Blow-up conditions for nonlinear Volterra integral equations with power nonlinearity

Abstract

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.

Blow-up collocation solutions of nonlinear homogeneous Volterra integral equations

“…. , Z n−1,1 fulfilling the equation (6). Note that these coefficients are strictly positive, and hence, it is guaranteed that the corresponding collocation solution z h is (strictly) positive in ]0, t n ].…”

Nonlinear Volterra integral equations and the Schröder functional equation

“…Among the many conditions obtained for the existence of nontrivial or blow-up solutions, we can distinguish in particular two kinds of such conditions: those expressed in terms of the convergence of some series [3,4] and those expressed in terms of the convergence of some integrals [5][6][7][8][9][10]. However, there is quite a big difference between these two classes of conditions.…”

“…If the integral ∫ t 0 K −1 (s 1−q ) ds s(− ln s)is convergent for t ∈ (0, δ), δ > 0, then Eq. (1) with g given by(8) has a nontrivial solution.Theorem 6.3. If Eq.…”

Blow-up time for solutions to some nonlinear Volterra integral equations

“…We can find such necessary and sufficient conditions e.g. in [2,4,6]. When the blow-up solution exists, an additional challenge lies in establishing the value of the critical time, when the explosion occurs [1,11,13].…”