### Abstract

We study the long-time behavior of positive solutions to the problem u _{t} - Δu = au - b(x)u^{p} in (0, ∞) × Ω, Bu = 0 on (0, ∞) × ∂Ω, where a is a real parameter, 6 ≥ 0 is in C^{μ}(Ω̄) and p > 1 is a constant, Ω is a C^{2+μ} bounded domain in R^{N} (N ≥ 2), the boundary operator B is of the standard Dirichlet, Neumann or Robyn type. Under the assumption that Ω̄_{o} := {x ∈ Ω : b(x) = 0} has non-empty interior, is connected, has smooth boundary and is contained in ω, it is shown in [8] that when a ≥ λ ^{d}
_{1}(Ω_{o}), for any fixed x ∈ Ω̄_{o}, lim_{t→∞} u(t, x) = ∞, and for any fixed x ∈ Ω̄ \ Ω̄_{o}, lim̄_{t→∞}u (t, x) ≤ Ū_{a}(x), lim _{t→∞}u(t, x) ≥ U_{a}(x), where U_{a} and Ū_{a} denote respectively the minimal and maximal positive solutions of the boundary blow-up problem -Δu = au - b(x)u^{p} in Ω \ Ω̄_{o}, Bu = 0 on ∂Ω, u = ∞ on ∂Ω_{o-} The main purpose of this paper is to show that, under the above assumptions, lim u(t, x) = U_{a}(x), ∀x ∈ Ω̄\Ω̄_{o}. t→∞ This proves a conjecture stated in [8]. Some extensions of this result are also discussed.

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

Pages (from-to) | 123-132 |

Number of pages | 10 |

Journal | Discrete and Continuous Dynamical Systems |

Volume | 25 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2009 Sep |

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### Keywords

- Asymptotic behavior
- Boundary blow-up
- Logistic equation

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics
- Analysis

### Cite this

*Discrete and Continuous Dynamical Systems*,

*25*(1), 123-132. https://doi.org/10.3934/dcds.2009.25.123

**On the long-time limit of positive solutions to the degenerate logistic equation.** / Du, Yihong; Yamada, Yoshio.

Research output: Contribution to journal › Article

*Discrete and Continuous Dynamical Systems*, vol. 25, no. 1, pp. 123-132. https://doi.org/10.3934/dcds.2009.25.123

}

TY - JOUR

T1 - On the long-time limit of positive solutions to the degenerate logistic equation

AU - Du, Yihong

AU - Yamada, Yoshio

PY - 2009/9

Y1 - 2009/9

N2 - We study the long-time behavior of positive solutions to the problem u t - Δu = au - b(x)up in (0, ∞) × Ω, Bu = 0 on (0, ∞) × ∂Ω, where a is a real parameter, 6 ≥ 0 is in Cμ(Ω̄) and p > 1 is a constant, Ω is a C2+μ bounded domain in RN (N ≥ 2), the boundary operator B is of the standard Dirichlet, Neumann or Robyn type. Under the assumption that Ω̄o := {x ∈ Ω : b(x) = 0} has non-empty interior, is connected, has smooth boundary and is contained in ω, it is shown in [8] that when a ≥ λ d 1(Ωo), for any fixed x ∈ Ω̄o, limt→∞ u(t, x) = ∞, and for any fixed x ∈ Ω̄ \ Ω̄o, lim̄t→∞u (t, x) ≤ Ūa(x), lim t→∞u(t, x) ≥ Ua(x), where Ua and Ūa denote respectively the minimal and maximal positive solutions of the boundary blow-up problem -Δu = au - b(x)up in Ω \ Ω̄o, Bu = 0 on ∂Ω, u = ∞ on ∂Ωo- The main purpose of this paper is to show that, under the above assumptions, lim u(t, x) = Ua(x), ∀x ∈ Ω̄\Ω̄o. t→∞ This proves a conjecture stated in [8]. Some extensions of this result are also discussed.

AB - We study the long-time behavior of positive solutions to the problem u t - Δu = au - b(x)up in (0, ∞) × Ω, Bu = 0 on (0, ∞) × ∂Ω, where a is a real parameter, 6 ≥ 0 is in Cμ(Ω̄) and p > 1 is a constant, Ω is a C2+μ bounded domain in RN (N ≥ 2), the boundary operator B is of the standard Dirichlet, Neumann or Robyn type. Under the assumption that Ω̄o := {x ∈ Ω : b(x) = 0} has non-empty interior, is connected, has smooth boundary and is contained in ω, it is shown in [8] that when a ≥ λ d 1(Ωo), for any fixed x ∈ Ω̄o, limt→∞ u(t, x) = ∞, and for any fixed x ∈ Ω̄ \ Ω̄o, lim̄t→∞u (t, x) ≤ Ūa(x), lim t→∞u(t, x) ≥ Ua(x), where Ua and Ūa denote respectively the minimal and maximal positive solutions of the boundary blow-up problem -Δu = au - b(x)up in Ω \ Ω̄o, Bu = 0 on ∂Ω, u = ∞ on ∂Ωo- The main purpose of this paper is to show that, under the above assumptions, lim u(t, x) = Ua(x), ∀x ∈ Ω̄\Ω̄o. t→∞ This proves a conjecture stated in [8]. Some extensions of this result are also discussed.

KW - Asymptotic behavior

KW - Boundary blow-up

KW - Logistic equation

UR - http://www.scopus.com/inward/record.url?scp=70349849523&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=70349849523&partnerID=8YFLogxK

U2 - 10.3934/dcds.2009.25.123

DO - 10.3934/dcds.2009.25.123

M3 - Article

VL - 25

SP - 123

EP - 132

JO - Discrete and Continuous Dynamical Systems- Series A

JF - Discrete and Continuous Dynamical Systems- Series A

SN - 1078-0947

IS - 1

ER -