### Abstract

We consider the asymptotic behaviour of solutions to the p-system with linear damping on the half-line R_{+}=(0, ∞), v_{t}-u_{x}=0,u_{t}+p(v)_{x}=-αu, with the Dirichlet boundary condition u single rule fence sign ;_{x=0}=0 or the Neumann boundary condition u_{x}single rule fence sign _{x=0}=0. The initial date (v_{0}, u_{0})(x) has the constant state (v_{+}, u_{+}) at x=∞. L. Hsiao and T.-P. Liu [Commun. Math. Phys.143 (1992), 599-605] have shown that the solution to the corresponding Cauchy problem behaves like diffusion wave, and K. Nishihara [J. Differential Equations131 (1996), 171-188; 137 (1997), 384-395] has proved its optimal convergence rate. Our main concern in this paper is the boundary effect. In the case of null-Dirichlet boundary condition on u, the solution (v, u) is proved to tend to (v_{+}, 0) as t tends to infinity. Its optimal convergence rate is also obtained by using the Green function of the diffusion equation with constant coefficients. In the case of null-Neumann boundary condition on u, v(0, t) is conservative and v(0, t)≡v_{0}(0) by virtue of the first equation, so that v(x, t) is expected to tend to the diffusion wave v(x, t) connecting v_{0}(0) and v_{+}. In fact the solution (v, u)(x, t) is proved to tend to (v(x, t), 0). In the special case v_{0}(0)=v_{+}, the optimal convergence rate is also obtained. However, this is not known in the case v_{0}(0)≠v_{+}.

Original language | English |
---|---|

Pages (from-to) | 439-458 |

Number of pages | 20 |

Journal | Journal of Differential Equations |

Volume | 156 |

Issue number | 2 |

Publication status | Published - 1999 Aug 10 |

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### ASJC Scopus subject areas

- Analysis

### Cite this

*Journal of Differential Equations*,

*156*(2), 439-458.

**Boundary effect on asymptotic behaviour of solutions to the p-system with linear damping.** / Nishihara, Kenji; Yang, Tong.

Research output: Contribution to journal › Article

*Journal of Differential Equations*, vol. 156, no. 2, pp. 439-458.

}

TY - JOUR

T1 - Boundary effect on asymptotic behaviour of solutions to the p-system with linear damping

AU - Nishihara, Kenji

AU - Yang, Tong

PY - 1999/8/10

Y1 - 1999/8/10

N2 - We consider the asymptotic behaviour of solutions to the p-system with linear damping on the half-line R+=(0, ∞), vt-ux=0,ut+p(v)x=-αu, with the Dirichlet boundary condition u single rule fence sign ;x=0=0 or the Neumann boundary condition uxsingle rule fence sign x=0=0. The initial date (v0, u0)(x) has the constant state (v+, u+) at x=∞. L. Hsiao and T.-P. Liu [Commun. Math. Phys.143 (1992), 599-605] have shown that the solution to the corresponding Cauchy problem behaves like diffusion wave, and K. Nishihara [J. Differential Equations131 (1996), 171-188; 137 (1997), 384-395] has proved its optimal convergence rate. Our main concern in this paper is the boundary effect. In the case of null-Dirichlet boundary condition on u, the solution (v, u) is proved to tend to (v+, 0) as t tends to infinity. Its optimal convergence rate is also obtained by using the Green function of the diffusion equation with constant coefficients. In the case of null-Neumann boundary condition on u, v(0, t) is conservative and v(0, t)≡v0(0) by virtue of the first equation, so that v(x, t) is expected to tend to the diffusion wave v(x, t) connecting v0(0) and v+. In fact the solution (v, u)(x, t) is proved to tend to (v(x, t), 0). In the special case v0(0)=v+, the optimal convergence rate is also obtained. However, this is not known in the case v0(0)≠v+.

AB - We consider the asymptotic behaviour of solutions to the p-system with linear damping on the half-line R+=(0, ∞), vt-ux=0,ut+p(v)x=-αu, with the Dirichlet boundary condition u single rule fence sign ;x=0=0 or the Neumann boundary condition uxsingle rule fence sign x=0=0. The initial date (v0, u0)(x) has the constant state (v+, u+) at x=∞. L. Hsiao and T.-P. Liu [Commun. Math. Phys.143 (1992), 599-605] have shown that the solution to the corresponding Cauchy problem behaves like diffusion wave, and K. Nishihara [J. Differential Equations131 (1996), 171-188; 137 (1997), 384-395] has proved its optimal convergence rate. Our main concern in this paper is the boundary effect. In the case of null-Dirichlet boundary condition on u, the solution (v, u) is proved to tend to (v+, 0) as t tends to infinity. Its optimal convergence rate is also obtained by using the Green function of the diffusion equation with constant coefficients. In the case of null-Neumann boundary condition on u, v(0, t) is conservative and v(0, t)≡v0(0) by virtue of the first equation, so that v(x, t) is expected to tend to the diffusion wave v(x, t) connecting v0(0) and v+. In fact the solution (v, u)(x, t) is proved to tend to (v(x, t), 0). In the special case v0(0)=v+, the optimal convergence rate is also obtained. However, this is not known in the case v0(0)≠v+.

UR - http://www.scopus.com/inward/record.url?scp=0033543057&partnerID=8YFLogxK

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M3 - Article

VL - 156

SP - 439

EP - 458

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 2

ER -