Blow-up and superexponential growth in superlinear Volterra equations

TL;DR

This paper investigates the conditions under which solutions to nonlinear Volterra equations blow up or grow superexponentially, providing sharp estimates on their growth rates and improving existing criteria for blow-up.

Contribution

It offers new sharp estimates on growth rates of solutions to superlinear Volterra equations with nonsingular kernels, enhancing understanding of blow-up behavior.

Findings

Established sharp growth rate estimates for solutions

Improved blow-up criteria using new methods

Unified analysis for explosive and nonexplosive solutions

Abstract

This paper concerns the finite-time blow-up and asymptotic behaviour of solutions to nonlinear Volterra integrodifferential equations. Our main contribution is to determine sharp estimates on the growth rates of both explosive and nonexplosive solutions for a class of equations with nonsingular kernels under weak hypotheses on the nonlinearity. In this superlinear setting we must be content with estimates of the form $\lim_{t\to\tau}A(x(t),t) = 1$, where $\tau$ is the blow-up time if solutions are explosive or $\tau = \infty$ if solutions are global. Our estimates improve on the sharpness of results in the literature and we also recover well-known blow-up criteria via new methods.