# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2022058
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## Global existence and blow up for systems of nonlinear wave equations related to the weak null condition

 1 Department of Mathematics, Faculty of Education, Mie University, 1577 Kurima-machiya-cho Tsu, Mie Prefecture 514-8507, Japan 2 Hokkaido University of Science, 7-Jo 15-4-1 Maeda, Teine, Sapporo, Hokkaido 006-8585, Japan

*Corresponding author: Kunio Hidano

Received  March 2021 Revised  October 2021 Early access April 2022

We discuss how the higher-order term
 $|u|^q$
 $(q>1+2/(n-1))$
has nontrivial effects in the lifespan of small solutions to the Cauchy problem for the system of nonlinear wave equations
 $\partial_t^2 u-\Delta u = |v|^p, \qquad \partial_t^2 v-\Delta v = |\partial_t u|^{(n+1)/(n-1)} +|u|^q$
in
 $n\,(\geq 2)$
space dimensions. We show the existence of a certain "critical curve" in the
 $pq$
-plane such that for any
 $(p,q)$
 $(p,q>1)$
lying below the curve, nonexistence of global solutions occurs, whereas for any
 $(p,q)$
 $(p>1+3/(n-1),\,q>1+2/(n-1))$
lying exactly on it, this system admits a unique global solution for small data. When
 $n = 3$
, the discussion for the above system with
 $(p,q) = (3,3)$
, which lies on the critical curve, has relevance to the study on systems satisfying the weak null condition, and we obtain a new result of global existence for such systems. Moreover, in the particular case of
 $n = 2$
and
 $p = 4$
it is observed that no matter how large
 $q$
is, the higher-order term
 $|u|^q$
never becomes negligible and it essentially affects the lifespan of small solutions.
Citation: Kunio Hidano, Kazuyoshi Yokoyama. Global existence and blow up for systems of nonlinear wave equations related to the weak null condition. Discrete and Continuous Dynamical Systems, doi: 10.3934/dcds.2022058
##### References:
 [1] R. Agemi, Y. Kurokawa and H. Takamura, Critical curve for p-q systems of nonlinear wave equations in three space dimensions, J. Differential Equations, 167 (2000), 87-133.  doi: 10.1006/jdeq.2000.3766. [2] S. Alinhac, Semilinear hyperbolic systems with blowup at infinity, Indiana Univ. Math. J., 55 (2006), 1209-1232.  doi: 10.1512/iumj.2006.55.2671. [3] S. Alinhac, Geometric Analysis of Hyperbolic Differential Equations: An Introduction, London Mathematical Society Lecture Note Series, 374, Cambridge University Press, Cambridge, 2010. doi: 10.1017/CBO9781139107198. [4] W. Dai, D. Fang and C. Wang, Global existence and lifespan for semilinear wave equations with mixed nonlinear terms, J. Differential Equations, 267 (2019), 3328-3354.  doi: 10.1016/j.jde.2019.04.007. [5] W. Dai, D. Fang and C. Wang, Lifespan of solutions to the Strauss type wave system on asymptotically flat space-times, Discrete Contin. Dyn. Syst., 40 (2020), 4985-4999.  doi: 10.3934/dcds.2020208. [6] D. Del Santo, V. Georgiev and E. Mitidieri, Global existence of the solutions and formation of singularities for a class of hyperbolic systems, Geometrical Optics and Related Topics, 1996 (Cortona), Progr. Nonlinear Differential Equations Appl., Birkhäuser Boston, Boston, MA, 32 (1997), 117–140. [7] K. Deng, Blow-up of solutions of some nonlinear hyperbolic systems, Rocky Mountain J. Math., 29 (1999), 807-820.  doi: 10.1216/rmjm/1181071610. [8] D. Fang and C. Wang, Weighted Strichartz estimates with angular regularity and their applications, Forum Math., 23 (2011), 181-205.  doi: 10.1515/FORM.2011.009. [9] W. Han and Y. Zhou, Blow up for some semilinear wave equations in multi-space dimensions, Comm. Partial Differential Equations, 39 (2014), 651-665.  doi: 10.1080/03605302.2013.863916. [10] K. Hidano, C. Wang and K. Yokoyama, The Glassey conjecture with radially symmetric data, J. Math. Pures Appl. $(9)$, 98 (2012), 518-541.  doi: 10.1016/j.matpur.2012.01.007. [11] K. Hidano, C. Wang and K. Yokoyama, Combined effects of two nonlinearities in lifespan of small solutions to semi-linear wave equations, Math. Ann., 366 (2016), 667-694.  doi: 10.1007/s00208-015-1346-1. [12] K. Hidano and K. Yokoyama, Life span of small solutions to a system of wave equations, Nonlinear Anal., 139 (2016), 106-130.  