**3. The Cauchy Problem**

First of all, we notice that, if the trace of the stress–energy tensor T is constant, *f*(*R*)-gravity with torsion reduces to GR with (or without) cosmological constant. Therefore, when this is the case, the Cauchy problem is well-formulated and well-posed [23,26,46]. For instance, this happens in vacuo and in the presence of electromagnetic (or also Yang–Mills) fields (if *f*(*R*) 6= *αR* 2 ).

The situation is more complicated if T 6= const.: in such a circumstance, the theory no longer amounts to GR, so the classical Bruhat results [46] do not apply, and the well-formulation and well-posedness of the Cauchy problem is not automatically assured.

To overcome this issue, the Cauchy problem could be addressed by exploiting the dynamical equivalence with scalar-tensor theories with Brans–Dicke parameter *ω*<sup>0</sup> = − 3 2 . Unfortunately, here a difficulty occurs: The d'Alembertian *g pq*∇*p*∇*q<sup>ϕ</sup>* disappears from Equation (18), and we no longer have the possibility of deriving the expression of the d'Alembertian as a function of the dynamical variables and their derivatives up to the first-order. In other words, we cannot eliminate the second-order derivatives of the scalar field *ϕ* from the Einstein-like Equation (16).

An alternative idea is to pass from the Jordan to the Einstein frame, making use of the conformal transformation technique. Following this approach, we can derive sufficient conditions for the well-posedness of the Cauchy problem for *f*(*R*)-gravity with torsion in the presence of a perfect fluid [44,47] or a Klein–Gordon scalar field [48]. Such conditions result in suitable requirements imposed on the function *f*(*R*), so they can be assumed as a sort of selection rule for viable *f*(*R*) models. In addition, we show that the function *f*(*R*) = *R* + *αR* 2 satisfies the above mentioned conditions, in such a way that the set of viable *f*(*R*) models is not empty.
