A generalization of Osgood's test and a comparison criterion for integral equations with noise

TL;DR

This paper extends Osgood's test to more general cases and introduces a comparison criterion for noisy integral equations, with applications to crack failure analysis involving Brownian motion noise.

Contribution

It generalizes Osgood's test for solution explosion and develops a new comparison criterion for integral equations with stochastic noise.

Findings

Established a generalized Osgood's test for explosion conditions.

Developed a comparison criterion for noisy integral equations.

Provided estimates for explosion times in crack failure models.

Abstract

In this work, we prove a generalization of Osgood's test for the explosion of the solutions of initial-value problems. We also establish a comparison criterion for the solution of integral equations with noise, and provide estimations of the time of explosion of problems arising in the investigation of crack failures where the noise is the absolute value of the Brownian motion