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Dang Duc Trong, Nguyen Huy Tuan (2008). Stabilized quasi-reversibility method for a class of nonlinear ill-posed problems. Electronic Journal of Differential Equations, Vol. 2008, No. 84, pp. 1-12.

Link:http://www.maths.soton.ac.uk/EMIS/journals/EJDE/Volumes/2008/84/trong.pdf#

Abstract. In this paper, we study a final value problem for the nonlinear parabolic equation

ut + Au = h(u(t), t), 0 < t < T

u(T) = ϕ,

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 € to setup an approximate nonlocal problem. We show that the approximate problems are well-posed under certain conditions and that their solutions con-verges 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.to nonlinear ill-posed problems.