Abstract
We prove the exponential decay of local energy for the Klein–Gordon equation with localized critical nonlinearity. The proof relies on generalized Strichartz estimates, and semi-group of Lax–Phillips.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aloui, L., Khenissi, M.: Stabilisation de l’équation des ondes dans un domaine éxterieur. Rev. Mat. Iberoam. 28, 1–16 (2002)
Bchatnia, A.: Exponential decay of the local energy for the solutions of critical wave equations outside a convex obstacle. Indag. Math. 23, 184–198 (2012)
Bchatnia, A.: Scattering for the critical and localised semilinear wave equation. Nonlinear Anal. 74(6), 2235–2242 (2011)
Bahouri, H., Gérard, P.: High frequency approximation of solutions to critical nonlineair wave equations. Am. J. Math. 121, 131–175 (1999)
Dehman, B., Gérard, P.: Stabilization for the nonlinear Klein-Gordon Equation with Critical exponent. Prépublication de l’Université de PARIS-SUD, Orsay, 2002–35
Folland, G.-B.: Introduction to Partial Differential Equations, 2nd edn. Prentice-Hall, New Delhi (2003)
Gallagher, I., Gérard, P.: Profile decomposition for the wave equation outside a convex obstacle. J. Math. Pures Appl. 80(1), 1–49 (2001)
Gradshteyn, I.-S., Ryzhik, I.-M.: Table of Integrals, Series and Products, 6th edn. Academic Press, San Diego (2000)
Grillakis, M.: Regularity and asymptotic behaviour of the wave equation with a critical nonlinearity. Ann. Math. 132, 485–509 (1990)
Grillakis, M.: Regularity for the wave equation with critical nonlinearity. Commun. Pure Appl. Math. 45, 749–774 (1992)
Lin, J.-E.: Superexponential decay of solutions of a semilinear wave equation. J. Funct. Anal. 31, 321–332 (1979)
Lax, P.-D., Phillips, R.-S.: Scattering Theory (Pure and Applied Mathematics 26). Academics Press, New York (1967)
Malloug, M.: Local energy decay for the damped Klein–Gordon equation in exterior domain. Appl. Anal. 96, 349–362 (2015)
Morawetz, C.: Time decay for the nonlinear Klein–Gordon equation. Proc. R. Soc. Lond. 306, 291–296 (1968)
Nunes, R.-S.-O., Bastos, W.-D.: Energy decay for the linear Klein–Gordon equation and boundary control. J. Math. Anal. Appl. 414, 934–944 (2014)
Polyanin, A.-D.: Handbook of Linear Partial Differential Equations for Engineers and Scientists. Chapman and Hall/CRC, Florida (2002)
Strauss, W.: Decay and asymptotics for \(\square u=F(u)\). J. Funct. Anal. 2, 409–457 (1968)
Shatah, J., Stuwe, M.: Regularity results for the nonlinear equations. Ann. Math. 138, 503–518 (1993)
Shatah, J., Stuwe, M.: Well-posedness in the energy space for semilinear wave equations with critical growth. Int. Math. Res. Not. 1994(7), 303–309 (1994)
Zhang, J., Zheng, J.: Strichartz estimate and nonlinear Klein–Gordon equation on nontrapping scattering space. J. Geom. Anal. 29(3), 2957–2984 (2019)
Zuily, C.: Solutions en grand temps d’équations d’ondes non linéaires. Astérisque 227 (1995), Séminaire Bourbaki, exp. no 779, pp. 107–144
Acknowledgements
The authors are very grateful to the anonymous referee for his/her helpful comments and suggestions that improved the manuscript.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Abdelaziz Rhandi.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Bchatnia, A., Mehenaoui, N. Decay of the local energy for the solutions of the critical Klein–Gordon equation. Semigroup Forum 100, 698–716 (2020). https://doi.org/10.1007/s00233-019-10075-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00233-019-10075-4