### Abstract

We study the large time behavior of small solutions to the Cauchy problem for the Vlasov-Poisson-Fokker-Planck equation, which is a degenerate parabolic equation with nonlocal nonlinearity. We construct finite dimensional invariant manifolds in a neighborhood of the origin in polynomially weighted Sobolev spaces, which enables us to compute systematically the long-time asymptotics for small solutions. To construct invariant manifolds, we make use of the "similarity variables" transformation as in C. E. Wayne's work in 1997, where invariant manifolds for parabolic equations in unbounded domains are constructed.

Original language | English |
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Pages (from-to) | 489-507 |

Number of pages | 19 |

Journal | SIAM Journal on Mathematical Analysis |

Volume | 33 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jan 1 2001 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Computational Mathematics
- Applied Mathematics

### Cite this

**Invariant manifolds and long-time asymptotics for the Vlasov-Poisson-Fokker-Planck equation.** / Kagei, Yoshiyuki.

Research output: Contribution to journal › Article

*SIAM Journal on Mathematical Analysis*, vol. 33, no. 2, pp. 489-507. https://doi.org/10.1137/S0036141000371368

}

TY - JOUR

T1 - Invariant manifolds and long-time asymptotics for the Vlasov-Poisson-Fokker-Planck equation

AU - Kagei, Yoshiyuki

PY - 2001/1/1

Y1 - 2001/1/1

N2 - We study the large time behavior of small solutions to the Cauchy problem for the Vlasov-Poisson-Fokker-Planck equation, which is a degenerate parabolic equation with nonlocal nonlinearity. We construct finite dimensional invariant manifolds in a neighborhood of the origin in polynomially weighted Sobolev spaces, which enables us to compute systematically the long-time asymptotics for small solutions. To construct invariant manifolds, we make use of the "similarity variables" transformation as in C. E. Wayne's work in 1997, where invariant manifolds for parabolic equations in unbounded domains are constructed.

AB - We study the large time behavior of small solutions to the Cauchy problem for the Vlasov-Poisson-Fokker-Planck equation, which is a degenerate parabolic equation with nonlocal nonlinearity. We construct finite dimensional invariant manifolds in a neighborhood of the origin in polynomially weighted Sobolev spaces, which enables us to compute systematically the long-time asymptotics for small solutions. To construct invariant manifolds, we make use of the "similarity variables" transformation as in C. E. Wayne's work in 1997, where invariant manifolds for parabolic equations in unbounded domains are constructed.

UR - http://www.scopus.com/inward/record.url?scp=0036054061&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036054061&partnerID=8YFLogxK

U2 - 10.1137/S0036141000371368

DO - 10.1137/S0036141000371368

M3 - Article

AN - SCOPUS:0036054061

VL - 33

SP - 489

EP - 507

JO - SIAM Journal on Mathematical Analysis

JF - SIAM Journal on Mathematical Analysis

SN - 0036-1410

IS - 2

ER -