Uniqueness theorems for solutions of differential equations (Q2901782)

The paper is devoted to the investigation of differential equations of special type. One of it is the so-called Darboux equation \(\frac{\partial^2u}{\partial \rho^2}+\frac{n}{\rho}\frac{\partial u}{\partial \rho}=\Delta_x u,\) where \(\Delta_x\) denotes the Laplace operator. Theorem 1 states the uniqueness of the solution of the above equation for \(u\in C^2\). Theorem 2 states the uniqueness of the so-called wave equation \({\partial^2u}/{\partial t^2}=\Delta_x u\) for some initial data and \(u\in C^2\). The results obtained are applicable to many other questions of analysis.