A singularly perturbed Cauchy problem for a parabolic equation in the presence of the “weak” turning point of the limit operator

The article is devoted to the development of S. A. Lomov's regularization method for singularly perturbed Cauchy problems in the case of violation of the stability conditions for the spectrum of the limit operator. In particular, the problem is considered in the presence of a “weak” turning point, in which the eigenvalues “stick together” at the initial moment of time. Problems with this kind of spectral features are well known to specialists in mathematical and theoretical physics, as well as in the theory of differential equations; however, they have not been previously considered from the point of view of the regularization method. Our work fills this gap. Based on the ideas of S. A. Lomov and A. G. Eliseev for asymptotic integration of problems with spectral features, it indicates how to introduce regularizing functions, describes in detail the algorithm of the regularization method in the case of a “weak” turning point, and justifies this algorithm; an asymptotic solution of any order with respect to a small parameter is constructed.