doi: 10.1016/j.na.2016.02.020. [13] K. Hidano and K. Yokoyama, Global existence for a system of quasi-linear wave equations in $3$D satisfying the weak null condition, Int. Math. Res. Not. IMRN, 2020 (2020), 39-70.  doi: 10.1093/imrn/rny024. [14] K. Hidano and D. Zha, Remarks on a system of quasi-linear wave equations in $3$D satisfying the weak null condition, Commun. Pure Appl. Anal., 18 (2019), 1735-1767.  doi: 10.3934/cpaa.2019082. [15] L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations, Mathématiques & Applications, 26, Springer-Verlag, Berlin, 1997. [16] T. Hoshiro, On weighted $L^2$ estimates of solutions to wave equations, J. Anal. Math., 72 (1997), 127-140.  doi: 10.1007/BF02843156. [17] M. Ikeda, M. Sobajima and K. Wakasa, Blow-up phenomena of semilinear wave equations and their weakly coupled systems, J. Differerential Equations, 267 (2019), 5165-5201.  doi: 10.1016/j.jde.2019.05.029. [18] F. John, Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math., 28 (1979), 235-268.  doi: 10.1007/BF01647974. [19] F. John, Blow-up for quasilinear wave equations in three space dimensions, Comm. Pure Appl. Math., 34 (1981), 29-51.  doi: 10.1002/cpa.3160340103. [20] F. John and S. Klainerman, Almost global existence to nonlinear wave equations in three space dimensions, Comm. Pure Appl. Math., 37 (1984), 443-455.  doi: 10.1002/cpa.3160370403. [21] S. Katayama, Lifespan of solutions for two space dimensional wave equations with cubic nonlinearity, Comm. Partial Differential Equations, 26 (2001), 205-232.  doi: 10.1081/PDE-100001753. [22] S. Katayama, Asymptotic pointwise behavior for systems of semilinear wave equations in three space dimensions, J. Hyperbolic Differ. Equ., 9 (2012), 263-323.  doi: 10.1142/S0219891612500099. [23] S. Katayama, T. Matoba and H. Sunagawa, Semilinear hyperbolic systems violating the null condition, Math. Ann., 361 (2015), 275-312.  doi: 10.1007/s00208-014-1071-1. [24] S. Klainerman, The null condition and global existence to nonlinear wave equations, Nonlinear Systems of Partial Differential Equations in Applied Mathematics, Part 1, 1984 (Santa Fe, N.M.), Lectures in Appl. Math., Amer. Math. Soc., Providence, RI, 23 (1986), 293–326. [25] 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. [26] H. Kubo and M. Ohta, Critical blowup for systems of semilinear wave equations in low space dimensions, J. Math. Anal. Appl., 240 (1999), 340-360.  doi: 10.1006/jmaa.1999.6585. [27] T.-T. Li and X. Yu, Life-span of classical solutions to fully nonlinear wave equations, Comm. Partial Differential Equations, 16 (1991), 909-940.  doi: 10.1080/03605309108820785. [28] T.-T. Li and Y. Zhou, A note on the life-span of classical solutions to nonlinear wave equations in four space dimensions, Indiana Univ. Math. J., 44 (1995), 1207-1248.  doi: 10.1512/iumj.1995.44.2026. [29] H. Lindblad, M. Nakamura and C. D. Sogge, Remarks on global solutions for nonlinear wave equations under the standard null conditions, J. Differential Equations, 254 (2013), 1396-1436.  doi: 10.1016/j.jde.2012.10.022. [30] 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. [31] H. Lindblad and I. Rodnianski, Global existence for the Einstein vacuum equations in wave coordinates, Comm. Math. Phys., 256 (2005), 43-110.  doi: 10.1007/s00220-004-1281-6. [32] M. A. Rammaha, Upper bounds for the life span of solutions to systems of nonlinear wave equations in two and three space dimensions, Nonlinear Anal., 25 (1995), 639-654.  doi: 10.1016/0362-546X(94)00155-B. [33] T. C. Sideris, Global behavior of solutions to nonlinear wave equations in three dimensions, Comm. Partial Differential Equations, 8 (1983), 1291-1323.  doi: 10.1080/03605308308820304. [34] T. C. Sideris, Nonresonance and global existence of prestressed nonlinear elastic waves, Ann. of Math. $(2)$, 151 (2000), 849-874.  doi: 10.2307/121050. [35] C. D. Sogge, Global existence for nonlinear wave equations with multiple speeds, Harmonic Analysis at Mount Holyoke, 2001 (South Hadley, MA), Contemp. Math., Amer. Math. Soc., Providence, RI, 320 (2003), 353–366. doi: 10.1090/conm/320/05618. [36] Y. Zhou, Blow up of solutions to the Cauchy problem for nonlinear wave equations, Chinese Ann. Math. Ser. B, 22 (2001), 275-280.  doi: 10.1142/S0252959901000280.

