Electron. J. Diff. Eqns., Vol. 2005(2005), No. 138, pp. 1-18.

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

Nguyen Thanh Long

Abstract:
In this paper we consider the nonlinear wave equation problem
$$\displaylines{ u_{tt}-B\big(\|u\|_0^2,\|u_{r}\|_0^2\big)(u_{rr}+\frac{1}{r}u_{r}) =f(r,t,u,u_{r}),\quad 0\hbox{ less than }r\hbox{ less than }1,\; 0\hbox{ less than }t\hbox{ less than } T, \cr \big|\lim_{r\to 0^+}\sqrt{r}u_{r}(r,t)\big|\hbox{ less than }\infty, \cr u_{r}(1,t)+hu(1,t)=0, \cr u(r,0)=\widetilde{u}_0(r), u_{t}(r,0)=\widetilde{u}_1(r). }$$
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 $b_0\hbox{ greater than }>0$.

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.

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Nguyen Thanh Long
Department of Mathematics and Computer Science
University of Natural Science
Vietnam National University HoChiMinh City
227 Nguyen Van Cu Str., Dist. 5
Hochiminh City, Vietnam
email: longnt@hcmc.netnam.vn   longnt2@gmail.com

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