Blow-up conditions for nonlinear Volterra integral equations with power nonlinearity
The authors study the existence of blow-up (i.e., nontrivial) solutions of nonlinear Volterra equations of the form \[ u(t)=\int_0^t k(t-s) g(u(s))\,ds, \quad t\geq 0,\tag{1} \] where \(K:(0,\infty)\to[0,\infty)\) is a locally integrable positive function and \(g:[0,\infty)\to[0,\infty)\) is a strictly increasing absolutely continuous function of the form \[ g(t)=\begin{cases} g_1(t)&\text{if}\;\;t\in[0,d),\\ bt^p&\text{if}\;\;t\geq d,\end{cases} \] where \(d>0,\;b>0,\;p>1\), \(g_1(0)=0\) and \(\lim_{u\to 0^+}(u/g_1(u))=0\). The following criterion is established: equation (1) has a blow-up solution if and only if the integral \[ \int_0^t K^{-1}(s)\frac{ds}{s(-\ln s)},\quad\text{where}\;\;K(t)=\int_0^t k(s)\,ds, \] is convergent for any \(t\in(0,1)\).
scientific article; zbMATH DE number 5380073