The purpose of this paper is to develop the well-known theory on the elliptic, hyperbolic and parabolic equations in nonsmooth domains that has been presented by such Russian mathematicians as V. A. Kondratiev, V. G. Mazya, B. A. Plamenevsky and S. A. Nazarov. We will obtain an asymptotic expansion of the generalized solutions of first initial boundaryvalue problems for strongly parabolic systems near the conic point on the boundary of the infinite cylinder.