American Institute of Mathematical Sciences

November  2014, 34(11): 4735-4749. doi: 10.3934/dcds.2014.34.4735

Stability of traveling wave solutions to Cauchy problem of diagnolizable quasilinear hyperbolic systems

 1 School of Mathematics, Taiyuan University of Technology, Shanxi, 030024, China 2 Department of Mathematics, Shanghai University, Shanghai 200444, China

Received  July 2013 Revised  March 2014 Published  May 2014

In this paper we consider the existence and stability of traveling wave solutions to Cauchy problem of diagonalizable quasilinear hyperbolic systems. Under the appropriate small oscillation assumptions on the initial traveling waves, we derive the stability result of the traveling wave solutions, especially for intermediate traveling waves. As the important examples, we will apply the results to some systems arising in fluid dynamics and elementary particle physics.
Citation: Cunming Liu, Jianli Liu. Stability of traveling wave solutions to Cauchy problem of diagnolizable quasilinear hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4735-4749. doi: 10.3934/dcds.2014.34.4735
References:
 [1] B. M. Barbashov, V. V. Nesterenko and A. M. Chervyakov, General solutions of nonlinear equations in the geometric theory of the relativistic string,, Commun. Math. Phys., 84 (1982), 471. doi: 10.1007/BF01209629. Google Scholar [2] G. Carbou, B. Hanouzet and R .Natalini, Semilinear behavior of totally linearly degenerate hyperbolic systems with relaxation,, J. Differential Equations, 246 (2009), 291. doi: 10.1016/j.jde.2008.05.015. Google Scholar [3] R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves,, Applied Mathematical Sciences, (1976). Google Scholar [4] W. R. Dai and D. X. Kong, Asymptotic behavior of global classical solutions of general quasilinear hyperbolic systems with weakly linear degeneracy,, Chinese Annals of Mathematics, 27B (2006), 263. Google Scholar [5] W. R. Dai and D. X. Kong, Global existence and asymptotic behavior of classical solutions of quasilnear hyperbolic systems with linear degenerate characteristic fields,, J.Differential Equations, 235 (2007), 127. doi: 10.1016/j.jde.2006.12.020. Google Scholar [6] D. X. Kong, Q. Zhang and Q. Zhou, The dynamics of relativistic strings moving in the Minkowski space $R^{1+n}$,, Commun. Math. Phys., 269 (2007), 153. doi: 10.1007/s00220-006-0124-z. Google Scholar [7] D. X. Kong and T. Yang, Asymptotic behavior of global classical solutions of quasilinear hyperbolic systems,, Comm. Partial Differemtial Equations, 28 (2003), 1203. doi: 10.1081/PDE-120021192. Google Scholar [8] D. X. Kong, Q. Y. Sun and Y. Zhou, The equation for time-like extremal surfaces in Minkowski space $R^{2+n}$,, Journal Math. Phy, 47 (2006). doi: 