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Equilibria and global dynamics of a problem with bifurcation from infinity

Arrieta Algarra, José María and Pardo San Gil, Rosa María and Rodríguez Bernal, Aníbal (2009) Equilibria and global dynamics of a problem with bifurcation from infinity. Journal of Differential Equations, 246 (5). 2055-2080 . ISSN 0022-0396

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Abstract

We consider a parabolic equation ut−Δu+u=0 with nonlinear boundary conditions , where as |s|→∞. In [J.M. Arrieta, R. Pardo, A. Rodríguez-Bernal, Bifurcation and stability of equilibria with asymptotically linear boundary conditions at infinity, Proc. Roy. Soc. Edinburgh Sect. A 137 (2) (2007) 225–252] the authors proved the existence of unbounded branches of equilibria for λ close to a Steklov eigenvalue of odd multiplicity. In this work, we characterize the stability of such equilibria and analyze several features of the bifurcating branches. We also investigate several question related to the global dynamical properties of the system for different values of the parameter, including the behavior of the attractor of the system when the parameter crosses the first Steklov eigenvalue and the existence of extremal equilibria. We include Appendix A where we prove a uniform antimaximum principle and several results related to the spectral behavior when the potential at the boundary is perturbed.


Item Type:Article
Uncontrolled Keywords:Stability; Uniqueness; Steklov eigenvalues; Bifurcation from infinity; Sublinear boundary conditions; Attractors; Extremal equilibria; Antimaximum principle
Subjects:Sciences > Mathematics > Differential equations
ID Code:13429
Deposited On:19 Oct 2011 07:22
Last Modified:06 Feb 2014 09:48

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