The rate of decay for solutions in nonlinear dissipative partial differential equations

Abstract

We show, applying the method used by Dyer and Edmunds [Proc. London Math. Soc. 18 (1968) 169]. for the Navier-Stokes equations, that the rate of decay for certain dissipative partial differential equations can be bounded below by an exponential. In particular we show that this holds true for a generalised diffusion equation closely linked to the complex Ginzburg-Landau equation [M.V. Bartuccelli et al., Phys. Scr, in press] and that a similar method can be applied to the Kuramoto-Sivashinsky equation and the Burgers equation. We also discuss the implications of these results for the estimates of the dissipative length scale derived for dissipative partial differential equations using ladder methods.