Non-autonomous Ginzburg--Landau equation and its attractors

M. I. Vishik ; V. V. Chepyzhov

Sbornik. Mathematics, Tome 196 (2005) no. 6, pp. 791-815

Résumé

The behaviour as $t\to+\infty$ of solutions $\{u(x,t),\ t\geqslant0\}$ of the non-autonomous Ginzburg–Landau (G.–L.) equation is studied. The main attention is focused on the case when the dispersion coefficient $\beta(t)$ in this equation satisfies the inequality $|\beta(t)|>\sqrt3$ for $t\in L$, where $L$ is an unbounded subset of $\mathbb R_+$. In this case the uniqueness theorem for the G.–L. equation is not proved. The trajectory attractor $\mathfrak A$ for this equation is constructed. If the coefficients and the exciting force are almost periodic (a.p.) in time and the uniqueness condition fails, then the trajectory attractor $\mathfrak A$ is proved to consist precisely of the solutions $\{u(x,t),\ t\geqslant0\}$ of the G.-L. equation that admit a bounded extension as solutions of this equation onto the entire time axis $\mathbb R$. The behaviour as $t\to+\infty$ of solutions of a perturbed G.–L. equation with coefficients and the exciting force that are sums of a.p. functions and functions approaching zero in the weak sense as $t\to+\infty$ is also studied.