Partial Differential Equations of Mathematical Physics in the expanding universe. Integral transform approach.

The Nobel Prize in Physics 2011 awarded for the discovery of the accelerating expansion of the Universe, emphasized once again an importance of the investigation of the classical equations of the mathematical physics, which live in the curved background. The partial differential equations of the mathematical physics in the accelerating universe have variable coefficients; even when they depend only on the time variable, this creates serious difficulties in the investigation of the nonlinear equations. Although the well-posedness of local in time initial value problems are known for many decades, existence of the global in time solutions is still open problem for equations arising in the models of the expanding universe. We will present an integral transform that maps solutions of some class of the partial differential equations with constant coefficients to solutions of more complicated equations, which have variable coefficients. We illustrate this transform by applications to several model equations. In particular, we give applications to the Klein-Gordon and wave equations in the space-times of the de Sitter and anti-de Sitter models of the expanding universe.