Higher-order asymptotic expansion for abstract linear second-order differential equations with time-dependent coefficients

Abstract

This paper is concerned with the asymptotic expansion of solutions to the initial-value problem of u″(t)+Au(t)+b(t)u′(t)=0 in a Hilbert space with a nonnegative selfadjoint operator A and a coefficient b(t)∼(1+t)−β(−1<β<1). In the case b(t)≡1, it is known that the higher-order asymptotic profiles are determined via a family of first-order differential equations of the form v′(t)+Av(t)=Fn(t) (Sobajima (2021) [10]). For the time-dependent case, it is only known that the asymptotic behavior of such a solution is given by the one of b(t)v′(t)+Av(t)=0. The result of this paper is to find the equations for all higher-order asymptotic profiles. It is worth noticing that the equation for n-th order profile u˜n is given via v′(t)+mn(t)Av(t)=Fn(t) which coefficient mn (time-scale) differs each other.