Czechoslovak Mathematical Journal, Vol. 69, No. 2, pp. 545-569, 2019
Truncated spectral regularization for an ill-posed non-linear parabolic problem
Ajoy Jana, M. Thamban Nair
Received September 18, 2017. Published online February 11, 2019.
Abstract: It is known that the nonlinear nonhomogeneous backward Cauchy problem $u_t(t)+Au(t)=f(t,u(t))$, $0\leq t<\tau$ with $u(\tau)=\phi$, where $A$ is a densely defined positive self-adjoint unbounded operator on a Hilbert space, is ill-posed in the sense that small perturbations in the final value can lead to large deviations in the solution. We show, under suitable conditions on $\phi$ and $f$, that a solution of the above problem satisfies an integral equation involving the spectral representation of $A$, which is also ill-posed. Spectral truncation is used to obtain regularized approximations for the solution of the integral equation, and error analysis is carried out with exact and noisy final value $\phi$. Also stability estimates are derived under appropriate parameter choice strategies. This work extends and generalizes many of the results available in the literature, including the work by Tuan (2010) for linear homogeneous final value problem and the work by Jana and Nair (2016b) for linear nonhomogeneous final value problem.
References: [1] N. Boussetila, F. Rebbani: A modified quasi-reversibility method for a class of ill-posed Cauchy problems. Georgian Math. J. 14 (2007), 627-642. DOI 10.1515/GMJ.2007.627 | MR 2389024 | Zbl 1132.35405
[2] G. W. Clark, S. F. Oppenheimer: Quasireversibility methods for non-well-posed problems. Electron. J. Differ. Equ. 1994 (1994), 9 pages. MR 1302574 | Zbl 0811.35157
[3] M. Denche, K. Bessila: 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
[4] M. Denche, S. Djezzar: A modified quasi-boundary value method for a class of abstract parabolic ill-posed problems. Bound. Value Probl. 2006 (2006), Article ID 37524, 8 pages. DOI 10.1155/BVP/2006/37524 | MR 2211398 | Zbl 1140.34397
[5] M. A. Fury: Modified quasi-reversibility method for nonautonomous semilinear problems. Electron. J. Diff. Eqns., Conf. 20 (2013), 65-78. MR 3128069 | Zbl 06283623
[6] J. A. Goldstein: Semigroups of Linear Operators and Applications. Oxford Mathematical Monographs, Oxford University Press, Oxford (1985). MR 0790497 | Zbl 0592.47034
[7] A. Jana, M. T. Nair: Quasi-reversibility method for an ill-posed nonhomogeneous parabolic problem. Numer. Funct. Anal. Optim. 37 (2016), 1529-1550. DOI 10.1080/01630563.2016.1216448 | MR 3579019 | Zbl 1377.35175
[8] A. Jana, M. T. Nair: Truncated spectral regularization for an ill-posed nonhomogeneous parabolic problem. J. Math. Anal. Appl. 438 (2016), 351-372. DOI 10.1016/j.jmaa.2016.01.069 | MR 3462582 | Zbl 1382.35144
[9] A. Jana, M. T. Nair: A truncated spectral regularization method for a source identification problem. To appear in J. Anal. DOI 10.1007/s41478-018-0080-y
[10] R. Lattès, J.-L. Lions: Méthode de quasi-réversibilité et applications. Travaux et Recherches Mathématiques 15, Dunod, Paris (1967). (In French.) MR 0232549 | Zbl 0159.20803
[11] K. Miller: Stabilized quasi-reversibility and other nearly-best-possible methods for non-well posed problems. Lect. Notes Math. 316, Springer, Berlin (1973), 161-176. DOI 10.1007/bfb0069627 | MR 0393903 | Zbl 0279.35004
[12] P. T. Nam: An approximate solution for nonlinear backward parabolic equations. J. Math. Anal. Appl. 367 (2010), 337-349. DOI 10.1016/j.jmaa.2010.01.020 | MR 2607262 | Zbl 1194.35494
[13] A. Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences 44, Springer, New York (1983). DOI 10.1007/978-1-4612-5561-1 | MR 0710486 | Zbl 0516.47023
[14] L. Perko: Differential Equations and Dynamical Systems. Texts in Applied Mathematics 7, Springer, New York (2001). DOI 10.1007/978-1-4613-0003-8 | MR 1801796 | Zbl 0973.34001
[15] R. E. Showalter: 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
[16] N. H. Tuan: Regularization for a class of backward parabolic problems. Bull. Math. Anal. Appl. 2 (2010), 18-26. MR 2658124 | Zbl 1312.35189
[17] N. H. Tuan, D. D. Trong: A simple regularization method for the ill-posed evolution equation. Czech. Math. J. 61 (2011), 85-95. DOI 10.1007/s10587-011-0019-9 | MR 2782761 | Zbl 1224.35165
[18] N. H. Tuan, D. D. Trong, P. H. Quan: On a backward Cauchy problem associated with continuous spectrum operator. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 73 (2010), 1966-1972. DOI 10.1016/j.na.2010.05.025 | MR 2674176 | Zbl 1197.35306
[19] K. Yosida: Functional Analysis. Grundlehren der Mathematischen Wissenschaften 123, Springer, Berlin (1980). DOI 10.1007/978-3-642-96208-0 | MR 0617913 | Zbl 0435.46002
Affiliations: Ajoy Jana, M. Thamban Nair, Department of Mathematics, I.I.T. Madras, Chennai-600 036, India, e-mail: janaajoy340@gmail.com, mtnair@iitm.ac.in
