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 ΠNj=1(d/dt - Aj)u = 0 as u(t) = ΣNj=1exp(tAj)fj. Here A1, . . . , AN are (linear) commuting semigroup generators, and Ai - Aj is 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) = ∫℘∫Xexp{∫t0Cv(r) dr}fμ(df)λ(dv), where v: [0, ∞) → {1, . . . , N} is locally Riemann integrable, Cv(r) = Av(r)(r), and μ (resp. λ) is a finite measure on X (resp. the space ℘ of these functions v). In addition we discuss the general solution of the inhomogeneous equation ΠNj=1(d/dt - Aj)u = h(t) for a rather general right-hand side h. © 1996 Academic Press, Inc.

Publication Title

Journal of Mathematical Analysis and Applications