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Decay rate for travelling waves of a relaxation model. (English) Zbl 0908.35074

The Cauchy problem for a hyperbolic system of two equations is considered. Initial conditions go to constants as \(x\rightarrow\infty\). This system contains the nonlinear term in its right-hand side and a small parameter on the left. So, the system models wave propagation with relaxation. The authors investigate the long time behavior of solution of the scaled system that corresponds to the original one provided that the small parameter goes to zero. They denote the travelling wave solutions with shock wave profile, discuss their existence, and prove convergence of disturbed solutions to the travelling wave one by estimating their decay rate.

MSC:

35L65 Hyperbolic conservation laws
35B40 Asymptotic behavior of solutions to PDEs
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[1] Clarke, J. F., Gas dynamics with relaxation effects, Rep. Prog. Phys., 41, 807-863 (1978)
[2] Chen, G.-Q.; Levermore, C. D.; Liu, T. P., Hyperbolic conservation laws with stiff relaxation terms and entropy, Comm. Pure Appl. Math., 47, 787-830 (1993) · Zbl 0806.35112
[3] Il’in, A. M.; Olienik, O. A., Asymptotic behavior of solutions of the Cauchy problem for certain quasilinear equations for large time, Mat. Sb., 51, 191-216 (1960)
[4] Jin, Shi; Xin, Zhouping, The relaxing schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math., 48, 555-563 (1995) · Zbl 0826.65078
[5] Jones, C.; Gardner, R.; Kapitula, T., Stability of travelling waves for non-convex scalar viscous conservation laws, Comm. Pure Appl. Math., 46, 505-526 (1993) · Zbl 0791.35078
[6] Liu, T. P., Hyperbolic conservation laws with relaxation, Comm. Math. Phys., 108, 153-175 (1987) · Zbl 0633.35049
[7] Whitham, G., Linear and Nonlinear Waves (1974), Wiley-Interscience: Wiley-Interscience New York · Zbl 0373.76001
[8] H. L. Liu, H. L. Liu, J. H. Wang, T. Yang, Asymptotic stability of shock profiles for non-convex convection-diffusion equation, Appl. Math. Lett.; H. L. Liu, H. L. Liu, J. H. Wang, T. Yang, Asymptotic stability of shock profiles for non-convex convection-diffusion equation, Appl. Math. Lett.
[9] H. L. Liu, J. Wang, 1995, Asymptotic stability of travelling wave solutions of a hyperbolic system with relaxation terms; H. L. Liu, J. Wang, 1995, Asymptotic stability of travelling wave solutions of a hyperbolic system with relaxation terms
[10] R. J. Leveque, J. Wang, A linear hyperbolic system with stiff source terms, Proc. Fourth Int. Conf. on Hyperbolic Problems, Taormina, Italy, 1992; R. J. Leveque, J. Wang, A linear hyperbolic system with stiff source terms, Proc. Fourth Int. Conf. on Hyperbolic Problems, Taormina, Italy, 1992 · Zbl 0921.35086
[11] Matsumura, A.; Nishihara, K., Asymptotic stability of travelling waves of scalar viscous conservation laws with non-convex nonlinearity, Comm. Math. Phys., 165, 83-96 (1994) · Zbl 0811.35080
[12] Nishida, T., Nonlinear Hyperbolic Equations and Related Topics in Fluid Dynamics. Nonlinear Hyperbolic Equations and Related Topics in Fluid Dynamics, Publ. Math. Orsay, Vol. 78 (1978), Univ. Paris-Sud · Zbl 0392.76065
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