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Fujita-type results for the degenerate parabolic equations on the Heisenberg groups

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Abstract

In this paper, we consider the Cauchy problem for the degenerate parabolic equations on the Heisenberg groups with power law non-linearities. We obtain Fujita-type critical exponents, which depend on the homogeneous dimension of the Heisenberg groups. The analysis includes the case of porous medium equations. Our proof approach is based on methods of nonlinear capacity estimates specifically adapted to the nature of the Heisenberg groups. We also use the Kaplan eigenfunctions method in combination with the Hopf-type lemma on the Heisenberg groups.

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References

  1. Adams, R.A., Fournier, J.J.F.: Sobolev Spaces, vol. 140. Elsevier, New York (2003)

    Google Scholar 

  2. Azman, I., Jleli, M., Samet, B.: Blow-up of solutions to parabolic inequalities in the Heisenberg group. Electron. J. Differ. Equ. 167, 1–9 (2015)

    MathSciNet  Google Scholar 

  3. Birindelli, I., Cutrì, A.: A semi-linear problem for the Heisenberg Laplacian. Rend. Semin. Mat. Univ. Padova 94, 137–153 (1995)

    MathSciNet  Google Scholar 

  4. Chen, H., Luo, P.: Lower bounds of Dirichlet eigenvalues for some degenerate elliptic operators. Calc. Var. Partial Differ. Equ. 54, 2831–2852 (2015)

    Article  MathSciNet  Google Scholar 

  5. Chen, X.P., Du, S.Z., Guo, T.P.: The Liouville theorem of a torsion system and its application to the symmetry group of a porous medium type equation on symmetric spaces. J. Lie Theory 31(2), 393–411 (2021)

    ADS  MathSciNet  Google Scholar 

  6. D’Ambrosio, L.: Critical degenerate inequalities on the Heisenberg group. Manuscr. Math. 106, 519–536 (2001)

    Article  MathSciNet  Google Scholar 

  7. Friedman, A., McLeod, B.: Blow-up of solutions of nonlinear degenerate parabolic equations. Arch. Ration. Mech. Anal. 96, 55–80 (1987)

    Article  MathSciNet  Google Scholar 

  8. 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 

  9. Galaktionov, V.A., Kurdyumov, S.P., Mikhailov, A.P., Samarskii, A.A.: Unbounded solutions of the Cauchy problem for the parabolic equation \(u_t=\nabla (u^\sigma \nabla u)+u^\beta \). Dokl. Akad. Nauk SSSR 252(6), 1362–1364 (1980)

    MathSciNet  Google Scholar 

  10. Galaktionov, V.A., Kurdyumov, S.P., Mikhailov, A.P., Samarskii, A.A.: Blow-Up in Quasilinear Parabolic Equations. De Gruyter Expositions in Mathematics. Springer, Berlin (1995)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  12. Georgiev, V., Palmieri, A.: Lifespan estimates for local in time solutions to the semilinear heat equation on the Heisenberg group. Ann. Mat. Pura Appl. 200(3), 999–1032 (2021)

    Article  MathSciNet  Google Scholar 

  13. Grätzer, G.: Math into LaTeX, 3rd edn. Birkhäuser, Basel (2000)

  14. Grillo, G., Muratori, M., Punzo, F.: Blow-up and global existence for the porous medium equation with reaction on a class of Cartan–Hadamard manifolds. J. Differ. Equ. 266(7), 4305–4336 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  15. Han, J.: Degenerate evolution inequalities on groups of Heisenberg type. J. Partial Differ. Equ. 18(4), 341–354 (2005)

    MathSciNet  Google Scholar 

  16. Hayakawa, K.: On Nonexistence of global solutions of some semilinear parabolic differential equations. Proc. Jpn. Acad. 49(7), 503–595 (1973)

