A generalization of the Einstein equations with the cosmological constant is considered for complex line elements. Several second order semilinear partial differential equations are derived from them as semilinear field equations in homogeneous and isotropic spaces. The nonrelativistic limits of the field equations are also considered. The properties of spatial expansion and contraction are studied based on energy estimates of the field equations. Several dissipative and anti-dissipative properties are remarked.
"Remarks on the derivation of several second order partial differential equations from a generalization of the Einstein equations." Osaka J. Math. 57 (2) 305 - 331, April 2020.