show all references

##### References:
 [1] R. Agemi, Y. Kurokawa and H. Takamura, Critical curve for p-q systems of nonlinear wave equations in three space dimensions, J. Differential Equations, 167 (2000), 87-133.  doi: 10.1006/jdeq.2000.3766. [2] S. Alinhac, Semilinear hyperbolic systems with blowup at infinity, Indiana Univ. Math. J., 55 (2006), 1209-1232.  doi: 10.1512/iumj.2006.55.2671. [3] S. Alinhac, Geometric Analysis of Hyperbolic Differential Equations: An Introduction, London Mathematical Society Lecture Note Series, 374, Cambridge University Press, Cambridge, 2010. doi: 10.1017/CBO9781139107198. [4] W. Dai, D. Fang and C. Wang, Global existence and lifespan for semilinear wave equations with mixed nonlinear terms, J. Differential Equations, 267 (2019), 3328-3354.  doi: 10.1016/j.jde.2019.04.007. [5] W. Dai, D. Fang and C. Wang, Lifespan of solutions to the Strauss type wave system on asymptotically flat space-times, Discrete Contin. Dyn. Syst., 40 (2020), 4985-4999.  doi: 10.3934/dcds.2020208. [6] D. Del Santo, V. Georgiev and E. Mitidieri, Global existence of the solutions and formation of singularities for a class of hyperbolic systems, Geometrical Optics and Related Topics, 1996 (Cortona), Progr. Nonlinear Differential Equations Appl., Birkhäuser Boston, Boston, MA, 32 (1997), 117–140. [7] K. Deng, Blow-up of solutions of some nonlinear hyperbolic systems, Rocky Mountain J. Math., 29 (1999), 807-820.  doi: 10.1216/rmjm/1181071610. [8] D. Fang and C. Wang, Weighted Strichartz estimates with angular regularity and their applications, Forum Math., 23 (2011), 181-205.  doi: 10.1515/FORM.2011.009. [9] W. Han and Y. Zhou, Blow up for some semilinear wave equations in multi-space dimensions, Comm. Partial Differential Equations, 39 (2014), 651-665.  doi: 10.1080/03605302.2013.863916. [10] K. Hidano, C. Wang and K. Yokoyama, The Glassey conjecture with radially symmetric data, J. Math. Pures Appl. $(9)$, 98 (2012), 518-541.  doi: 10.1016/j.matpur.2012.01.007. [11] K. Hidano, C. Wang and K. Yokoyama, Combined effects of two nonlinearities in lifespan of small solutions to semi-linear wave equations, Math. Ann., 366 (2016), 667-694.  doi: 10.1007/s00208-015-1346-1. [12] K. Hidano and K. Yokoyama, Life span of small solutions to a system of wave equations, Nonlinear Anal., 139 (2016), 106-130.  doi: 10.1016/j.na.2016.02.020. [13] K. Hidano and K. Yokoyama, Global existence for a system of quasi-linear wave equations in $3$D satisfying the weak null condition, Int. Math. Res. Not. IMRN, 2020 (2020), 39-70.  doi: 10.1093/imrn/rny024. [14] K. Hidano and D. Zha, Remarks on a system of quasi-linear wave equations in $3$D satisfying the weak null condition, Commun. Pure Appl. Anal., 18 (2019), 1735-1767.  doi: 10.3934/cpaa.2019082. [15] L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations, Mathématiques & Applications, 26, Springer-Verlag, Berlin, 1997. [16] T. Hoshiro, On weighted $L^2$ estimates of solutions to wave equations, J. Anal. Math., 72 (1997), 127-140.  doi: 10.1007/BF02843156. [17] M. Ikeda, M. Sobajima and K. Wakasa, Blow-up phenomena of semilinear wave equations and their weakly coupled systems, J. Differerential Equations, 267 (2019), 5165-5201.  doi: 10.1016/j.jde.2019.05.029. [18] F. John, Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math., 28 (1979), 235-268.  doi: 10.1007/BF01647974. [19] F. John, Blow-up for quasilinear wave equations in three space dimensions, Comm. Pure Appl. Math., 34 (1981), 29-51.  