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Acta Mathematica Vietnamica

, Volume 42, Issue 1, pp 99–111 | Cite as

Stability Results for Semi-linear Parabolic Equations Backward in Time

  • Nguyen Van DucEmail author
  • Nguyen Van Thang
Article

Abstract

Let H be a Hilbert space with the norm ∥⋅∥, and let A:D(A) ⊂ HH be a positive self-adjoint unbounded linear operator on H such that −A generates a C 0 semi-group on H. Let φ be in H, E > ε a given positive number and let f : [0, THH satisfy the Lipschitz condition ∥f(t, w 1)−f(t, w 2)∥ ≤ kw 1w 2∥,w 1,w 2H, for some non-negative constant k independent of t, w 1 and w 2. It is proved that if u 1 and u 2 are two solutions of the ill-posed semi-linear parabolic equation backward in time u t + A u = f(t, u), 0 < tT,∥u(T)−φ∥ ≤ ε and ∥u i (0)∥ ≤ E, i = 1,2, then
$$\|u_{1}(t)-u_{2}(t)\| \leq 2\varepsilon^{t/T} E^{1-t/T}\exp\Big[\Big(2k+\frac{1}{4}k^{2}(T+t)\Big)\frac{t(T-t)}{T}\Big] \quad \forall t \in [0,T]. $$
The ill-posed problem is stabilized by a modification of Tikhonov regularization which yields an error estimate of Hölder type.

Keywords

Semi-linear parabolic equations backward in time Ill-posed problems Stability estimate Log-convexity method Tikhonov regularization 

Mathematics Subject Classification (2010)

35K58 

Notes

Acknowledgments

This research was supported by Vietnam Ministry of Education and Training under grant number B2013-27-09.

References

  1. 1.
    Agoshkov, V.I.: Optimal Control Methods and the Method of Adjoint Equations in Problems of Mathematical Physics. Russian Academy of Sciences, Institute for Numerical Mathematics, Moscow (2003). (Russian)zbMATHGoogle Scholar
  2. 2.
    Alifanov, O.M.: Inverse Heat Transfer Problems. Wiley, New York (1994)CrossRefzbMATHGoogle Scholar
  3. 3.
    Aubert, G., Kornprobst, P.: Mathematical Problems in Image Processing. Springer, New York (2006)zbMATHGoogle Scholar
  4. 4.
    Beck, J.V., Blackwell, B., Clair, St.C.R.: Inverse Heat Conduction, Ill-Posed Probl. Wiley, New York (1985)zbMATHGoogle Scholar
  5. 5.
    Carasso, A.S.: Hazardous continuation backward in time in nonlinear parabolic equations, and an experiment in deblurring nonlinearly blurred imagery. J. Res. Natl. Inst. Stand. Technol. 118, 199–217 (2013)CrossRefGoogle Scholar
  6. 6.
    Cartan, H.: Differential Calculus. Kershaw publishing company Ltd (1971)Google Scholar
  7. 7.
    Ghidaglia, J.M.: Some backward uniqueness results. Nonlinear Anal. 8, 777–790 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hào, D.N.: Methods for Inverse Heat Conduction Problems. Peter Lang Verlag, Frankfurt/Main, Bern, New York, Paris (1998)Google Scholar
  9. 9.
    Hào, D.N., Duc, N.V., Lesnic, D.: Regularization of parabolic equations backwards in time by a non-local boundary value problem method. IMA J. Appl. Math. 75, 291–315 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hào, D.N., Duc, N.V.: A non-local boundary value problem method for semi-linear parabolic equations backward in time. Appl. Anal. 94, 446–463 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kukavica, I.: Log-log convexity and backward uniqueness. Proc. Am. Math. Soc. 135, 2415–2421 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lavrent’ev, M.M., Romanov, V.G., Shishatskii, G.P.: Ill-posed Problems in Mathematical Physics and Analysis. Am. Math. Soc., Providence, R. I. (1986)Google Scholar
  13. 13.
    Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)CrossRefzbMATHGoogle Scholar
  14. 14.
    Long, N.T., Dinh, A.P.N.: Approximation of a parabolic non-linear evolution equation backward in time. Inverse Probl. 10, 905–914 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Long, N.T., Dinh, A.P.N.: Note on a regularization of a parabolic nonlinear evolution equation backwards in time. Inverse Probl. 12, 455–462 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Nam, P.T.: An approximate solution for nonlinear backward parabolic equations. J. Math. Anal. Appl. 367, 337–349 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Quan, P.H., Trong, D.D., Triet, L.M.: On a backward nonlinear parabolic equation with time and space dependent thermal conductivity: regularization and error estimates. J. Inverse Ill-Posed Probl. 22, 375–401 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Payne, L.: Improperly Posed Problems in Partial Differential Equations. SIAM, Philadelphia (1975)Google Scholar
  19. 19.
    Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences 44 Springer-Verlag (1983)Google Scholar
  20. 20.
    Shutyaev, V.P.: Control Operators and Iterative Algorithms in Variational Data Assimilation Problems. Nauka, Moscow. (Russian) (2001)Google Scholar
  21. 21.
    Tautenhahn, U.: Optimality for ill-posed problems under general source conditions. Numer. Funct. Anal. Optim. 19, 377–398 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Tautenhahn, U., Schröter, T.: On optimal regularization methods for the backward heat equation. Z. Anal. Anwend. 15, 475–493 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Trong, D.D., Tuan, N.H.: Regularization and error estimate for the nonlinear backward heat problem using a method of integral equation. Nonlinear Anal. 71, 4167–4176 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2016

Authors and Affiliations

  1. 1.Department of MathematicsVinh UniversityVinh CityVietnam

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