Archive for Rational Mechanics and Analysis

, Volume 169, Issue 2, pp 147–157

Backward Uniqueness for Parabolic Equations

Article

Abstract.

It is shown that a function u satisfying |∂tu|≦M(|u|+|∇u|), |u(x, t)|≦MeM|x|2 in (ℝn \ (BR) × [0, T] and u(x, 0) = 0 for xℝn \ BR must vanish identically in ℝn \ BR×[0, T].

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References

  1. 1.
    Ahlfors, L.V.: Complex Analysis. McGraw-Hill, 1966Google Scholar
  2. 2.
    Chen, X.Y.: A strong unique continuation theorem for parabolic equations. Math. Ann. 311, 603–630 (1996)CrossRefMATHGoogle Scholar
  3. 3.
    Escauriaza, L.: Carleman inequalities and the heat operator. Duke Math. J. 104, 113–127 (2000)MathSciNetMATHGoogle Scholar
  4. 4.
    Escauriaza, L., Vega, L.: Carleman inequalities and the heat operator II. Indiana U. Math. J. 50, 1149–1169 (2001)MathSciNetMATHGoogle Scholar
  5. 5.
    Escauriaza, L., Fernández, F.J.: Unique continuation for parabolic operators. (to appear)Google Scholar
  6. 6.
    Fernández, F.J.: Unique continuation for parabolic operators II. (to appear)Google Scholar
  7. 7.
    Hopf, E.: Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr. 4, 213–231 (1950)MATHGoogle Scholar
  8. 8.
    Hörmander, L.: Linear Partial Differential Operators. Springer, 1963Google Scholar
  9. 9.
    Hörmander, L.: Uniqueness theorems for second order elliptic differential equations. Communications in PDE 8, 21–64 (1983)Google Scholar
  10. 10.
    Jones, B.F.: A fundamental solution of the heat equation which is supported in a strip. J. Math. Anal. Appl. 60, 314–324 (1977)MATHGoogle Scholar
  11. 11.
    Kato, T.: Strong L p-solutions of the Navier-Stokes equations in ℝm with applications to weak solutions. Math. Zeit. 187, 471–480 (1984)MathSciNetMATHGoogle Scholar
  12. 12.
    Ladyzhenskaya, O.A.: Mathematical problems of the dynamics of viscous incompressible fluids. Gordon and Breach, 1969Google Scholar
  13. 13.
    Ladyženskaja, O.A., Solonnikov, V.A., Uralceva, N.N.: Linear and quasilinear equations of parabolic type. Translations of Mathematical Monographs, Amer. Math. Soc., 1968Google Scholar
  14. 14.
    Lin, F.H.: A uniqueness theorem for parabolic equations. Comm. Pure Appl. Math. 42, 125–136 (1988)Google Scholar
  15. 15.
    Littman, W.: Boundary control theory for hyperbolic and parabolic partial differential equations with constant coefficients. Annali Scuola Norm. Sup. Pisa Serie IV 3, 567–580 (1978)Google Scholar
  16. 16.
    Micu, S., Zuazua, E.: On the lack of null-controllability of the heat equation on the half space. Portugaliae Mathematica 58, 1–24 (2001)MathSciNetMATHGoogle Scholar
  17. 17.
    Poon, C.C.: Unique continuation for parabolic equations. Comm. Partial Differential Equations 21, 521–539 (1996)MathSciNetMATHGoogle Scholar
  18. 18.
    Saut, J.C., Scheurer, E.: Unique continuation for evolution equations. J. Differential Equations 66, 118–137 (1987)MathSciNetMATHGoogle Scholar
  19. 19.
    Seregin, G., Šverák, V.: The Navier-Stokes equations and backward uniqueness. (to appear)Google Scholar
  20. 20.
    Sogge, C.D.: A unique continuation theorem for second order parabolic differential operators. Ark. Mat. 28, 159–182 (1990)MathSciNetMATHGoogle Scholar
  21. 21.
    Treves, F.: Linear Partial Differential Equations. Gordon and Breach, 1970Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  1. 1.Dpto. de MatemáticasUPV/EHUBilbaoSpain
  2. 2.Steklov Institute of MathematicsSt.PeterburgRussia
  3. 3.School of MathematicsUniversity of MinnesotaMinneapolisUSA

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