Abstract
This article is involved with the asymptotic behavior of solutions for nonlinear hyperbolic system with external friction. The global existence of classical solutions is proven, and Lp convergence rates are obtained. Compared with the results obtained by Hsiao and Liu, better convergence rates are obtained in this article.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Cao W, Huang F, Li T, et al. Global entropy solutions to an inhomogeneous isentropic compressible Euler system. Acta Math Sci, 2016, 36B: 1215–1224
Geng S, Zhang L. L p-convergence rates to nonlinear diffusion waves for quasilinear equations with nonlinear damping. Z Angew Math Phys, 2015, 66: 31–50
He C, Huang F, Yong Y. Stability of planar diffusion wave for nonlinear evolution equation. Science China Mathematics, 2012, 55: 337–352
Hsiao L, Liu T. Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping. Comm Math Phys, 1992, 143: 599–605
Hsiao L, Liu T. Nonlinear diffusion phenomena of nonlinear hyperbolic system. Chin Ann Math, Ser B, 1993, 14: 465–480
Ma H, Mei M. Best asymptotic profile for linear damped p-system with boundary effect. J Differential Equations, 2010, 249: 446–484
Marcati P, Mei M, Rubino B. Optimal convergence rates to diffusion waves for solutions of the hyperbolic conservation laws with damping. J Math Fluid Mech, 2005, 7: S224–S240
Marcati P, Nishihara K. The L p - L q estimates of solutions to one-dimensional damped wave equations and their application to the compressible flow through porous media. J Differential Equations, 2003, 191: 445–469
Matsumura A. Global existence and asymptotics of the solutions of the second-order quasilinear hyperbolic equations with the first-order dissipation. Publ RIMS, Kyoto Univ, 1977, 13: 349–379
Mei M. Nonlinear diffusion waves for hyperbolic p-system with nonlinear damping. J Differential Equations, 2009, 247: 1275–1296
Mei M. Best asymptotic profile for hyperbolic p-system with damping. SIAM J Math Anal, 2010, 42: 1–23
Nishida T. Nonlinear hyperbolic equations and related topics in fluid dynamics. Publications Mathématique D’ orsay 78–02, Départment de Mathématique, Université de Paris-Sud, 1978
Nishihara K. Convergence rates to nonlinear diffusion waves for solutions of system of hyperbolic conservation laws with damping. J Differential Equations, 1996, 131: 171–188
Nishihara K. Asymptotic behavior of solutions of quasilinear hyperbolic equations with linear damping. J Differential Equations, 1997, 137: 384–396
Nishihara K, Wang W, Yang T. L p-convergence rate to nonlinear diffusion waves for p-system with damping. J Differential Equations, 2000, 161: 191–218
Zhao H. Convergence to strong nonlinear diffusion waves for solutions of p-system with damping. J Differential Equations, 2001, 174: 200–236
Zhu C, Jiang M. Lp-decay rates to nonlinear diffusion waves for p-system with nonlinear damping. Sci Chin, Ser A, 2006, 49: 721–739
Author information
Authors and Affiliations
Corresponding author
Additional information
The first author was supported by the National Natural Science Foundation of China (11701489, 11871412), the Hunan Provincial Natural Science Foundation of China (2018JJ2373, 2018JJ3481).
Rights and permissions
About this article
Cite this article
Geng, S., Tang, Y. Convergence Rates to Nonlinear Diffusive Waves for Solutions to Nonlinear Hyperbolic System. Acta Math Sci 39, 46–56 (2019). https://doi.org/10.1007/s10473-019-0105-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10473-019-0105-x