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<∞}{U(t,s):0≤s≤t<∞} associated with a semilinear evolution equation ∂u/∂t=∂2u/∂x2+f(u),u(0)=u0∈X∂u/∂t=∂2u/∂x2+f(u),u(0)=u0∈X with periodic boundary conditions, it is proved that, under suitable conditions, the unique (mild) solution u(t)=U(t,0)u(0),t≥0u(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.