    ADS  MathSciNet  Google Scholar 

  17. Jleli, M., Kirane, M., Samet, B.: A Fujita-type theorem for a multitime evolutionary p-Laplace inequality in the Heisenberg group. Electron. J. Differ. Equ. 2016, 1–8 (2016)

    MathSciNet  Google Scholar 

  18. Mastrolia, P., Monticelli, D.D., Punzo, F.: Nonexistence of solutions to parabolic differential inequalities with a potential on Riemannian manifolds. Math. Ann. 367, 929–963 (2017)

    Article  MathSciNet  Google Scholar 

  19. Meglioli, G., Punzo, F.: Blow-up and global existence for solutions to the porous medium equation with reaction and slowly decaying density. J. Differ. Equ. 269(10), 8918–8958 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  20. Mochizuki, K., Suzuki, R.: Critical exponent and critical blow-up for quasilinear parabolic equations. Isr. J. Math. 98, 141–156 (1997)

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  22. Pascucci, A.: Fujita type results for a class of degenerate parabolic operators. Adv. Differ. Equ. 4(5), 755–776 (1999)

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  24. Ruzhansky, M., Yessirkegenov, M.: Existence and non-existence of global solutions for semilinear heat equations and inequalities on sub-Riemannian manifolds, and Fujita exponent on unimodular Lie groups. J. Differ. Equ. 308, 455–473 (2022)

    Article  ADS  MathSciNet  Google Scholar 

  25. Ruzhansky, M., Sabitbek, B., Torebek, B.: Global existence and blow-up of solutions to porous medium equation and pseudo-parabolic equation. I. Stratified Groups. Manuscr. Math. (2022). https://doi.org/10.1007/s00229-022-01390-2

    Article  Google Scholar 

  26. Ruzhansky, M., Suragan, D.: Layer potentials, Kac’s problem, and refined Hardy inequality on homogeneous Carnot groups. Adv. Math. 308, 483–528 (2017)

    Article  MathSciNet  Google Scholar 

  27. Test, A.B.C.: On a Test. J. Test. 88, 100–120 (2000)

    Google Scholar 

  28. Winkler, M.: A critical exponent in a degenerate parabolic equation. Math. Methods Appl. Sci. 25, 911–925 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  29. Yang, Zh.: Fujita exponent and nonexistence result for the Rockland heat equation. Appl. Math. Lett. 121, 107386 (2021)

    Article  MathSciNet  Google Scholar 

  30. Zhang, Q.S.: The critical exponent of a reaction diffusion equation on some Lie groups. Math. Z. 228, 51–72 (1998)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

Ahmad Fino is supported by the Research Group Unit, College of Engineering and Technology, American University of the Middle East. Berikbol Torebek is supported by the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan (Grant No. AP14869090), by the FWO Odysseus 1 grant G.0H94.18N: Analysis and Partial Differential Equations, and by the Methusalem programme of the Ghent University Special Research Fund (BOF) (Grant Number 01M01021). Michael Ruzhansky is also supported by EPSRC Grants EP/R003025/2 and EP/V005529/1.

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Authors and Affiliations

  1. College of Engineering and Technology, American University of the Middle East, Egaila, 54200, Kuwait

    Ahmad Z. Fino

  2. Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Ghent, Belgium

    Michael Ruzhansky & Berikbol T. Torebek

  3. School of Mathematical Sciences, Queen Mary University of London, London, UK

    Michael Ruzhansky

  4. Institute of Mathematics and Mathematical Modeling, 125 Pushkin Str., 050010, Almaty, Kazakhstan

    Berikbol T. Torebek

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  1. Ahmad Z. Fino
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  2. Michael Ruzhansky
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Correspondence to Ahmad Z. Fino.

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Fino, A.Z., Ruzhansky, M. & Torebek, B.T. Fujita-type results for the degenerate parabolic equations on the Heisenberg groups. Nonlinear Differ. Equ. Appl. 31, 19 (2024). https://doi.org/10.1007/s00030-023-00907-2

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