Memory-induced blow-up solutions and their dynamical transitions in distributed delay differential equations
Abstract
In this paper, we investigate finite-time blow-up solutions for distributed delay differential equations incorporating memory effects with a specific gamma distribution kernel. Focusing on typical nonlinear terms that cause finite-time singularities in ordinary differential equations (ODEs), we examine how memory effects change the existence, rate, and qualitative properties of blow-up. By comparing these behaviors with those of the corresponding non-delayed and memory-free ODEs, we show that time delay originating from memory not only essentially induces finite-time blow-up (``memory-induced blow-up''), but also drastically alters the blow-up profile, such as accelerating algebraic blow-up rates or driving a qualitative transition from ODE quenching to logarithmic blow-up. Furthermore, by incorporating a self-inhibitory term, we reveal a threshold phenomenon in the phase space that governs the occurrence and non-occurrence of blow-up depending on the initial conditions and parameters. These results are established by reducing the system to a two-dimensional ODE via the linear chain trick, and analyzing the dynamics at infinity using Poincaré-type compactification, blow-up techniques, and the center manifold theorem.
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BibTeX
@misc{ichida2026memoryinduced,
title = {{Memory-induced blow-up solutions and their dynamical transitions in distributed delay differential equations}},
author = {Yu Ichida},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15470},
eprint = {2609.15470},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15470},
}
RIS
TY - UNPB
TI - Memory-induced blow-up solutions and their dynamical transitions in distributed delay differential equations
AU - Yu Ichida
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15470
UR - https://arxiv.org/abs/2609.15470
AB - In this paper, we investigate finite-time blow-up solutions for distributed delay differential equations incorporating memory effects with a specific gamma distribution kernel. Focusing on typical nonlinear terms that cause finite-time singularities in ordinary differential equations (ODEs), we examine how memory effects change the existence, rate, and qualitative properties of blow-up. By comparing these behaviors with those of the corresponding non-delayed and memory-free ODEs, we show that time delay originating from memory not only essentially induces finite-time blow-up (``memory-induced blow-up''), but also drastically alters the blow-up profile, such as accelerating algebraic blow-up rates or driving a qualitative transition from ODE quenching to logarithmic blow-up. Furthermore, by incorporating a self-inhibitory term, we reveal a threshold phenomenon in the phase space that governs the occurrence and non-occurrence of blow-up depending on the initial conditions and parameters. These results are established by reducing the system to a two-dimensional ODE via the linear chain trick, and analyzing the dynamics at infinity using Poincaré-type compactification, blow-up techniques, and the center manifold theorem.
ER -