Notes on ultraslow nonlocal telegraph evolution equations

Thang, Nguyen Nhu

doi:10.1090/proc/15877

Notes on ultraslow nonlocal telegraph evolution equations
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by Nguyen Nhu Thang;

Proc. Amer. Math. Soc. 151 (2023), 583-593

DOI: https://doi.org/10.1090/proc/15877

Published electronically: November 10, 2022

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Abstract:

This paper provides a refinement of the study of asymptotic behaviour for a class of nonlocal in time telegraph equations with positively singular kernels. Based on fundamental properties of relaxation functions and recent representation of the fundamental solution in [Nonlinear Anal. 193 (2020), 111411], we establish the asymptotic expansions of the variance of the stochastic process for both long-time and short-time, which sharply improves the main result in [Proc. Amer. Math. Soc. 149 (2021), 2067–2080] by removing their technical conditions on the regularly varying behaviours and reformulating the asymptotic expansion in a more natural form. By analysing a new noncommutative operation on a subclass of completely positive functions, we provide a new way to construct finitely many ultraslow subdiffusion processes that are rapidly slower than a given ultraslow kernel. Consequently, we show that for a given completely monotonic ultraslow kernel, there is an induced kernel whose corresponding mean square displacement is logarithmic.

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Notes on ultraslow nonlocal telegraph evolution equations
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Abstract: