American Institute of Mathematical Sciences

Advanced
March  2003, 2(1): 77-89. doi: 10.3934/cpaa.2003.2.77

Breakdown of $C^1$ solution to the Cauchy problem for quasilinear hyperbolic systems with characteristics with constant multiplicity

Libin Wang 1,

1. 

Institute of Mathematics, Fudan University, Shanghai 200433, China

Received  July 2002 Revised  October 2002 Published  December 2002

In this paper we consider the Cauchy problem for quasilinear hyperbolic systems with characteristics with constant multiplicity. Suppose that characteristics with constant multiplicity ($>$1) are linearly degenerate only at $u=0$, if there is a genuinely nonlinear simple characteristic which does not have certain "monotonicity" and the initial data possess some decaying properties, we obtain the blow-up result for the $C^1$ solution to the Cauchy problem.
Keywords: Quasilinear hyperbolic system, life-span., Cauchy problem, blow up.
Mathematics Subject Classification: 35A21, 35L45, 35L60, 35Q7.
Citation: Libin Wang. Breakdown of $C^1$ solution to the Cauchy problem for quasilinear hyperbolic systems with characteristics with constant multiplicity. Communications on Pure and Applied Analysis, 2003, 2 (1) : 77-89. doi: 10.3934/cpaa.2003.2.77
[1]

Xiuting Li, Lei Zhang. The Cauchy problem and blow-up phenomena for a new integrable two-component peakon system with cubic nonlinearities. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 3301-3325. doi: 10.3934/dcds.2017140
[2]

Lingwei Ma, Zhong Bo Fang. A new second critical exponent and life span for a quasilinear degenerate parabolic equation with weighted nonlocal sources. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1697-1706. doi: 10.3934/cpaa.2017081
[3]

Cunming Liu, Jianli Liu. Stability of traveling wave solutions to Cauchy problem of diagnolizable quasilinear hyperbolic systems. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4735-4749. doi: 10.3934/dcds.2014.34.4735
[4]

Mohammad Kafini. On the blow-up of the Cauchy problem of higher-order nonlinear viscoelastic wave equation. Discrete and Continuous Dynamical Systems - S, 2022, 15 (5) : 1221-1232. doi: 10.3934/dcdss.2021093
[5]

C. Brändle, F. Quirós, Julio D. Rossi. Non-simultaneous blow-up for a quasilinear parabolic system with reaction at the boundary. Communications on Pure and Applied Analysis, 2005, 4 (3) : 523-536. doi: 10.3934/cpaa.2005.4.523
[6]

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
[7]

Juntang Ding, Chenyu Dong. Blow-up results of the positive solution for a weakly coupled quasilinear parabolic system. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021222
[8]

Yuya Tanaka, Tomomi Yokota. Finite-time blow-up in a quasilinear degenerate parabolic–elliptic chemotaxis system with logistic source and nonlinear production. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022075
[9]

Meng-Rong Li. Estimates for the life-span of the solutions for some semilinear wave equations. Communications on Pure and Applied Analysis, 2008, 7 (2) : 417-432. doi: 10.3934/cpaa.2008.7.417
[10]

Miaoqing Tian, Sining Zheng. Global boundedness versus finite-time blow-up of solutions to a quasilinear fully parabolic Keller-Segel system of two species. Communications on Pure and Applied Analysis, 2016, 15 (1) : 243-260. doi: 10.3934/cpaa.2016.15.243
[11]

Todor Gramchev, Nicola Orrú. Cauchy problem for a class of nondiagonalizable hyperbolic systems. Conference Publications, 2011, 2011 (Special) : 533-542. doi: 10.3934/proc.2011.2011.533
[12]

Yoshikazu Giga. Interior derivative blow-up for quasilinear parabolic equations. Discrete and Continuous Dynamical Systems, 1995, 1 (3) : 449-461. doi: 10.3934/dcds.1995.1.449
[13]

Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete and Continuous Dynamical Systems, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71
[14]

Marina Chugunova, Chiu-Yen Kao, Sarun Seepun. On the Benilov-Vynnycky blow-up problem. Discrete and Continuous Dynamical Systems - B, 2015, 20 (5) : 1443-1460. doi: 10.3934/dcdsb.2015.20.1443
[15]

Victor A. Galaktionov, Juan-Luis Vázquez. The problem Of blow-up in nonlinear parabolic equations. Discrete and Continuous Dynamical Systems, 2002, 8 (2) : 399-433. doi: 10.3934/dcds.2002.8.399
[16]

Tong Li, Kun Zhao. On a quasilinear hyperbolic system in blood flow modeling. Discrete and Continuous Dynamical Systems - B, 2011, 16 (1) : 333-344. doi: 10.3934/dcdsb.2011.16.333
[17]

Felipe Linares, M. Panthee. On the Cauchy problem for a coupled system of KdV equations. Communications on Pure and Applied Analysis, 2004, 3 (3) : 417-431. doi: 10.3934/cpaa.2004.3.417
[18]

Juliana Fernandes, Liliane Maia. Blow-up and bounded solutions for a semilinear parabolic problem in a saturable medium. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1297-1318. doi: 10.3934/dcds.2020318
[19]

Pierpaolo Esposito, Maristella Petralla. Symmetries and blow-up phenomena for a Dirichlet problem with a large parameter. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1935-1957. doi: 10.3934/cpaa.2012.11.1935
[20]

Yihong Du, Zongming Guo, Feng Zhou. Boundary blow-up solutions with interior layers and spikes in a bistable problem. Discrete and Continuous Dynamical Systems, 2007, 19 (2) : 271-298. doi: 10.3934/dcds.2007.19.271

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (65)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy