Coupled Volterra integral equations with blowing up solutions (Q1751563)

The following system of nonlinear integral equations related to combustion problems is considered \[ \begin{cases} u(x)=\int\limits_{0}^{x} (x-s)^{\alpha-1}[v(s)]^{\gamma}ds, \\ v(x)=\int\limits_{0}^{x}k(x-s,u(s))ds, \end{cases}\tag{1} \] where \[ k(x-s,u(s))=\sum\limits_{i=1}^{m}(x-s)^{\beta_i-1}g_i(u(s)). \] Here \(\alpha,\beta_i>1\), \(i=1,2,\dots,m\), \(\gamma>0\), and the functions \(g_i\), \(i=1,2,\dots,m\), are nondecreasing continuous functions such that \(g_i(0) = 0\). The author studies ignition and blow-up criteria for the system of Equations (1). Necessary and sufficient conditions for the existence and explosion of positive solutions are given. In addition, the uniqueness of the positive solutions is shown. The main results are obtained by monotonicity methods.