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Capacity estimate for the blow-up set of parabolic equations

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Abstract

A capacity estimate for the blow-up set of parabolic equations is derived. It refines the Lebesgue measure estimate (Sakaguchi and Suzuki in Arch Rational Mech Anal 142:143–153, 1998), includes the result on the elliptic case (T. Sato, T. Suzuki, F. Takahashi, in p-capacity of the singular set of p-harmonic function vanishes, preprint), and provides information on the profile of any post-blow-up solution.

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References

  1. Baras P. and Cohen L. (1987). Complete blow-up after T max for the solution of a semilinear heat equation. J. Funct. Anal. 71: 142–174

    Article  MATH  MathSciNet  Google Scholar 

  2. Bebernes J. and Eberly D. (1989). Mathematical Problems from Combustion Theory. Springer, New York

    MATH  Google Scholar 

  3. DiBenedetto E. (1993). Degenerate Parabolic Equations. Springer, New York

    MATH  Google Scholar 

  4. Evans L.C. and Gariepy R.F. (1992). Measure Theory and Fine Properties of Functions. CRC, Boca Raton

    MATH  Google Scholar 

  5. Fila M. (2005) Blow-up of solutions of supercritical parabolic equations. In: Dafermos C.M., Feireisl E. (eds.) Handbook of Differential Equations, Evolutionary Equations, vol 2, pp. 105–158. Amsterdam: Elsevier

    Google Scholar 

  6. Galaktionov V.A. and Vázquez L.L. (1997). Continuation of blow-up solutions of nonlinear heat equations in several space dimensions. Commun. Pure Appl. Math. 50: 1–67

    Article  MATH  Google Scholar 

  7. Lacey A.A. and Tzanetis D. (1988). Complete blow-up for a semilinear diffusion equation with a sufficiently large initial condition. IMA J. Appl. Math. 41: 207–215

    Article  MATH  MathSciNet  Google Scholar 

  8. Masuda K. (1984). Analytic solutions of some nonlinear diffusion equations. Math. Z. 187: 61–73

    Article  MATH  MathSciNet  Google Scholar 

  9. Ni W.M., Sacks P. and Tavantzis J. (1984). On the asymptotic behavior of solutions of certain quasilinear parabolic equations. J. Diff. Eqaut. 54: 97–120

    Article  MATH  MathSciNet  Google Scholar 

  10. Sakaguchi S. and Suzuki T. (1998). Interior imperfect ignition cannot occur on a set of positive measure. Arch. Rational Mech. Anal. 142: 143–153

    Article  MATH  MathSciNet  Google Scholar 

  11. Sakaguchi S. and Suzuki T. (1998). Nonexistence of solutions for a degenerate parabolic equation describing imperfect ignition. Nonlinear Anal. 31: 665–669

    Article  MATH  MathSciNet  Google Scholar 

  12. Sohr H. (2001). The Navier–Stokes Equations: an Elementary functional analytic approach. Birkhäuser, Boston

    MATH  Google Scholar 

  13. Temam R. (1977). Navier–Stokes Equations. North-Holland, Amsterdam

    MATH  Google Scholar 

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Authors and Affiliations

  1. Division of Mathematical Science, Department of System Innovation, Graduate School of Engineering Science, Osaka University, Machikaneyamacho 1-3, Toyonakashi, 560-8531, Japan

    Takashi Suzuki

  2. Department of Mathematics, Graduate School of Science, Osaka City University, Sugimoto 3-3-138, Sumiyoshiku, Osakashi, 558-8585, Japan

    Futoshi Takahashi

Authors
  1. Takashi Suzuki
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  2. Futoshi Takahashi
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Correspondence to Futoshi Takahashi.

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Suzuki, T., Takahashi, F. Capacity estimate for the blow-up set of parabolic equations. Math. Z. 259, 867–878 (2008). https://doi.org/10.1007/s00209-007-0252-y

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  • DOI: https://doi.org/10.1007/s00209-007-0252-y

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