On resolvability of a parabolic problem on the graph

An initial-boundary problem of a parabolic type given on a geometrical graph (a spatial network) is considered. It is supposed that coefficients of the equation satisfy the Hölder condition with respect to spatial and time variables on edges of the graph. Non-uniform conditions of the first, second or third kind are set on the boundary of the network. The solution of the equation satisfies the consistency condition for derivatives at nodes of the graph and can be discontinuous. Meanwhile, it is supposed that coefficients from conditions on the boundary and in nodes of the graph satisfy the Hölder condition with respect to a time variable. The theorem of existence of a mixed problem is proved. It gives representation of the solution through thermal potentials.