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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Positive unstable periodic solutions for superlinear parabolic equations


Authors: Norimichi Hirano and Noriko Mizoguchi
Journal: Proc. Amer. Math. Soc. 123 (1995), 1487-1495
MSC: Primary 35K55; Secondary 35B10, 35B35
MathSciNet review: 1234627
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Abstract: In this paper, we are concerned with a superlinear parabolic equation

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {\frac{{\partial u}}{{\partial t... ...in {{\mathbf{R}}_ + } \times \partial \Omega ,} \hfill \\ \end{array} } \right.$

where $ \Omega \subset {{\mathbf{R}}^N}$ is a bounded domain with smooth boundary $ \partial \Omega $, h is T-periodic with respect to the first variable, and $ 1 < p < \frac{{N + 2}}{{N - 2}}$ if $ N \geq 3$ and $ 1 < p < + \infty $ if $ N \leq 2$. It is shown that there exist a stable and an unstable positive T-periodic solution for this problem if h is sufficiently small in $ {L^\infty }$.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1995-1234627-2
PII: S 0002-9939(1995)1234627-2
Keywords: Parabolic nonlinear problem, periodic solutions, stable solutions, unstable solutions
Article copyright: © Copyright 1995 American Mathematical Society