Cauchy Problem for the Euler–Poisson–Darboux Equation with a Dirac Potential Concentrated at Finitely Many Given Points

Abstract

In a strip, we consider an equation with the Euler–Poisson–Darboux operator containing a real positive parameter. We prove an energy inequality and the uniqueness of the classical solution of the Cauchy problem for the homogeneous equation, derive a formula for the solution, and establish its continuous dependence on the parameter. For the equation with a free term that is a linear combination of solution values at finitely many given points (a Dirac potential), we prove the uniqueness of the classical solution of the Cauchy problem and obtain a solution formula.