Inhomogeneous nonlinear high-order evolution equations


Well-posedness issues of some high-order evolution equations with inhomogeneous exponential non-linearity are investigated in even space dimensions.

This is a preview of subscription content, access via your institution.


  1. 1.

    Adachi, S., Tanaka, K.: Trudinger type inequalities in \({{\mathbb{R}}}^{N}\) and their best exponent. Proc. Am. Math. Soc. 128(7), 2051–2057 (1999)

    Article  Google Scholar 

  2. 2.

    Adams, D.R.: Sobolev spaces. Academic Press, New York (1975)

    MATH  Google Scholar 

  3. 3.

    Caristi, G., Mitidieri, E.: Existence and nonexistence of global solutions of higher-order parabolic problems with slow decay initial data. J. Math. Anal. Appl. 279, 710–722 (2003)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Colliander, J., Ibrahim, S., Majdoub, M., Masmoudi, N.: Energy critical NLS in two space dimensions. J. Hyperbolic Differ. Equ 6(3), 549–575 (2009)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Constantine, G.M., Savitis, T.H.: A multivariate Faa Di Bruno formula with applications. Trans. Am. Math. Soc. 348(2), 503–520 (1996)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Egorov, Y.V., Galaktionov, V.A., Kondratiev, V.A., Pohozaev, S.I.: On the asymptotics of global solutions of higher-order semi-linear parabolic equations in the super-critical range. C. R. Acad. Sci. Paris, Ser. I 335, 805–810 (2002)

    Article  Google Scholar 

  7. 7.

    Galaktionov, V.A.: Critical global asymptotics in higher-order semi-linear parabolic equations. Int. J. Math. Math. Sci. 60, 3809–3825 (2003)

    Article  Google Scholar 

  8. 8.

    Galaktionov, V.A., Pohozaev, S.I.: Blow-up and critical exponents for nonlinear hyperbolic equations. Nonl. Anal. 53, 453–466 (2003)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Ibrahim, S., Majdoub, M., Masmoudi, N.: Global solutions for a semi-linear \(2D\) Klein-Gordon equation with exponential type non-linearity. Comm. Pure App. Math. 59(11), 1639–1658 (2006)

    Article  Google Scholar 

  10. 10.

    Ibrahim, S., Masmoudi, N., Nakanishi, K.: Scattering threshold for the focusing nonlinear Klein-Gordon equation. Anal. PDE 4(3), 405–460 (2011)

    MathSciNet  Article  Google Scholar 

  11. 11.

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

    MathSciNet  Article  Google Scholar 

  12. 12.

    Ioku, N.: The Cauchy problem for heat equations with exponential non-linearity. J. Diff. Equ. 251(4), 1172–1194 (2011)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Kim, J., Arnold, A., Yao, X.: Global estimates of fundamental solutions for higher-order Schrödinger equations. Monatshefte für Mathematik 168(2), 253–266 (2012)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Kim, J., Arnold, A., Yao, X.: Estimates for a class of oscillatory integrals and decay rates for wave-type equations. J. Math. Anal. Appl. 394(1), 139–151 (2012)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Lions, J.L.: Symètrie et compacité dans les espaces de Sobolev. J. Funct. Anal. 49, 315–334 (1982)

    Article  Google Scholar 

  16. 16.

    Moser, J.: A sharp form of an inequality of N. Trudinger. Ind. Univ. Math. J. 20, 1077–1092 (1971)

    MathSciNet  Article  Google Scholar 

  17. 17.

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

    MathSciNet  Article  Google Scholar 

  18. 18.

    Nakamura, M., Ozawa, T.: Global solutions in the critical Sobolev space for the wave equations with non-linearity of exponential growth. Math. Z 231, 479–487 (1999)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Ogawa, T., Ozawa, T.: Trudinger type inequalities and uniqueness of weak solutions for the nonlinear Schrödinger mixed problem. J. Math. Anal. Appl. 155(2), 531–540 (1991)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Ozawa, T.: On critical cases of Sobolev’s inequalities. J. Funct. Anal. 127, 259–269 (1995)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Payne, L.E., Sattinger, D.H.: Saddle points and instability of nonlinear hyperbolic equations. Isr. J. Math. 22, 273–303 (1975)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Peletier, L., Troy, W.C.: Higher order models in Physics and Mechanics. Prog. Non Diff. Eq. App. 45, (2001)

  23. 23.

    Ruf, B.: A sharp Moser-Trudinger type inequality for unbounded domains in \({{\mathbb{R}}}^{2}\). J. Funct. Anal. 219, 340–367 (2004)

    Article  Google Scholar 

  24. 24.

    Ruf, B., Sani, S.: sharp Adams-type inequalities in \({{\mathbb{R}}}^{n}\). Trans. Am. Math. Soc. 365, 645–670 (2013)

    Article  Google Scholar 

  25. 25.

    Saanouni, T.: Remarks on the semi-linear Schrödinger equation. J. Math. Anal. Appl. 400, 331–344 (2013)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Saanouni, T.: Global well-posedness of some high-order semi-linear wave and Schrödinger type equations with exponential non-linearity. Comm. Pur. Appl. Anal. 13(1), 273–291 (2014)

    Article  Google Scholar 

  27. 27.

    Saanouni, T.: Global well-posedness and instability of an inhomogeneous nonlinear Schrödinger equation. Mediterr. J. Math. 12(2), 387–417 (2015)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Saanouni, T.: A note on the inhomogeneous nonlinear heat equation in two space dimensions. Mediterr. J. Math. 13, 3651–3672 (2016)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Saanouni, T.: A note on the critical nonlinear high-order Schrödinger equation. J. Math. Anal. Appl 451, 536–756 (2017)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Saanouni, T.: Global well-posedness of some high-order focusing semi-linear evolution equations with exponential non-linearity. Adv. Nonl. Anal. 7(1), 67–84 (2018)

    MathSciNet  MATH  Google Scholar 

  31. 31.

    Cui, Shangbin, Guo, Cuihua: Well-posedness of higher-order nonlinear Schrödinger equations in Sobolev spaces \(H^s({\mathbb{R}}^n)\) and applications. Nonl. Anal. 67, 687–707 (2007)

    Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Tarek Saanouni.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Saanouni, T. Inhomogeneous nonlinear high-order evolution equations. Bol. Soc. Mat. Mex. 26, 1063–1095 (2020).

Download citation


  • Inhomogeneous high-order evolution equations
  • Ground state
  • Moser–Trudinger inequality
  • Well-posedness

Mathematics Subject Classification

  • Primary 35Q55
  • Secondary 35B30