Abstract
In this paper, we show that one-dimension systems of quasilinear wave equations with null conditions admit global classical solutions for small initial data. This result extends Luli, Yang and Yu’s seminal work (Luli et al. in Adv Math 329:174–188, 2018) from the semilinear case to the quasilinear case. Furthermore, we also prove that the global solution is asymptotically free in the energy sense. In order to achieve these goals, we will employ Luli, Yang and Yu’s weighted energy estimates with positive weights, introduce some space-time weighted energy estimates and pay some special attentions to the highest order energies, then use some suitable bootstrap process to close the argument.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abbrescia, L., Wong, W.: Geometric analysis of 1+1 dimensional quasilinear wave equations, arXiv:1912.04692v1 (2019)
Alinhac, S.: The null condition for quasilinear wave equations in two space dimensions I. Invent. Math. 145, 597–618 (2001)
Christodoulou, D.: Global solutions of nonlinear hyperbolic equations for small initial data. Commun. Pure Appl. Math. 39, 267–282 (1986)
Dai, W., Kong, D.: Global existence and asymptotic behavior of classical solutions of quasilinear hyperbolic systems with linearly degenerate characteristic fields. J. Differ. Equ. 235, 127–165 (2007)
Gu, C.: On the Cauchy problem for harmonic maps defined on two-dimensional Minkowski space. Commun. Pure Appl. Math. 33, 727–737 (1980)
John, F.: Blow-up for quasilinear wave equations in three space dimensions. Commun. Pure Appl. Math. 34, 29–51 (1981)
Katayama, S.: Global Solutions and the Asymptotic Behavior for Nonlinear Wave Equations with Small Initial Data, MSJ Memoirs, vol. 36. Mathematical Society of Japan, Tokyo (2017)
Klainerman, S.: Global existence for nonlinear wave equations. Commun. Pure Appl. Math. 33, 43–101 (1980)
Klainerman, S.: Long time behaviour of solutions to nonlinear wave equations. In: Proceedings of the International Congress of Mathematicians, Vol. 1, 2, pp. 1209–1215. (Warsaw, 1983), PWN, Warsaw (1984)
Klainerman, S.: Uniform decay estimates and the Lorentz invariance of the classical wave equation. Commun. Pure Appl. Math. 38, 321–332 (1985)
Klainerman, S.: The null condition and global existence to nonlinear wave equations. In: Nonlinear Systems of Partial Differential Equations in Applied Mathematics, Part 1 (Santa Fe, N.M., 1984), Lectures in Application Mathematics, vol. 23, American Mathematical Society, Providence, RI, pp. 293–326 (1986)
Li, T., Yu, X., Zhou, Y.: Life-span of classical solutions to one-dimensional nonlinear wave equations. Chin. Ann. Math. Ser. B 13, 266–279 (1992)
Li, T., Zhou, Y., Kong, D.: Weak linear degeneracy and global classical solutions for general quasilinear hyperbolic systems. Commun. Partial Differ. Equ. 19, 1263–1317 (1994)
Lindblad, H.: A remark on global existence for small initial data of the minimal surface equation in Minkowskian space time. Proc. Am. Math. Soc. 132, 1095–1102 (2004)
Lindblad, H., Rodnianski, I.: The global stability of Minkowski space-time in harmonic gauge. Ann. Math. (2) 171, 1401–1477 (2010)
Liu, C., Liu, J.: Stability of traveling wave solutions to Cauchy problem of diagnolizable quasilinear hyperbolic systems. Discrete Contin. Dyn. Syst. 34, 4735–4749 (2014)
Liu, C., Qu, P.: Existence and stability of traveling wave solutions to first-order quasilinear hyperbolic systems. J. Math. Pures Appl. (9) 100, 34–68 (2013)
Luli, G., Yang, S., Yu, P.: On one-dimension semi-linear wave equations with null conditions. Adv. Math. 329, 174–188 (2018)
Nakamura, M.: Remarks on a weighted energy estimate and its application to nonlinear wave equations in one space dimension. J. Differ. Equ. 256, 389–406 (2014)
Wong, W.: Global existence for the minimal surface equation on \({\mathbb{R}}^{1,1}\). Proc. Am. Math. Soc. Ser. B 4, 47–52 (2017)
Zha, D.: Global and almost global existence for general quasilinear wave equations in two space dimensions. J. Math. Pures Appl. (9) 123, 270–299 (2019)
Zha, D.: Remarks on energy approach for global existence of some one-dimension quasilinear hyperbolic systems. J. Differ. Equ.tions 267, 6125–6132 (2019)
Zhou, Y.: Point-wise decay estimate for the global classical solutions to quasilinear hyperbolic systems. Math. Methods Appl. Sci. 32, 1669–1680 (2009)
Acknowledgements
The author would like to express his sincere gratitude to Prof. Arick Shao, Prof. Willie Wai-Yeung Wong and Prof. Shiwu Yang for their helpful comments on the topic in this paper. The author was supported by National Natural Science Foundation of China (No. 11801068) and the Fundamental Research Funds for the Central Universities.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by M. Struwe.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
