## Abstract

We consider the asymptotic behavior of the solution of quasilinear hyperbolic equation with linear dampingV_{tt}-a(V_{x})_{x}+αV _{t}=0,(x,t)∈R×(0,∞), ((*))subsequent to [K. Nishihara,J. Differential Equations131(1996), 171-188]. In that article, the system with dampingv_{t}-u_{x}=0,u_{t}+p(v) _{x}=-αu,p′(v)<0(v>0) was treated, and the convergence rates to the diffusion wave by [L. Hsiao and T.-P. Liu,Comm. Math. Phys.143(1992), 599-605 and L. Hsiao and T.-P. Liu,Chinese Ann. Math. Ser. B14(1993), 465-480] were improved. For initial data (v_{0},u_{0})(x) satisfying (v_{0},u_{0})(±∞)=(v_{0},0) and ∫^{∞}
_{-∞}(v_{0}(x)-v_{0})dx=0 this system is reduced to the quasilinear hyperbolic equation with linear dampingV_{tt}+(p(v_{0}+V_{x})-p(v_{0})) _{x}+αV_{t}=0, and the decay rates V(t)_{L}
^{∞}=O(t^{-1/2}), V_{x}(t)_{L}
^{∞}=O(t^{-1}), V_{t}(t)_{L}
^{∞}=O(t^{-3/2}), were obtained provided that (V,V_{t}) _{t=0}∈(H^{3}×H^{2}) is small and inL^{1}×L^{1}. From these decay rates,Vis expected to behave as a solution to the parabolic equation. The aims of this paper are to seek the asymptotic profileφ(x,t) satisfying φ(t)_{L}
^{∞}=O(t^{-1/2}), φ_{x}(t)_{L}
^{∞}=O(t^{-1}), φ_{t}(t)_{L}
^{∞}=O(t^{-3/2}), and to show that the solution of (*) satisfies (V-φ)(t)_{L}
^{∞}=O(t^{-1}), (V-φ)_{x}(t)_{L}
^{∞}=O(t^{-3/2}), (V-φ)_{t}(t)_{L}
^{∞}=O(t^{-2}). The proof is based on the elementary energy method and the Green function method for the parabolic equation.

Original language | English |
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Pages (from-to) | 384-395 |

Number of pages | 12 |

Journal | Journal of Differential Equations |

Volume | 137 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1997 Jul 1 |

## ASJC Scopus subject areas

- Analysis