### Abstract

The initial-boundary value problem on the negative half-line R_{-}[formula]is considered, subsequently to T.-P. Liu and K. Nishihara (1997, J. Differential Equations133, 296-320). Here, the flux f is a smooth function satisfying f(u_{±})=0 and the Oleinik shock condition f(φ)<0 for u_{+}<φ<u_{-} if u_{+}<u_{-} or f(φ)0 for u_{+}φu_{-} if u_{+}<u_{-}. In this situation the corresponding Cauchy problem on the whole line R=(-∞,∞) to (*) has a stationary viscous shock wave φ(x+x_{0}) for any fixed x_{0}. Our aim in this paper is to show that the solution u(x,t) to (*) behaves as φ(x+d(t)) with d(t)=O(lnt) as t→∞ under the suitable smallness conditions. When f=u^{2}/2, the fact was shown by T.-P. Liu and S.-H. Yu (1997, Arch. Rational Mech. Anal.139, 57-82), based on the Hopf-Cole transformation. Our proof is based on the weighted energy method.

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

Pages (from-to) | 535-550 |

Number of pages | 16 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 255 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2001 Mar 15 |

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### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

**Boundary effect on a stationary viscous shock wave for scalar viscous conservation laws.** / Nishihara, Kenji.

Research output: Contribution to journal › Article

*Journal of Mathematical Analysis and Applications*, vol. 255, no. 2, pp. 535-550. https://doi.org/10.1006/jmaa.2000.7255

}

TY - JOUR

T1 - Boundary effect on a stationary viscous shock wave for scalar viscous conservation laws

AU - Nishihara, Kenji

PY - 2001/3/15

Y1 - 2001/3/15

N2 - The initial-boundary value problem on the negative half-line R-[formula]is considered, subsequently to T.-P. Liu and K. Nishihara (1997, J. Differential Equations133, 296-320). Here, the flux f is a smooth function satisfying f(u±)=0 and the Oleinik shock condition f(φ)<0 for u+<φ- if u+- or f(φ)0 for u+φu- if u+-. In this situation the corresponding Cauchy problem on the whole line R=(-∞,∞) to (*) has a stationary viscous shock wave φ(x+x0) for any fixed x0. Our aim in this paper is to show that the solution u(x,t) to (*) behaves as φ(x+d(t)) with d(t)=O(lnt) as t→∞ under the suitable smallness conditions. When f=u2/2, the fact was shown by T.-P. Liu and S.-H. Yu (1997, Arch. Rational Mech. Anal.139, 57-82), based on the Hopf-Cole transformation. Our proof is based on the weighted energy method.

AB - The initial-boundary value problem on the negative half-line R-[formula]is considered, subsequently to T.-P. Liu and K. Nishihara (1997, J. Differential Equations133, 296-320). Here, the flux f is a smooth function satisfying f(u±)=0 and the Oleinik shock condition f(φ)<0 for u+<φ- if u+- or f(φ)0 for u+φu- if u+-. In this situation the corresponding Cauchy problem on the whole line R=(-∞,∞) to (*) has a stationary viscous shock wave φ(x+x0) for any fixed x0. Our aim in this paper is to show that the solution u(x,t) to (*) behaves as φ(x+d(t)) with d(t)=O(lnt) as t→∞ under the suitable smallness conditions. When f=u2/2, the fact was shown by T.-P. Liu and S.-H. Yu (1997, Arch. Rational Mech. Anal.139, 57-82), based on the Hopf-Cole transformation. Our proof is based on the weighted energy method.

UR - http://www.scopus.com/inward/record.url?scp=0035869884&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0035869884&partnerID=8YFLogxK

U2 - 10.1006/jmaa.2000.7255

DO - 10.1006/jmaa.2000.7255

M3 - Article

AN - SCOPUS:0035869884

VL - 255

SP - 535

EP - 550

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 2

ER -