Criterion for the solvability of the cauchy problem for an abstract Euler–Poisson–Darboux equation

A Family of Singular Differential Equations

“…and is the Bessel operator, and the validity of equality ( 10) is established earlier in [4], see also Chapter 1 in [5,[6][7][8][9][10][11].…”

Solvability of Degenerating Hyperbolic Differential Equations with Unbounded Operator Coefficients

“…We consider a more general form of the differential equation and the operator coefficient (which, in addition, has a variable component) than in [4][5][6]. The research is also conducted by reduction to a Euler-Poisson-Darboux equation, whereby we need to use the operator Bessel function introduced by the present author [8] and studied in [8,9].…”

“…Let us describe the class of operators A to be dealt with below when considering various equations. The papers [8,9] studied the well-posed solvability of the Cauchy problem…”

“…A necessary and sufficient solvability condition is stated in [8] in terms of estimates for the norm of the resolvent (λI − A) −1 and its weighted derivatives, while the paper [9], unlike [8], states a criterion for the uniform well-posedness of this problem in terms of a fractional power of the resolvent and its nonweighted derivatives.…”

“…The class G k will be referred to as the well-posed solvability class and the number k, as its index . Examples of operators A ∈ G k and the Bessel operator functions Y k (t) generated by them can be found in [9]. In particular, if…”