`International Journal of Mathematics and Mathematical SciencesVolume 2012 (2012), Article ID 943621, 20 pageshttp://dx.doi.org/10.1155/2012/943621`
Research Article

## A Regularization of the Backward Problem for Nonlinear Parabolic Equation with Time-Dependent Coefficient

1Department of Mathematics and Applications, Sai Gon University, 273 An Duong Vuong, District 5, Ho Chi Minh City, Vietnam
2Department of Mathematics, University of Natural Science, Vietnam National University, 227 Nguyen Van Cu, District 5, Ho Chi Minh City, Vietnam

Received 31 March 2012; Revised 21 June 2012; Accepted 11 August 2012

Copyright © 2012 Pham Hoang Quan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Z. Qian, C.-L. Fu, and R. Shi, “A modified method for a backward heat conduction problem,” Applied Mathematics and Computation, vol. 185, no. 1, pp. 564–573, 2007.
2. R. E. Ewing, “The approximation of certain parabolic equations backward in time by Sobolev equations,” SIAM Journal on Mathematical Analysis, vol. 6, pp. 283–294, 1975.
3. X.-L. Feng, Z. Qian, and C.-L. Fu, “Numerical approximation of solution of nonhomogeneous backward heat conduction problem in bounded region,” Mathematics and Computers in Simulation, vol. 79, no. 2, pp. 177–188, 2008.
4. C.-L. Fu, X.-T. Xiong, and Z. Qian, “Fourier regularization for a backward heat equation,” Journal of Mathematical Analysis and Applications, vol. 331, no. 1, pp. 472–480, 2007.
5. D. N. Hào, “A mollification method for ill-posed problems,” Numerische Mathematik, vol. 68, no. 4, pp. 469–506, 1994.
6. A. Hasanov and J. L. Mueller, “A numerical method for backward parabolic problems with non-selfadjoint elliptic operators,” Applied Numerical Mathematics, vol. 37, no. 1-2, pp. 55–78, 2001.
7. D. D. Trong, P. H. Quan, and N. H. Tuan, “A quasi-boundary value method for regularizing nonlinear ill-posed problems,” Electronic Journal of Differential Equations, vol. 2009, no. 109, pp. 1–16, 2009.
8. D. T. Dang and H. T. Nguyen, “Regularization and error estimates for nonhomogeneous backward heat problems,” Electronic Journal of Differential Equations, vol. 2006, no. 4, pp. 1–10, 2006.
9. I. V. Mel'nikova and A. I. Filinkov, The Cauchy Problem. Three Approaches, vol. 120 of Monograph and Surveys in Pure and Applied Mathematics, Chapman & Hall, London, UK, 2001.
10. D. D. Trong and N. H. Tuan, “A nonhomogeneous backward heat problem: regularization and error estimates,” Electronic Journal of Differential Equations, vol. 2008, no. 33, pp. 1–14, 2008.
11. D. D. Trong, P. H. Quan, T. V. Khanh, and N. H. Tuan, “A nonlinear case of the 1-D backward heat problem: regularization and error estimate,” Zeitschrift für Analysis und ihre Anwendungen, vol. 26, no. 2, pp. 231–245, 2007.
12. P. H. Quan, D. D. Trong, L. M. Triet, and N. H. Tuan, “A modified quasi-boundary value method for regularizing of a backward problem with time-dependent coefficient,” Inverse Problems in Science and Engineering, vol. 19, no. 3, pp. 409–423, 2011.
13. M. Lees and M. H. Protter, “Unique continuation for parabolic differential equations and inequalities,” Duke Mathematical Journal, vol. 28, pp. 369–382, 1961.