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

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3 Citations (Scopus)

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 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.

Original languageEnglish
Pages (from-to)678-692
Number of pages15
JournalSIAM Journal on Mathematical Analysis
Volume31
Issue number3
Publication statusPublished - 2000 Feb
Externally publishedYes

Keywords

  • Comparison theorem
  • Degenerate parabolic equation
  • Flat hat
  • Intersection comparison
  • p-Laplace operator

ASJC Scopus subject areas

  • Mathematics(all)
  • Analysis
  • Applied Mathematics

Cite this

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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",
year = "2000",
month = "2",
language = "English",
volume = "31",
pages = "678--692",
journal = "SIAM Journal on Mathematical Analysis",
issn = "0036-1410",
publisher = "Society for Industrial and Applied Mathematics Publications",
number = "3",

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TY - JOUR

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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VL - 31

SP - 678

EP - 692

JO - SIAM Journal on Mathematical Analysis

JF - SIAM Journal on Mathematical Analysis

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