# Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## Positive unstable periodic solutions for superlinear parabolic equationsHTML articles powered by AMS MathViewer

by Norimichi Hirano and Noriko Mizoguchi
Proc. Amer. Math. Soc. 123 (1995), 1487-1495 Request permission

## Abstract:

In this paper, we are concerned with a superlinear parabolic equation $\left \{ {\begin {array}{*{20}{c}} {\frac {{\partial u}}{{\partial t}} - \Delta u = {u^p} + h(t,x),} \hfill & {(t,x) \in {{\mathbf {R}}_ + } \times \Omega ,} \hfill \\ {{u = 0,}} \hfill & {(t,x) \in {{\mathbf {R}}_ + } \times \partial \Omega ,} \hfill \\ {{u > 0,}} \hfill & {(t,x) \in {{\mathbf {R}}_ + } \times \partial \Omega ,} \hfill \\ \end {array} } \right .$ 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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