10.1063/1.2158435. Google Scholar [9] P. D. Lax, Hyperbolic systems of conservation laws $\mbox{I\!I}$,, Comm. Pure Appl. Math., 10 (1957), 537. doi: 10.1002/cpa.3160100406. Google Scholar [10] T. T. Li, Global Classical Solutions for Quasilinear Hyperbolic Systems,, Research in Applied Mathematics, (1994). Google Scholar [11] T. T. Li and W. C. Yu, Boundary Value Problems for Quasilinear Hyperbolic Systems,, Duke University Mathematics Series V, (1985). Google Scholar [12] T. T. Li, Y. Zhou and D. X. Kong, Weak linear degeneracy and global classical solutions for general quasilinear hyperbolic systems,, Comm. Partial Differential Equations, 19 (1994), 1263. doi: 10.1080/03605309408821055. Google Scholar [13] T. T. Li, Y. Zhou and D. X. Kong, Global classical solutions for general quasilinear hyperbolic systems with decay initial data,, Nonlinear Analysis, 28 (1997), 1299. doi: 10.1016/0362-546X(95)00228-N. Google Scholar [14] C. M. Liu and P. Qu, Existence and stability of traveling wave solutions to first-order quasilinear hyperbolic systems,, J. Math. Pures Appl., 100 (2013), 34. doi: 10.1016/j.matpur.2012.10.011. Google Scholar [15] J. L. Liu and Y. Zhou, Asymptotic behaviour of global classical solutions of diagonalizable quasilinear hyperbolic systems,, Math. Meth. Appl. Sci., 30 (2007), 479. doi: 10.1002/mma.797. Google Scholar [16] J. L. Liu and Y. Zhou, Initial-boundary value problem for the equation of time-like extremal surfaces in Minkowski space,, J. Math. Phys., 49 (2008). doi: 10.1063/1.2890393. Google Scholar [17] J. L. Liu and Y. Zhou, The initial-boundary value problem on a strip for the equation of time-like extremal surfaces in Minkowski space,, Discrete Contin. Dyn. Syst., 23 (2009), 381. doi: 10.3934/dcds.2009.23.381. Google Scholar [18] A. Majda, Compressible Fluid Flow and System of Conservation Laws in Several Space Variables,, Volume 53, (1984). doi: 10.1007/978-1-4612-1116-7. Google Scholar [19] Y. J. Peng and Y. F. Yang, Well-posedness and long-time behavior of Lipschitz solutions to generalized extremal surface equations,, Journal of Mathematical Physics, 52 (2011). doi: 10.1063/1.3591133. Google Scholar [20] B. L. Rozdestvenkii and N. N. Janenko, Systems of Quasilinear Equations and Their Applications to Gas Dynamics,, Translated mathematical monographs 55, (1981). Google Scholar [21] Z. Q. Shao, A note on the asymptotic behavior of global classical solutions of diagonalizable quasilinear hyperbolic systems,, Nonlinear Analysis, 73 (2010), 600. doi: 10.1016/j.na.2010.03.029. Google Scholar [22] Y. Zhou, Global classical solutions to quasilinear hyperbolic systems with weak linear degeneracy,, Chin.Ann.Math., 25 (2004), 37. doi: 10.1142/S0252959904000044. Google Scholar

