On local solvability and blow-up of solutions of an abstract nonlinear Volterra integral equation

Abstract

A theorem on noncontinuable solutions is proved for abstract Volterra integral equations with operator-valued kernels (continuous and polar). It is shown that if there is no global solvability, then the C-norm of the solution is unbounded but does not tend to infinity in general. An example of Volterra equations whose noncontinuable solutions are unbounded but not infinitely large is constructed. It is shown that the theorems on noncontinuable solutions of the Cauchy problem for abstract equations of the first and nth kind (with a linear leading part) are special cases of the theorems proved in this paper.Obviously, this space is complete. We assume that the mapping A(t, u): R + × B → B has the following properties A1 and A2. Property A1. The mapping A(t, u) is continuous in the sense of the metric (2.2). Property A2. There is a function μ(t, s): RWe immediately note that property A1 implies the following property A3.Indeed, it follows from A1 that the numerical function A(t, θ) is continuous for all t ≥ 0. Further, properties A2 and A3 imply the following property.Property A4. There is a function λ(t, s): R 2 + → R + , bounded on each rectangle [0; T ] × [0; S], T, S > 0, such that ∀ t ≥ 0 ∀ u ∈ B A(t, u) ≤ λ(t, u ).

Instantaneous blow‐up versus local solvability of solutions to the Cauchy problem for the equation of a semiconductor in a magnetic field

“…This process can be continued. Using arguments similar to the reasoning of Panin, we can prove that there exists maximal T 0 = T 0 ( u 0 ) > 0 such that either T 0 = + ∞ or T 0 < + ∞ , and in the latter case, we have □…”

On critical exponents for weak solutions to the Cauchy problem for one nonlinear equation with gradient nonlinearity

“…Since (20), it follows that for any 𝜀 > 0, there is N ∈ N and that for all n, m ⩾ N, the following inequality holds:…”

“…For any function u 0 (x), which fulfills assumptions of Lemma 5, with 𝛼 ∈ (0, 1] and q = 2 or q ⩾ 4, there is T 𝛼 = T 𝛼 (u 0 ) > 0, such that for all T ∈ (0, T 𝛼 ), there exists a unique solution v(x, t) of the integral equation (191) Proof. To prove the theorem, one should use the compression mappings method and implement the standard algorithm for continuation of the local-in-time solution v(x, t) ∈ C([0, T]; C 1+𝛼 ((1 + |x| 2 ) 1∕2 ; R 3 )) (as described, for example, in Panin 20 ). Using (75), (76), results of lemmas 5, (7), and the estimate (187), one can easily arrive at the theorem statement.…”

“…Proof To prove the theorem, one should use the compression mappings method and implement the standard algorithm for continuation of the local‐in‐time solution $$$ v\left(x,t\right)\in \mathbb{C}\left(\left[0,T\right];{\mathbb{C}}&amp;amp;#x0005E;{1&amp;amp;#x0002B;\alpha}\left({\left(1&amp;amp;#x0002B;{\left&amp;amp;#x0007C;x\right&amp;amp;#x0007C;}&amp;amp;#x0005E;2\right)}&amp;amp;#x0005E;{1/2};{\mathbb{R}}&amp;amp;#x0005E;3\right)\right) $$$ (as described, for example, in Panin 20 ). Using (), (), results of lemmas 5, (7), and the estimate (), one can easily arrive at the theorem statement.…”

An Investigation of an Integral Equation Involving Convex–Concave Nonlinearities

“…Furthermore, a method for computing the boundary of an interval outside of which the solution can blow up has been proposed (see also [24]). In [25], the local solvability and blow-up of solutions to an abstract nonlinear Volterra integral equation have been investigated. Recently, in [26], the authors proposed a new method and a tool to validate the numerical results of Volterra integral equations with discontinuous kernels in linear and nonlinear forms obtained from the Adomian decomposition method.…”