  In the present paper we derive Liouville type results and existence of
periodic solutions for $\chi^{(2)}$ type systems with non-homogeneous
nonlinearities. Moreover, we prove both universal bounds as well as singularity
and decay estimates for this class of problems. In this study, we have to face
new difficulties due to the non-homogenous nonlinearities. To overcome this
issue, we carry out delicate integral estimates for this class of
nonlinearities and modify the usual scaling and blow up arguments. This seems
to be the first result for parabolic systems with non-homogeneous
nonlinearities.
