Structure of Solutions to Linear Evolution Equations: Extensions of d’Alembert’s Formula

The d’Alembert formula expresses the general solution of the factored equation jNs1 drdtyAj.us0 as u t.s Njs1exp tAj.fj. Here A1, . . . , AN are linear. commuting semigroup generators, and AiyAjis injective for i/j. The analogue of this fails when Aj depends on t. But in this nonautonomous case we show that the general solution has the form u t.sH H exp H tC r. dr4fm df.l dn., where P X 0 n n: w0,`.ª 1, . . . , N4is locally Riemann integrable, C r.sA r., and m resp. n n r . l.is a finite measure on X resp. the space P of these functions n .. In addition we discuss the general solution of the inhomogeneous equation jNs1 drdtyAj.u sh t. for a rather general right-hand side h.