International Journal of Differential Equations

Improved Regularization Method for Backward Cauchy Problems Associated with Continuous Spectrum Operator

Abstract

We consider in this paper an abstract parabolic backward Cauchy problem associated with an unbounded linear operator in a Hilbert space $H$, where the coefficient operator in the equation is an unbounded self-adjoint positive operator which has a continuous spectrum and the data is given at the final time $t=T$ and a solution for $0\le t is sought. It is well known that this problem is illposed in the sense that the solution (if it exists) does not depend continuously on the given data. The method of regularization used here consists of perturbing both the equation and the final condition to obtain an approximate nonlocal problem depending on two small parameters. We give some estimates for the solution of the regularized problem, and we also show that the modified problem is stable and its solution is an approximation of the exact solution of the original problem. Finally, some other convergence results including some explicit convergence rates are also provided.

Article information

Source
Int. J. Differ. Equ., Volume 2011 (2011), Article ID 913125, 11 pages.

Dates
Revised: 17 August 2011
Accepted: 26 September 2011
First available in Project Euclid: 25 January 2017

https://projecteuclid.org/euclid.ijde/1485313261

Digital Object Identifier
doi:10.1155/2011/913125

Mathematical Reviews number (MathSciNet)
MR2854951

Zentralblatt MATH identifier
1242.34107

Citation

Djezzar, Salah; Teniou, Nihed. Improved Regularization Method for Backward Cauchy Problems Associated with Continuous Spectrum Operator. Int. J. Differ. Equ. 2011 (2011), Article ID 913125, 11 pages. doi:10.1155/2011/913125. https://projecteuclid.org/euclid.ijde/1485313261

References

• R. Lattès and J.-L. Lions, Méthode de Quasi-Réversibilité et Applications, Travaux et Recherches Mathématiques, no. 15, Dunod, Paris, France, 1967.
• M. M. Lavrentiev, Some Improperly Posed Problems of Mathematical Physics, vol. 11 of Springer Tracts in Natural Philosophy, Springer, Berlin, Germany, 1967.
• K. Miller, “Stabilized quasi-reversibility and other nearly-best-possible methods for non- well-posed problems,” in Symposium on Non-Well Posed Problems and Logarithmic Convexity (Heriot-Watt University, Edinburgh, Scotland, 1972), vol. 316 of Lecture Notes in Mathematics, pp. 161–176, Springer, Berlin, Germany, 1973.
• L. E. Payne, “Some general remarks on improperly posed problems for partial differential equations,” in Symposium on Non-Well Posed Problems and Logarithmic Convexity (Heriot-Watt University, Edinburgh, Scotland, 1972), vol. 316 of Lecture Notes in Mathematics, pp. 1–30, Springer, Berlin, Germany, 1973.
• R. E. Showalter, “The final value problem for evolution equations,” Journal of Mathematical Analysis and Applications, vol. 47, no. 3, pp. 563–572, 1974.
• M. Ababna, “Regularization by nonlocal conditions of the problem of the control of the initial condition for evolution operator-differential equations,” Vestnik Belorusskogo Gosudarstvennogo Uni-versiteta. Seriya 1. Fizika, Matematika, Informatika, vol. 81, no. 2, pp. 60–63, 1998 (Russian).
• G. W. Clark and S. F. Oppenheimer, “Quasi-reversibility methods for non-well posed problems,” Electronic Journal of Differential Equations, no. 8, pp. 1–9, 1994.
• R. E. Showalter, “Cauchy problem for hyper-parabolic partial differential equations,” North-Holland Mathematics Studies, vol. 110, no. C, pp. 421–425, 1985.
• V. K. Ivanov, I. V. Mel'nikova, and A. I. Filinkov, Differentsialno-Operatornye Uravneniya i Nekorrektnye Zadachi, Fizmatlit “Nauka”, Moscow, Russia, 1995.
• I. V. Mel'nikova, “Regularization of ill-posed differential problems,” Sibirskii Matematicheskii Zhurnal, vol. 33, no. 2, pp. 125–134, 1992 (Russian).
• I. V. Mel'nikova, “Regularization of ill-posed differential problems,” Siberian Mathematical Journal, vol. 33, no. 2, pp. 289–298, 1992.
• K. A. Ames and L. E. Payne, “Asymptotic behavior for two regularizations of the Cauchy problem for the backward heat equation,” Mathematical Models & Methods in Applied Sciences, vol. 8, no. 1, pp. 187–202, 1998.
• K. A. Ames, L. E. Payne, and P. W. Schaefer, “Energy and pointwise bounds in some non-standard parabolic problems,” Proceedings of the Royal Society of Edinburgh. Section A. Mathematics, vol. 134, no. 1, pp. 1–9, 2004.
• M. Denche and S. Djezzar, “A modified quasi-boundary value method for a class of abstract parabolic ill-posed problems,” Boundary Value Problems, vol. 2006, Article ID 37524, 8 pages, 2006.
• S. Djezzar, “Regularization method for an abstract backward Cauchy problem,” in Proceedings of the 3rd International Conference on Mathematical Sciences (ICM '08), vol. 3, pp. 1116–1125, March 2008.
• B. M. Campbell Hetrick and R. J. Hughes, “Continuous dependence results for inhomogeneous ill-posed problems in Banach space,” Journal of Mathematical Analysis and Applications, vol. 331, no. 1, pp. 342–357, 2007.
• N. H. Tuan, “Regularization for a class of backward parabolic problems,” Bulletin of Mathematical Analysis and Applications, vol. 2, no. 2, pp. 18–26, 2010.
• N. H. Tuan, D. D. Trong, and P. H. Quan, “On a backward Cauchy problem associated with continuous spectrum operator,” Nonlinear Analysis, vol. 73, no. 7, pp. 1966–1972, 2010.
• N. Boussetila and F. Rebbani, “Optimal regularization method for ill-posed Cauchy problems,” Electronic Journal of Differential Equations, no. 147, pp. 1–15, 2006.
• M. Denche and K. Bessila, “A modified quasi-boundary value method for ill-posed problems,” Journal of Mathematical Analysis and Applications, vol. 301, no. 2, pp. 419–426, 2005.
• S. Djezzar and N. Teniou, “Modified regularization method for backward Cauchy problems,” in Proceedings of the 3rd Conference on Mathematical Sciences (CMS '11), pp. 1512–1519, Zarqa, Jordan, April 2011.
• A. Pazy, “Semi-groups of linear operators and applications to partial differential equations,” in Applied Mathematical Sciences, vol. 44, Springer, New York, NY, USA, 1983.