The stable manifold of the standing wave of the Nagumo equation (Q910550)

The stable manifold of the standing wave of the Nagumo equation \(u_ t=u_{xx}+u(1-u)(u-a)\) on the real line, is described as a level surface of a smooth functional defined on the Sobolev space \(H^ 1\). The global description of the qualitative behavior of the flow given in the paper, shows that this hypersurface is a threshold: solutions starting below the stable manifold decay to 0, while above the manifold, solutions expand to the constant state 1.