Article

 Title: A simple regularization method for the ill-posed evolution equation (English) Author: Tuan, Nguyen Huy Author: Trong, Dang Duc Language: English Journal: Czechoslovak Mathematical Journal ISSN: 0011-4642 (print) ISSN: 1572-9141 (online) Volume: 61 Issue: 1 Year: 2011 Pages: 85-95 Summary lang: English . Category: math . Summary: The nonhomogeneous backward Cauchy problem $$u_t +Au(t) = f(t),\quad u(T) = \varphi$$, where $A$ is a positive self-adjoint unbounded operator which has continuous spectrum and $f$ is a given function being given is regularized by the well-posed problem. New error estimates of the regularized solution are obtained. This work extends earlier results by N. Boussetila and by M. Denche and S. Djezzar. (English) Keyword: nonlinear parabolic problem Keyword: backward problem Keyword: semigroup of operators Keyword: ill-posed problem Keyword: contraction principle MSC: 35K05 MSC: 35K99 MSC: 47H10 MSC: 47J06 idZBL: Zbl 1224.35165 idMR: MR2782761 DOI: 10.1007/s10587-011-0019-9 . Date available: 2011-05-23T12:33:17Z Last updated: 2016-04-07 Stable URL: http://hdl.handle.net/10338.dmlcz/141520 . Reference: [1] Ames, K. A., Hughes, R. J.: Structural stability for ill-posed problems in Banach space.Semigroup Forum 70 (2005), 127-145. Zbl 1109.34041, MR 2107199, 10.1007/s00233-004-0153-x Reference: [2] Boussetila, N., Rebbani, F.: Optimal regularization method for ill-posed Cauchy problems.Electron. J. Differ. Equ. 147 (2006), 1-15. Zbl 1112.35336, MR 2276572 Reference: [3] Clark, G. W., Oppenheimer, S. F.: Quasireversibility methods for non-well posed problems.Electron. J. Diff. Eqns. 1994 (1994), 1-9. Zbl 0811.35157, MR 1302574 Reference: [4] Denche, M., Bessila, K.: A modified quasi-boundary value method for ill-posed problems.J. Math. Anal. Appl. 301 (2005), 419-426. Zbl 1084.34536, MR 2105682, 10.1016/j.jmaa.2004.08.001 Reference: [5] Denche, M., Djezzar, S.: A modified quasi-boundary value method for a class of abstract parabolic ill-posed problems.Bound. Value Probl. 2006, Article ID 37524 (2006), 1-8. Zbl 1140.34397, MR 2211398 Reference: [6] Eldén, L., Berntsson, F., Reginska, T.: Wavelet and Fourier methods for solving the sideways heat equation.SIAM J. Sci. Comput. 21 (2000), 2187-2205. MR 1762037, 10.1137/S1064827597331394 Reference: [7] Fu, C.-L., Xiong, X.-T., Fu, P.: Fourier regularization method for solving the surface heat flux from interior observations.Math. Comput. Modelling 42 (2005), 489-498. Zbl 1122.80016, MR 2173470, 10.1016/j.mcm.2005.08.003 Reference: [8] Fu, C.-L.: Simplified Tikhonov and Fourier regularization methods on a general sideways parabolic equation.J. Comput. Appl. Math. 167 (2004), 449-463. Zbl 1055.65106, MR 2064702, 10.1016/j.cam.2003.10.011 Reference: [9] Fu, C.-L., Xiang, X.-T., Qian, Z.: Fourier regularization for a backward heat equation.J. Math. Anal. Appl. 331 (2007), 472-480. MR 2306017, 10.1016/j.jmaa.2006.08.040 Reference: [10] Gajewski, H., Zaccharias, K.: Zur regularisierung einer klass nichtkorrekter probleme bei evolutiongleichungen.J. Math. Anal. Appl. 38 (1972), 784-789. MR 0308625, 10.1016/0022-247X(72)90083-2 Reference: [11] Hào, D. N., Duc, N. Van, Sahli, H.: A non-local boundary value problem method for parabolic equations backward in time.J. Math. Anal. Appl. 345 (2008), 805-815. MR 2429181, 10.1016/j.jmaa.2008.04.064 Reference: [12] Huang, Y., Zheng, Q.: Regularization for a class of ill-posed Cauchy problems.Proc. Am. Math. Soc. 133 (2005), 3005-3012. Zbl 1073.47016, MR 2159779, 10.1090/S0002-9939-05-07822-6 Reference: [13] Lattès, R., Lions, J.-L.: Méthode de Quasi-réversibilité et Applications.Dunod Paris (1967), French. MR 0232549 Reference: [14] Long, N. T., Ding, A. Pham Ngoc: Approximation of a parabolic nonlinear evolution equation backwards in time.Inverse Probl. 10 (1994), 905-914. MR 1286629 Reference: [15] Mel'nikova, I. V., Filinkov, A. I.: Abstract Cauchy problems: Three approaches. Monograph and Surveys in Pure and Applied Mathematics, Vol. 120.Chapman & Hall/CRC London-New York/Boca Raton (2001). MR 1823612 Reference: [16] Miller, K.: Stabilized quasi-reversibility and other nearly-best-possible methods for non-well posed problems. Sympos. non-well posed probl. logarithmic convexity.Lect. Notes Math. Vol. 316 Springer Berlin (1973), 161-176. MR 0393903, 10.1007/BFb0069627 Reference: [17] Payne, L. E.: Improperly Posed Problems in Partial Differential Equations.SIAM Philadelphia (1975). Zbl 0302.35003, MR 0463736 Reference: [18] Pazy, A.: Semigroups of Linear Operators and Application to Partial Differential Equations.Springer New York (1983). MR 0710486 Reference: [19] Showalter, R. E.: The final value problem for evolution equations.J. Math. Anal. Appl. 47 (1974), 563-572. Zbl 0296.34059, MR 0352644, 10.1016/0022-247X(74)90008-0 Reference: [20] Showalter, R. E.: Quasi-reversibility of first and second order parabolic evolution equations. Improp. Posed Bound. Value Probl. (Conf. Albuquerque, 1974).Res. Notes in Math., No. 1 Pitman London (1975), 76-84. MR 0477359 Reference: [21] Tautenhahn, U., Schröter, T.: On optimal regularization methods for the backward heat equation.Z. Anal. Anwend. 15 (1996), 475-493. MR 1394439, 10.4171/ZAA/711 Reference: [22] Tautenhahn, U.: Optimality for ill-posed problems under general source conditions.Numer. Funct. Anal. Optimization 19 (1998), 377-398. Zbl 0907.65049, MR 1624930, 10.1080/01630569808816834 Reference: [23] Trong, D. D., Tuan, N. H.: Stabilized quasi-reversibility method for a class of nonlinear ill-posed problems.Electron. J. Differ. Equ. No 84 (2008). Zbl 1171.35485, MR 2411080 .

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