Finite dimensional aspects of semilinear parabolic equations (Q1915999)

The semiflow \(\Phi\) generated by the parabolic equation: \[ \partial_t u-\partial^2_{xx} u=f(x,u) \quad\text{ in } [0;1] \times [0,\infty), \quad u(0,t) =u(1,t)=0, \] where \(f\in C^2\) has uniformly bounded derivatives and satisfies \(f(x,0)=0\) for \(0\leq x\leq 1\), is studied. It is proved that this semiflow \(\Phi\) on the Banach space \(X=L^2([0;1])\) is a product of finite dimensional flows, it means, there is a decomposition of \(X\) into finite dimensional subspaces \(X_n\) together with projections to these subspaces which commute with the flow \(\Phi\). More precisely, the following notation is used: \[ X_n= \bigoplus^n_{j=1} \langle \sin j \pi x \rangle,\;X^0_n= \langle\sin n \pi x\rangle, \text{ and } Y_n= X^\perp_n, \] \(P_n\) is the orthogonal projector to \(X_n\), \(P^0_n\) is the projector to \(X^0_n\), and \(Q_n= Id-P_n\). For \(n\) big enough there are invariant manifolds \(M_n\) and \(M^0_{n+i}\), \(0<i\), and there exist also projections \(\pi_n : L^2\to M_n\) and \(\pi^0_n: L^2 \to M^0_n\) which commute with the semiflow. It is proved that the map \(h:u\to P_n \pi_n(u)+ \sum^\infty_{i=1} P^0_{n+i} \pi^0_{n+i} (u)\) is a conjugacy to a product of finite dimensional flows.