Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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

 
 

 

Semilinear Neumann boundary value problems on a rectangle


Author: Junping Shi
Journal: Trans. Amer. Math. Soc. 354 (2002), 3117-3154
MSC (2000): Primary 35J25, 35B32; Secondary 35J60, 34C11
DOI: https://doi.org/10.1090/S0002-9947-02-03007-6
Published electronically: April 2, 2002
MathSciNet review: 1897394
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: 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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 35J25, 35B32, 35J60, 34C11

Retrieve articles in all journals with MSC (2000): 35J25, 35B32, 35J60, 34C11


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: shij@math.wm.edu

Keywords: Semilinear elliptic equations, secondary bifurcations, global bifurcation diagrams, asymptotic behavior of solutions
Received by editor(s): April 17, 2001
Published electronically: April 2, 2002
Article copyright: © Copyright 2002 American Mathematical Society