Space-Semidiscrete Approximation and Quenching-Time Convergence for a Nonlocal Neumann Problem with Power-Law Singular Reaction

Abstract

We study a spatially semidiscrete approximation of a nonlocal Neumann-type quenching problem with a power-law singular reaction. We first present continuous extremal and Kaplan–Jensen estimates as analytical benchmarks. The main analysis focuses on a quadrature-based cooperative system with nonnegative weights, for which we establish positivity, comparison, finite-time quenching, remaining-time estimates, and a discrete average bound. Under a reaction-dominance condition, a transformed-gap estimate for the semidiscrete flow yields quenching-time bounds that explicitly incorporate the quadrature kernel mass. Assuming second-order consistency of the trapezoidal quadrature, we prove that the semidiscrete solution converges uniformly on compact pre-quenching intervals and that the semidiscrete quenching time converges to its continuous counterpart. We also establish convergence of subcritical hitting times under a strict-crossing assumption. Adaptive explicit and implicit–explicit (IMEX) Euler computations illustrate these results.