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September  2022, 21(9): 3117-3139. doi: 10.3934/cpaa.2022092

## Systems of semilinear wave equations with multiple speeds in two space dimensions and a weaker null condition

 Department of Mathematics, Graduate School of Science, Osaka University, 1-1 Machikaneyama-cho, Toyonaka, Osaka 560-0043, Japan

* Corresponding author

Received  February 2022 Revised  April 2022 Published  September 2022 Early access  May 2022

Fund Project: The work of the second author is partially supported by JSPS KAKENHI Grant No. 18H01128

We consider the Cauchy problem for systems of semilinear wave equations with multiple propagation speeds in two space dimensions. The nonlinear terms are assumed to be cubic and to depend only on first derivatives of the unknowns. The null condition is one of the sufficient condition for the small data global existence for the single-speed case, and it is extended to the multiple-speed case by many authors. Recently, various weaker sufficient conditions for the small data global existence were obtained for the single-speed case. In this paper, we introduce the multiple-speed version of one of such weaker null conditions, and we show the small data global existence. The asymptotic behavior of global solutions is also investigated through associated systems of ODEs called the reduced systems.

Citation: Minggang Cheng, Soichiro Katayama. Systems of semilinear wave equations with multiple speeds in two space dimensions and a weaker null condition. Communications on Pure and Applied Analysis, 2022, 21 (9) : 3117-3139. doi: 10.3934/cpaa.2022092
##### References:
 [1] R. Agemi and K. Yokoyama, The null condition and global existence of solutions to systems of wave equations with different speeds, in Advances in Nonlinear Partial Differential Equations and Stochastics (eds. S. Kawahara and T. Yanagisawa), Ser. on Adv. Math. for Appl. Sci., Vol. 48, World Scientific, (1998), 43–86. [2] S. Alinhac, The null condition for quasilinear wave equations in two space dimensions I, Invent. Math., 145 (2001), 597-618.  doi: 10.1007/s002220100165. [3] S. Alinhac, Semilinear hyperbolic systems with blowup at infinity, Indiana Univ. Math. J., 55 (2006), 1209-1232.  doi: 10.1512/iumj.2006.55.2671. [4] D. Christodoulou, Global solutions of nonlinear hyperbolic equations for small initial data, Comm. Pure Appl. Math., 39 (1986), 267-282.  doi: 10.1002/cpa.3160390205. [5] P. Godin, Lifespan of solutions of semilinear wave equations in two space dimensions, Comm. Partial Differ. Equ., 18 (1993), 895-916.  doi: 10.1080/03605309308820955. [6] A. Hoshiga, The initial value problems for quasi-linear wave equations in two space dimensions with small data, Adv. Math. Sci. Appl., 5 (1995), 67-89. [7] A. Hoshiga, The existence of global solutions to systems of quasilinear wave equations with quadratic nonlinearities in $2$-dimensional space, Funkcial. Ekvac., 49 (2006), 357-384.  doi: 10.1619/fesi.49.357. [8] A. Hoshiga, The existence of the global solutions to semilinear wave equations with a class of cubic nonlinearities in $2$-dimensional space, Hokkaido Math. J., 37 (2008), 669-688.  doi: 10.14492/hokmj/1249046363. [9] A. Hoshiga and H. Kubo, Global small amplitude solutions of nonlinear hyperbolic systems with a critical exponent under the null condition, SIAM J. Math. Anal., 31 (2000), 486-513.  