Abstract
This paper is concerned with the asymptotic behavior of the solution for quasilinear hyperbolic equations with nonlinear damping. The main novelty in this paper is that we obtain the L p(2 ≤ p ≤ +∞) convergence rates of the solution to the quasilinear hyperbolic equations, and we need none of the additional technical assumptions for the nonlinear damping f(v) given by Li and Saxton (Q Appl Math 61:295–313, 2003).
Similar content being viewed by others
References
Duyn C.T., Van Peletier L.A.: A class of similarity solutions of the nonlinear diffusion equation. Nonlinear Anal. 1, 223–233 (1977)
Hsiao L.: Quasilinear Hyperbolic System an Dissipative Mechanisms. Word Scientific, Singapore (1997)
Hsiao L., Liu T.-P.: Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping. Commun. Math. Phys. 143, 599–605 (1992)
Hsiao L., Liu T.-P.: Nonlinear diffusion phenomena of nonlinear hyperbolic system. Chin. Ann. Math. Ser. B 14, 465–480 (1993)
Huang F., Li J., Matsumura A.: Asymptotic stability of combination of viscous contact wave with rarefaction waves for one-dimensional compressible Navier–Stokes system. Arch. Ration. Mech. Anal. 197, 89–116 (2010)
Jiang M., Zhu C.: Convergence rates to nonlinear diffusion waves for p-system with nonlinear damping on quadrant. Discrete Contin. Dyn. Syst. Ser. A 23, 887–918 (2009)
Li H.-L., Saxton K.: Asymptotic behavior of solutions to quasilinear hyperbolic equations with nonlinear damping. Q. Appl. Math. 61, 295–313 (2003)
Lin C.-K., Lin C.-T., Mei M.: Asymptotic behavior of solution to nonlinear damped p-system with boundary effect. Int. J. Numer. Anal. Model. Ser. B 1, 70–92 (2010)
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. 7, S224–S240 (2005)
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. 13, 349–379 (1977)
Mei M.: Nonlinear diffusion waves for hyperbolic p-system with nonlinear damping. J. Differ. Equ. 247, 1275–1296 (2009)
Mei M.: Best asymptotic profile for hyperbolic p-system with damping. SIAM J. Math. Anal. 42, 1–23 (2010)
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. Differ. Equ. 131, 171–188 (1996)
Nishihara K., Wang W.-K., Yang T.: L p -convergence rate to nonlinear diffusion waves for p-system with damping. J. Differ. Equ. 161, 191–218 (2000)
Saxton K., Saxton R., Kosinski W.: On second sound at the critical temperature. Q. Appl. Math. 57, 723–740 (1999)
Saxton K., Saxton R.: Nonlinearity and memory effects in low temperature heat propagation. Arch. Mech. 52, 127–142 (2000)
Zhao H.-J.: Convergence to strong nonlinear diffusion waves for solutions of p-system with damping. J. Differ. Equ. 174, 200–236 (2001)
Zhu C., Jiang M.: L p-decay rates to nonlinear diffusion waves for p-system with nonlinear damping. Sci. China Ser. A 49, 721–739 (2006)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Geng, S., Zhang, L. L p-convergence rates to nonlinear diffusion waves for quasilinear equations with nonlinear damping. Z. Angew. Math. Phys. 66, 31–50 (2015). https://doi.org/10.1007/s00033-013-0392-3
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00033-013-0392-3