-
Previous Article
On the decay and stability of global solutions to the 3D inhomogeneous MHD system
- CPAA Home
- This Issue
- Next Article
Blowup time and blowup mechanism of small data solutions to general 2-D quasilinear wave equations
1. | School of Mathematical Sciences, Jiangsu Provincial Key Laboratory for Numerical, Simulation of Large Scale Complex Systems, Nanjing Normal University, Nanjing, 210023, China |
2. | Mathematical Institute, University of Göttingen, Bunsenstr. 3-5, Göttingen, D-37073, Germany |
For the 2-D quasilinear wave equation $\sum\nolimits_{i,j = 0}^2 {{g_{ij}}} (\nabla u)\partial _{ij}^2u = 0$, whose coefficients are independent of the solution $u$, the blowup result of small data solution has been established in [
References:
[1] |
S. Alinhac,
Blowup of small data solutions for a class of quasilinear wave equations in two space dimensions, Ann. of Math., 149 (1999), 97-127.
doi: 10.2307/121020. |
[2] |
S. Alinhac,
Blowup of small data solutions for a class of quasilinear wave equations in two space dimensions. Ⅱ, Acta Math., 182 (1999), 1-23.
doi: 10.1007/BF02392822. |
[3] |
S. Alinhac,
The null condition for quasilinear wave equations in two space dimensions. Ⅱ, Amer. J. Math., 123 (2001), 1071-1101.
|
[4] |
S. Alinhac,
An example of blowup at infinity for quasilinear wave equations, Asterisque, 284 (2003), 1-91.
|
[5] |
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. |
[6] |
D. Christodoulou and S. Miao, Compressible Flow and Euler's Equations, Surveys of Modern Mathematics, 9. International Press, Somerville, MA; Higher Education Press, Beijing, 2014. |
[7] |
Binbing Ding, Yingbo Liu and Huicheng Yin,
The small data solutions of general 3-D quasilinear wave equations. Ⅰ, SIAM Journal on Mathematical Analysis, 47 (2015), 4192-4228.
doi: 10.1137/151004793. |
[8] |
Binbing Ding, Ingo Witt and Huicheng Yin,
The small data solutions of general 3-D quasilinear wave equations. Ⅱ, J. Differential Equations, 261 (2016), 1429-1471.
doi: 10.1016/j.jde.2016.04.002. |
[9] |
Binbing Ding, Ingo Witt and Huicheng Yin,
Blowup of classical solutions for 2-D quasilinear wave equations with small initial data, Quart. Appl. Math., 73 (2015), 773-796.
doi: 10.1090/qam/1410. |
[10] |
Binbing Ding, Ingo Witt and Huicheng Yin,
On the blowup of classical solutions to the 3-D pressure-gradient systems, J. Differential Equations, 252 (2012), 3608-3629.
doi: 10.1016/j.jde.2011.11.018. |
[11] |
P. Godin,
Lifespan of solutions of semilinear wave equations in two space dimensions, Comm. Partial Differential Equations, 18 (1993), 895-916.
doi: 10.1080/03605309308820955. |
[12] |
Fei Guo and phenomena Wave-breaking,
decay properties and limit behaviour of solutions of the Degasperis-Procesi equation, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 805-824.
doi: 10.1017/S0308210511000321. |
[13] |
Fei Guo and Weiwei Peng,
Blowup solutions for the generalized two-component CamassaHolm system on the circle, Nonlinear Anal., 105 (2014), 120-133.
doi: 10.1016/j.na.2014.03.021. |
[14] |
L. Hömander, The Lifespan of Classical Solutions of Nonlinear Hyperbolic Equations, MittagLeffler report No. 5,1985. |
[15] |
L. Hömander, Lectures on Nonlinear Hyperbolic Equations, Mathematiques & Applications 26, Springer Verlag, Heidelberg, 1997. |
[16] |
A. Hoshiga,
The asymptotic behaviour of the radially symmetric solutions to quasilinear wave equations in two space dimensions, Hokkaido Math. J., 24 (1995), 575-615.
doi: 10.14492/hokmj/1380892610. |
[17] |
F. John,
Blow-up of radial solutions of utt = c2(ut)∆u in three space dimensions, Mat. Apl. Comput., 4 (1985), 3-18.
