Abstract Cauchy problems that involve a product of two Euler-Poisson-Darboux operators

Let X be a Banach space, let B be the generator of a continuous group in X, and let A = B². Assume that D(A^r) is dense in X for r an arbitrarily large positive integer and that a and b are non-negative reals. Solution representations are developed for the abstract differential equation

$\text{(D}{}^2{}_{\mathrm{t}}\text{+}\text{b}\text{t}\text{D}{}_{\mathrm{t}}\text{? A) · (D}{}^2{}_{\mathrm{t}}\text{+}\text{a}\text{t}\text{D}{}_{\mathrm{t}}\text{? A) u(t) = 0, t > 0}$

corresponding to initial conditions of the form: (i) u(0+) = φ, u^(j)(0+) = 0, j = 1, 2, 3 and (ii) u²(0+) = φ, u^j(0+) = 0, j = 0, 1, 3 (with φ∈D(A^r)) for all choices of a and b.