Backward heat equation with time dependent variable coefficient is severely ill‐posed in the sense of Hadamard, so we need regularization. In this paper, we consider Backward heat equation with time dependent variable coefficient, and by small perturbing, we obtain an approximation problem. We show this approximation problem is well‐posed with small parameter. Also, we show this approximation system converges to the original problem when parameter goes to zero. Here, we use modified‐quasi boundary value method to regularize this problem. Copyright © 2016 John Wiley & Sons, Ltd.

Inverse problems in partial deferential equations are severely ill posed in the sense of Hadamard. So the heat equation with a terminal condition problem is ill posed even in the sobolev space so regularization is needed. In this paper, we discuss about the convergence result of the approximation problem in the Sobolev space H 2 ðR n Þ, which is well posed. By using a small parameter, we construct an approximation problem and use a quasi-boundary value method to regularize nonlinear heat equation. Finally, we prove the approximation solution converges to the original solution whenever the parameter goes to zero in H 2 ðR n Þ. KEYWORDS backward heat equation, regularization MSC CLASSIFICATION 47A52

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