Electronic Journal of Differential Equations,
Vol. 2008(2008), No. 84, pp. 1-12.

Title: Stabilized quasi-reversibility method for 
       a class of nonlinear ill-posed problems
  
Authors: Dang Duc Trong  (Hochiminh City National Univ., Vietnam)
         Nguyen Huy Tuan (Ton Duc Thang Univ., Vietnam)

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 08, 2008.
Math Subject Classifications: 35K05, 35K99, 47J06, 47H10.
Key Words: Ill-posed problem; nonlinear parabolic equation;
           quasi-reversibility methods; 
	   stabilized quasi-reversibility methods.