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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.

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.

Reviewer: P. A. Bois (Villeneuve d’Ascq)

### 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) |

### Keywords:

semilinear; attractors; existence; stability; stable equilibrium; asymptotically stable; transition layers; dynamical systems
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\textit{S. B. Angenent} et al., J. Differ. Equations 67, 212--242 (1987; Zbl 0634.35041)

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