Uniqueness and representation of solutions of the generalized Euler-Poisson-Darboux equation

Abstract

Let $\beta\geq\alpha>-1/2$ and $F$ be an even function of class $C^2(\mathbb{R})$. The paper studies the properties of solutions to the Cauchy problem

$$\frac{\partial^2U}{\partial x^2}+\frac{(2\alpha+1)}{x}\frac{\partial U}{\partial x}=   \frac{\partial^2U}{\partial t^2}+\frac{(2\beta+1)}{t} \frac{\partial U}{\partial t}, \quad x>0,\,\, t>0,$$

$$U(x,0)=F(x), \quad \frac{\partial U}{\partial t}(x,0)=0, \quad x\geq 0$$

related to the structure of the kernel of the operator

$$\mathcal{A}F(t)=\int\limits_{0}^{\pi}F(\sqrt{r^2+t^2-2rt\cos\theta})\sin^{2\alpha}\theta d\theta$$

for a fixed $r>0$. It is shown that the functions from $\mathrm{Ker}\, \mathcal{A}$ 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 $\mathrm{Ker}\, \mathcal{A}$ 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.