|
[18] |
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. |
[19] |
M. Keel, H. Smith and C. D. Sogge,
Almost global existence for quasilinear wave equations in three space dimensions, J. Amer. Math. Soc., 17 (2004), 109-153.
doi: 10.1090/S0894-0347-03-00443-0. |
[20] |
S. Klainerman, The Null Condition and Global Existence to Nonlinear Wave Equations, Lectures in Appl. Math. , 23, Amer. Math. Soc. , Providence, RI, 1986. |
[21] |
S. Klainerman,
Remarks on the global Sobolev inequalities in the Minkowski space ℝn+1, Comm. Pure Appl. Math., 40 (1987), 111-117.
doi: 10.1002/cpa.3160400105. |
[22] |
Yutian Lei,
Singularity analysis of Ginzburg-Landau energy related to p-wave superconductivity, Z. Angew. Math. Phys., 64 (2013), 1249-1266.
doi: 10.1007/s00033-012-0284-y. |
[23] |
Ta-tsien Li and Yun-mei Chen,
Initial value problems for nonlinear wave equations, Comm. Partial Differential Equations, 13 (1988), 383-422.
doi: 10.1080/03605308808820547. |
[24] |
H. Lindblad,
On the lifespan of solutions of nonlinear wave equations with small initial data, Comm. Pure Appl. Math., 43 (1990), 445-472.
doi: 10.1002/cpa.3160430403. |
[25] |
H. Lindblad,
Global solutions of nonlinear wave equations, Comm. Pure Appl. Math., 45 (1992), 1063-1096.
doi: 10.1002/cpa.3160450902. |
[26] |
H. Lindblad,
Global solutions of quasilinear wave equations, Amer. J. Math., 130 (2008), 115-157.
doi: 10.1353/ajm.2008.0009. |
[27] |
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. |
[28] |
J. Speck, Shock formation in small-data solutions to 3D quasilinear wave equations, preprint, arXiv: 1407.6320. |
[29] |
Sijue Wu,
Global wellposedness of the 3-D full water wave problem, Invent. Math., 184 (2011), 125-220.
doi: 10.1007/s00222-010-0288-1. |
[30] |
Sijue Wu,
Almost global wellposedness of the 2-D full water wave problem, Invent. Math., 177 (2009), 45-135.
doi: 10.1007/s00222-009-0176-8. |
show all references
References:
[1] |
S. Alinhac,
Blowup of small data solutions for a class of quasilinear wave equations in two space dimensions, Ann. of Math., 149 (1999), 97-127.
doi: 10.2307/121020. |
[2] |
S. Alinhac,
Blowup of small data solutions for a class of quasilinear wave equations in two space dimensions. Ⅱ, Acta Math., 182 (1999), 1-23.
doi: 10.1007/BF02392822. |
[3] |
S. Alinhac,
The null condition for quasilinear wave equations in two space dimensions. Ⅱ, Amer. J. Math., 123 (2001), 1071-1101.
|
[4] |
S. Alinhac,
An example of blowup at infinity for quasilinear wave equations, Asterisque, 284 (2003), 1-91.
|
[5] |
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. |
[6] |
D. Christodoulou and S. Miao, Compressible Flow and Euler's Equations, Surveys of Modern Mathematics, 9. International Press, Somerville, MA; Higher Education Press, Beijing, 2014. |
[7] |
Binbing Ding, Yingbo Liu and Huicheng Yin,
The small data solutions of general 3-D quasilinear wave equations. Ⅰ, SIAM Journal on Mathematical Analysis, 47 (2015), 4192-4228.
doi: 10.1137/151004793. |
[8] |
Binbing Ding, Ingo Witt and Huicheng Yin,
The small data solutions of general 3-D quasilinear wave equations. Ⅱ, J. Differential Equations, 261 (2016), 1429-1471.
doi: 10.1016/j.jde.2016.04.002. |
[9] |
Binbing Ding, Ingo Witt and Huicheng Yin,
Blowup of classical solutions for 2-D quasilinear wave equations with small initial data, Quart. Appl. Math., 73 (2015), 773-796.
doi: 10.1090/qam/1410. |
[10] |
Binbing Ding, Ingo Witt and Huicheng Yin,
On the blowup of classical solutions to the 3-D pressure-gradient systems, J. Differential Equations, 252 (2012), 3608-3629.
