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A Global Existence Result for a Semilinear Wave Equation with Lower Order Terms on Compact Lie Groups

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Abstract

In this paper, we study the semilinear wave equation with lower order terms (damping and mass) and with power type nonlinearity \(|u|^p\) on compact Lie groups. We will prove the global in time existence of small data solutions in the evolution energy space without requiring any lower bounds for \(p>1\). In our approach, we employ some results from Fourier analysis on compact Lie groups.

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References

  1. Bourgain, J.: Fourier transform restriction phenomena for certain lattice subsets and application to nonlinear evolution equations. Part I: Schrödinger equations. Geom. Funct. Anal. 3(2), 107–156 (1993)

    Article  MathSciNet  Google Scholar 

  2. Bourgain, J.: Exponential sums and nonlinear Schrödinger equations. Geom. Funct. Anal. 3(2), 157–178 (1993)

    Article  MathSciNet  Google Scholar 

  3. Burq, N., Gerard, P., Tzvetkov, N.: An instability property of the nonlinear Schrödinger equation on \(S^d\). Math. Res. Lett. 9, 323–335 (2002)

    Article  MathSciNet  Google Scholar 

  4. Burq, N., Gerard, P., Tzvetkov, N.: Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds. Am. J. Math. 126(3), 569–605 (2004)

    Article  Google Scholar 

  5. Burq, N., Gerard, P., Tzvetkov, N.: Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces. Invent. Math. 159, 187–223 (2005)

    Article  MathSciNet  Google Scholar 

  6. Burq, N., Gerard, P., Tzvetkov, N.: Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear Schrödinger equations. Ann. Scient. Éc. Norm. Sup. 38(2), 255–301 (2005)

    Article  Google Scholar 

  7. D’Abbicco, M.: Asymptotics for damped evolution operators with mass-like terms. Compl. Anal. Dyn. Syst. VI 653, 93–116 (2015)

    MathSciNet  MATH  Google Scholar 

  8. Ebert, M.R., Reissig, M.: Methods for Partial Differential Equations. Birkhäuser, Basel (2018)

    Book  Google Scholar 

  9. Fischer, V., Ruzhansky, M.: Quantization on Nilpotent Lie Groups. Progr. Math., vol. 314, Birkhäuser/Springer (2016)

  10. Garetto, C., Ruzhansky, M.: Wave equation for sums of squares on compact Lie groups. J. Differ. Equ. 258(12), 4324–4347 (2015)

    Article  MathSciNet  Google Scholar 

  11. Georgiev, V., Palmieri, A.: Critical exponent of Fujita-type for the semilinear damped wave equation on the Heisenberg group with power nonlinearity. J. Differ. Equ. 269(1), 420–448 (2020). https://doi.org/10.1016/j.jde.2019.12.009

    Article  MathSciNet  MATH  Google Scholar 

  12. Georgiev, V., Lindblad, H., Sogge, C.D.: Weighted Strichartz estimates and global existence for semi-linear wave equations. Am. J. Math. 119(6), 1291–1319 (1997)

    Article  Google Scholar 

  13. Ginibre, J., Velo, G.: The global Cauchy problem for the non linear Schrödinger equation revisited. Annales de l’I.H.P. Analyse non linéaire 2(4), 307–327 (1985)

    MATH  Google Scholar 

  14. Ginibre, J., Velo, G.: Smoothing properties and retarded estimates for some dispersive evolution equations. Commun. Math. Phys. 144(1), 163–188 (1992)

    Article  MathSciNet  Google Scholar 

  15. Girardi, G.: Semilinear damped Klein-Gordon models with time-dependent coefficients. In: D’Abbicco, M., Ebert, M.R., Georgiev, V., Ozawa, T. (eds.) New Tools for Nonlinear PDEs and Application. Trends in Mathematics, Springer, Cham (2019)

    Google Scholar 

  16. Glassey, R.T.: Finite-time blow-up for solutions of nonlinear wave equations. Math. Z. 177(3), 323–340 (1981)

    Article  MathSciNet  Google Scholar 

  17. Glassey, R.T.: Existence in the large for \(\square u = F(u)\) in two space dimensions. Math. Z. 178(2), 233–261 (1981)

    Article  MathSciNet  Google Scholar 

  18. Ikehata, R., Tanizawa, K.: Global existence of solutions for semilinear damped wave equations in \(\mathbb{R}^N\) with noncompactly supported initial data. Nonlinear Anal. 61(7), 1189–1208 (2005)

    Article  MathSciNet  Google Scholar 

  19. John, F.: Blow-up of solutions of nonlinear wave equations in three space dimensions. Manuscripta Math. 28(1–3), 235–268 (1979)

    Article  MathSciNet  Google Scholar 

  20. Kapitanski, L.: Minimal compact global attractor for a damped semilinear wave equation. Commun. Partial. Differ. Equ. 20(7/8), 1303–1323 (1995)

