### Abstract

The behavior of solutions u of the degenerate parabolic equation Ut = λ(|uχ|^{p-2} uχ)χ + u |^{q-2}u(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 U_{o} 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 language | English |
---|---|

Pages (from-to) | 678-692 |

Number of pages | 15 |

Journal | SIAM Journal on Mathematical Analysis |

Volume | 31 |

Issue number | 3 |

Publication status | Published - 2000 Feb |

Externally published | Yes |

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)
- Analysis
- Applied Mathematics

### Cite this

**Behavior of solutions near the flat hats of stationary solutions for a degenerate parabolic equation.** / Takeuchi, Shingo.

Research output: Contribution to journal › Article

*SIAM Journal on Mathematical Analysis*, vol. 31, no. 3, pp. 678-692.

}

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

UR - http://www.scopus.com/inward/record.url?scp=0034370909&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0034370909&partnerID=8YFLogxK

M3 - Article

VL - 31

SP - 678

EP - 692

JO - SIAM Journal on Mathematical Analysis

JF - SIAM Journal on Mathematical Analysis

SN - 0036-1410

IS - 3

ER -