Gevrey Asymptotics for Logarithmic-Type Solutions to Singularly Perturbed Problems with Nonlocal Nonlinearities

Abstract

We investigate a family of nonlinear partial differential equations which are singularly perturbed in a complex parameter ϵ and singular in a complex time variable t at the origin. These equations combine differential operators of Fuchsian type in time t and space derivatives on horizontal strips in the complex plane with a nonlocal operator acting on the parameter ϵ known as the formal monodromy around 0. Their coefficients and forcing terms comprise polynomial and logarithmic-type functions in time and are bounded holomorphic in space. A set of logarithmic-type solutions are shaped by means of Laplace transforms relatively to t and ϵ and Fourier integrals in space. Furthermore, a formal logarithmic-type solution is modeled which represents the common asymptotic expansion of the Gevrey type of the genuine solutions with respect to ϵ on bounded sectors at the origin.