### Abstract

We investigate the asymptotic behaviors of solutions of the initial-boundary value problem to the generalized Burgers equation u_{t} + f(u)_{x} = u_{xx} on the half-line with the conditions u(0, t) = u_{-}, u(∞, t) = u_{+}, where the corresponding Cauchy problem admits the rarefaction wave as an asymptotic state. In the present problem, because of the Dirichlet boundary, the asymptotic states are divided into five cases dependent on the signs of the characteristic speeds f′(u±) of the boundary state u_{-} = u(0) and the far field state u_{+} = u(∞). In all cases both global existence of the solution and the asymptotic behavior are shown without smallness conditions. New wave phenomena are observed. For instance, when f′(u_{-}) < 0 < f′(u_{+}), the solution behaves as the superposition of (a part of) a viscous shock wave as boundary layer and a rarefaction wave propagating away from the boundary.

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

Pages (from-to) | 293-308 |

Number of pages | 16 |

Journal | SIAM Journal on Mathematical Analysis |

Volume | 29 |

Issue number | 2 |

Publication status | Published - 1998 Mar |

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### Keywords

- Asymptotic behavior
- Rarefaction wave
- Viscous shock wave

### ASJC Scopus subject areas

- Mathematics(all)
- Analysis
- Applied Mathematics

### Cite this

*SIAM Journal on Mathematical Analysis*,

*29*(2), 293-308.

**Behaviors of solutions for the Burgers equation with boundary corresponding to rarefaction waves.** / Liu, Tai Ping; Matsumura, Akitaka; Nishihara, Kenji.

Research output: Contribution to journal › Article

*SIAM Journal on Mathematical Analysis*, vol. 29, no. 2, pp. 293-308.

}

TY - JOUR

T1 - Behaviors of solutions for the Burgers equation with boundary corresponding to rarefaction waves

AU - Liu, Tai Ping

AU - Matsumura, Akitaka

AU - Nishihara, Kenji

PY - 1998/3

Y1 - 1998/3

N2 - We investigate the asymptotic behaviors of solutions of the initial-boundary value problem to the generalized Burgers equation ut + f(u)x = uxx on the half-line with the conditions u(0, t) = u-, u(∞, t) = u+, where the corresponding Cauchy problem admits the rarefaction wave as an asymptotic state. In the present problem, because of the Dirichlet boundary, the asymptotic states are divided into five cases dependent on the signs of the characteristic speeds f′(u±) of the boundary state u- = u(0) and the far field state u+ = u(∞). In all cases both global existence of the solution and the asymptotic behavior are shown without smallness conditions. New wave phenomena are observed. For instance, when f′(u-) < 0 < f′(u+), the solution behaves as the superposition of (a part of) a viscous shock wave as boundary layer and a rarefaction wave propagating away from the boundary.

AB - We investigate the asymptotic behaviors of solutions of the initial-boundary value problem to the generalized Burgers equation ut + f(u)x = uxx on the half-line with the conditions u(0, t) = u-, u(∞, t) = u+, where the corresponding Cauchy problem admits the rarefaction wave as an asymptotic state. In the present problem, because of the Dirichlet boundary, the asymptotic states are divided into five cases dependent on the signs of the characteristic speeds f′(u±) of the boundary state u- = u(0) and the far field state u+ = u(∞). In all cases both global existence of the solution and the asymptotic behavior are shown without smallness conditions. New wave phenomena are observed. For instance, when f′(u-) < 0 < f′(u+), the solution behaves as the superposition of (a part of) a viscous shock wave as boundary layer and a rarefaction wave propagating away from the boundary.

KW - Asymptotic behavior

KW - Rarefaction wave

KW - Viscous shock wave

UR - http://www.scopus.com/inward/record.url?scp=0032395447&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0032395447&partnerID=8YFLogxK

M3 - Article

VL - 29

SP - 293

EP - 308

JO - SIAM Journal on Mathematical Analysis

JF - SIAM Journal on Mathematical Analysis

SN - 0036-1410

IS - 2

ER -