Publication Cover
Applicable Analysis
An International Journal
Volume 93, 2014 - Issue 10
148
Views
5
CrossRef citations to date
0
Altmetric
Articles

Conditional stability in a backward parabolic system

&
Pages 2174-2198 | Received 15 Nov 2013, Accepted 03 Dec 2013, Published online: 05 Feb 2014

References

  • Brezis H. Functional analysis, Sobolev spaces and partial differential equations. Berlin: Springer-Verlag; 2011.
  • Tanabe H. Equations of evolution. London: Pitman; 1979.
  • Ames KA, Straughan B. Non-standard and improperly posed problems. San Diego (CA): Academic Press; 1997.
  • Baumeister J. Stable solution of inverse problems. Braunschweig: Vieweg; 1987.
  • Carasso AS. Logarithmic convexity and the “slow evolution” constraints in ill-posed initial value problems. SIAM J. Math. Anal. 1999;30:479–496.
  • Colton DL. Analytic theory of partial differential equations. Boston: Pitman; 1980.
  • Eldén L. Time discretization in the backward solution of parabolic equations I. Math. Compt. 1982;39:53–84.
  • Ewing RE. The approximation of certain parabolic equation backward in time by Sobolev equations. SIAM J. Math. Anal. 1975;6:283–294.
  • Lattés R, Lions J-L. The method of quasireversibility: applications to partial differential equations. New York (NY): Elsevier; 1969.
  • Liu J. Numerical solutions of forward and backward problem for 2-D heat conduction equation. J. Comput. Appl. Math. 2002;145:459–482.
  • Miller K. Nonunique continuation for uniformly parabolic and elliptic equations in self-adjoint divergence form with Hölder continuous coefficients. Arch. Rat. Mech. Anal. 1974;54:105–117.
  • Kreĭn SG, Prozorovskaya OI. Analytic semigroups and incorrect problems for evolutionary equations. Soviet Math. Dokl. 1960;133:35–38.
  • Lavrent’ev MM, Romanov VG, Shishat.skiĭ SP. Ill-posed problems of mathematical physics and analysis. Providence (RI): American Mathematical Society; 1986.
  • Payne LE. Improperly posed problems in partial differential equations. Philadelphia (PA): SIAM; 1975.
  • Isakov V. Inverse problems for partial differential equations. Berlin: Springer-Verlag; 1998.
  • Klibanov MV. Estimates of initial conditions of parabolic equations and inequalities via lateral Cauchy data. Inverse Probl. 2006;22:495–514.
  • Agmon S, Nirenberg L. Properties of solutions of ordinary differential equations in Banach spaces. Commun. Pure Appl. Math. 1963;16:121–239.
  • Lees M, Protter MH. Unique continuation for parabolic differential equations and inequalities. Duke Math. J. 1961;28:369–382.
  • Yamamoto M. Carleman estimates for parabolic equations and applications. Inverse Prob. 2009;25: 123013 (75pp).
  • Xu D, Yamamoto M. Stability estimates in state-estimation for a heat process. In: Proceedings of the Second ISAAC Congress, Vol. 1 (Fukuoka, 1999). Dordrecht: Kluwer; 2000. p. 193–198.
  • Yamamoto M, Zou J. Simultaneous reconstruction of the initial temperature and heat radiative coefficient. Inverse Prob. 2001;17:1181–1202.
  • Hào DN, Duc NV. Stability results for backward parabolic equations with time-dependent coefficients. Inverse Prob. 2011;27: 025003 (20pp).
  • Giaquinta M. Introduction to regularity theory for nonlinear elliptic systems. Basel: Birkhäuser; 1993.
  • Kythe PK. Fundamental solutions for differential operators and applications. Boston: Birkhäuser; 1996.
  • Klibanov MV, Timonov AA. Carleman estimates for coefficient inverse problems and numerical applications. Utrecht: VSP; 2004.
  • Adams RA. Sobolev spaces. New York (NY): Academic Press; 1975.
  • Ladyzhenskaya OA. The boundary value problems of mathematical physics. New York (NY): Springer-Verlag; 1985.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.