Abstract initial boundary value problems

We consider abstract initial boundary value problems in a spirit similar to that of the classical theory of linear semigroups. We assume that the solution u at time t is given by u(t) = S(t) ξ + V(t)g, where ξ and g are respectively the initial and boundary data and S(t) and V(t) are linear operators. We take as a departing point the functional equations satisfied by the propagators S and V. We discuss conditions under which a pair (S, V) describes the solution of an abstract differential initial boundary value problem. Several examples are provided of parabolic and hyperbolic problems that can be accommodated within the abstract theory. We study the backward Euler's method for the time integration of the problems considered.