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On refined blowup estimates for the exponential reaction-diffusion equation

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Abstract

We consider radial decreasing solutions of the semilinear heat equation with exponential nonlinearity. We provide a relatively simple proof of the sharp upper estimates for the final blowup profile and for the refined space-time behavior. We actually establish a global, upper space-time estimate, which contains those of the final and refined profiles as special cases.

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Notes

  1. Recall that, under the assumptions of Theorem 1, since 0 is the only blowup point (this follows from (1.4)), the space-profile \(u(x,T)= \lim _{t\rightarrow T}u(x,t)\) exists for all \(x\ne 0\) by standard parabolic estimates.

References

  1. Bebernes, J., Bressan, A., Eberly, D.: A description of blow-up for the solid fuel ignition model. Indiana Univ. Math. J. 36, 131–136 (1987)

    Article  MathSciNet  Google Scholar 

  2. Bebernes, J., Bricher, S.: Final time blowup profiles for semilinear parabolic equations via center manifold theory. SIAM J. Math. Anal. 23, 852–869 (1992)

    Article  MathSciNet  Google Scholar 

  3. Bebernes, J., Eberly, D.: Mathematical Problems from Combustion Theory, Applied Mathematical Sciences Series Profile, vol. 83. Springer, New York (1989)

    MATH  Google Scholar 

  4. Bressan, A.: On the asymptotic shape of blow-up. Indiana Univ. Math. J. 39, 947–960 (1990)

    Article  MathSciNet  Google Scholar 

  5. Bressan, A.: Stable blow-up patterns. J. Differ. Equ. 98, 57–75 (1992)

    Article  MathSciNet  Google Scholar 

  6. Chen, Y.-G.: Blow-up solutions of a semilinear parabolic equation with the Neumann and Robin boundary conditions. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 37, 537–574 (1990)

    MathSciNet  MATH  Google Scholar 

  7. Dold, J.W.: Analysis of the early stage of thermal runaway. Q. J. Mech. Appl. Math. 38, 361–387 (1985)

    Article  Google Scholar 

  8. Eberly, D., Troy, W.C.: Existence of logarithmic-type solutions to the Kapila-Kassoy problem in dimensions \(3\) through \(9\). J. Differ. Equ. 70, 309–324 (1987)

    Article  MathSciNet  Google Scholar 

  9. Fila, M., Pulkkinen, A.: Nonconstant selfsimilar blow-up profile for the exponential reaction-diffusion equation. Tohoku Math. J. 60, 303–328 (2008)

    Article  MathSciNet  Google Scholar 

  10. Friedman, A., McLeod, B.: Blow-up of positive solutions of semilinear heat equations. Indiana Univ. Math. J. 34, 425–447 (1985)

    Article  MathSciNet  Google Scholar 

  11. Guo, J.-S., Souplet, Ph.: Fast rate of formation of dead-core for the heat equation with strong absorption and applications to fast blow-up. Math. Ann. 331, 651–667 (2005)

    Article  MathSciNet  Google Scholar 

  12. Herrero, M.A., Velázquez, J.J.L.: Blow-up behaviour of one-dimensional semilinear parabolic equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 10, 131–189 (1993)

    Article  MathSciNet  Google Scholar 

  13. Herrero, M.A., Velázquez, J.J.L.: Plane structures in thermal runaway. Israel J. Math. 81, 321–341 (1993)

    Article  MathSciNet  Google Scholar 

  14. Lacey, A.A.: Mathematical analysis of thermal runaway for spatially inhomogeneous reactions. SIAM J. Appl. Math. 43, 1350–1366 (1983)

    Article  MathSciNet  Google Scholar 

  15. Lacey, A.A., Tzanetis, D.E.: Global, unbounded solutions to a parabolic equation. J. Differ. Equ. 101, 80–102 (1993)

    Article  MathSciNet  Google Scholar 

  16. Pulkkinen, A.: Blow-up profiles of solutions for the exponential reaction–diffusion equation. Math. Methods Appl. Sci. 34, 2011–2030 (2011)

    Article  MathSciNet  Google Scholar 

  17. Quittner, P., Souplet, Ph.: Superlinear Parabolic Problems. Blow-Up, Global Existence and Steady States, Birkhäuser Advanced Texts, 2nd edn. Springer/Birkhäuser, Cham (2019)

  18. Souplet, Ph.: A simplified approach to the refined blowup behavior for the nonlinear heat equation. SIAM J. Math. Anal. 51, 991–1013 (2019)

    Article  MathSciNet  Google Scholar 

  19. Souplet, Ph., Winkler, M.: Blow-up profiles for the parabolic-elliptic Keller-Segel system in dimensions \(n\ge 3\). Commun. Math. Phys. 367, 665–681 (2019)

    Article  Google Scholar 

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Authors and Affiliations

  1. Université Sorbonne Paris Nord, CNRS UMR 7539, Laboratoire Analyse, Géométrie et Applications, 93430, Villetaneuse, France

    Philippe Souplet

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  1. Philippe Souplet
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Correspondence to Philippe Souplet.

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Dedicated to Professor Eiji Yanagida on the occasion of his 65th birthday.

This article is part of the topical collection dedicated to Professor Eiji Yanagida on the occasion of his 65th birthday, edited by Senjo Shimizu, Tohru Ozawa, Kazuhiro Ishige, Marek Fila.

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Souplet, P. On refined blowup estimates for the exponential reaction-diffusion equation. Partial Differ. Equ. Appl. 3, 16 (2022). https://doi.org/10.1007/s42985-022-00152-9

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  • DOI: https://doi.org/10.1007/s42985-022-00152-9

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