On the behavior of the solutions of the Kuramoto-Sivashinsky equation for negative time (Q1191816)

It is proved that for the Kuramoto-Sivashinsky equation the set \(M\) of all initial data such that the corresponding solution \(u\) exists for every \(t \in \mathbb{R}\) and satisfies \[ \limsup_{t \to-\infty} {\log | u(t) |_{L^ 2} \over t}<\infty, \] reduces to the universal attractor \(A\). Therefore a solution \(u\) outside the attractor either can not be extended for \(t \in \mathbb{R}\) or satisfies \(\lim_{t \to-\infty} \log| u(t)|_{L^ 2}/ | t |=\infty\).