Resolvability of the Cauchy problem for a linear equation of second order with non-densely defined operator coefficients which generate semigroups with singularities (Q1902790)

The author considers an abstract Cauchy problem (1) \(v'' + A(t)v' + B(t)v = f(t)\) \((0 < t < 1)\), \(v(0) = v_0\), \(v'(0) = v_1\) in a Banach space \(E\) where \(A(t)\), \(B(t)\) are given linear operators for every \(t \in [0,1]\) with stationary domains \(D = D(A(t))\) and \(D_B = D (B(t))\), and \(f(t)\) is a continuous function. Under quite general assumptions, admitting that the sets \(D\) and \(D_B\) are not dense in \(E\) and the semigroups generated by \(A(t)\) and \(B(t) A^{-1} (t)\) have singularities, he obtains sufficient conditions for (1) to have a unique solution. As an application, he proves a result on the solvability of a boundary value problem for a partial differential equation describing sound propagation in a viscous gas.