Criterion for the Solvability of the Weighted Cauchy Problem for an Abstract Euler–Poisson–Darboux Equation

Abstract

In a Banach space E, we consider the abstract Euler–Poisson–Darboux equation u″(t) + kt⁻¹u′(t) = Au(t) on the half-line. (Here k ∈ ℝ is a parameter, and A is a closed linear operator with dense domain on E.) We obtain a necessary and sufficient condition for the solvability of the Cauchy problem u(0) = 0, lim _t→0+t^ku′(t) = u₁, k < 0, for this equation. The condition is stated in terms of an estimate for the norms of the fractional power of the resolvent of A and its derivatives. We introduce the operator Bessel function with negative index and study its properties.