Electron. J. Diff. Eqns., Vol. 2008(2008), No. 84, pp. 1-12.

Stabilized quasi-reversibility method for a class of nonlinear ill-posed problems

Dang Duc Trong, Nguyen Huy Tuan

In this paper, we study a final value problem for the nonlinear parabolic equation
 u_t+Au =h(u(t),t),\quad  0<t<T\cr
 u(T)= \varphi ,
where $A$ is a non-negative, self-adjoint operator and $h$ is a Lipchitz function. Using the stabilized quasi-reversibility method presented by Miller, we find optimal perturbations, of the operator $A$, depending on a small parameter $\epsilon $ to setup an approximate nonlocal problem. We show that the approximate problems are well-posed under certain conditions and that their solutions converges if and only if the original problem has a classical solution. We also obtain estimates for the solutions of the approximate problems, and show a convergence result. This paper extends the work by Hetrick and Hughes [11] to nonlinear ill-posed problems.

Submitted April 28, 2008. Published June 8, 2008.
Math Subject Classifications: 35K05, 35K99, 47J06, 47H10.
Key Words: Ill-posed problem; nonlinear parabolic equation; quasi-reversibility methods; stabilized quasi-reversibility methods.

Show me the PDF file (243 KB), TEX file, and other files for this article.

Dang Duc Trong
Department of Mathematics and Informatics
Hochiminh City National University
227 Nguyen Van Cu, Q. 5, Hochiminh City, Vietnam
email: ddtrong@mathdep.hcmuns.edu.vn
Nguyen Huy Tuan
Department of Mathematics and Informatics
Ton Duc Thang University
98 Ngo Tat To street , Binh Thanh district Hochiminh City, Vietnam
email: tuanhuy_bs@yahoo.com

Return to the EJDE web page