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

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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
DOIs
Publication statusPublished - 2000 Jan 1

Keywords

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

ASJC Scopus subject areas

  • Analysis
  • Computational Mathematics
  • Applied Mathematics

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