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Stability analysis for positive solutions for classes of semilinear elliptic boundary-value problems with nonlinear boundary conditions

Part of: Partial differential equations Elliptic equations and systems

Published online by Cambridge University Press:  28 June 2017

Jerome Goddard II  and
Ratnasingham Shivaji
Jerome Goddard II
Affiliation:
Department of Mathematics, Auburn University at Montgomery, Montgomery, AL 36124, USA (jgoddard@aum.edu)
Ratnasingham Shivaji
Affiliation:
Department of Mathematics and Statistics, University of North Carolina Greensboro, Greensboro, NC 27402, USA (r_shivaj@uncg.edu)
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We investigate the stability properties of positive steady-state solutions of semilinear initial–boundary-value problems with nonlinear boundary conditions. In particular, we employ a principle of linearized stability for this class of problems to prove sufficient conditions for the stability and instability of such solutions. These results shed some light on the combined effects of the reaction term and the boundary nonlinearity on stability properties. We also discuss various examples satisfying our hypotheses for stability results in dimension 1. In particular, we provide complete bifurcation curves for positive solutions for these examples.

Keywords

nonlinear boundary conditionssemipositonestabilityprinciple of linearized stability

MSC classification

Secondary: 35J66: Nonlinear boundary value problems for nonlinear elliptic equations 35B09: Positive solutions 35B35: Stability
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 147 , Issue 5 , October 2017 , pp. 1019 - 1040
Copyright
Copyright © Royal Society of Edinburgh 2017