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Global existence for abstract nonlinear Volterra-Fredholm functional integrodifferential equation

Dhakne M. B., Kucche K. D.

Demonstratio Mathematica

2012

Vol. 45, nr 1

117-127

In the present paper, we investigate the global existence of solutions to initial value problem for nonlinear mixed Volterra-Fredholm functional integrodifferential equations in Banach spaces. The technique used in our analysis is based on an application of the topological transversality theorem known as Leray-Schauder alternative and rely on a priori bounds of solution.

Uniform stabilization of the quasi-linear Kirchhoff wave equation with a nonlinear boundary feedback

Lasiecka I.

Control and Cybernetics

2000

Vol. 29, no 1

179-197

An n-dimensional quasi-linear wave equation defined on bounded domain Omega with Neumann boundary conditions imposed on the boundary Gamma and with a nonlinear boundary feedback acting on a portion of the boundary [Gamma sup 1 is a subset of Gamma] is considered. Global existence, uniqueness and uniform decay rates are established for the model, under the assumption that the H[sup 1](Omega) x L[sub 2](Omega) norms of the initial data are sufficiently small. The result presented in this paper extends these obtained recently in Lasiecka and Ong (1999), where the Dirichlet boundary conditions are imposed on the uncontrolled portion of the boundary Gamma[sub o] = Gamma \ [closure of a set Gamma sub 1], and the two portions of the boundary are assumed disjoint, i.e. [... ]. The goal of this paper is to remove this restriction. This is achieved by considering the "pure" Neumann problem subject to convexity assumption imposed on Gamma[sub o]. \@eng\\