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Dang Duc Trong, Nguyen Huy Tuan (2009). Regularization and error estimate for the nonlinear backward heat problem using a method of integral equation. Nonlinear Analysis: Theory, Methods & Applications, Volume 71, Issue 9, pp. 4167-4176.

Abstract. We consider the inverse time problem for the nonlinear heat equation in the form

The nonlinear problem is severely ill-posed. We shall use the method of integral equation to regularize the problem and to get some error estimates. We show that the approximate problems are well-posed and that their solution u (x,t) converges on [0,T] if and only if the original problem has a unique solution. We obtain several other results, including some explicit convergence rates. Some numerical tests illustrate that the proposed method is feasible and effective.