A system of nonlinear Volterra equations with blow-up solutions (Q388711)

A system of two nonlinear convolution-type Volterra equations (of the second kind) NEWLINE\[NEWLINEu_1(t)=\int_0^tk_1(t-s)F_1(u_2(s)+h_2(s))\;ds,NEWLINE\]NEWLINE NEWLINE\[NEWLINEu_2(t)=\int_0^tk_2(t-s)F_2(u_1(s)+h_1(s))\;dsNEWLINE\]NEWLINE with nonnegative kernels and positive, increasing and convex nonlinearities is considered. It is shown that a unique local solution exists which is global if the \(k_i\) decay quickly enough. It is shown that the solution is strictly increasing, and sufficient conditions for a blow-up together with a blow-up rate are obtained. The resuls are applied to fractional integral kernels with power nonlinearities, and for \(k_i(t)=e^{-ct}t^{\beta-1}\).