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arXiv · 1710.07583

Blow-up and superexponential growth in superlinear Volterra equations

John A. D. Appleby·Denis D. Patterson
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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.

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BibTeXRIS

John A. D. Appleby, Denis D. Patterson. 2017-10-20. Blow-up and superexponential growth in superlinear Volterra equations. https://doi.org/10.3934/dcds.2018174

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