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

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) similar to (1 + t)(-beta )(-1 < beta < 1). In the case b(t) equivalent to 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. (C) 2022 Elsevier Inc. All rights reserved.