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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Semilinear Neumann boundary value problems on a rectangle
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by Junping Shi PDF
Trans. Amer. Math. Soc. 354 (2002), 3117-3154 Request permission


We consider a semilinear elliptic equation \begin{equation*} \Delta u+\lambda f(u)=0, \;\; \mathbf {x}\in \Omega ,\;\; \frac {\partial u}{\partial n }=0, \;\; \mathbf {x}\in \partial \Omega , \end{equation*} where $\Omega$ is a rectangle $(0,a)\times (0,b)$ in $\mathbf {R}^2$. For balanced and unbalanced $f$, we obtain partial descriptions of global bifurcation diagrams in $(\lambda ,u)$ space. In particular, we rigorously prove the existence of secondary bifurcation branches from the semi-trivial solutions, which is called dimension-breaking bifurcation. We also study the asymptotic behavior of the monotone solutions when $\lambda \to \infty$. The results can be applied to the Allen-Cahn equation and some equations arising from mathematical biology.
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Additional Information
  • Junping Shi
  • Affiliation: Department of Mathematics, College of William and Mary, Williamsburg, Virginia 23187, and Department of Mathematics, Harbin Normal University, Harbin, Heilongjiang, P. R. China 150080
  • MR Author ID: 616436
  • ORCID: 0000-0003-2521-9378
  • Email:
  • Received by editor(s): April 17, 2001
  • Published electronically: April 2, 2002
  • © Copyright 2002 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 354 (2002), 3117-3154
  • MSC (2000): Primary 35J25, 35B32; Secondary 35J60, 34C11
  • DOI:
  • MathSciNet review: 1897394