show all references

References:
 [1] B. M. Barbashov, V. V. Nesterenko and A. M. Chervyakov, General solutions of nonlinear equations in the geometric theory of the relativistic string,, Commun. Math. Phys., 84 (1982), 471. doi: 10.1007/BF01209629. Google Scholar [2] G. Carbou, B. Hanouzet and R .Natalini, Semilinear behavior of totally linearly degenerate hyperbolic systems with relaxation,, J. Differential Equations, 246 (2009), 291. doi: 10.1016/j.jde.2008.05.015. Google Scholar [3] R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves,, Applied Mathematical Sciences, (1976). Google Scholar [4] W. R. Dai and D. X. Kong, Asymptotic behavior of global classical solutions of general quasilinear hyperbolic systems with weakly linear degeneracy,, Chinese Annals of Mathematics, 27B (2006), 263. Google Scholar [5] W. R. Dai and D. X. Kong, Global existence and asymptotic behavior of classical solutions of quasilnear hyperbolic systems with linear degenerate characteristic fields,, J.Differential Equations, 235 (2007), 127. doi: 10.1016/j.jde.2006.12.020. Google Scholar [6] D. X. Kong, Q. Zhang and Q. Zhou, The dynamics of relativistic strings moving in the Minkowski space $R^{1+n}$,, Commun. Math. Phys., 269 (2007), 153. doi: 10.1007/s00220-006-0124-z. Google Scholar [7] D. X. Kong and T. Yang, Asymptotic behavior of global classical solutions of quasilinear hyperbolic systems,, Comm. Partial Differemtial Equations, 28 (2003), 1203. doi: 10.1081/PDE-120021192. Google Scholar [8] D. X. Kong, Q. Y. Sun and Y. Zhou, The equation for time-like extremal surfaces in Minkowski space $R^{2+n}$,, Journal Math. Phy, 47 (2006). doi: 10.1063/1.2158435. Google Scholar [9] P. D. Lax, Hyperbolic systems of conservation laws $\mbox{I\!I}$,, Comm. Pure Appl. Math., 10 (1957), 537. doi: 10.1002/cpa.3160100406. Google Scholar [10] T. T. Li, Global Classical Solutions for Quasilinear Hyperbolic Systems,, Research in Applied Mathematics, (1994). Google Scholar [11] T. T. Li and W. C. Yu, Boundary Value Problems for Quasilinear Hyperbolic Systems,, Duke University Mathematics Series V, (1985). Google Scholar [12] T. T. Li, Y. Zhou and D. X. Kong, Weak linear degeneracy and global classical solutions for general quasilinear hyperbolic systems,, Comm. Partial Differential Equations, 19 (1994), 1263. doi: 10.1080/03605309408821055. Google Scholar [13] T. T. Li, Y. Zhou and D. X. Kong, Global classical solutions for general quasilinear hyperbolic systems with decay initial data,, Nonlinear Analysis, 28 (1997), 1299. doi: 10.1016/0362-546X(95)00228-N. Google Scholar [14] C. M. Liu and P. Qu, Existence and stability of traveling wave solutions to first-order quasilinear hyperbolic systems,, J. Math. Pures Appl., 100 (2013), 34. doi: 10.1016/j.matpur.2012.10.011. Google Scholar [15] J. L. Liu and Y. Zhou, Asymptotic behaviour of global classical solutions of diagonalizable quasilinear hyperbolic systems,, Math. Meth. Appl. Sci., 30 (2007), 479. doi: 10.1002/mma.797. Google Scholar [16] J. L. Liu and Y. Zhou, Initial-boundary value problem for the equation of time-like extremal surfaces in Minkowski space,, J. Math. Phys., 49 (2008). doi: 10.1063/1.2890393. Google Scholar [17] J. L. Liu and Y. Zhou, The initial-boundary value problem on a strip for the equation of time-like extremal surfaces in Minkowski space,, Discrete Contin. Dyn. Syst., 23 (2009), 381. doi: 10.3934/dcds.2009.23.381. Google Scholar [18] A. Majda, Compressible Fluid Flow and System of Conservation Laws in Several Space Variables,, Volume 53, (1984). doi: 10.1007/978-1-4612-1116-7. Google Scholar [19] Y. J. Peng and Y. F. Yang, Well-posedness and long-time behavior of Lipschitz solutions to generalized extremal surface equations,, Journal of Mathematical Physics, 52 (2011). doi: 10.1063/1.3591133. Google Scholar [20] B. L. Rozdestvenkii and N. N. Janenko, Systems of Quasilinear Equations and Their Applications to Gas Dynamics,, Translated mathematical monographs 55, (1981). Google Scholar [21] Z. Q. Shao, A note on the asymptotic behavior of global classical solutions of diagonalizable quasilinear hyperbolic systems,, Nonlinear Analysis, 73 (2010), 600. doi: 10.1016/j.na.2010.03.029. Google Scholar [22] Y. Zhou, Global classical solutions to quasilinear hyperbolic systems with weak linear degeneracy,, Chin.Ann.Math., 25 (2004), 37. doi: 10.1142/S0252959904000044. Google Scholar
