## 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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### References:

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