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Lifespan estimates for local in time solutions to the semilinear heat equation on the Heisenberg group

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Abstract

In this paper, we consider the semilinear Cauchy problem for the heat equation with power nonlinearity in the Heisenberg group \(\mathbf {H}_n\). The heat operator is given in this case by \(\partial _t-\varDelta _{{{\,\mathrm{H}\,}}}\), where \(\varDelta _{{{\,\mathrm{H}\,}}}\) is the so-called sub-Laplacian on \(\mathbf {H}_n\). We prove that the Fujita exponent \(1+2/Q\) is critical, where \(Q=2n+2\) is the homogeneous dimension of \(\mathbf {H}_n\). Furthermore, we prove sharp lifespan estimates for local in time solutions in the subcritical case and in the critical case. In order to get the upper bound estimate for the lifespan (especially, in the critical case), we employ a revisited test function method developed recently by Ikeda–Sobajima. On the other hand, to find the lower bound estimate for the lifespan, we prove a local in time result in weighted \(L^\infty \) space.

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References

  1. Beals, R., Gaveau, B., Greiner, P.C.: Hamilton-Jacobi theory and the heat kernel on Heisenberg groups. J. Math. Pures Appl. 79, 633–689 (2000)

    Article  MathSciNet  Google Scholar 

  2. Bonfiglioli, A., Lanconelli, E., Uguzzoni, F.: Stratified Lie Groups and Potential Theory for Their Sub-Laplacians. Springer Monographs in Mathematics. Springer, Berlin (2007)

    MATH  Google Scholar 

  3. Fischer, V., Ruzhansky, M.: Quantization on Nilpotent Lie Groups. Volume 314 of Progress in Mathematics. Birkhäuser/Springer, Berlin (2016)

    Book  Google Scholar 

  4. Folland, G.B.: Subelliptic estimates and function spaces on nilpotent Lie groups. Ark. Mat. 13(2), 161–207 (1975)

    Article  MathSciNet  Google Scholar 

  5. Folland, G.B., Stein, E.M.: Hardy Spaces on Homogeneous Groups. Volume 28 of Mathematical Notes. Princeton University Press, University of Tokyo Press, Princeton, Tokyo (1982)

    MATH  Google Scholar 

  6. Fujita, H.: On the blowing up of solutions of the Cauchy problem for \(u_t=\Delta u+u^{1+\alpha }\). J. Fac. Sci. Univ. Tokyo Sect. I(13), 109–124 (1966)

    Google Scholar 

  7. Fujiwara, K., Georgiev, V., Ozawa, T.: Note for global existence of semilinear heat equation in weighted \(L^{\infty }\) space. Pliska Stud. Math. 30, 7–20 (2019)

    MathSciNet  MATH  Google Scholar 

  8. Gaveau, B.: Principe de moindre action, propagation de la chaleur et estimées sous elliptiques sur certains groupes nilpotents. Acta Math. 139, 95–153 (1977)

    Article  MathSciNet  Google Scholar 

  9. Georgiev, V., Palmieri, A.: Upper bound estimates for local in time solutions to the semilinear heat equation on stratified lie groups in the sub-Fujita case. AIP Conf. Proc. 2159, 020003 (2019). https://doi.org/10.1063/1.5127465

    Article  Google Scholar 

  10. Greiner, P., Li, Y.: Heat kernels old and new. Bull. Inst. Math. Acad. Sin. (N.S.) 12, 1–37 (2017)

    MathSciNet  MATH  Google Scholar 

  11. Hayakawa, K.: On the nonexistence of global solutions of some semilinear parabolic differential equations. Proc. Jpn. Acad. 49, 503–505 (1974)

    MathSciNet  MATH  Google Scholar 

  12. Hörmander, L.: Hypoelliptic second order differential equations. Acta Math. 119, 147–171 (1967)

    Article  MathSciNet  Google Scholar 

  13. Hulanicki, A.: The distribution of energy in the Brownian motion in the Gaussian field and analytic-hypoellipticity of certain subelliptic operators on the Heisenberg group. Stud. Math. 56, 165–173 (1976)

    Article  MathSciNet  Google Scholar 

  14. Ikeda, M., Sobajima, M.: Life-span of solutions to semilinear wave equation with time-dependent critical damping for specially localized initial data. Math. Ann. 372(3/4), 1017–1040 (2018)

