Explicit construction of an inertial manifold for a reaction diffusion equation (Q582458)

An inertial manifold is constructed for the scalar reaction-diffusion equation \(u_ t=vu_{xx}+f(u)\) with a cubic nonlinearity. Uniform bounds are obtained for the number of zeros along solutions to the variational equations satisfied by the difference of two elements on the unstable manifolds of equilibria. This uniformity leads to the global parametrization of the attractor as a function defined in the linear unstable manifold of the least stable equilibrium. By the introduction of local techniques near each equilibrium, we succeed in constructing an inertial manifold of lowest possible dimension.