Klein-Gordon Thermal Equation for a Preplanckian Universe

In the paper the quantum hyperbolic equation formulated in [M. Kozlowski, J. Marciak-Kozlowska, From Quarks to Bulk Matter, Hadronic Press, 2001], is appled to the study of the propagation of the initial thermal state of the Universe. It is shown that the propagation depends on the barrier height. The Planck wall potential is introduced, V_P=hbar/8t_P=1.25 10^18 GeV where t_P is a Planck time. For the barrier height V<V_P the master thermal equation is the modified Klein-Gordon equation and for barrier height V>V_P the master equation is the Klein-Gordon equation. The solutions of both type equations for Cauchy boundary conditions are discussed.