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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
DOI: https://doi.org/10.1090/S0002-9939-1995-1234627-2
MathSciNet review: 1234627
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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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Keywords: Parabolic nonlinear problem, periodic solutions, stable solutions, unstable solutions
Article copyright: © Copyright 1995 American Mathematical Society