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Decay in the one dimensional generalized Improved Boussinesq equation

  • Christopher Maulén
  • Claudio MuñozEmail author
Original Paper
Part of the following topical collections:
  1. Theory of PDEs

Abstract

We consider the decay problem for the generalized improved (or regularized) Boussinesq model with power type nonlinearity, a modification of the originally ill-posed shallow water waves model derived by Boussinesq. This equation has been extensively studied in the literature, describing plenty of interesting behavior, such as global existence in the space \(H^1\times H^2\), existence of super luminal solitons, and lack of a standard stability method to describe perturbations of solitons. The associated decay problem has been studied by Liu, and more recently by Cho–Ozawa, showing scattering in weighted spaces provided the power of the nonlinearity p is sufficiently large. In this paper we remove that condition on the power p and prove decay to zero in terms of the energy space norm \(L^2\times H^1\), for any \(p>1\), in two almost complementary regimes: (1) outside the light cone for all small, bounded in time \(H^1\times H^2\) solutions, and (2) decay on compact sets of arbitrarily large bounded in time \(H^1\times H^2\) solutions. The proof consists in finding two new virial type estimates, one for the exterior cone problem based in the energy of the solution, and a more subtle virial identity for the interior cone problem, based in a modification of the momentum.

Mathematics Subject Classification

35Q35 35Q53 

Notes

Acknowledgements

We thank C. Kwak and M. A. Alejo for several important remarks that helped to improve a first draft of this paper.

References

  1. 1.
    Alejo, M.A., Cortez, F., Kwak, C., Muñoz, C.: On the dynamics of zero-speed solutions for Camassa–Holm type equations (2019). arXiv:1810.09594 (preprint; to appear in IMRN)
  2. 2.
    Alejo, M.A., Muñoz, C.: Almost sharp nonlinear scattering in one-dimensional Born-Infeld equations arising in nonlinear Electrodynamics. Proc. AMS 146(5), 2225–2237 (2018)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bona, J., Souganidis, P., Strauss, W.: Stability and instability of solitary waves of Korteweg–de Vries type. Proc. R. Soc. Lond. A 411, 395–412 (1987)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Boussinesq, J.: Théorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond. J. Math. Pure Appl. 17, 55–108 (1872)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Cerpa, E., Crépeau, E.: On the control of the improved Boussinesq equation. SIAM J. Control Optim. 56(4), 3035–3049 (2018)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chree, C.: Longitudinal vibrations of a Corcablar Bar. Q. J. Pure Appl. Math. 21, 287–298 (1886)zbMATHGoogle Scholar
  7. 7.
    Grillakis, M., Shatah, J., Strauss, W.: Stability theory of solitary waves in the presence of symmetry. I. J. Funct. Anal. 74(1), 160–197 (1987)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Linares, F.: Notes on Boussinesq Equation, p. 71 (2005). http://preprint.impa.br/FullText/Linares__Fri_Dec_23_09_48_59_BRDT_2005/beq.pdf
  9. 9.
    Kishimoto, N.: Sharp local well-posedness for the “good” Boussinesq equation. J. Differ. Equations 254, 2393–2433 (2013)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Wang, S., Chen, G.: Small amplitude solutions of the generalized IMBq equation. J. Math. Anal. Appl. 274, 846–866 (2002)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Cho, Y., Ozawa, T.: Remarks on modified improved Boussinesq equations in one space dimension, proceedings: mathematical. Phys. Eng. Sci. 462(2071), 1949–1963 (2006)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kwak, C., Muñoz, C.: Extended decay properties for generalized BBM equations Fields Institute Comm. (2018) (preprint) Google Scholar
  13. 13.
    Kwak, C., Muñoz, C.: Asymptotic dynamics for the small data weakly dispersive one-dimensional hamiltonian ABCD system. arXiv:1902.00454 (preprint; to appear in T. of the AMS)
  14. 14.
    Kwak, C., Muñoz, C., Poblete, F., Pozo, J.C.: The scattering problem for the Hamiltonian abcd Boussinesq system in the energy space. J. Math. Pures Appl. (9) 127, 121–159 (2019)MathSciNetCrossRefGoogle Scholar
  15. 15.
    El Dika, K.: Smoothing effect of the generalized BBM equation for locelized solutions moving to the right. Discrete Contin. Dyn. Syst. 12(5), 973–982 (2005)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kowalczyk, M., Martel, Y., Muñoz, C.: Kink dynamics in the \(\phi ^4\) model: asymptotic stability for odd perturbations in the energy space. J. AMS 30, 769–798 (2017)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Kowalczyk, M., Martel, Y., Muñoz, C.: Nonexistence of small, odd breathers for a class of nonlinear wave equations. Lett. Math. Phys. 107(5), 921–931 (2017)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Kowalczyk, M., Martel, Y., Muñoz, C.: Soliton dynamics for the 1D NLKG equation with symmetry and in the absence of internal modes (2019). arXiv:1903.12460 (preprint)
  19. 19.
    Liu, Y.: Existence and blow up of solutions of a nonlinear Pochhammer–Chree equation. Indiana Univ. Math. J. 45(3), 797–816 (1996) (Fall) Google Scholar
  20. 20.
    Martel, Y., Merle, F.: Asymptotic stability of solitons for subcritical generalized KdV equations. Arch. Ration. Mech. Anal. 157(3), 219–254 (2001)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Martel, Y., Merle, F.: Asymptotic stability of solitons for subcritical gKdV equations revisited. Nonlinearity 18(1), 55–80 (2005)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Mizumachi, T.: Stability of line solitons for the KP-II equation in \(\mathbb{R}^2\). Mem. Am. Math. Soc. 238(1125), vii+95 (2015)zbMATHGoogle Scholar
  23. 23.
    Mizumachi, T.: Stability of line solitons for the KP-II equation in \(\mathbb{R}^2\). II. Proc. R. Soc. Edinburgh Sect. A 148(1), 149–198 (2018)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Muñoz, C., Poblete, F., Pozo, J.C.: Scattering in the energy space for Boussinesq equations. Commun. Math. Phys. 361(1), 127–141 (2018)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Pochhammer, L.: Ueber die Fortpflanzungsgeschwindigkeiten kleiner Schwingungen in einem unbegrenzten isotropen Kreiscylinder. J. für die Reine Angewandte Math. 81, 324–336 (1876)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Pego, R., Weinstein, M.: Eigenvalues, and instabilities of solitary waves. Philos. Trans. R. Soc. Lond. Ser. A 340(1656), 47–94 (1992)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Pego, R., Weinstein, M.: Convective linear stability of solitary waves for Boussinesq equations. Stud. Appl. Math. 99, 311–375 (1997)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Smereka, P.: A Remark on the Solitary Wave Stability for a Boussinesq Equation. Nonlinear Dispersive Wave Systems (Orlando, FL, 1991), pp. 255–263. World Scientific Publishing, River Edge (1992)Google Scholar
  29. 29.
    Whitham, G.B.: Linear and Nonlinear Waves, Pure and Applied Mathematics, p. 636. Wiley, New York (1974)Google Scholar

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Authors and Affiliations

  1. 1.Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático (UMI 2807 CNRS)Universidad de ChileSantiagoChile
  2. 2.CNRS and Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático (UMI 2807 CNRS)Universidad de ChileSantiagoChile

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