-
Previous Article
Dimension and measure of baker-like skew-products of $\boldsymbol{\beta}$-transformations
- DCDS Home
- This Issue
-
Next Article
A Sharkovsky theorem for non-locally connected spaces
Formation of singularities to quasi-linear hyperbolic systems with initial data of small BV norm
1. | Department of Mathematics, Shanghai Normal University, Shanghai 200234, China |
References:
[1] |
S. Bianchini and A. Bressan, Vanishing viscousity solutions to nonlinear hyperbolic systems, Ann. Math., 161 (2005), 223-242.
doi: 10.4007/annals.2005.161.223. |
[2] |
A. Bressan, Contractive metrices for nonlinear hyperbolic systems, Indiana U. Math. J., 37 (1988), 409-421.
doi: 10.1512/iumj.1988.37.37021. |
[3] |
A. Bressan, "Hyerbolic Systems of Conservation Laws: The One Dimensional Cauchy Problem," Oxford University Press, 2000. |
[4] |
Wen-Rong Dai, Asymptotic behavior of classical solutions of quasilinear non-strictly hyperbolic systems with weakly linear degeneracy, Chinese Ann. Math. B, 27 (2006), 263-286.
doi: 10.1007/s11401-004-0523-4. |
[5] |
Wen-Rong Dai, Geometry of quasilinear hyperbolic systems with characteristic fields of constant multiplicity, J. Math. Anal. Appl., 327 (2007), 188-202.
doi: 10.1016/j.jmaa.2006.04.014. |
[6] |
Wen-Rong Dai, Analysis of singularities for one-dimensional quasilinear hyperbolic systems, J. Math. Anal. Appl., 362 (2010), 72-89.
doi: 10.1016/j.jmaa.2009.08.035. |
[7] |
Wen-Rong Dai and De-Xing Kong, Global existence and asymptotic behavior of classical solutions of quasi-linear hyperbolic systems with linearly degenerate characteristic fields, J. Differ. Equations, 235 (2007), 127-165.
doi: 10.1016/j.jde.2006.12.020. |
[8] |
L. Hörmander, The lifespan of classical solutions of nonlinear hyperbolic equations, Lecture Notes in Mathematics, 1256 (1987), 214-280.
doi: 10.1007/BFb0077745. |
[9] |
F. John, Formation of singularities in one-dimensional nonlinear wave propagation, Commun. Pur. Appl. Math., 27 (1974), 377-405.
doi: 10.1002/cpa.3160270307. |
[10] |
De-xing Kong, Lifespan of classical solutions to quasilinear hyperbolic systems with slow decay initial data, Chinese Ann. Math. B, 21 (2000), 413-440.
doi: 10.1142/S0252959900000431. |
[11] |
De-xing Kong, "Cauchy Problem for Quasilinear Hyperbolic Systems," MSJ Memoirs, 6, Mathematical Society of Japan, Tokyo, 2000. |
[12] |
De-xing Kong, Formation and propagation of singularities for $2\times 2$ quasilinear hyperbolic systems, T. Am. Math. Soc., 354 (2002), 3155-3179.
doi: 10.1090/S0002-9947-02-02982-3. |
[13] |
De-xing Kong and Ta-tsien Li, A note on blow-up phenomenon of classical solutions to quasilinear hyperbolic systems, Nonlinear Anal., 49 (2002), 535-539.
doi: 10.1016/S0362-546X(01)00121-3. |
[14] |
De-xing Kong and Tong Yang, Asymtotic behavior of global classical solutions of quasilinear hyperbolic systems, Commun. Part. Diff. Eq., 28 (2003), 1203-1220.
doi: 10.1081/PDE-120021192. |
[15] |
Ta-Tsien Li and De-xing Kong, Global classical solutions with small amplitude for general quasi-linear hyperbolic systems, in "New Approaches in Nonlinear Analysis," Hardronic Press, (1999), 203-237. |
[16] |
Ta-Tsien Li, De-Xing Kong and Yi Zhou, Global classical solutions for general quasilinear nonstrictly hyperbolic systems, Nonlinear Studies, 3 (1996), 203-229. |
[17] |
Ta-Tsien Li, Yi Zhou and De-Xing Kong, Weak linear degeneracy and global classical solutions for general quasi-linear hyperbolic systems, Commun. Part. Diff. Eq., 19 (1994), 1263-1317.
