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Existence, Non-existence, and Uniqueness for a Heat Equation with Exponential Nonlinearity in ℝ2

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Abstract

We consider a semilinear heat equation with exponential nonlinearity in ℝ2. We prove that local solutions do not exist for certain data in the Orlicz space exp L 2(ℝ2), even though a small data global existence result holds in the same space exp L 2(ℝ2). Moreover, some suitable subclass of exp L 2(ℝ2) for local existence and uniqueness is proposed.

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References

  1. Adachi, S., Tanaka, K.: Trudinger type inequalities in R N and their best exponents. Proc. Amer. Math. Soc. 128, 2051–2057 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Adams, R.A., Fournier, J.J.F.: “Sobolev Spaces,” Second edition. Academic Press, Pure and applied mathematics (2003)

    Google Scholar 

  3. Bennett, C., Sharpley, R.: “Interpolation of operators,” Pure and applied mathematics. Academic Press (1988)

  4. Brezis, H., Cazenave, T.: A nonlinear heat equation with singular initial data. Journ. d’anal. Math. 68, 186–212 (1996)

    MathSciNet  Google Scholar 

  5. Colliander, J., Ibrahim, S., Majdoub, M., Masmoudi, N.: Energy critical NLS in two space dimensions. J. Hyperbolic Differ. Equ. 6, 549–575 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Furioli, G., Lemarié-Rieusset, P. G., Terraneo, E.: Unicité dans \(L^{3(\mathbb {R}^{3})}\) et d’autres espaces fonctionnels limites pour Navier-Stokes. Rev. Mat. Iberoamericana 16, 605–667 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  7. Haraux, A., Weissler, F.B.: Nonuniqueness for a semilinear initial value problem. Indiana Univ. Math. J 31, 167–189 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ibrahim, S., Majdoub, M., Masmoudi, N., Nakanishi, K.: Scattering for the two-dimensional NLS with exponential nonlinearity. Nonlinearity 25, 1843–1849 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Ibrahim, S., Jrad, R., Majdoub, M., Saanouni, T.: Local well posedness of a 2D semilinear heat equation. Bull. Belg. Math. Soc. Simon Stevin 21(3), 535–551 (2014)

    MathSciNet  MATH  Google Scholar 

  10. Ibrahim, S., Majdoub, M., Masmoudi, N.: Global solutions for a semilinear 2D Klein-Gordon equation with exponential type nonlinearity. Comm. Pure App. Math. 59, 1639–1658 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ioku, N.: Journal of Differential Equations 251, 1172–1194 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Ladyzhenskaya, O.A., Solonnikov, V.A., Ural’tseva, N.N.: Linear and quasilinear equations of parabolic type, Amer. Math. Soc., Transl. Math. Monographs, Providence, P.I. (1968)

  13. Nakamura, M., Ozawa, T.: Nonlinear Schrödinger equations in the Sobolev space of critical order. J. Funct. Anal. 155, 364–380 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ni, W.-M., Sacks, P.: Trans. Amer. Math. Soc. 287, 657–671 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ogawa, T.: A proof of Trudinger’s inequality and its application to nonlinear Schrödinger equations. Nonlinear Anal. 14, 765–769 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  16. Quittner, P., Souplet, P.: Superlinear parabolic problems, Birkhäuser Advanced Texts: Basler Lehrbücher. Birkhäuser Verlag, Basel (2007)

    Google Scholar 

  17. Rao, M.M., Ren, Z.D.: Applications of Orlicz spaces, Monographs and Textbooks in Pure and Applied Mathematics, vol. 250. Marcel Dekker, Inc., New York (2002)

    Google Scholar 

  18. Ruf, B.: A sharp Trudinger-Moser type inequality for unbounded domains in \(\mathbb {R}^{2}\). J. Funct. Anal. 219, 340–367 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ruf, B., Terraneo, E.: The Cauchy problem for a semilinear heat equation with singular initial data. Progr. Nonlinear Differential Equations Appl. 50, 295–309 (2002)

    MathSciNet  Google Scholar 

  20. Snoussi, S., Tayachi, S., Weissler, F.B.: Asymptotically self-similar global solutions of a semilinear parabolic equation with a nonlinear gradient term. A 129, 1291–1307 (1999)

    MathSciNet  MATH  Google Scholar 

  21. Tayachi, S.: Forward self-similar solutions of a semilinear parabolic equation with a nonlinear gradient term. Differential Integral Equations 9, 1107–1117 (1996)

    MathSciNet  MATH  Google Scholar 

  22. Terraneo, E.: Non-uniqueness for a critical non-linear heat equation. Comm. Partial Differential Equations 27, 185–218 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  23. Trudinger, N.S.: On imbeddings into Orlicz spaces and some applications. J. Math. Mech. 17, 473–483 (1967)

    MathSciNet  MATH  Google Scholar 

  24. Weissler, F.B.: Local existence and nonexistence for semilinear parabolic equations in L p. Indiana Univ. Math. J 29, 79–102 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  25. Weissler, F.B.: Existence and nonexistence of global solutions for a semilinear heat equation. Israel Math. J 38, 29–40 (1981)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Graduate School of Science Engineering, Ehime University, 2-5 Bunkyo-cho, Matsuyama, Ehime, 790-8577, Japan

    Norisuke Ioku

  2. Dipartimento di Matematica “F. Enriques”, Università degli Studi di Milano, via C. Saldini 50, Milano, 20133, Italy

    Bernhard Ruf & Elide Terraneo

Authors
  1. Norisuke Ioku
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  2. Bernhard Ruf
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  3. Elide Terraneo
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Correspondence to Norisuke Ioku.

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Ioku, N., Ruf, B. & Terraneo, E. Existence, Non-existence, and Uniqueness for a Heat Equation with Exponential Nonlinearity in ℝ2 . Math Phys Anal Geom 18, 29 (2015). https://doi.org/10.1007/s11040-015-9199-0

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  • DOI: https://doi.org/10.1007/s11040-015-9199-0

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