### Abstract

In this paper we are concerned with the reaction-diffusion equation u_{t} = Δu + f(u) in a ball of R^{N} with Dirichlet boundary condition. We assume that f satisfies the concave-convex condition. A typical example is f(u) = |u|^{q-1}u + |u|^{p-1}u (0 < q < 1 < p < (N+2)/(N-2)). First we obtain the complete structure of positive solutions to the stationary problem; Δφ + f(φ) = 0. Next we state the relations between this structure and time-depending behaviors of nonnegative solutions (global existence or blow up) to the non-stationary problem.

Original language | English |
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Pages (from-to) | 789-800 |

Number of pages | 12 |

Journal | Nonlinear Analysis, Theory, Methods and Applications |

Volume | 47 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2001 Aug 1 |

Event | 3rd World Congres of Nonlinear Analysts - Catania, Sicily, Italy Duration: 2000 Jul 19 → 2000 Jul 26 |

### Fingerprint

### Keywords

- Blow up
- Comparison theorem
- Global solution
- Non-Lipschitzian nonlinearity
- Radially symmetric solution
- Reaction-diffusion equation

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

**Stabilization of solutions of the diffusion equation with a non-lipschitz reaction term.** / Kuto, Kousuke.

Research output: Contribution to journal › Conference article

}

TY - JOUR

T1 - Stabilization of solutions of the diffusion equation with a non-lipschitz reaction term

AU - Kuto, Kousuke

PY - 2001/8/1

Y1 - 2001/8/1

N2 - In this paper we are concerned with the reaction-diffusion equation ut = Δu + f(u) in a ball of RN with Dirichlet boundary condition. We assume that f satisfies the concave-convex condition. A typical example is f(u) = |u|q-1u + |u|p-1u (0 < q < 1 < p < (N+2)/(N-2)). First we obtain the complete structure of positive solutions to the stationary problem; Δφ + f(φ) = 0. Next we state the relations between this structure and time-depending behaviors of nonnegative solutions (global existence or blow up) to the non-stationary problem.

AB - In this paper we are concerned with the reaction-diffusion equation ut = Δu + f(u) in a ball of RN with Dirichlet boundary condition. We assume that f satisfies the concave-convex condition. A typical example is f(u) = |u|q-1u + |u|p-1u (0 < q < 1 < p < (N+2)/(N-2)). First we obtain the complete structure of positive solutions to the stationary problem; Δφ + f(φ) = 0. Next we state the relations between this structure and time-depending behaviors of nonnegative solutions (global existence or blow up) to the non-stationary problem.

KW - Blow up

KW - Comparison theorem

KW - Global solution

KW - Non-Lipschitzian nonlinearity

KW - Radially symmetric solution

KW - Reaction-diffusion equation

UR - http://www.scopus.com/inward/record.url?scp=0035425713&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0035425713&partnerID=8YFLogxK

U2 - 10.1016/S0362-546X(01)00223-1

DO - 10.1016/S0362-546X(01)00223-1

M3 - Conference article

AN - SCOPUS:0035425713

VL - 47

SP - 789

EP - 800

JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

SN - 0362-546X

IS - 2

ER -