doi: 10.1016/j.jde.2011.11.018. |
[11] |
P. Godin,
Lifespan of solutions of semilinear wave equations in two space dimensions, Comm. Partial Differential Equations, 18 (1993), 895-916.
doi: 10.1080/03605309308820955. |
[12] |
Fei Guo and phenomena Wave-breaking,
decay properties and limit behaviour of solutions of the Degasperis-Procesi equation, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 805-824.
doi: 10.1017/S0308210511000321. |
[13] |
Fei Guo and Weiwei Peng,
Blowup solutions for the generalized two-component CamassaHolm system on the circle, Nonlinear Anal., 105 (2014), 120-133.
doi: 10.1016/j.na.2014.03.021. |
[14] |
L. Hömander, The Lifespan of Classical Solutions of Nonlinear Hyperbolic Equations, MittagLeffler report No. 5,1985. |
[15] |
L. Hömander, Lectures on Nonlinear Hyperbolic Equations, Mathematiques & Applications 26, Springer Verlag, Heidelberg, 1997. |
[16] |
A. Hoshiga,
The asymptotic behaviour of the radially symmetric solutions to quasilinear wave equations in two space dimensions, Hokkaido Math. J., 24 (1995), 575-615.
doi: 10.14492/hokmj/1380892610. |
[17] |
F. John,
Blow-up of radial solutions of utt = c2(ut)∆u in three space dimensions, Mat. Apl. Comput., 4 (1985), 3-18.
|
[18] |
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. |
[19] |
M. Keel, H. Smith and C. D. Sogge,
Almost global existence for quasilinear wave equations in three space dimensions, J. Amer. Math. Soc., 17 (2004), 109-153.
doi: 10.1090/S0894-0347-03-00443-0. |
[20] |
S. Klainerman, The Null Condition and Global Existence to Nonlinear Wave Equations, Lectures in Appl. Math. , 23, Amer. Math. Soc. , Providence, RI, 1986. |
[21] |
S. Klainerman,
Remarks on the global Sobolev inequalities in the Minkowski space ℝn+1, Comm. Pure Appl. Math., 40 (1987), 111-117.
doi: 10.1002/cpa.3160400105. |
[22] |
Yutian Lei,
Singularity analysis of Ginzburg-Landau energy related to p-wave superconductivity, Z. Angew. Math. Phys., 64 (2013), 1249-1266.
doi: 10.1007/s00033-012-0284-y. |
[23] |
Ta-tsien Li and Yun-mei Chen,
Initial value problems for nonlinear wave equations, Comm. Partial Differential Equations, 13 (1988), 383-422.
doi: 10.1080/03605308808820547. |
[24] |
H. Lindblad,
On the lifespan of solutions of nonlinear wave equations with small initial data, Comm. Pure Appl. Math., 43 (1990), 445-472.
doi: 10.1002/cpa.3160430403. |
[25] |
H. Lindblad,
Global solutions of nonlinear wave equations, Comm. Pure Appl. Math., 45 (1992), 1063-1096.
doi: 10.1002/cpa.3160450902. |
[26] |
H. Lindblad,
Global solutions of quasilinear wave equations, Amer. J. Math., 130 (2008), 115-157.
doi: 10.1353/ajm.2008.0009. |
[27] |
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. |
[28] |
J. Speck, Shock formation in small-data solutions to 3D quasilinear wave equations, preprint, arXiv: 1407.6320. |
[29] |
Sijue Wu,
Global wellposedness of the 3-D full water wave problem, Invent. Math., 184 (2011), 125-220.
doi: 10.1007/s00222-010-0288-1. |
[30] |
Sijue Wu,
Almost global wellposedness of the 2-D full water wave problem, Invent. Math., 177 (2009), 45-135.
