Previous |  Up |  Next

Article

Keywords:
nonlinear parabolic problem; backward problem; semigroup of operators; ill-posed problem; contraction principle
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.
References:
[1] Ames, K. A., Hughes, R. J.: Structural stability for ill-posed problems in Banach space. Semigroup Forum 70 (2005), 127-145. DOI 10.1007/s00233-004-0153-x | MR 2107199 | Zbl 1109.34041
[2] Boussetila, N., Rebbani, F.: Optimal regularization method for ill-posed Cauchy problems. Electron. J. Differ. Equ. 147 (2006), 1-15. MR 2276572 | Zbl 1112.35336
[3] Clark, G. W., Oppenheimer, S. F.: Quasireversibility methods for non-well posed problems. Electron. J. Diff. Eqns. 1994 (1994), 1-9. MR 1302574 | Zbl 0811.35157
[4] Denche, M., Bessila, K.: A modified quasi-boundary value method for ill-posed problems. J. Math. Anal. Appl. 301 (2005), 419-426. DOI 10.1016/j.jmaa.2004.08.001 | MR 2105682 | Zbl 1084.34536
[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. MR 2211398 | Zbl 1140.34397
[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. DOI 10.1137/S1064827597331394 | MR 1762037
[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. DOI 10.1016/j.mcm.2005.08.003 | MR 2173470 | Zbl 1122.80016
[8] Fu, C.-L.: Simplified Tikhonov and Fourier regularization methods on a general sideways parabolic equation. J. Comput. Appl. Math. 167 (2004), 449-463. DOI 10.1016/j.cam.2003.10.011 | MR 2064702 | Zbl 1055.65106
[9] Fu, C.-L., Xiang, X.-T., Qian, Z.: Fourier regularization for a backward heat equation. J. Math. Anal. Appl. 331 (2007), 472-480. DOI 10.1016/j.jmaa.2006.08.040 | MR 2306017
[10] Gajewski, H., Zaccharias, K.: Zur regularisierung einer klass nichtkorrekter probleme bei evolutiongleichungen. J. Math. Anal. Appl. 38 (1972), 784-789. DOI 10.1016/0022-247X(72)90083-2 | MR 0308625
[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. DOI 10.1016/j.jmaa.2008.04.064 | MR 2429181
[12] Huang, Y., Zheng, Q.: Regularization for a class of ill-posed Cauchy problems. Proc. Am. Math. Soc. 133 (2005), 3005-3012. DOI 10.1090/S0002-9939-05-07822-6 | MR 2159779 | Zbl 1073.47016
[13] Lattès, R., Lions, J.-L.: Méthode de Quasi-réversibilité et Applications. Dunod Paris (1967), French. MR 0232549
[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
[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
[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. DOI 10.1007/BFb0069627 | MR 0393903
[17] Payne, L. E.: Improperly Posed Problems in Partial Differential Equations. SIAM Philadelphia (1975). MR 0463736 | Zbl 0302.35003
[18] Pazy, A.: Semigroups of Linear Operators and Application to Partial Differential Equations. Springer New York (1983). MR 0710486
[19] Showalter, R. E.: The final value problem for evolution equations. J. Math. Anal. Appl. 47 (1974), 563-572. DOI 10.1016/0022-247X(74)90008-0 | MR 0352644 | Zbl 0296.34059
[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
[21] Tautenhahn, U., Schröter, T.: On optimal regularization methods for the backward heat equation. Z. Anal. Anwend. 15 (1996), 475-493. DOI 10.4171/ZAA/711 | MR 1394439
[22] Tautenhahn, U.: Optimality for ill-posed problems under general source conditions. Numer. Funct. Anal. Optimization 19 (1998), 377-398. DOI 10.1080/01630569808816834 | MR 1624930 | Zbl 0907.65049
[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). MR 2411080 | Zbl 1171.35485
Partner of
EuDML logo