### Abstract

Global existence and asymptotic behavior of solutions for degenerate parabolic equations including u_{t} = λ div (|∇u|^{p-2} ∇u) + |u|^{q-2}u(1 - |u|^{r}) are studied, where λ is a positive parameter; p > 2, q ≥ 2 and r > 0 are constants. In particular, the behavior of solutions for the initial data close to a maximal stationary solution is discussed. It is shown that the maximal stationary solution is asymptotically stable if p ≥ q and stable if p < q. For the latter case, some remarks on the attractivity of maximal stationary solution are also given.

Original language | English |
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Pages (from-to) | 1715-1724 |

Number of pages | 10 |

Journal | Nonlinear Analysis, Theory, Methods and Applications |

Volume | 47 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2001 Aug |

Externally published | Yes |

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### Keywords

- Asymptotic behavior
- Degenerate parabolic equations
- Logistic reaction
- p-Laplacian
- Stability

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics
- Mathematics(all)

### Cite this

**Asymptotic behavior of solutions for partial differential equations with degenerate diffusion and logistic reaction.** / Takeuchi, Shingo.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Asymptotic behavior of solutions for partial differential equations with degenerate diffusion and logistic reaction

AU - Takeuchi, Shingo

PY - 2001/8

Y1 - 2001/8

N2 - Global existence and asymptotic behavior of solutions for degenerate parabolic equations including ut = λ div (|∇u|p-2 ∇u) + |u|q-2u(1 - |u|r) are studied, where λ is a positive parameter; p > 2, q ≥ 2 and r > 0 are constants. In particular, the behavior of solutions for the initial data close to a maximal stationary solution is discussed. It is shown that the maximal stationary solution is asymptotically stable if p ≥ q and stable if p < q. For the latter case, some remarks on the attractivity of maximal stationary solution are also given.

AB - Global existence and asymptotic behavior of solutions for degenerate parabolic equations including ut = λ div (|∇u|p-2 ∇u) + |u|q-2u(1 - |u|r) are studied, where λ is a positive parameter; p > 2, q ≥ 2 and r > 0 are constants. In particular, the behavior of solutions for the initial data close to a maximal stationary solution is discussed. It is shown that the maximal stationary solution is asymptotically stable if p ≥ q and stable if p < q. For the latter case, some remarks on the attractivity of maximal stationary solution are also given.

KW - Asymptotic behavior

KW - Degenerate parabolic equations

KW - Logistic reaction

KW - p-Laplacian

KW - Stability

UR - http://www.scopus.com/inward/record.url?scp=0035424378&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0035424378&partnerID=8YFLogxK

U2 - 10.1016/S0362-546X(01)00304-2

DO - 10.1016/S0362-546X(01)00304-2

M3 - Article

AN - SCOPUS:0035424378

VL - 47

SP - 1715

EP - 1724

JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

SN - 0362-546X

IS - 3

ER -