Differential and Integral Equations
- Differential Integral Equations
- Volume 14, Number 6 (2001), 751-768.
The uniqueness of the stable positive solution for a class of superlinear indefinite reaction diffusion equations
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.
Differential Integral Equations, Volume 14, Number 6 (2001), 751-768.
First available in Project Euclid: 21 December 2012
Permanent link to this document
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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
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