Differential and Integral Equations

The uniqueness of the stable positive solution for a class of superlinear indefinite reaction diffusion equations

Rosa Gómez-Reñasco and Julián López-Gómez

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Abstract

In this paper we characterize the existence and prove the uniqueness of the stable positive steady-state for a general class of superlinear indefinite reaction diffusion equations in the absence of $L_\infty$ a priori bounds. More precisely, it will be shown that the model possesses a linearly stable positive steady-state if, and only if, the trivial solution is linearly unstable and the model possesses some positive steady-state. Moreover, it is unique if it exists. Actually, the minimal positive steady-state provides us with the unique linearly stable positive steady-state of the model. This is an extremely striking result since these problems can have an arbitrarily large number of positive steady-states as a result of having spatial inhomogeneities or varying the geometry of the support domain where the reaction takes place.

Article information

Source
Differential Integral Equations, Volume 14, Number 6 (2001), 751-768.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356123245

Mathematical Reviews number (MathSciNet)
MR1826959

Zentralblatt MATH identifier
1161.35420

Subjects
Primary: 35K57: Reaction-diffusion equations
Secondary: 35B60: Continuation and prolongation of solutions [See also 58A15, 58A17, 58Hxx] 35K15: Initial value problems for second-order parabolic equations

Citation

Gómez-Reñasco, Rosa; López-Gómez, Julián. The uniqueness of the stable positive solution for a class of superlinear indefinite reaction diffusion equations. Differential Integral Equations 14 (2001), no. 6, 751--768. https://projecteuclid.org/euclid.die/1356123245


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