Calculus of Variations and Geometric Measure Theory

A. Jevnikar - J. Wang - W. Yang

Liouville type theorems and periodic solutions for the nonhomogeneous parabolic systems

created by jevnikar on 30 Aug 2020
modified on 25 Jun 2023


Accepted Paper

Inserted: 30 aug 2020
Last Updated: 25 jun 2023

Journal: Z. Angew. Math. Phys.
Year: 2023


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.

Keywords: A priori estimates, Liouville type results, Parabolic system, $\chi^{(2)}$ system, Non-homogeneous nonlinearity, Periodic solutions