Blow-up and superexponential growth in superlinear Volterra equations

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 $\tau =\mathrm{\infty }$ 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.