Abstract
Let \(\beta \geqslant \alpha > - {\text{1/2}}\) and \(F\) be an even function of class \({{C}^{2}}(\mathbb{R})\) . The paper studies the properties of solutions to the Cauchy problem \(\frac{{{{\partial }^{2}}U}}{{\partial {{x}^{2}}}} + \frac{{(2\alpha + 1)}}{x}\frac{{\partial U}}{{\partial x}}\frac{{{{\partial }^{2}}U}}{{\partial {{t}^{2}}}} + \frac{{(2\beta + 1)}}{t}\frac{{\partial U}}{{\partial t}},\quad x > 0,\;\;t > 0,\) \(U(x,0) = F(x),\quad \frac{{\partial U}}{{\partial t}}(x,0) = 0,\quad x \geqslant 0\) related to the structure of the kernel of the operator \(\mathcal{A}F(t) = \int\limits_0^\pi F\left( {\sqrt {{{r}^{2}} + {{t}^{2}} - 2rt\cos \theta } } \right){{\sin }^{{2\alpha }}}\theta d\theta \) for a fixed \(r > 0\) . It is shown that the functions from \(\operatorname{Ker} \mathcal{A}\) are uniquely determined by their values on \((0,r)\) and this interval cannot be replaced by the interval \((0,\rho )\) with \(\rho < r\) . A description of \(\operatorname{Ker} \mathcal{A}\) is found in the form of series of normalized Bessel functions \({{j}_{\alpha }}(\lambda x)\) , \(\lambda \in {{\mathcal{N}}_{r}}\) , where \({{\mathcal{N}}_{r}} = \{ x > 0:{{j}_{\alpha }}(rx) = 0\} \) . With the help of these results, new uniqueness theorems for solutions to the indicated Cauchy problem are established, theorems on the representation of solutions satisfying the condition \(U(\xi ,t) = 0\) , \(\xi \in E\) , \(t > 0\) are obtained, where the set \(E\) consists of one positive number or \(E\) coincides with the set of positive zeros of the function \({{j}_{\alpha }}\) , and a new theorem on two radii is proved.
