The Cauchy problem in $q$-theories and quadratic gravity

The present talk is focused on two models of different type.
The first are the so called $q$-models, which are scenarios that may explain the smallness of the cosmological constant.
The second are the quadratic gravity models of Stelle, which are renormalizable higher order derivative models of gravity, but they possess ghostly states propagating in the phase space.
The Cauchy formulation for these models will be discussed.
The present talk shows that the Cauchy problem is well posed even not reaching this limit. Furthermore, given smooth initial conditions the solutions are smooth, under certain restriction of the potential function. A similar analysis is performed for quadratic gravity.
It is important to mention that there exist previous work of David Noakes about the Cauchy problem of quadratic gravity but these results are obtained at a linearized level. In the present work these results are extended beyond perturbation. An important technical condition, namely x-compactness plays an important role in our discussion. The seminar will not be intended only to specialist, and some basic concepts will be discussed as well.