    Article  MathSciNet  Google Scholar 

  15. Ikeda, M., Sobajima, M.: Sharp upper bound for lifespan of solutions to some critical semilinear parabolic, dispersive and hyperbolic equations via a test function method. Nonlinear Anal. 182, 57–74 (2019)

    Article  MathSciNet  Google Scholar 

  16. Ikeda, M., Sobajima, M., Wakasa, K.: Blow-up phenomena of semilinear wave equations and their weakly coupled systems. J. Differ. Equ. (2019). https://doi.org/10.1016/j.jde.2019.05.029

    Article  MathSciNet  MATH  Google Scholar 

  17. Kobayashi, K., Sirao, T., Tanaka, H.: On the growing up problem for semilinear heat equations. J. Math. Soc. Jpn. 29, 407–424 (1977)

    MathSciNet  MATH  Google Scholar 

  18. Kusuoka, S., Stroock, D.: Applications of the Malliavin calculus, Part III. J. Fac. Sci. Univ. Tokyo Sec. IA Math. 34, 391–442 (1987)

    MATH  Google Scholar 

  19. Lee, T.Y., Ni, W.N.: Global existence, large time behavior and life span of solutions of a semilinear parabolic Cauchy problem. Trans. Am. Math. Soc. 333, 365–378 (1992)

    Article  MathSciNet  Google Scholar 

  20. Mitidieri, E., Pohozaev, S.: A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities. Proc. Steklov Inst. Math. 234, 1–362 (2001)

    MathSciNet  MATH  Google Scholar 

  21. Müller, D., Stein, E.M.: \(L^p\)-estimates for the wave equation on the Heisenberg group. Rev. Mat. Iberoamericana 15, 297–334 (1999)

    Article  MathSciNet  Google Scholar 

  22. Nachman, A.I.: The wave equation on the Heisenberg group. Commun. Par. Differ. Equ. 7, 675–714 (1982)

    Article  MathSciNet  Google Scholar 

  23. Pascucci, A.: Semilinear equations on nilpotent Lie groups: global existence and blow-up of solutions. Le Matematiche 53(2), 345–357 (1998)

    MathSciNet  MATH  Google Scholar 

  24. Pohozaev, S., Véron, L.: Nonexistence results of solutions of semilinear differential inequalities on the Heisenberg group. Manuscripta Math. 102, 85–99 (2000)

    Article  MathSciNet  Google Scholar 

  25. Ruzhansky, M., Yessirkegenov, N.: Existence and non-existence of global solutions for semilinear heat equations and inequalities on sub-Riemannian manifolds, and Fujita exponent on unimodular Lie groups . Preprint arXiv:1812.01933v2 (2018)

  26. Sugitani, S.: On nonexistence of global solutions for some nonlinear integral equations. Osaka J. Math. 12, 45–51 (1975)

    MathSciNet  MATH  Google Scholar 

  27. Varopoulos, N.T., Saloff-Coste, L., Coulhon, T.: Analysis and Geometry on Groups. Cambridge Tracts in Mathematics, vol. 100. Cambridge University Press, Cambridge (1992)

    MATH  Google Scholar 

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Acknowledgements

V. Georgiev is supported in part by GNAMPA - Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni, by Institute of Mathematics and Informatics, Bulgarian Academy of Sciences and Top Global University Project, Waseda University and by the University of Pisa, Project PRA 2018 49. A. Palmieri is supported by the University of Pisa, Project PRA 2018 49. Both authors thank the anonymous referee for pointing out reference [23] and for her/his valuable suggestions.

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Authors and Affiliations

  1. Department of Mathematics, University of Pisa, Largo B. Pontecorvo 5, 56127, Pisa, Italy

    Vladimir Georgiev & Alessandro Palmieri

  2. Faculty of Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo, 169-8555, Japan

    Vladimir Georgiev

  3. Institute of Mathematics and Informatics-BAS, Acad. G. Bonchev Str., Block 8, 1113, Sofia, Bulgaria

    Vladimir Georgiev

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  1. Vladimir Georgiev
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  2. Alessandro Palmieri
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Correspondence to Alessandro Palmieri.

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Georgiev, V., Palmieri, A. Lifespan estimates for local in time solutions to the semilinear heat equation on the Heisenberg group. Annali di Matematica 200, 999–1032 (2021). https://doi.org/10.1007/s10231-020-01023-z

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