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

K. Kuto

doi:10.1016/S0362-546X(01)00223-1

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

K. Kuto^*

^*Corresponding author for this work

School of Fundamental Science and Engineering

Research output: Contribution to journal › Conference article › peer-review

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-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.

Original language	English
Pages (from-to)	789-800
Number of pages	12
Journal	Nonlinear Analysis, Theory, Methods and Applications
Volume	47
Issue number	2
DOIs	https://doi.org/10.1016/S0362-546X(01)00223-1
Publication status	Published - 2001 Aug 1
Event	3rd World Congres of Nonlinear Analysts - Catania, Sicily, Italy Duration: 2000 Jul 19 → 2000 Jul 26

Keywords

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

ASJC Scopus subject areas

Analysis
Applied Mathematics

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

Cite this

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abstract = "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.",

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AU - Kuto, K.

PY - 2001/8/1

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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

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

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JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

IS - 2

T2 - 3rd World Congres of Nonlinear Analysts

Y2 - 19 July 2000 through 26 July 2000

ER -

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

Abstract

Keywords

ASJC Scopus subject areas

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