R. Teman

Application: The Chaffee—Infante Reaction—Diffusion Equation

As an example of a parabolic reaction—diffusion equation with less stringent conditions than in Chapter 18, we briefly outline the construction of an inertial manifold for the Chaffee—Infante equation [H] in two dimensions:

Stability with Respect to Perturbations

\frac{{\partial u}}{{\partial t}} - \Delta u + \lambda \left( {{u^3} - u} \right) = 0,\;\lambda > {\text{ }}0,\Omega = {\left[ { - \pi , + \pi } \right]^2} = {T^2},{\text{ periodic boundary conditions, }}u\left( 0 \right) = {u_0}

Presentation of the Approach and of the Main Results

(19.1) (we do not restrict ourselves to odd periodic functions). For λ > 1, this equation admits multiple nonconstant steady states besides u = 0 and u = ± 1.

Spectral Blocking Property

We recall that in the case of the Kuramoto—Sivashinsky [HN, HN1,HNZ, NSh] equation on the space H of odd L-periodic functions, (du/dt)+Au+R(u)=0, we have

Cone Invariance Properties

R\left( u \right)B\left( {u,u} \right) + Cu + f,

Construction of the Inertial Manifold

(15.1) with