Quenching analysis for nonlinear Volterra integral equations

We study nonlinear second-kind Volterra integral equations (VIEs) of Hammerstein type with convolution kernels where the nonlinear kernel function has a singular point. Quenching occurs only when solutions approach the singularity point in finite time, but this is not sufficient for the blow-up of the first derivative at that time. For weakly singular convolution kernels, we give some sharp conditions for the quenching of the solution that are similar to the ones provided in our blow-up analysis for Hammerstein-type VIEs (J. Integral Equations Appl. 24 (2012), 487-512). However, for smooth convolution kernels we require new techniques for analyzing the blow-up of the first derivative of the solution. Finally, we apply our quenching conditions to some VIEs that correspond to applications modelled by parabolic differential equations that are driven by moving concentrated nonlinear sources, nonlocal effects or nonlinear boundary conditions.