The Snapshot Problem for the Euler-Poisson-Darboux Equation

Abstract

The generalized Euler-Poisson-Darboux (EPD) equation with complex parameter $\alpha$ is given by $$ \Delta_x u=\frac{\partial^2 u}{\partial t^2}+\frac{n-1+2\alpha}{t}\,\frac{\partial u}{\partial t}, $$ where $u(x,t)\in \mathscr E(\mathbb R^n\times \mathbb R)$, with $u$ even in $t$. For $\alpha=0$ and $\alpha=1$ the solution $u(x,t)$ represents a mean value over spheres and balls, respectively, of radius $|t|$ in $\mathbb R^n$. In this paper we consider existence and uniqueness results for the following two-snapshot problem: for fixed positive real numbers $r$ and $s$ and smooth functions $f$ and $g$ on $\mathbb R^n$, what are the conditions under which there is a solution $u(x,t)$ to the generalized EPD equation such that $u(x,r)=f(x)$ and $u(x,s)=g(x)$? The answer leads to a discovery of Liouville-like numbers related to Bessel functions, and we also study the properties of such numbers.