doi: 10.1007/s00222-009-0176-8. |
[1] |
Jong-Shenq Guo, Satoshi Sasayama, Chi-Jen Wang. Blowup rate estimate for a system of semilinear parabolic equations. Communications on Pure and Applied Analysis, 2009, 8 (2) : 711-718. doi: 10.3934/cpaa.2009.8.711 |
[2] |
Xuan Liu, Ting Zhang. $ H^2 $ blowup result for a Schrödinger equation with nonlinear source term. Electronic Research Archive, 2020, 28 (2) : 777-794. doi: 10.3934/era.2020039 |
[3] |
Kunio Hidano, Dongbing Zha. Remarks on a system of quasi-linear wave equations in 3D satisfying the weak null condition. Communications on Pure and Applied Analysis, 2019, 18 (4) : 1735-1767. doi: 10.3934/cpaa.2019082 |
[4] |
María José Beltrán, José Bonet, Carmen Fernández. Classical operators on the Hörmander algebras. Discrete and Continuous Dynamical Systems, 2015, 35 (2) : 637-652. doi: 10.3934/dcds.2015.35.637 |
[5] |
Baoquan Yuan, Xiaokui Zhao. Blowup of smooth solutions to the full compressible MHD system with compact density. Kinetic and Related Models, 2014, 7 (1) : 195-203. doi: 10.3934/krm.2014.7.195 |
[6] |
Lawrence Ein, Wenbo Niu, Jinhyung Park. On blowup of secant varieties of curves. Electronic Research Archive, 2021, 29 (6) : 3649-3654. doi: 10.3934/era.2021055 |
[7] |
Yanghong Huang, Andrea Bertozzi. Asymptotics of blowup solutions for the aggregation equation. Discrete and Continuous Dynamical Systems - B, 2012, 17 (4) : 1309-1331. doi: 10.3934/dcdsb.2012.17.1309 |
[8] |
Elio E. Espejo, Masaki Kurokiba, Takashi Suzuki. Blowup threshold and collapse mass separation for a drift-diffusion system in space-dimension two. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2627-2644. doi: 10.3934/cpaa.2013.12.2627 |
[9] |
Qunying Zhang, Zhigui Lin. Blowup, global fast and slow solutions to a parabolic system with double fronts free boundary. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 429-444. doi: 10.3934/dcdsb.2012.17.429 |
[10] |
Xiaofeng Hou, Limei Zhu. Serrin-type blowup criterion for full compressible Navier-Stokes-Maxwell system with vacuum. Communications on Pure and Applied Analysis, 2016, 15 (1) : 161-183. doi: 10.3934/cpaa.2016.15.161 |
[11] |
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, 2022 doi: 10.3934/dcds.2022058 |
[12] |
Valerii Los, Vladimir A. Mikhailets, Aleksandr A. Murach. An isomorphism theorem for parabolic problems in Hörmander spaces and its applications. Communications on Pure and Applied Analysis, 2017, 16 (1) : 69-98. doi: 10.3934/cpaa.2017003 |
[13] |
Ping Lin. Feedback controllability for blowup points of semilinear heat equations. Discrete and Continuous Dynamical Systems - B, 2017, 22 (4) : 1425-1434. doi: 10.3934/dcdsb.2017068 |
[14] |
Piotr Biler, Grzegorz Karch, Jacek Zienkiewicz. Morrey spaces norms and criteria for blowup in chemotaxis models. Networks and Heterogeneous Media, 2016, 11 (2) : 239-250. doi: 10.3934/nhm.2016.11.239 |
[15] |
Zhengce Zhang, Bei Hu. Gradient blowup rate for a semilinear parabolic equation. Discrete and Continuous Dynamical Systems, 2010, 26 (2) : 767-779. doi: 10.3934/dcds.2010.26.767 |
[16] |
Chi-Cheung Poon. Blowup rate of solutions of a degenerate nonlinear parabolic equation. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5317-5336. doi: 10.3934/dcdsb.2019060 |
[17] |
Piotr Biler, Elio E. Espejo, Ignacio Guerra. Blowup in higher dimensional two species chemotactic systems. Communications on Pure and Applied Analysis, 2013, 12 (1) : 89-98. doi: 10.3934/cpaa.2013.12.89 |
[18] |
Jong-Shenq Guo, Bei Hu. Blowup rate estimates for the heat equation with a nonlinear gradient source term. Discrete and Continuous Dynamical Systems, 2008, 20 (4) : 927-937. doi: 10.3934/dcds.2008.20.927 |
[19] |
Shaohua Chen, Runzhang Xu, Hongtao Yang. Global and blowup solutions for general Lotka-Volterra systems. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1757-1768. doi: 10.3934/cpaa.2016012 |
[20] |
Grzegorz Karch, Kanako Suzuki, Jacek Zienkiewicz. Finite-time blowup of solutions to some activator-inhibitor systems. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 4997-5010. doi: 10.3934/dcds.2016016 |
2020 Impact Factor: 1.916
Tools
Metrics
Other articles
by authors
[Back to Top]