the method of reduced instability allows us to analyze the linearized problem. We close with an outlook on situations, where diffusive behavior is expected, but where a proof is still missing.