Problem without initial conditions for a class of inverse parabolic operator-differential equations of third order (Q1673689)

Consider a third-order differential operator equation of the form \[ \left (- \frac{d}{dt} +A \right )^3 u(t) + \sum \limits_{j=1}^3 A_j \frac{d^{3-j} u(t)}{dt^{3-j}} =f(t), \;\;t \in \mathbb R_+, \eqno{(1)} \] where \(A_j \) \((j=1,2,3)\) are linear unbounded operators, \(A\) is a self-adjoint positive definite operator in a separable Hilbert space \(H\). The well-posedness (the regular solvability) of the equation (1) in a weighted Sobolev-type space is established. The solvability conditions are formulated in terms of the operator coefficients of the equation. Additionally, the norms of the operators of intermediate derivatives closely related to the solvability conditions are estimated. The relation between the weight exponent and the lower boundary of the spectrum of the basic operator involved in the principal part of the equation is established.