A non-local boundary value problem method for semi-linear parabolic equations backward in time

References

• Lattes R, Lions J-L. Méthode de Quasi-Réversibilité et Applications [The method of quasi-reversibility. Applications to partial differential equations]. Paris: Dunod; 1967; [ English translation New York (NY): Elsevier; 1969].
   Google Scholar
• Ewing RE. The approximation of certain parabolic equations backward in time by Sobolev equations. SIAM J. Math. Anal. 1975;6:283–294.
   Web of Science ®Google Scholar
• Gajewski H, Zaccharias K. Zur Ruguliarisierung einer nichtkorrekter Probleme bei Evolutionsgleichungen. J. Math. Anal. Appl. 1972;38:784–789.
   Web of Science ®Google Scholar
• Huang Y, Quan Z. Regularization for a class of ill-posed Cauchy problem. Proc. Amer. Math. Soc. 2005;133:3005–3012.
   Web of Science ®Google Scholar
• Showalter RE. The final value problem for evolution equations. J. Math. Anal. Appl. 1974;47:563–572.
   Web of Science ®Google Scholar
• Miller K. Stabilized quasireversibility and other nearly best possible methods for non-well-posed problems. In: Knops RJ, editor. Symposium on non-well-posed problems and logarithmic convexity. Vol. 316, Lecture notes in mathematics. Berlin: Springer-Verlag; 1973. p. 161–176.
   Google Scholar
• Tikhonov AN, Arsenin VY. Solutions of ill-posed problems. Washington (DC): Winston; 1977.
   Google Scholar
• Ames KA, Clark GW, Epperson JF, Oppenheimer SF. A comparison of regularizations for an ill-posed problem. Math. Comput. 1998;224:1451–1471.
   Web of Science ®Google Scholar
• Clark GW, Oppenheimer SF. Quasireversiblity methods for non-well-posed problems. Electron. J. Differ. Equ. 1994;8:1–9.
   Google Scholar
• Denche SM, Bessila K. A modified quasi-boundary value method for ill-posed problems. J. Math. Anal. Appl. 2005;301:419–426.
   Web of Science ®Google Scholar
• Hào DN, Duc NV, Sahli H. A non-local boundary value problem method for parabolic equations backward in time. J. Math. Anal. Appl. 2008;345:805–815.
   Web of Science ®Google Scholar
• DN Hào, Duc NV, Lesnic D. A non-local boundary value problem method for the Cauchy problem for elliptic equations. Inverse Probl. 2009;25:055002. 27 p.
   Web of Science ®Google Scholar
• Hào DN, Duc NV, Lesnic D. Regularization of parabolic equations backwards in time by a non-local boundary value problem method. IMA J. Appl. Math. 2010;75:291–315.
   Web of Science ®Google Scholar
• Hào DN, Duc NV. Stability results for backward parabolic equations with time dependent coefficients. Inverse Probl. 2011;27:025003. 20 p.
   Web of Science ®Google Scholar
• Hào DN, Duc NV. Regularization of backward parabolic equations in Banach spaces. J. Inverse Ill-Posed Probl. 2012;20:745–763.
   Web of Science ®Google Scholar
• Mel’nikova IV. Regularization of ill-posed differential problems. Siberian Math. J. 1992;33:289–298.
   Web of Science ®Google Scholar
• Showalter RE. Cauchy problem for hyper-parabolic partial differential equations. In: Lakshmikantham V, editor. Trends in the theory and practice of non-linear analysis, (Arlington, Tex., 1984). Vol. 110, North-Holland mathematics studies. Amsterdam: North-Holland; 1985. p. 421–425.
   Google Scholar
• Boussetila N, Rebbani F. Optimal regularization method for ill-posed Cauchy problems. Electron. J. Differ. Equ. 2006;147:1–15.
   Google Scholar
• Long N-T, Pham Ngoc Dinh A. Approximation of a parabolic non-linear evolution equation backward in time. Inverse Probl. 1994;10:905–914.
   Web of Science ®Google Scholar
• Long N-T, Pham Ngoc Dinh A. Note on a regularization of a parabolic nonlinear evolution equation backwards in time. Inverse Probl. 1996;12:455–462.
   Web of Science ®Google Scholar
• Nam PT. An approximate solution for nonlinear backward parabolic equations. J. Math. Anal. Appl. 2010;367:337–349.
   Web of Science ®Google Scholar
• Trong DD, Tuan NH. Regularization and error estimate for the nonlinear backward heat problem using a method of integral equation. Nonlinear Anal. 2009;71:4167–4176.
   Web of Science ®Google Scholar
• Tautenhahn U. Optimality for ill-posed problems under general source conditions. Numer. Funct. Anal. Optim. 1998;19:377–398.
   Web of Science ®Google Scholar
• Tautenhahn U, Schröter T. On optimal regularization methods for the backward heat equation. Z. Anal. Anwendungen. 1996;15:475–493.
   Google Scholar
• Hào DN, Duc NV, Van Thang N. Stability results for semi-linear parabolic equations backward in time. Forthcoming 2013.
   Google Scholar