# American Institute of Mathematical Sciences

August  2018, 38(8): 3993-4017. doi: 10.3934/dcds.2018174

## Blow-up and superexponential growth in superlinear Volterra equations

 School of Mathematical Sciences, Dublin City University, Dublin, Ireland

* Corresponding author

Received  October 2017 Revised  March 2018 Published  May 2018

This paper concerns the finite-time blow-up and asymptotic behaviour of solutions to nonlinear Volterra integro-differential 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 $\lim_{t\toτ}A(x(t), t) = 1$, where $τ$ is the blow-up time if solutions are explosive or $τ = ∞$ 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.

Citation: John A. D. Appleby, Denis D. Patterson. Blow-up and superexponential growth in superlinear Volterra equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3993-4017. doi: 10.3934/dcds.2018174
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