We consider the boundary value problems in a quarter-plane for a loaded heat conduction operator (one-dimensional in the space variable). A peculiarity of the operator in question is as follows: first, the spectral parameter is the coefficient of the loaded summand; second, the order of the derivative in the loaded summand is equal to that of the differential part of the operator, and third, the load point moves with a variable velocity. We demonstrate that the boundary value problem under study is Noetherian.Keywords: loaded heat conduction operator, boundary value problem, adjoint operator, spectrum, resolvent set, Noetherian operator, index of an operatorIn the article we consider the boundary value problems for a loaded heat conduction operator (onedimensional in the space variable) in a quarter-plane which relates to the class of functional-differential operators and has the form Lu + λBu, where L is the differential part and B is the loaded part. The operator in question is peculiar since the spectral parameter λ is the coefficient of the loaded summand, the order of the derivative in the loaded summand is equal to that of the differential part of the operator, and the load point defined by the functionx(t) moves with a variable velocity; i.e., the derivativex (t) is not always constant. Moreover, the load operator B in the generalized spectral problem Lu + λBu = 0 of this article is not invertible (usually, B −1 is supposed to be exist; for example, see [1, pp. 520-523]). Such operator is called spectrally loaded. Unlike the loaded differential equations studied early in [2][3][4][5], in our case the loaded summand in the equation is not a weak perturbation of the differential part [6,7]. Here the loaded differential operator reveals some new properties not enjoyed by the operators with weak perturbation. We demonstrate that the boundary value problem under consideration is Noetherian and has finite positive index for some strictly described values of the spectral parameter λ in the complex plane. Observe also the article [8] devoted to solving the boundary value problem for a parabolic equation when its loaded part enters in the coefficient of the lower-order summand of the equation in question.The article comprises eight sections and the conclusion. After stating the problems (Section 1), we reduce the initial adjoint boundary value problems to a pair of adjoint Volterra-type integral equations (Section 2). Then we introduce a pair of adjoint characteristic integral equations (Section 3) and give their solutions (Section 4). After that we solve the adjoint Volterra-type integral equations by regularization using the characteristic equations (Section 5) and the initial boundary value problem (Section 6). Finally, Sections 7 and 8 are devoted to spectral problems. In conclusion, we present a series of problems which can be studied by this scheme.1. Statements of the problems. Let R + = (0, ∞), R − = (−∞, 0), and R = (−∞, ∞). In the domain Q = {x ∈ R + , t ∈ R + }, we consider the following boundary value p...
