This paper studies the Klein-Gordon equation and two modifications in an infinite Cantor set a fractal space-time. Their variational formulations are established discussed, spatio-temporal discontinuity requires both spatio-fractal derivative temporal for practical applications. Some basic properties of local fractional two-scale elucidated, derivation Euler-Lagrange is illustrated.

<p style='text-indent:20px;'>We consider a damped Klein-Gordon equation with variable diffusion coefficient. This problem is challenging because of the equation's unbounded nonlinearity. First, we study nonlinearity's continuity properties. Then existence and uniqueness solutions established. The main result solution map on set admissible parameters. Its application to parameter identific...