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

Abstract:
In this paper, we study a final value problem for the nonlinear parabolic equation
$$\displaylines{ u_t+Au =h(u(t),t),\quad 0<t<T\cr u(T)= \varphi , }$$
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.

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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

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