Nonlinear Kirchhoff-Carrier wave equation in a unit membrane with mixed homogeneous boundary conditions

Abstract

In this paper we consider the nonlinear wave equation problem utt - B(∥u∥²₀, ∥ur∥²₀ (urr + 1/r ur) = ƒ(r, t, u, ur), 0 < r < 1, 0 < t < T, | limr→0+ √rur(r, t)| < ∞, ur(1, t) + hu(1, t) = 0, u(r, 0) = ũ0(r), ut(r, 0) = ũ1(r). To this problem, we associate a linear recursive scheme for which the existence of a local and unique weak solution is proved, in weighted Sobolev using standard compactness arguments. In the latter part, we give sufficient conditions for quadratic convergence to the solution of the original problem, for an autonomous right-hand side independent on u_r and a coefficient function B of the form B = B(∥u∥²₀) = b₀ + ∥u∥²₀ with b₀ > 0.