On Equations with Infinitely Many Derivatives: Integral Transforms and the Cauchy Problem

We analyze initial value problems for ordinary differential equations with infinitely many derivatives such as (linearized versions of) nonlocal field equations of motion appearing in particle physics, nonlocal cosmology and string theory. We show that the corresponding initial value problem on a half line is well-posed and that it requires only a finite number of initial conditions. We also investigate nonlinear pseudo-equations defined by functions of the Laplace operator, that is, nonlinear partial differential equations with infinitely many derivatives. © Published under licence by IOP Publishing Ltd.