doi: 10.1137/S0036141097326064. [10] S. Katayama, Global existence and asymptotic behavior of solutions to systems of semilinear wave equations in two space dimensions, Hokkaido J. Math., 37 (2008), 689-714.  doi: 10.14492/hokmj/1249046364. [11] S. Katayama, Asymptotic pointwise behavior for systems of semilinear wave equations in three space dimensions, J. Hyperbolic Differ. Equ., 9 (2012), 263-323. [12] S. Katayama, Global Solutions and the Asymptotic Behavior for Nonlinear Wave Equations with Small Initial Data, MSJ Memoirs, Math. Soc. Japan, Tokyo, 2017. [13] S. Katayama, Remarks on the asymptotic behavior of global solutions to systems of semilinear wave equations, in Asymptotic Analysis for Nonlinear Dispersive and Wave Equations, Math. Soc. Japan, Tokyo, (2019), 55–84. doi: 10.2969/aspm/08110055. [14] S. Katayama, T. Matoba and H. Sunagawa, Semilinear hyperbolic systems violating the null condition, Math. Ann., 31 (2015), 275-312.  doi: 10.1007/s00208-014-1071-1. [15] S. Katayama, A. Matsumura and H. Sunagawa, Energy decay for systems of semilinear wave equations with dissipative structure in two space dimensions, NoDEA Nonlinear Differ. Equ. Appl., 22 (2015), 601-628.  doi: 10.1007/s00030-014-0297-7. [16] S. Katayama, D. Murotani and H. Sunagawa, The energy decay and asymptotics for a class of semilinear wave equations in two space dimensions, J. Evol. Equ., 12 (2012), 891-916.  doi: 10.1007/s00028-012-0160-4. [17] S. Klainerman, The null condition and global existence to nonlinear wave equations, in Nonlinear Systems of Partial Differential Equations in Applied Mathematics, Part 1, Lectures in Appl. Math., AMS, Providence, (1986), 293-326. [18] S. Klainerman, Remarks on the global Sobolev inequalities in the Minkowski space ${\mathbb R}^{n+1}$, Comm. Pure Appl. Math., 40 (1987), 111-117.  doi: 10.1002/cpa.3160400105. [19] M. Kovalyov, Resonance-type behaviour in a system of nonlinear wave equations, J. Differ. Equ., 77 (1989), 73-83.  doi: 10.1016/0022-0396(89)90157-5. [20] H. Kubo, Asymptotic behavior of solutions to semilinear wave equations with dissipative structure, Discrete Contin. Dynam. Systems, Supplement Volume, 2007 (2007), 602–613. [21] H. Kubo, Modification of the vector-field method related to quadratically perturbed wave equations in two space dimensions, in Asymptotic Analysis for Nonlinear Dispersive and Wave Equations, Math. Soc. Japan, Tokyo, (2019), 139–172 doi: 10.2969/aspm/08110139. [22] H. Lindblad and I. Rodnianski, The weak null condition for Einstein's equations, C. R. Math. Acad. Sci. Paris, 336 (2003), 901-906.  doi: 10.1016/S1631-073X(03)00231-0. [23] Y. Nishii and H. Sunagawa, Agemi type structural condition for systems of semilinear wave equations, J. Hyperbolic Differ. Equ., 17 (2020), 459-473.  doi: 10.1142/S0219891620500125. [24] Y. Nishii, H. Sunagawa and H. Terashita, Energy decay for small solutions to semilinear wave equations with weakly dissipative structure, J. Math. Soc. Japan, 73 (2021), 767–779. . doi: 10.2969/jmsj/84148414. [25] T. C. Sideris and S. Tu, Global existence for systems of nonlinear wave equations in $3$D with multiple speeds, SIAM J. Math. Anal., 33 (2001), 477-488.  doi: 10.1137/S0036141000378966. [26] K. Yokoyama, Global existence of classical solutions to systems of wave equations with critical nonlinearity in three space dimensions, J. Math. Soc. Japan, 52 (2000), 609-632.  doi: 10.2969/jmsj/05230609.

