Absorbing estimate implies trajectory statistical solutions for nonlinear elliptic equations in half-cylindrical domains

In this article, we reveal an essential property for a certain model nonlinear second-order elliptic equations in half-cylindrical domains, which is that absorbing estimate implies trajectory statistical solutions. The obtained results indicate that the trajectory statistical solutions associated with the elliptic equations satisfy the spatial Liouville theorem. The proofs rely on the theory of infinite-dimensional dynamical system and Kakutani-Riesz Representation Theorem. The approach and idea for construction trajectory statistical solutions presented here could be extended and developed to investigate the statistical solutions of elliptic equations on unbounded domains, which should open up the frontier of the study of statistical property of solutions for elliptic equations.