Abstract
Recent results on the stability of equilibrium solutions of a parabolic equation are given with indications of the proofs. Particular attention is devoted to dependence of the stability properties on the shape of the domain and the manner in which nonhomogeneous stable equilibria can occur through a bifurcation induced by varying the domain.
This research was supported in part by the National Science Foundation under MCS-79-0-774, in part by the United States Army under AROD DAAG 27-79-C-0161, and in part by the United States Air Force under AF-AFOSR 76-3092C.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
C. Bardos, H. Matano & J. Smoller, Some results on the instability of the solutions of reaction diffusion equations. Preprint.
P. Brunovsky & S.N. Chow, Generic properties of stationary states of reaction diffusion equations. To be submitted.
R.G. Casten & C.J. Holland, Instability results for a reaction diffusion equation with Neumann boundary conditions. J. Differential Equations 27 (1978), 266–273.
N. Chafee, Asymptotic behaviour of a one-dimensional heat equation with homogeneous Neumann boundary conditions. J. Differential Equations 18 (1975), 111–134.
S.N. Chow, J.K. Hale & J. Mallet-Paret, Applications of generic bifurcation, II. Arch. Rat. Mech. Anal. 62 (1976), 209–236.
J.K. Hale & P. Massatt, Convergence of solutions in gradient-like systems. To be submitted.
J.K. Hale & A. Stokes, Behaviour of solutions near integral manifolds. Arch. Rat. Mech. Anal. 6 (1960), 133–170.
J.K. Hale & J. Vegas, Bifurcation with respect to domain in a parabolic equation. Submitted to Arch. Rat. Mech. Anal.
J.K. Hale, Functional Differential Equations. Appl. Math. Sci. Vol. 3, Springer-Verlag, 1977.
J.K. Hale, Some recent results on dissipative processes. Proc. Symp. on Functional Differential equations and Dynamical Systems. São Carlos, Brazil, (1979). To appear in Lecture Notes in Math., Springer-Verlag.
D. Henry, Geometric Theory of Semilinear Parabolic Equations. To appear in Lecture Notes in Math., Springer-Verlag.
I.G. Malkin, Theory of Stability of Motion. Moscow 1952.
H. Matano, Convergence of solutions of one-dimensional semi-linear parabolic equations. J. Math. Kyoto University 18 (1978), 221–227.
H. Matano, Asymptotic behaviour and stability of solutions of semilinear diffusion equations. Res. Inst. Math. Sci., Kyoto 15 (1979), 401–454.
J. Smoller & A. Wasserman, Global bifurcation of steady-state solutions. J. Differential Equations. To appear, 12/1980.
M. Urabe, Nonlinear Autonomous Oscillations, Academic Press, 1967.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1981 Springer-Verlag
About this paper
Cite this paper
Hale, J.K. (1981). Stability and bifurcation in a parabolic equation. In: Rand, D., Young, LS. (eds) Dynamical Systems and Turbulence, Warwick 1980. Lecture Notes in Mathematics, vol 898. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0091911
Download citation
DOI: https://doi.org/10.1007/BFb0091911
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-11171-9
Online ISBN: 978-3-540-38945-3
eBook Packages: Springer Book Archive