STRUCTURE OF SOLUTIONS TO LINEAR EVOLUTION-EQUATIONS - EXTENSIONS OF DALEMBERTS FORMULA

The d'Alembert formula expresses the general solution of the factored equation Pi(j=1)(N)(d/dt - A(j))u = 0 as u(t) = Sigma(j=1)(N) exp(tA(j ))f(j). Here A(1), ..., A(N) are (linear) commuting semigroup generato rs, and A(i) - A(j) is injective for i not equal j. The analogue of th is fails when A(j) depends on t. But in this nonautonomous case we sho w that the general solution has the form u(t)= integral(P) integral(X) exp{integral(0)(t)C(nu)(r)dr}f mu(df)lambda(d nu), where nu: [0, infi nity) --> {1,..., N) is locally Riemann integrable, C-nu(r) = A(nu(r)) (r), and mu (resp. lambda) is a finite measure on X (resp. the space P of these functions nu). In addition we discuss the general solution o f the inhomogeneous equation Pi(j=1)(N)(d/dt - A(j))u = h(t) for a rat her general right-hand side h. (C) 1996 Academic Press, Inc.