The problem of the estimating of a blow-up time for solutions of Volterra nonlinear integral equation with convolution kernel is studied. New estimates, lower and upper, are found and, moreover, the procedure for the improvement of the lower estimate is presented. Main results are illustrated by examples. The new estimates are also compared with some earlier ones related to a shear band model.
In this work we consider nonlinear Volterra equations of a special type. We find necessary and sufficient conditions for blow-up of solutions to this class of equations.
Walter and Weckesser's result (Aequationes Math 46:212-219, 1993), extending the Bushell-Okrasiński convolution type inequality (Bushell and Okrasiński in J Lond Math Soc (2) 41:503-510, 1990), gave some general conditions on the functions k : [0, d) → R and g : [0, ∞) → R under which, for every increasing function f : [0, d) → [0, ∞), the inequality x 0 k (x − s) g (f (s)) ds ≤ g x 0 f (s) ds , x ∈ (0, d) , is satisfied. Applying the result on a simultaneous system of functional inequalities, we prove that if d > 1, then, in general, both k and g must be power functions.
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