Global existence and blow-up of solutions for coupled bi-harmonic nonlinear wave equations

Radhia Ghanmi; Tarek Saanouni

doi:10.1515/anly-2022-1055

Article

Global existence and blow-up of solutions for coupled bi-harmonic nonlinear wave equations

Radhia Ghanmi and Tarek Saanouni

Published/Copyright: September 30, 2022

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

From the journal Analysis Volume 43 Issue 1

Abstract

This work studies the coupled nonlinear fourth-order wave system

u ¨ i + Δ 2 ⁢ u i + u i = ± ( ∑ 1 ≤ j ≤ m a i ⁢ j ⁢ | u j | p ) ⁢ | u i | p - 2 ⁢ u i .

The main goal is to develop a local theory in the energy space and to investigate some issues of the global theory. Indeed, using a standard contraction argument coupled with Strichartz estimates, one obtains a local solution in the inhomogeneous Sobolev space ( H 1 ) m for the energy sub-critical regime. Then the local solution extends to a global one in the attractive regime; also in the energy critical case there is a global solution with small data. For a repulsive source term, by using the potential well theory with a concavity argument, the local solution may concentrate in finite time or extend to a global one. Finally, in the inter-critical regime, one proves the existence of infinitely many non-global solutions with data near to the stationary solution. Here, one needs to deal with the coupled source term which gives some technical restrictions such as p ≥ 2 in order to avoid a singularity. This assumption in the inter-critical regime gives a restriction on the space dimension.

Keywords: Nonlinear wave equation; global well-posedness; blow-up

MSC 2010: 35L75

References

[1] H. Berestycki and T. Cazenave, Instabilité des états stationnaires dans les équations de Schrödinger et de Klein–Gordon non linéaires, C. R. Acad. Sci. Paris 293 (1981), no. 9, 489–492. Search in Google Scholar

[2] F. P. Bretherton, Resonant interactions between waves. The case of discrete oscillations, J. Fluid Mech. 20 (1964), 457–479. 10.1017/S0022112064001355Search in Google Scholar

[3] J. Ferreira and G. P. Menzala, Decay of solutions of a system of nonlinear Klein–Gordon equations, Int. J. Math. Math. Sci. 9 (1986), no. 3, 471–483. 10.1155/S0161171286000601Search in Google Scholar

[4] R. Ghanmi, Asymptotics for a class of coupled fourth-order Schrödinger equations, Mediterr. J. Math. 15 (2018), no. 3, Paper No. 109. 10.1007/s00009-018-1153-5Search in Google Scholar

[5] E. Hebey and B. Pausader, An introduction to fourth-order nonlinear wave equations, 2008. Search in Google Scholar

[6] S. Ibrahim, N. Masmoudi and K. Nakanishi, Scattering threshold for the focusing nonlinear Klein–Gordon equation, Anal. PDE 4 (2011), no. 3, 405–460. 10.2140/apde.2011.4.405Search in Google Scholar

[7] S. Levandosky, Stability and instability of fourth-order solitary waves, J. Dynam. Differential Equations 10 (1998), no. 1, 151–188. 10.1023/A:1022644629950Search in Google Scholar

[8] S. P. Levandosky, Decay estimates for fourth order wave equations, J. Differential Equations 143 (1998), no. 2, 360–413. 10.1006/jdeq.1997.3369Search in Google Scholar

[9] S. P. Levandosky and W. A. Strauss, Time decay for the nonlinear beam equation, Methods Appl. Anal. 7 (2000), 479–487. 10.4310/MAA.2000.v7.n3.a5Search in Google Scholar

[10] M.-R. Li and L.-Y. Tsai, Existence and nonexistence of global solutions of some system of semilinear wave equations, Nonlinear Anal. 54 (2003), no. 8, 1397–1415. 10.1016/S0362-546X(03)00192-5Search in Google Scholar

[11] A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity, Dover, New York, 1944. Search in Google Scholar

[12] L. A. Medeiros and M. M. Miranda, Weak solutions for a system of nonlinear Klein–Gordon equations, Ann. Mat. Pura Appl. (4) 146 (1987), 173–183. 10.1007/BF01762364Search in Google Scholar

[13] L. A. Medeiros and G. Perla Menzala, On a mixed problem for a class of nonlinear Klein–Gordon equations, Acta Math. Hungar. 52 (1988), no. 1–2, 61–69. 10.1007/BF01952481Search in Google Scholar

[14] M. M. Miranda and L. A. Medeiros, On the existence of global solutions of a coupled nonlinear Klein–Gordon equations, Funkcial. Ekvac. 30 (1987), no. 1, 147–161. Search in Google Scholar

[15] L. Nirenberg, Remarks on strongly elliptic partial differential equations, Comm. Pure Appl. Math. 8 (1955), 649–675. 10.1002/cpa.3160080414Search in Google Scholar

[16] B. Pausader, Scattering and the Levandosky–Strauss conjecture for fourth-order nonlinear wave equations, J. Differential Equations 241 (2007), no. 2, 237–278. 10.1016/j.jde.2007.06.001Search in Google Scholar

[17] L. A. Peletier and W. C. Troy, Spatial Patterns Higher Order Models in Physics and Mechanics, Progr. Nonlinear Differential Equations Appl. 45, Birkhäuser, Boston, 2001. Search in Google Scholar

[18] I. E. Segal, Nonlinear partial differential equations in quantum field theory, Proc. Sympos. Appl. Math. 17 (1965), 210–226. 10.1090/psapm/017/0202406Search in Google Scholar

[19] J. Shatah, Stable standing waves of nonlinear Klein–Gordon equations, Comm. Math. Phys. 91 (1983), no. 3, 313–327. 10.1007/BF01208779Search in Google Scholar

[20] J. Shatah and W. Strauss, Instability of nonlinear bound states, Comm. Math. Phys. 100 (1985), no. 2, 173–190. 10.1007/BF01212446Search in Google Scholar

[21] Y. Wang, Non-existence of global solutions of a class of coupled non-linear Klein–Gordon equations with non-negative potentials and arbitrary initial energy, IMA J. Appl. Math. 74 (2009), no. 3, 392–415. 10.1093/imamat/hxp004Search in Google Scholar

[22] W. Xiao and Y. Ping, Global solutions and finite time blow up for some system of nonlinear wave equations, Appl. Math. Comput. 219 (2012), no. 8, 3754–3768. 10.1016/j.amc.2012.10.005Search in Google Scholar

Received: 2022-03-26

Accepted: 2022-07-04

Published Online: 2022-09-30

Published in Print: 2023-02-01

You are currently not able to access this content.

Articles in the same Issue

https://doi.org/10.1515/anly-2022-1055

Keywords for this article

Nonlinear wave equation; global well-posedness; blow-up