We consider the initial-boundary value problems on $\mathbb{R}^{+}\times \mathbb{R}^{+}$ for one-dimension systems of quasilinear wave equations with null conditions. We first show that for homogeneous Dirichlet boundary values and sufficiently small initial data, classical solutions always globally exist. Then we prove that the global solution will scatter, i.e., it will converge to some solution of one dimensional homogeneous linear wave equations as time tends to infinity, in the energy sense. Finally we show the inverse scattering result: the scattering data can determine the global solution uniquely.
Dongbing ZHA
,
Yitong SUN
,
Tingqiang HOU
. GLOBAL EXISTENCE AND ASYMPTOTIC BEHAVIOR FOR INITIAL-BOUNDARY VALUE PROBLEMS OF ONE-DIMENSION QUASILINEAR WAVE EQUATIONS WITH NULL CONDITIONS[J]. Acta mathematica scientia, Series B, 2026
, 46(2)
: 568
-594
.
DOI: 10.1007/s10473-026-0204-4
[1] Alejo M A, Muñoz C. Almost sharp nonlinear scattering in one-dimensional Born-Infeld equations arising in nonlinear electrodynamics. Proc Amer Math Soc,2018, 146: 2225-2237
[2] Alinhac S. The null condition for quasilinear wave equations in two space dimensions I. Invent Math2001, 145: 597-618
[3] Born M, Infeld L. Foundations of the new field theory. Proceedings of the Royal Society of London, Series A, Mathematical and Physical Character, 1934, 144: 425-451
[4] Christodoulou D. Global solutions of nonlinear hyperbolic equations for small initial data. Comm Pure Appl Math, 1986, 39: 267-282
[5] Geba D A, Grillakis M G.An Introduction to the Theory of Wave Maps and Related Geometric Problems. Hackensack, NJ: World Scientific Publishing, 2017
[6] Gu C H. On the Cauchy problem for harmonic maps defined on two-dimensional Minkowski space. Comm Pure Appl Math, 1980, 33: 727-737
[7] Katayama S.Global Solutions and the Asymptotic Behavior for Nonlinear Wave Equations with Small Initial Data. Tokyo: Mathematical Society of Japan, 2017
[8] Keel M, Smith H F, Sogge C D. Global existence for a quasilinear wave equation outside of star-shaped domains. J Funct Anal, 2002, 189: 155-226
[9] Klainerman S.Long time behaviour of solutions to nonlinear wave equations//Proceedings of the International Congress of Mathematicians, Vol 1, 2 (Warsaw, 1983), PWN, Warsaw. 1984: 1209-1215
[10] Klainerman S.The null condition and global existence to nonlinear wave equations//Nonlinear systems of partial differential equations in applied mathematics, Part 1 (Santa Fe, NM, 1984), Lectures in Appl Math, vol~23. Providence, RI: Amer Math Soc, 1986: 293-326
[11] Li J, Yin H. Global smooth solutions of 3-D null-form wave equations in exterior domains with Neumann boundary conditions. J Differential Equations, 2018, 264: 5577-5628
[12] Li M. Asymptotic behavior of global solutions to one-dimension quasilinear wave equations. Dyn Partial Differ Equ, 2021, 18: 81-100
[13] Lindblad H. A remark on global existence for small initial data of the minimal surface equation in Minkowskian space time. Proc Amer Math Soc, 2004, 132: 1095-1102
[14] Lindblad H. Global solutions of quasilinear wave equations. Amer J Math, 2008, 130: 115-157
[15] Lindblad H, Rodnianski I. The global stability of Minkowski space-time in harmonic gauge. Ann of Math, 2010, 171(2): 1401-1477
[16] Luli G K, Yang S, Yu P. On one-dimension semi-linear wave equations with null conditions. Adv Math, 2018, 329: 174-188
[17] Metcalfe J, Nakamura M, Sogge C D. Global existence of solutions to multiple speed systems of quasilinear wave equations in exterior domains. Forum Math, 2005, 17: 133-168
[18] Metcalfe J, Sogge C D. Hyperbolic trapped rays and global existence of quasilinear wave equations. Invent Math, 2005, 159: 75-117
[19] Nakamura M. Remarks on a weighted energy estimate and its application to nonlinear wave equations in one space dimension. J Differential Equations, 2014, 256: 389-406
[20] Roždestvenskiǐ B L, Janenko N N. Systems of Quasilinear Equations and Their Applications to Gas Dynamics. Translated from the second Russian edition by J. R. Schulenberger. Providence, RI: American Mathematical Society, 1983
[21] Trespalacios J. Global existence and long-time behavior in the $1 + 1$-dimensional principal chiral model with applications to solitons. Ann Henri Poincaré, 2024, 25: 4671-4712
[22] Wang J, Wei C. A globally smooth solution to the relativistic string equation. J Geom Anal, 2023, 33: Paper 205
[23] Zha D. Global and almost global existence for general quasilinear wave equations in two space dimensions. J Math Pures Appl, 2019, 123(9): 270-299
[24] Zha D. On one-dimension quasilinear wave equations with null conditions. Calc Var Partial Differential Equations, 2020, 59: Paper 94
[25] Zha D.Global existence for initial-boundary value problems of one-dimension quasilinear wave equations with null conditions. arXiv: 2408.05898
[26] Zha D, Wang F. On initial-boundary value problems for one-dimension semilinear wave equations with null conditions. J Differential Equations, 2021, 275: 638-651
[27] Zha D, Xue X. On initial-boundary value problems for some one dimensional quasilinear wave equations: Global existence, scattering and rigidity. J Geom Anal, 2023, 33: Paper 302
