Asymptotic expansions with exponential, power, and logarithmic functions for non-autonomous nonlinear differential equations

Abstract

This paper develops further and systematically the asymptotic expansion theory that was initiated by Foias and Saut in [11]. We study the long-time dynamics of a large class of dissipative systems of nonlinear ordinary differential equations with time-decaying forcing functions. The nonlinear term can be, but not restricted to, any smooth vector field which, together with its first derivative, vanishes at the origin. The forcing function can be approximated, as time tends to infinity, by a series of functions which are coherent combinations of exponential, power and iterated logarithmic functions. We prove that any decaying solution admits an asymptotic expansion, as time tends to infinity, corresponding to the asymptotic structure of the forcing function. Moreover, these expansions can be generated by more than two base functions and go beyond the polynomial formulation imposed in previous work.

45 pages. Accepted to Journal of Evolution Equations

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• Asymptotic expansions for the Lagrangian trajectories from solutions of the Navier-Stokes equations
• A new form of asymptotic expansion for non-smooth differential equations with time-decaying forcing functions