 [1] Jong-Shenq Guo, Ying-Chih Lin. Traveling wave solution for a lattice dynamical system with convolution type nonlinearity. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 101-124. doi: 10.3934/dcds.2012.32.101 [2] Anupam Sen, T. Raja Sekhar. Structural stability of the Riemann solution for a strictly hyperbolic system of conservation laws with flux approximation. Communications on Pure & Applied Analysis, 2019, 18 (2) : 931-942. doi: 10.3934/cpaa.2019045 [3] Xi Wang, Zuhan Liu, Ling Zhou. Asymptotic decay for the classical solution of the chemotaxis system with fractional Laplacian in high dimensions. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 4003-4020. doi: 10.3934/dcdsb.2018121 [4] Yue-Jun Peng, Yong-Fu Yang. Long-time behavior and stability of entropy solutions for linearly degenerate hyperbolic systems of rich type. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3683-3706. doi: 10.3934/dcds.2015.35.3683 [5] Alain Hertzog, Antoine Mondoloni. Existence of a weak solution for a quasilinear wave equation with boundary condition. Communications on Pure & Applied Analysis, 2002, 1 (2) : 191-219. doi: 10.3934/cpaa.2002.1.191 [6] H. M. Yin. Optimal regularity of solution to a degenerate elliptic system arising in electromagnetic fields. Communications on Pure & Applied Analysis, 2002, 1 (1) : 127-134. doi: 10.3934/cpaa.2002.1.127 [7] Kun Li, Jianhua Huang, Xiong Li. Traveling wave solutions in advection hyperbolic-parabolic system with nonlocal delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2091-2119. doi: 10.3934/dcdsb.2018227 [8] Chunhua Jin. Global classical solution and stability to a coupled chemotaxis-fluid model with logistic source. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3547-3566. doi: 10.3934/dcds.2018150 [9] Libin Wang. Breakdown of $C^1$ solution to the Cauchy problem for quasilinear hyperbolic systems with characteristics with constant multiplicity. Communications on Pure & Applied Analysis, 2003, 2 (1) : 77-89. doi: 10.3934/cpaa.2003.2.77 [10] Weiguo Zhang, Yan Zhao, Xiang Li. Qualitative analysis to the traveling wave solutions of Kakutani-Kawahara equation and its approximate damped oscillatory solution. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1075-1090. doi: 10.3934/cpaa.2013.12.1075 [11] Út V. Lê. Regularity of the solution of a nonlinear wave equation. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1099-1115. doi: 10.3934/cpaa.2010.9.1099 [12] GUANGBING LI. Positive solution for quasilinear Schrödinger equations with a parameter. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1803-1816. doi: 10.3934/cpaa.2015.14.1803 [13] Zhaoquan Xu, Jiying Ma. Monotonicity, asymptotics and uniqueness of travelling wave solution of a non-local delayed lattice dynamical system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 5107-5131. doi: 10.3934/dcds.2015.35.5107 [14] Bernard Brighi, S. Guesmia. Asymptotic behavior of solution of hyperbolic problems on a cylindrical domain. Conference Publications, 2007, 2007 (Special) : 160-169. doi: 10.3934/proc.2007.2007.160 [15] Tong Li, Kun Zhao. On a quasilinear hyperbolic system in blood flow modeling. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 333-344. doi: 10.3934/dcdsb.2011.16.333 [16] Claudianor Oliveira Alves, Paulo Cesar Carrião, Olímpio Hiroshi Miyagaki. Signed solution for a class of quasilinear elliptic problem with critical growth. Communications on Pure & Applied Analysis, 2002, 1 (4) : 531-545. doi: 10.3934/cpaa.2002.1.531 [17] Xiang-Dong Fang. A positive solution for an asymptotically cubic quasilinear Schrödinger equation. Communications on Pure & Applied Analysis, 2019, 18 (1) : 51-64. doi: 10.3934/cpaa.2019004 [18] Manh Hong Duong, Hoang Minh Tran. On the fundamental solution and a variational formulation for a degenerate diffusion of Kolmogorov type. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3407-3438. doi: 10.3934/dcds.2018146 [19] Vladimir E. Fedorov, Natalia D. Ivanova. Identification problem for a degenerate evolution equation with overdetermination on the solution semigroup kernel. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 687-696. doi: 10.3934/dcdss.2016022 [20] Dominika Pilarczyk. Asymptotic stability of singular solution to nonlinear heat equation. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 991-1001. doi: 10.3934/dcds.2009.25.991

2018 Impact Factor: 1.143