doi: 10.1080/03605309408821055. |
[18] |
Ta-Tsien Li, Yi Zhou and De-Xing Kong, Global classical solutions for general quasi-linear hyperbolic systems with decay initial data, Nonlinear Anal., 28 (1997), 1299-1332.
doi: 10.1016/0362-546X(95)00228-N. |
[19] |
Tai-Ping Liu, Development of singularities in the nonlinear waves for quasi-linear hyperbolic partial differential equations, J. Differ. Equations, 33 (1979), 92-111.
doi: 10.1016/0022-0396(79)90082-2. |
[20] |
M. Schatzman, Continuous Glimm functional and uniqueness of the solution of Riemann problem, Indiana U. Math. J., 34 (1985), 533-589.
doi: 10.1512/iumj.1985.34.34030. |
[21] |
M. Schatzman, The geometry of continuous Glimm functionals, Lectures in Applied Mathematics, 23 (1986), 417-439. |
[22] |
Yi Zhou, Global classical solutions to quasilinear hyperbolic systems with weak linear degeneracy, Chinese Ann. Math. Ser. B, 25 (2004), 37-56.
doi: 10.1142/S0252959904000469. |
[23] |
Yi Zhou and Yong-Fu Yang, Global classical solutions of mixed initial-boundary value problem for quasilinear hyperbolic systems, Nonlinear Anal., 73 (2010), 1543-1561.
doi: 10.1016/j.na.2010.04.057. |
show all references
References:
[1] |
S. Bianchini and A. Bressan, Vanishing viscousity solutions to nonlinear hyperbolic systems, Ann. Math., 161 (2005), 223-242.
doi: 10.4007/annals.2005.161.223. |
[2] |
A. Bressan, Contractive metrices for nonlinear hyperbolic systems, Indiana U. Math. J., 37 (1988), 409-421.
doi: 10.1512/iumj.1988.37.37021. |
[3] |
A. Bressan, "Hyerbolic Systems of Conservation Laws: The One Dimensional Cauchy Problem," Oxford University Press, 2000. |
[4] |
Wen-Rong Dai, Asymptotic behavior of classical solutions of quasilinear non-strictly hyperbolic systems with weakly linear degeneracy, Chinese Ann. Math. B, 27 (2006), 263-286.
doi: 10.1007/s11401-004-0523-4. |
[5] |
Wen-Rong Dai, Geometry of quasilinear hyperbolic systems with characteristic fields of constant multiplicity, J. Math. Anal. Appl., 327 (2007), 188-202.
doi: 10.1016/j.jmaa.2006.04.014. |
[6] |
Wen-Rong Dai, Analysis of singularities for one-dimensional quasilinear hyperbolic systems, J. Math. Anal. Appl., 362 (2010), 72-89.
doi: 10.1016/j.jmaa.2009.08.035. |
[7] |
Wen-Rong Dai and De-Xing Kong, Global existence and asymptotic behavior of classical solutions of quasi-linear hyperbolic systems with linearly degenerate characteristic fields, J. Differ. Equations, 235 (2007), 127-165.
doi: 10.1016/j.jde.2006.12.020. |
[8] |
L. Hörmander, The lifespan of classical solutions of nonlinear hyperbolic equations, Lecture Notes in Mathematics, 1256 (1987), 214-280.
doi: 10.1007/BFb0077745. |
[9] |
F. John, Formation of singularities in one-dimensional nonlinear wave propagation, Commun. Pur. Appl. Math., 27 (1974), 377-405.
doi: 10.1002/cpa.3160270307. |
[10] |
De-xing Kong, Lifespan of classical solutions to quasilinear hyperbolic systems with slow decay initial data, Chinese Ann. Math. B, 21 (2000), 413-440.
