Problem without Initial Conditions for a Class of Inverse Parabolic Operator-Differential Equations of Third Order

Abstract—In a weighted Sobolev-type space, the well-posedness and unique solvability of a problem without initial conditions for a third-order operator-differential equation with an inverse parabolic principal part are established. The solvability conditions are expressed 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. Note that the principal part of the equation has a multiple characteristic.