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.