doi: 10.1142/S0252959900000431. |
[11] |
De-xing Kong, "Cauchy Problem for Quasilinear Hyperbolic Systems," MSJ Memoirs, 6, Mathematical Society of Japan, Tokyo, 2000. |
[12] |
De-xing Kong, Formation and propagation of singularities for $2\times 2$ quasilinear hyperbolic systems, T. Am. Math. Soc., 354 (2002), 3155-3179.
doi: 10.1090/S0002-9947-02-02982-3. |
[13] |
De-xing Kong and Ta-tsien Li, A note on blow-up phenomenon of classical solutions to quasilinear hyperbolic systems, Nonlinear Anal., 49 (2002), 535-539.
doi: 10.1016/S0362-546X(01)00121-3. |
[14] |
De-xing Kong and Tong Yang, Asymtotic behavior of global classical solutions of quasilinear hyperbolic systems, Commun. Part. Diff. Eq., 28 (2003), 1203-1220.
doi: 10.1081/PDE-120021192. |
[15] |
Ta-Tsien Li and De-xing Kong, Global classical solutions with small amplitude for general quasi-linear hyperbolic systems, in "New Approaches in Nonlinear Analysis," Hardronic Press, (1999), 203-237. |
[16] |
Ta-Tsien Li, De-Xing Kong and Yi Zhou, Global classical solutions for general quasilinear nonstrictly hyperbolic systems, Nonlinear Studies, 3 (1996), 203-229. |
[17] |
Ta-Tsien Li, Yi Zhou and De-Xing Kong, Weak linear degeneracy and global classical solutions for general quasi-linear hyperbolic systems, Commun. Part. Diff. Eq., 19 (1994), 1263-1317.
doi: 10.1080/03605309408821055. |
[18] |
Ta-Tsien Li, Yi Zhou and De-Xing Kong, Global classical solutions for general quasi-linear hyperbolic systems with decay initial data, Nonlinear Anal., 28 (1997), 1299-1332.
doi: 10.1016/0362-546X(95)00228-N. |
[19] |
Tai-Ping Liu, Development of singularities in the nonlinear waves for quasi-linear hyperbolic partial differential equations, J. Differ. Equations, 33 (1979), 92-111.
doi: 10.1016/0022-0396(79)90082-2. |
[20] |
M. Schatzman, Continuous Glimm functional and uniqueness of the solution of Riemann problem, Indiana U. Math. J., 34 (1985), 533-589.
doi: 10.1512/iumj.1985.34.34030. |
[21] |
M. Schatzman, The geometry of continuous Glimm functionals, Lectures in Applied Mathematics, 23 (1986), 417-439. |
[22] |
Yi Zhou, Global classical solutions to quasilinear hyperbolic systems with weak linear degeneracy, Chinese Ann. Math. Ser. B, 25 (2004), 37-56.
doi: 10.1142/S0252959904000469. |
[23] |
Yi Zhou and Yong-Fu Yang, Global classical solutions of mixed initial-boundary value problem for quasilinear hyperbolic systems, Nonlinear Anal., 73 (2010), 1543-1561.
