Blow–up and superexponential growth in superlinear volterra equations

Abstract

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 limt→τ 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.