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Analysis of explosion for nonlinear Volterra equations. (English) Zbl 0932.45007

The author gives a survey of results on Volterra integral equations of the form \[ u(t) = \int_0^t k(t-s)r(s)g(u(s)+h(s)) ds,\quad t \geq 0, \] where the solution becomes infinite in finite time. The questions studied in the papers covered by the survey are sufficient conditions to ensure the existence of a blow-up solution, estimates for the time to the singularity and the asymptotic behavior at the singularity. The connections to the related partial differential equations describing various combustion or explosion processes are discussed and some results on the related problem of quenching are mentioned as well.

MSC:

45G05 Singular nonlinear integral equations
80A25 Combustion
45M05 Asymptotics of solutions to integral equations
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