A Volterra series representation for a class of nonlinear infinite dimensional systems with periodic boundary conditions

This paper proves the existence of a Volterra series representation for the mild solutions of a class of nonlinear infinite dimensional systems. More specifically, given the evolutionary system/operator { U ( t , s ) : 0 ≤ s ≤ t < ∞ } associated with a semilinear evolution equation ∂ u / ∂ t = ∂ 2 u / ∂ x 2 + f ( u ) , u ( 0 ) = u 0 ∈ X with periodic boundary conditions, it is proved that, under suitable conditions, the unique (mild) solution u ( t ) = U ( t , 0 ) u ( 0 ) , t ≥ 0 can be expanded by a Volterra series. A recursive algorithm is given to construct the Volterra kernels/series terms and a nonlinear heat equation is discussed to illustrate the proposed method.