doi: 10.1016/j.na.2010.04.057. |
[1] |
Jaakko Kultima, Valery Serov. Reconstruction of singularities in two-dimensional quasi-linear biharmonic operator. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022011 |
[2] |
Qiong Chen, Chunlai Mu, Zhaoyin Xiang. Blow-up and asymptotic behavior of solutions to a semilinear integrodifferential system. Communications on Pure and Applied Analysis, 2006, 5 (3) : 435-446. doi: 10.3934/cpaa.2006.5.435 |
[3] |
Huiling Li, Mingxin Wang. Properties of blow-up solutions to a parabolic system with nonlinear localized terms. Discrete and Continuous Dynamical Systems, 2005, 13 (3) : 683-700. doi: 10.3934/dcds.2005.13.683 |
[4] |
Hongwei Chen. Blow-up estimates of positive solutions of a reaction-diffusion system. Conference Publications, 2003, 2003 (Special) : 182-188. doi: 10.3934/proc.2003.2003.182 |
[5] |
Evgeny Galakhov, Olga Salieva. Blow-up for nonlinear inequalities with gradient terms and singularities on unbounded sets. Conference Publications, 2015, 2015 (special) : 489-494. doi: 10.3934/proc.2015.0489 |
[6] |
Satyanad Kichenassamy. Control of blow-up singularities for nonlinear wave equations. Evolution Equations and Control Theory, 2013, 2 (4) : 669-677. doi: 10.3934/eect.2013.2.669 |
[7] |
Xueqin Peng, Gao Jia. Existence and asymptotical behavior of positive solutions for the Schrödinger-Poisson system with double quasi-linear terms. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 2325-2344. doi: 10.3934/dcdsb.2021134 |
[8] |
C. Y. Chan. Recent advances in quenching and blow-up of solutions. Conference Publications, 2001, 2001 (Special) : 88-95. doi: 10.3934/proc.2001.2001.88 |
[9] |
Marek Fila, Hiroshi Matano. Connecting equilibria by blow-up solutions. Discrete and Continuous Dynamical Systems, 2000, 6 (1) : 155-164. doi: 10.3934/dcds.2000.6.155 |
[10] |
Petri Juutinen. Convexity of solutions to boundary blow-up problems. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2267-2275. doi: 10.3934/cpaa.2013.12.2267 |
[11] |
Hua Chen, Nian Liu. Asymptotic stability and blow-up of solutions for semi-linear edge-degenerate parabolic equations with singular potentials. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 661-682. doi: 10.3934/dcds.2016.36.661 |
[12] |
Min Zhu, Shuanghu Zhang. Blow-up of solutions to the periodic modified Camassa-Holm equation with varying linear dispersion. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 7235-7256. doi: 10.3934/dcds.2016115 |
[13] |
Min Zhu, Ying Wang. Blow-up of solutions to the periodic generalized modified Camassa-Holm equation with varying linear dispersion. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 645-661. doi: 10.3934/dcds.2017027 |
[14] |
Yu-Zhu Wang, Weibing Zuo. On the blow-up criterion of smooth solutions for Hall-magnetohydrodynamics system with partial viscosity. Communications on Pure and Applied Analysis, 2014, 13 (3) : 1327-1336. doi: 10.3934/cpaa.2014.13.1327 |
[15] |
Yūki Naito, Takasi Senba. Blow-up behavior of solutions to a parabolic-elliptic system on higher dimensional domains. Discrete and Continuous Dynamical Systems, 2012, 32 (10) : 3691-3713. doi: 10.3934/dcds.2012.32.3691 |
[16] |
Shu-Xiang Huang, Fu-Cai Li, Chun-Hong Xie. Global existence and blow-up of solutions to a nonlocal reaction-diffusion system. Discrete and Continuous Dynamical Systems, 2003, 9 (6) : 1519-1532. doi: 10.3934/dcds.2003.9.1519 |
[17] |
Hayato Miyazaki. Strong blow-up instability for standing wave solutions to the system of the quadratic nonlinear Klein-Gordon equations. Discrete and Continuous Dynamical Systems, 2021, 41 (5) : 2411-2445. doi: 10.3934/dcds.2020370 |
[18] |
Johannes Lankeit. Infinite time blow-up of many solutions to a general quasilinear parabolic-elliptic Keller-Segel system. Discrete and Continuous Dynamical Systems - S, 2020, 13 (2) : 233-255. doi: 10.3934/dcdss.2020013 |
[19] |
Yongsheng Mi, Boling Guo, Chunlai Mu. Well-posedness and blow-up scenario for a new integrable four-component system with peakon solutions. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 2171-2191. doi: 10.3934/dcds.2016.36.2171 |
[20] |
Vo Anh Khoa, Le Thi Phuong Ngoc, Nguyen Thanh Long. Existence, blow-up and exponential decay of solutions for a porous-elastic system with damping and source terms. Evolution Equations and Control Theory, 2019, 8 (2) : 359-395. doi: 10.3934/eect.2019019 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]