Liouville type theorems and periodic solutions for the nonhomogeneous parabolic systems

TL;DR

This paper establishes Liouville type theorems and periodic solutions for nonhomogeneous parabolic systems, introducing new techniques to handle nonlinearities and providing bounds, decay, and singularity estimates.

Contribution

It presents the first results for parabolic systems with non-homogeneous nonlinearities, using novel integral estimates and modified scaling methods.

Findings

Liouville type results for nonhomogeneous systems

Existence of periodic solutions

Universal bounds and decay estimates

Abstract

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.