Integral Transform Approach to the Cauchy Problem for the Evolution Equations

Abstract

In this note we describe some integral transform that allows to write solutions of the Cauchy problem for one partial differential equation via solution of another one. It was suggested by author in J. Differ. Equ. 206:227–252, 2004 in the case when the last equation is a wave equation, and then used in the series of articles (see, e.g., Yagdjian in J. Differ. Equ. 206:227–252, 2004, Yagdjian and Galstian in J. Math. Anal. Appl. 346(2):501–520, 2008, Yagdjian and Galstian in Commun. Math. Phys. 285:293–344, 2009, Yagdjian in Rend. Ist. Mat. Univ. Trieste 42:221–243, 2010, Yagdjian in J. Math. Anal. Appl. 396(1):323–344, 2012, Yagdjian in Commun. Partial Differ. Equ. 37(3):447–478, 2012, Yagdjian in Semilinear Hyperbolic Equations in Curved Spacetimepp, pp. 391–415, 2014 and Yagdjian in J. Math. Phys. 54(9):091503, 2013) to investigate several well-known equations such as Tricomi-type equation, the Klein–Gordon equation in the de Sitter and Einstein–de Sitter spacetimes. The generalization given in this note allows us to consider also evolution equations with x-dependent coefficients.