Behavior of solutions near the flat hats of stationary solutions for a degenerate parabolic equation

研究成果: Article

3 引用 (Scopus)

抄録

The behavior of solutions u of the degenerate parabolic equation Ut = λ(|uχ|p-2 uχ)χ + u |q-2u(1-|u| r), defined in (0,1) X (0, +∞), is discussed. It is well known that there exists a stationary solution ø which has an open set Ω where it is identically ±1. We call a graph {(χ, ø (χ));χ ∈ Ω} flat hats. We investigate the behavior of u(χ,t) near (χ,t) ∈ Ω x [0,+∞) where |u(χ, t) - ø(χ)| is very small. We will give a sufficient condition for initial data Uo that the intersection points between the flat hats of ø and u never change as a function of t along u(',t;uo). Even if the condition failed, it is also proved that the changing area of the intersection points is uniformly bounded for t. Moreover we study stability properties for the positive stationary solution and the sign-changing stationary solutions.

元の言語English
ページ(範囲)678-692
ページ数15
ジャーナルSIAM Journal on Mathematical Analysis
31
発行部数3
出版物ステータスPublished - 2000 2
外部発表Yes

ASJC Scopus subject areas

  • Mathematics(all)
  • Analysis
  • Applied Mathematics

これを引用

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title = "Behavior of solutions near the flat hats of stationary solutions for a degenerate parabolic equation",
abstract = "The behavior of solutions u of the degenerate parabolic equation Ut = λ(|uχ|p-2 uχ)χ + u |q-2u(1-|u| r), defined in (0,1) X (0, +∞), is discussed. It is well known that there exists a stationary solution {\o} which has an open set Ω where it is identically ±1. We call a graph {(χ, {\o} (χ));χ ∈ Ω} flat hats. We investigate the behavior of u(χ,t) near (χ,t) ∈ Ω x [0,+∞) where |u(χ, t) - {\o}(χ)| is very small. We will give a sufficient condition for initial data Uo that the intersection points between the flat hats of {\o} and u never change as a function of t along u(',t;uo). Even if the condition failed, it is also proved that the changing area of the intersection points is uniformly bounded for t. Moreover we study stability properties for the positive stationary solution and the sign-changing stationary solutions.",
keywords = "Comparison theorem, Degenerate parabolic equation, Flat hat, Intersection comparison, p-Laplace operator",
author = "Shingo Takeuchi",
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pages = "678--692",
journal = "SIAM Journal on Mathematical Analysis",
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T1 - Behavior of solutions near the flat hats of stationary solutions for a degenerate parabolic equation

AU - Takeuchi, Shingo

PY - 2000/2

Y1 - 2000/2

N2 - The behavior of solutions u of the degenerate parabolic equation Ut = λ(|uχ|p-2 uχ)χ + u |q-2u(1-|u| r), defined in (0,1) X (0, +∞), is discussed. It is well known that there exists a stationary solution ø which has an open set Ω where it is identically ±1. We call a graph {(χ, ø (χ));χ ∈ Ω} flat hats. We investigate the behavior of u(χ,t) near (χ,t) ∈ Ω x [0,+∞) where |u(χ, t) - ø(χ)| is very small. We will give a sufficient condition for initial data Uo that the intersection points between the flat hats of ø and u never change as a function of t along u(',t;uo). Even if the condition failed, it is also proved that the changing area of the intersection points is uniformly bounded for t. Moreover we study stability properties for the positive stationary solution and the sign-changing stationary solutions.

AB - The behavior of solutions u of the degenerate parabolic equation Ut = λ(|uχ|p-2 uχ)χ + u |q-2u(1-|u| r), defined in (0,1) X (0, +∞), is discussed. It is well known that there exists a stationary solution ø which has an open set Ω where it is identically ±1. We call a graph {(χ, ø (χ));χ ∈ Ω} flat hats. We investigate the behavior of u(χ,t) near (χ,t) ∈ Ω x [0,+∞) where |u(χ, t) - ø(χ)| is very small. We will give a sufficient condition for initial data Uo that the intersection points between the flat hats of ø and u never change as a function of t along u(',t;uo). Even if the condition failed, it is also proved that the changing area of the intersection points is uniformly bounded for t. Moreover we study stability properties for the positive stationary solution and the sign-changing stationary solutions.

KW - Comparison theorem

KW - Degenerate parabolic equation

KW - Flat hat

KW - Intersection comparison

KW - p-Laplace operator

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M3 - Article

AN - SCOPUS:0034370909

VL - 31

SP - 678

EP - 692

JO - SIAM Journal on Mathematical Analysis

JF - SIAM Journal on Mathematical Analysis

SN - 0036-1410

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ER -