Coupled Volterra equations with blow-up solutions

A pair of coupled nonlinear Volterra equations are examined for possible blow-up solutions. The system is motivated by certain models of explosion phenomena in a diffusive medium. Criteria for a blow-up to occur as well as bounds on the time of its occurrence are derived for a general class of nonlinearities. Specific results are obtained for two special cases involving power law and exponential nonlinearities. Also, the asymptotic growth rate near blow-up is determined for these two special cases when the kernel behaves like that of the one-dimension heat equation.