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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 equations
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by Norimichi Hirano and Noriko Mizoguchi PDF
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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Additional Information
  • © Copyright 1995 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 123 (1995), 1487-1495
  • MSC: Primary 35K55; Secondary 35B10, 35B35
  • DOI: https://doi.org/10.1090/S0002-9939-1995-1234627-2
  • MathSciNet review: 1234627