On behavior of the Cauchy-type integral at the endpoints of the integration contour and its application to boundary value problems for parabolic equations with changing direction of time

We consider N. I. Muskhelishvili's theorem about the behavior of Cauchy-type integrals at the endpoints of the integration contour and the discontinuity points of the density and its application to boundary value problems for $2n$-parabolic equations with changing direction of time. For parabolic equations with changing direction of time, the smoothness of initial and boundary data does not imply in general that the solution belongs to the Holder spaces. Application of the theory of singular equations makes it possible to specify necessary and sufficient conditions for the solution to belong to the Hölder spaces. Moreover, under general gluing conditions, using unified approach we can show that for such equations the nonintegral exponent of the space may essentially affect both the number of solvability conditions and the smoothness of the solutions. To prove the solvability of boundary value problems for such equations, we consider continuous bonding gluing conditions with the $(2n-1)$-th derivative. Note that in the case of $n = 3$ the smoothness of the initial data and solvability conditions determine that the solution belongs to smoother Holder spaces near the endpoints of the integration contour.