Abstract
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.
| 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
On the long-time limit of positive solutions to the degenerate logistic equation. / Du, Yihong; Yamada, Yoshio.
In: Discrete and Continuous Dynamical Systems, Vol. 25, No. 1, 09.2009, p. 123-132.Research output: Contribution to journal › Article
}
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 -