On the global behaviour of solutions of some fourth order nonlinear equations

V. K. Kalantarov

Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 19, Tome 163 (1987), pp. 66-75

Résumé

Zbere are considered two classes of fourth order nonlinear evolution equations, for first class, included the well known Hahn–Hillard equation, it is proved that there exists a global minimal $B$-attractor, and it is compact and connected, for the second class, included Sivashinsky equation, it is proved a blow-up theorem. In addition, for the Kuramoto–Sivashinsky equation, in one-dimensional case, for even solutions it is prouved the existence of a global minimal $B$-attractor in the fase-space $W_2^1$. Xhis attraetor is compact and connected. In the multi-dimensional case $(n=2,3)$ under some assumption, it is proved the existence of compact attractors for some bounded sets.