Endpoint-Controlled Two-Term Asymptotics for Nonlinear Caputo Relaxation: Memory–Constitutive Competition and Hitting Times

Grienggrai Rajchakit

doi:10.53941/bci.2026.100014

Abstract

We derive two-term asymptotic expansions for nonlinear Caputo relaxation while isolating the recent-history concentration that first‑order regular variation does not control. Under a terminal uniform‑integrability condition, the Caputo dissipation retains its finite‑initial‑value term and has an explicit memory correction. Ultimate convexity and a local derivative bound are sufficient for this condition, whereas a plateau–catch‑up construction proves that bare regular variation is insufficient. For a restoring law \(\phi(x) = cx^{\beta}\{1 + ax^{\sigma} + o(x^{\sigma})\}\), the state satisfies \(u(t) \sim K t^{-\alpha/\beta}\). Its first relative correction has order \(t^{-\alpha\sigma/\beta}\) when \(\sigma < 1\) and order \(t^{-\alpha/\beta}\) when \(\sigma \ge 1\), separating constitutive dominance, resonance, and memory dominance. For the exact‑power law, the coefficient changes sign at \(\alpha = \beta/(\beta + 1)\); the same sign determines whether the corrected residual hitting time is earlier or later than its leading estimate. At \(T = 3000\), implicit L1 diagnostics recover the subcritical and supercritical state coefficients within \(0.85\%\) and \(4.6\%\), respectively, and the resonant coefficient within \(1\%\). These computations support the predicted signs and scales but do not replace the analytical endpoint hypothesis.

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