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In this paper, we study the inverse problem for a class of abstract ultraparabolic equations which is well known to be ill-posed. We employ some elementary results of semi-group theory to present the formula of solution, then show the instability cause. Since the solution exhibits unstable dependence on the given data functions, we propose a regularization method to stabilize the solution, then obtain the error estimate. A numerical example shows that the method is efficient and feasible. This work slightly extends the earlier results in Zouyed and Rebbani (J. Inverse Ill-Posed Probl. 22(4):449-466, 2014).
Tuan, N. H., Khoa, V. A., Lan, L. T., & Hung, T. T. (2015). On the regularization of solution of an inverse ultraparabolic equation associated with perturbed final data. Journal of Inequalities and Applications, 2015(1). https://doi.org/10.1186/s13660-014-0526-y