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Stable transition layers in a semilinear boundary value problem. (English) Zbl 0634.35041

This paper is devoted to the proof of the existence and the stability of the solutions of the equation: \[ u_ t=\varepsilon^2u_{xx}+u(1-u)(u-a(x)) \] with some boundary conditions, for \((x,t)\in [0,1]\times [0,+\infty [\). It is a classical mathematical model for several problems of mathematical physics.
Seven theorems are successively established. The principal assumption is that \(a(x)-1/2\) possesses a finite number of simple zeros in the interval \(]0,1[\). The principal results obtained by the authors are: (i) the existence of stable equilibrium solutions; (ii) if \(\varepsilon\to 0\), the solution of the problem goes to a unique asymptotically stable solution; (iii) if \(\bar u(x)\) is this stable solution, the main part of the variation of \(\bar u(x)\) occurs through “transition layers” which are located near the zeroes of \(a(x)-1/2\).
A very elegant interpretation of these solutions is, finally, given to dynamical systems.

MSC:

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35B35 Stability in context of PDEs
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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