    Article  MathSciNet  Google Scholar 

  21. Keel, M., Tao, T.: Endpoint Strichartz estimates. Am. J. Math. 120(5), 955–980 (1998)

    Article  MathSciNet  Google Scholar 

  22. Lindblad, H., Sogge, C.: On existence and scattering with minimal regularity for semilinear wave equations. J. Funct. Anal. 130(2), 357–426 (1995)

    Article  MathSciNet  Google Scholar 

  23. Matsumura, A.: On the asymptotic behavior of solutions of semi-linear wave equations. Publ. Res. Inst. Math. Sci. 12(1), 169–189 (1976)

    Article  MathSciNet  Google Scholar 

  24. Palmieri, A.: Decay estimates for the linear damped wave equation on the Heisenberg group. J. Funct. Anal. 279(9), 108721 (2020). https://doi.org/10.1016/j.jfa.2020.108721

    Article  MathSciNet  MATH  Google Scholar 

  25. Palmieri, A.: On the blow-up of solutions to semilinear damped wave equations with power nonlinearity in compact Lie groups. J. Differ. Equ. 281, 85–104 (2021). https://doi.org/10.1016/j.jde.2021.02.002

    Article  MathSciNet  MATH  Google Scholar 

  26. Palmieri, A.: Semilinear wave equation on compact Lie groups. J. Pseudo-Differ. Oper. Appl. 12, 43 (2021). https://doi.org/10.1007/s11868-021-00414-x

    Article  MathSciNet  MATH  Google Scholar 

  27. Pizzichillo, F.: Linear and non-linear damped wave equations. Master thesis, University of Bari, (2014), 62pp

  28. Ruzhansky, M., Tokmagambetov, N.: Nonlinear damped wave equations for the sub-Laplacian on the Heisenberg group and for Rockland operators on graded Lie groups. J. Differ. Equ. 265(10), 5212–5236 (2018)

    Article  MathSciNet  Google Scholar 

  29. Ruzhansky, M., Turunen, V.: Pseudo-differential operators and symmetries. Background analysis and advanced topics, volume 2 of Pseudo-Differential Operators. Theory and Applications. Birkhäuser Verlag, Basel (2010)

  30. Ruzhansky, M., Yessirkegenov, N.: Hardy, Hardy-Sobolev, Hardy–Littlewood–Sobolev and Caffarelli–Kohn–Nirenberg inequalities on general Lie groups. arXiv:1810.08845v2 (2018)

  31. Schaeffer, J.: The equation \(u_{tt}-\Delta u = |u|^p\) for the critical value of \(p\). Proc. R. Soc. Edinburgh Sect. A 101(1–2), 31–44 (1985)

    Article  Google Scholar 

  32. Sideris, T.C.: Nonexistence of global solutions to semilinear wave equations in high dimensions. J. Differ. Equ. 52(3), 378–406 (1984)

    Article  MathSciNet  Google Scholar 

  33. Strichartz, R.: Restriction of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. Duke Math. J. 44(3), 705–714 (1977)

    Article  MathSciNet  Google Scholar 

  34. Todorova, G., Yordanov, B.: Critical exponent for a nonlinear wave equation with damping. J. Differ. Equ. 174(2), 464–489 (2001)

    Article  MathSciNet  Google Scholar 

  35. Velicu, A., Yessirkegenov, N.: Rellich, Gagliardo-Nirenberg, Trudinger and Caffarelli–Kohn–Nirenberg inequalities for Dunkl operators and applications. Isr. J. Math. (2021). https://doi.org/10.1007/s11856-021-2261-7

    Article  Google Scholar 

  36. Yordanov, B.T., Zhang, Q.S.: Finite time blow up for critical wave equations in high dimensions. J. Funct. Anal. 231(2), 361–374 (2006)

    Article  MathSciNet  Google Scholar 

  37. Zhang, Q.S.: A blow-up result for a nonlinear wave equation with damping: The critical case. C. R. Acad. Sci. Paris, Ser. I 333(2), 109–114 (2001)

    Article  MathSciNet  Google Scholar 

  38. Zhou, Y.: Blow up of solutions to semilinear wave equations with critical exponent in high dimensions. Chin. Ann. Math. Ser. B 28(2), 205–212 (2007)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

A. Palmieri is member of the Gruppo Nazionale per L’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Instituto Nazionale di Alta Matematica (INdAM). The author thanks Vladimir Georgiev (University of Pisa) for his helpful comments.

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  1. Alessandro Palmieri

    Present address: Department of Mathematics, University of Bari, Via E. Orabona 4, 70125, Bari, Italy

Authors and Affiliations

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

    Alessandro Palmieri

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  1. Alessandro Palmieri
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Communicated by Michael Ruzhansky.

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Palmieri, A. A Global Existence Result for a Semilinear Wave Equation with Lower Order Terms on Compact Lie Groups. J Fourier Anal Appl 28, 21 (2022). https://doi.org/10.1007/s00041-022-09915-9

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  • DOI: https://doi.org/10.1007/s00041-022-09915-9

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