Electronic Journal of Differential Equations,
Vol. 2005(2005), No. 138, pp. 1-18.

Title: Nonlinear Kirchhoff-Carrier wave equation in a unit membrane with 
       mixed homogeneous boundary conditions
       
Author: Nguyen Thanh Long (Vietnam National Univ., Hochiminh, Vietnam)

Abstract: 
 In this paper we consider the nonlinear wave equation problem
 $$\displaylines{
 u_{tt}-B\big(\|u\|_0^2,\|u_{r}\|_0^2\big)(u_{rr}+\frac{1}{r}u_{r})
 =f(r,t,u,u_{r}),\quad 0<r<1,\; 0<t<T, \\
 \big|\lim_{r\to 0^+}\sqrt{r}u_{r}(r,t)\big|<\infty, \\
 u_{r}(1,t)+hu(1,t)=0, \\
 u(r,0)=\widetilde{u}_0(r), u_{t}(r,0)=\widetilde{u}_1(r).
 }$$
 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  $B=B(\|u\|_0^2)=b_0+\|u\|_0^2$
 with $b_0>0$.
 
Submitted August 3, 2004. Published December 1, 2005.
Math Subject Classifications: 35L70, 35Q72.
Key Words: Nonlinear wave equation; Galerkin method; quadratic convergence;
           weighted Sobolev spaces.