TY - JOUR
T1 - Existence of secondary bifurcations or isolas for PDEs
AU - Gameiro, Marcio
AU - Lessard, Jean Philippe
N1 - Funding Information:
The first author was partially supported by the JSPS Postdoctoral Fellowship No. P08016 and by the JSPS Grant-in-Aid for Scientific Research No. 2008016 , Ministry of Education, Science, Technology, Culture and Sports, Japan and by Fapesp Processo 2010/00875-9 and CNPq Processo 306453/2009-6.
Funding Information:
The second author was partially supported by NSF grant DMS-0511115 , by DARPA, and by DOE grant DE-FG02-05ER25711 .
PY - 2011/8
Y1 - 2011/8
N2 - In this paper, we introduce a method to conclude about the existence of secondary bifurcations or isolas of steady state solutions for parameter dependent nonlinear partial differential equations. The technique combines the Global Bifurcation Theorem, knowledge about the non-existence of nontrivial steady state solutions at the zero parameter value and explicit information about the coexistence of multiple nontrivial steady states at a positive parameter value. We apply the method to the two-dimensional SwiftHohenberg equation.
AB - In this paper, we introduce a method to conclude about the existence of secondary bifurcations or isolas of steady state solutions for parameter dependent nonlinear partial differential equations. The technique combines the Global Bifurcation Theorem, knowledge about the non-existence of nontrivial steady state solutions at the zero parameter value and explicit information about the coexistence of multiple nontrivial steady states at a positive parameter value. We apply the method to the two-dimensional SwiftHohenberg equation.
KW - Bifurcations of equilibria for PDEs
KW - Global Bifurcation Theorem
KW - Rigorous numerics
KW - Secondary bifurcation
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U2 - 10.1016/j.na.2011.03.046
DO - 10.1016/j.na.2011.03.046
M3 - Article
AN - SCOPUS:79955760238
SN - 0362-546X
VL - 74
SP - 4131
EP - 4137
JO - Nonlinear Analysis, Theory, Methods and Applications
JF - Nonlinear Analysis, Theory, Methods and Applications
IS - 12
ER -