Blow-up solutions in one-dimensional diffusion models

A sufficient and necessary condition for the existence of blow-up solutions of a certain class of nonlinear Volterra integral equations with kernels arising from various diffusion models and with nonlinearities satisfying the condition g(0)=0 is given in the form of the so-called generalized Osgood condition. Such nonlinearities were not permitted in the original articles of Olmstead and Roberts where these diffusion models were introduced.