On a method of solving of Cauchy problem for partial differential equations on a half-plane in the class of functions of polynomial growth (Q1311155)

With help of the Fourier transform the author proves existence, uniqueness and an explicit solution formula to the Cauchy problem for the equation \[ {\partial ^nu \over \partial t^n} + \sum^n_{k = 1} P_k \left( i {\partial \over \partial x} \right) {\partial^{n - k} u \over \partial t^{n - k}} = 0, \;t > 0, \;x \in\mathbb{R}^1, \] where \(P_k\) are polynomials. Solutions are found in a class of functions with polynomial growth in both variables.