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Sets of Unique Continuation for Heat Equation

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Abstract

We study nodal lines of solutions to the heat equations. We interested in the global geometry of nodal sets, in the whole domain of definition of the solution. The local structure of nodal sets is a well understander subject, while the global geometry of nodal lines is much less clear. We give a detailed analysis of a simple component of a nodal set of a solution of the heat equation. Our results are motivated by some applied problems. They related to classical problems of the unique continuation and the backward uniqueness for parabolic equations.

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References

  1. Varkentina, N.: PhD thesis, Aix-Marseille Université (2012)

  2. Jones, B. F.: A fundamental solution of the heat equation which is supported in a strip. J. Math. Anal. Appl. 60, 314–324 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  3. Littman, V.: Boundary control theory for hyperbolic and parabolic partial differential equations with constant coefficients. Annli. Scuola. Norm. Sup. Pisa Serie IV 3, 567–580 (1978)

    MathSciNet  Google Scholar 

  4. Escauriaza, L., Seregin, G., S̆verák, V.: Backward uniqueness for parabolic equations. Arch. Ration. Mech. Anal. 169, 147–157 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Evans, L. C.: Partial Differential Equations. AMS, Providence (1998)

    MATH  Google Scholar 

  6. Sogge, C. D.: A unique continuation theorem for second order parabolic differential operators. Ark. Mat. 28, 159–82 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  7. Lin, F.-H.: Nodal sets of solutions of elliptic and parabolic equations. Comm. Pure Appl. Math. 44 (287)

  8. Han, Q., Lin, F.-H.: Nodal sets of solutions of parabolic equations: II Comm. Pure Appl. Math. 47, 1219–1238 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  9. Angenent, S.B.: Nodal properties of solutions of parabolic equations, Rocky Mountain. J. Math 21, 585–592 (1991)

    MathSciNet  MATH  Google Scholar 

  10. Matano, H: Nonincrease of the lap number of solution for a one-dimensional semi-linear parabolic equation. J. Fac. Sci. Univ. Tokyo Sect. 1A Math. 29, 410–441 (1982)

    MathSciNet  Google Scholar 

  11. Eremenko, A., Jakobson, D., Nadirashvili, N.: On nodal sets and nodal domains on S 2 and R 2. Ann. Inst. Fourier 57, 2345–2360 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. John, F.: Partial Differential Equations, 4th ed. Springer (1982)

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Authors and Affiliations

  1. Aix-Marseille Université, I2M, 39, rue F. Joliot-Curie, 13453, Marseille, France

    Nikolai Nadirashvili

  2. Université Bordeaux 1, 351, Cours de la Libération, 33405, Talence, Cedex, France

    Nadezda Varkentina

Authors
  1. Nikolai Nadirashvili
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  2. Nadezda Varkentina
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Correspondence to Nikolai Nadirashvili.

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Nadirashvili, N., Varkentina, N. Sets of Unique Continuation for Heat Equation. Potential Anal 41, 1267–1272 (2014). https://doi.org/10.1007/s11118-014-9419-4

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  • DOI: https://doi.org/10.1007/s11118-014-9419-4

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