On the effects of spatial expansion and contraction on several semilinear partial differential equations (Nonlinear Wave and Dispersive Equations)

Abstract

The derivation of several second order partial differential equations is considered based on the scalar-field equation and its non-relativistic limit in the uniform and isotropic space. The field equation is derived as the Euler-Lagrange equation for a Lagrangian given in the spacetime which is a solution of the Einstein equation for non-Hermitian line elements. Some results on the Cauchy problem of the limit equation are introduced. The derivation of some equations for vectors and their energy estimates are also introduced. A dissipative property of the spatial expansion is remarked.