Existence of global solutions and global attractor for the third order Lugiato–Lefever equation on T

Miyaji, Tomoyuki; Tsutsumi, Yoshio

Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 7, pp. 1707-1725.

We show the existence of global solution and the global attractor in $L^2(\mathbf{T})$ for the third order Lugiato–Lefever equation on T. Without damping and forcing terms, it has three conserved quantities, that is, the $L^2(\mathbf{T})$ norm, the momentum and the energy, but the leading term of the energy functional is not positive definite. So only the $L^2$ norm conservation is useful for the third order Lugiato–Lefever equation unlike the KdV and the cubic NLS equations. Therefore, it seems important and natural to construct the global attractor in $L^2(\mathbf{T})$. For the proof of the global attractor, we use the smoothing effect of cubic nonlinearity for the reduced equation.

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