doi: 10.1002/cpa.3160340103. [20] F. John and S. Klainerman, Almost global existence to nonlinear wave equations in three space dimensions, Comm. Pure Appl. Math., 37 (1984), 443-455.  doi: 10.1002/cpa.3160370403. [21] S. Katayama, Lifespan of solutions for two space dimensional wave equations with cubic nonlinearity, Comm. Partial Differential Equations, 26 (2001), 205-232.  doi: 10.1081/PDE-100001753. [22] S. Katayama, Asymptotic pointwise behavior for systems of semilinear wave equations in three space dimensions, J. Hyperbolic Differ. Equ., 9 (2012), 263-323.  doi: 10.1142/S0219891612500099. [23] S. Katayama, T. Matoba and H. Sunagawa, Semilinear hyperbolic systems violating the null condition, Math. Ann., 361 (2015), 275-312.  doi: 10.1007/s00208-014-1071-1. [24] S. Klainerman, The null condition and global existence to nonlinear wave equations, Nonlinear Systems of Partial Differential Equations in Applied Mathematics, Part 1, 1984 (Santa Fe, N.M.), Lectures in Appl. Math., Amer. Math. Soc., Providence, RI, 23 (1986), 293–326. [25] 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. [26] H. Kubo and M. Ohta, Critical blowup for systems of semilinear wave equations in low space dimensions, J. Math. Anal. Appl., 240 (1999), 340-360.  doi: 10.1006/jmaa.1999.6585. [27] T.-T. Li and X. Yu, Life-span of classical solutions to fully nonlinear wave equations, Comm. Partial Differential Equations, 16 (1991), 909-940.  doi: 10.1080/03605309108820785. [28] T.-T. Li and Y. Zhou, A note on the life-span of classical solutions to nonlinear wave equations in four space dimensions, Indiana Univ. Math. J., 44 (1995), 1207-1248.  doi: 10.1512/iumj.1995.44.2026. [29] H. Lindblad, M. Nakamura and C. D. Sogge, Remarks on global solutions for nonlinear wave equations under the standard null conditions, J. Differential Equations, 254 (2013), 1396-1436.  doi: 10.1016/j.jde.2012.10.022. [30] 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. [31] H. Lindblad and I. Rodnianski, Global existence for the Einstein vacuum equations in wave coordinates, Comm. Math. Phys., 256 (2005), 43-110.  doi: 10.1007/s00220-004-1281-6. [32] M. A. Rammaha, Upper bounds for the life span of solutions to systems of nonlinear wave equations in two and three space dimensions, Nonlinear Anal., 25 (1995), 639-654.  doi: 10.1016/0362-546X(94)00155-B. [33] T. C. Sideris, Global behavior of solutions to nonlinear wave equations in three dimensions, Comm. Partial Differential Equations, 8 (1983), 1291-1323.  doi: 10.1080/03605308308820304. [34] T. C. Sideris, Nonresonance and global existence of prestressed nonlinear elastic waves, Ann. of Math. $(2)$, 151 (2000), 849-874.  doi: 10.2307/121050. [35] C. D. Sogge, Global existence for nonlinear wave equations with multiple speeds, Harmonic Analysis at Mount Holyoke, 2001 (South Hadley, MA), Contemp. Math., Amer. Math. Soc., Providence, RI, 320 (2003), 353–366. doi: 10.1090/conm/320/05618. [36] Y. Zhou, Blow up of solutions to the Cauchy problem for nonlinear wave equations, Chinese Ann. Math. Ser. B, 22 (2001), 275-280.  doi: 10.1142/S0252959901000280.
Critical curves in the $pq$-plane related to nonexistence of global solutions when $n = 3$. $\max \left\{\frac{p+2+q^{-1}}{p q-1}, \frac{q+2+p^{-1}}{p q-1}\right\}-1 = 0$ is the equation of the critical curve for (4)
The tail of each arrow represents the point $(p,q) = ({\hat p},{\hat q})$. The head of the arrow $({\rm ii})$ represents the point $({\hat p},{\tilde q})$ on the critical curve
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