# Asymptotic stability via the Krein–Rutman theorem for singularly perturbed parabolic periodic Dirichlet problems

*2010, Volume 15, Numbers 2-3, pp. 382-389*

Author(s):

**Nefedov N. N., Recke L., Schneider K. R.**

We consider singularly perturbed semilinear parabolic periodic problems and assume the existence of a family of solutions. We present an approach to establish the exponential asymptotic stability of these solutions by means of a special class of lower and upper solutions. The proof is based on a corollary of the Krein–Rutman theorem.

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