Fourier parameterization of attractors for dissipative equations in one space dimension
The author considers dissipative equations of the form \[ u_t+(-1)^m u^{(2m)}+ f(x,u,\dots, u^{(2m-1)})= 0 \] and proves that the global attractor is indeed a graph over a sufficiently large number of modes, that is, there exists a spectral projector with finite rank which is one-to-one on the global attractor. The method is based on the fact that for equations of the type \[ V_t- V_{xx}+\beta V_x+ \alpha V= 0 \] there is a change of variables which eliminates the gradient term. A similar fact holds for the equations of order \(2m\).
scientific article; zbMATH DE number 2057859