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CPAA

We study local and global well-posedness of the initial value
problem for the Schrödinger-Debye equation in the

*periodic case*. More precisely, we prove local well-posedness for the periodic Schrödinger-Debye equation with subcritical nonlinearity in arbitrary dimensions. Moreover, we derive a new*a priori*estimate for the $H^1$ norm of solutions of the periodic Schrödinger-Debye equation. A novel phenomenon obtained as a by-product of this*a priori*estimate is the global well-posedness of the periodic Schrödinger-Debye equation in dimensions $1$ and $2$*without*any smallness hypothesis of the $H^1$ norm of the initial data in the "focusing" case.
DCDS

In this note, we consider a partially hyperbolic horseshoe and prove uniqueness of equilibrium states for a class of potentials. In particular we obtain that there exists a unique maximal entropy measure.

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