Semigroup methods to solve non-autonomous evolution equations

Under regularity conditions on the family of (unbounded, linear, closed) operators (L(t))(t is an element of(0,T]) (T > 0) on a Banach space X, there exists an evolution family (V(t, s))(T greater than or equal to t greater than or equal to s>0) on X such that U(t, s)x = L(t)V-1(t, s)L(s)x is the unique classical solution of the non-autonomous evolution equation [GRAPHICS] for x is an element of D(L(s)). Moreover, the evolution semigroup associated to the evolution family (V(t, s))(T greater than or equal to t greater than or equal to s>0) on C-0((0,T]; X), the Banach space of continuous functions f from [0, T] into X satisfying f(0) = 0, is generated by the closure of -L(.)(d/dt + L(.))L(.)(-1). An application to parabolic partial differential equations is given.