On the first mixed problem in banach spaces for the degenerate parabolic equations with changing time direction

The paper is devoted to the study of a section of nonclassical differential equations, namely, the solvability problems for second-order parabolic equations with changing time direction. It is well known that in ordinary boundary value problems for strictly parabolic equations the smoothness of the initial and boundary conditions completely ensures that the solutions belong to Hölder spaces, but in the case of equations with changing time direction the smoothness of the initial and boundary conditions does not ensure that the solutions belong to such spaces. S. A. Tersenov, for a model parabolic equation with changing time direction, and S. G. Pyatkov, for a more general second-order equation, both obtained necessary and sufficient conditions for solvability of the corresponding mixed problems in Holder spaces, while the initial and boundary conditions were always assumed to be zero. In this paper, we consider cases where the initial and boundary conditions belong to Banach spaces. We introduce functional spaces in which solutions must be sought and obtain appropriate a priori estimates that make it possible to find solvability conditions for these problems. We also study the properties of the solutions obtained. In particular, we establish the equivalence of the Riesz and Littlewood–Paley conditions analogous to those for solutions of strictly elliptic and strictly parabolic equations of the second order. The unique solvability of the first mixed problem with boundary and initial functions from a Banach space is proved.