show all references

##### References:
 [1] R. Agemi and K. Yokoyama, The null condition and global existence of solutions to systems of wave equations with different speeds, in Advances in Nonlinear Partial Differential Equations and Stochastics (eds. S. Kawahara and T. Yanagisawa), Ser. on Adv. Math. for Appl. Sci., Vol. 48, World Scientific, (1998), 43–86. [2] S. Alinhac, The null condition for quasilinear wave equations in two space dimensions I, Invent. Math., 145 (2001), 597-618.  doi: 10.1007/s002220100165. [3] S. Alinhac, Semilinear hyperbolic systems with blowup at infinity, Indiana Univ. Math. J., 55 (2006), 1209-1232.  doi: 10.1512/iumj.2006.55.2671. [4] D. Christodoulou, Global solutions of nonlinear hyperbolic equations for small initial data, Comm. Pure Appl. Math., 39 (1986), 267-282.  doi: 10.1002/cpa.3160390205. [5] P. Godin, Lifespan of solutions of semilinear wave equations in two space dimensions, Comm. Partial Differ. Equ., 18 (1993), 895-916.  doi: 10.1080/03605309308820955. [6] A. Hoshiga, The initial value problems for quasi-linear wave equations in two space dimensions with small data, Adv. Math. Sci. Appl., 5 (1995), 67-89. [7] A. Hoshiga, The existence of global solutions to systems of quasilinear wave equations with quadratic nonlinearities in $2$-dimensional space, Funkcial. Ekvac., 49 (2006), 357-384.  doi: 10.1619/fesi.49.357. [8] A. Hoshiga, The existence of the global solutions to semilinear wave equations with a class of cubic nonlinearities in $2$-dimensional space, Hokkaido Math. J., 37 (2008), 669-688.  doi: 10.14492/hokmj/1249046363. [9] A. Hoshiga and H. Kubo, Global small amplitude solutions of nonlinear hyperbolic systems with a critical exponent under the null condition, SIAM J. Math. Anal., 31 (2000), 486-513.  doi: 10.1137/S0036141097326064. [10] S. Katayama, Global existence and asymptotic behavior of solutions to systems of semilinear wave equations in two space dimensions, Hokkaido J. Math., 37 (2008), 689-714.  doi: 10.14492/hokmj/1249046364. [11] S. Katayama, Asymptotic pointwise behavior for systems of semilinear wave equations in three space dimensions, J. Hyperbolic Differ. Equ., 9 (2012), 263-323. [12] S. Katayama, Global Solutions and the Asymptotic Behavior for Nonlinear Wave Equations with Small Initial Data, MSJ Memoirs, Math. Soc. Japan, Tokyo, 2017. [13] S. Katayama, Remarks on the asymptotic behavior of global solutions to systems of semilinear wave equations, in Asymptotic Analysis for Nonlinear Dispersive and Wave Equations, Math. Soc. Japan, Tokyo, (2019), 55–84. doi: 10.2969/aspm/08110055. [14] S. Katayama, T. Matoba and H. Sunagawa, Semilinear hyperbolic systems violating the null condition, Math. Ann., 31 (2015), 275-312.  doi: 10.1007/s00208-014-1071-1. [15] S. Katayama, A. Matsumura and H. Sunagawa, Energy decay for systems of semilinear wave equations with dissipative structure in two space dimensions, NoDEA Nonlinear Differ. Equ. Appl., 22 (2015), 601-628.  doi: 10.1007/s00030-014-0297-7. [16] S. Katayama, D. Murotani and H. Sunagawa, The energy decay and asymptotics for a class of semilinear wave equations in two space dimensions, J. Evol. Equ., 12 (2012), 891-916.  doi: 10.1007/s00028-012-0160-4. [17] S. Klainerman, The null condition and global existence to nonlinear wave equations, in Nonlinear Systems of Partial Differential Equations in Applied Mathematics, Part 1, Lectures in Appl. Math., AMS, Providence, (1986), 293-326. [18] S. Klainerman, Remarks on the global Sobolev inequalities in the Minkowski space ${\mathbb R}^{n+1}$, Comm. Pure Appl. Math., 40 (1987), 111-117.  doi: 10.1002/cpa.3160400105. [19] M. Kovalyov, Resonance-type behaviour in a system of nonlinear wave equations, J. Differ. Equ., 77 (1989), 73-83.  doi: 10.1016/0022-0396(89)90157-5. [20] H. Kubo, Asymptotic behavior of solutions to semilinear wave equations with dissipative structure, Discrete Contin. Dynam. Systems, Supplement Volume, 2007 (2007), 602–613. [21] H. Kubo, Modification of the vector-field method related to quadratically perturbed wave equations in two space dimensions, in Asymptotic Analysis for Nonlinear Dispersive and Wave Equations, Math. Soc. Japan, Tokyo, (2019), 139–172 doi: 10.2969/aspm/08110139. [22] H. Lindblad and I. Rodnianski, The weak null condition for Einstein's equations, C. R. Math. Acad. Sci. Paris, 336 (2003), 901-906.  doi: 10.1016/S1631-073X(03)00231-0. [23] Y. Nishii and H. Sunagawa, Agemi type structural condition for systems of semilinear wave equations, J. Hyperbolic Differ. Equ., 17 (2020), 459-473.  doi: 10.1142/S0219891620500125. [24] Y. Nishii, H. Sunagawa and H. Terashita, Energy decay for small solutions to semilinear wave equations with weakly dissipative structure, J. Math. Soc. Japan, 73 (2021), 767–779. . doi: 10.2969/jmsj/84148414. [25] T. C. Sideris and S. Tu, Global existence for systems of nonlinear wave equations in $3$D with multiple speeds, SIAM J. Math. Anal., 33 (2001), 477-488.  doi: 10.1137/S0036141000378966. [26] K. Yokoyama, Global existence of classical solutions to systems of wave equations with critical nonlinearity in three space dimensions, J. Math. Soc. Japan, 52 (2000), 609-632.  doi: 10.2969/jmsj/05230609.
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