A nonlinear parabolic equation with varying domain. (English) Zbl 0569.35048

Hale, Jack K.; Vegas, Jose

Arch. Ration. Mech. Anal. 86, 99-123 (1984).

This paper considers the problem \(u_ t=\Delta u+f(\lambda,u)\) in \(\Omega_{\epsilon}\) \(\partial u/\partial n=0\) on \(\partial \Omega_{\epsilon}\) where \(0<\epsilon \leq 1\), \(\Omega_{\epsilon}\) is a domain in \(R^ n\) and \(\lambda\) a small real parameter. The objective is to discuss the equilibrium solutions and their stability as a function of (\(\lambda\),\(\epsilon)\) in a neighborhood of (0,0), when the domain \(\Omega_{\epsilon}\) increases in \(\epsilon\), \(| \Omega_{\epsilon}|\) is continuous in \(\epsilon\), \(| \Omega_{\epsilon}\setminus \Omega_ 0| \to 0\) as \(\epsilon\) \(\to 0\) and \(\Omega_ 0\) is the union of connected domains \(\Omega^ L_ 0\) and \(\Omega^ R_ 0\) with disjoint closures. The approach is a Lyapunov-Schmidt method devised as follows. For \(\Omega_ 0\), 0 is a double eigenvalue of the Laplacian with Neumann boundary conditions. Using some more assumptions on \(\Omega_{\epsilon}\), the discussion of the equilibrium solutions is reduced to a pair of bifurcation equations depending upon the parameter \(\epsilon\). The technical difficulties occur in showing the smoothness properties of the bifurcation equation.

Reviewer: J.Mawhin

MSC:

35K55	Nonlinear parabolic equations
35B32	Bifurcations in context of PDEs
35B40	Asymptotic behavior of solutions to PDEs
35B30	Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs

Keywords:

small real parameter; equilibrium solutions; stability; Lyapunov-Schmidt method; Laplacian; Neumann boundary conditions; bifurcation equations

References:

[1]	Adams, R. A.: Sobolev Spaces, Academic Press, New York, 1975. · Zbl 0314.46030
[2]	Babu?ka, I. and Výborný, R.: Continuous dependence of the eigenvalues on the domain, Czechoslovak Math. J., 15, 169-178 (1965). · Zbl 0137.32302
[3]	Casten, R. G., and Holland, C. J.: Instability results for reaction diffusion equations with Neumann boundary conditions, Differential Equations, 27, 266-273 (1978). · Zbl 0359.35039
[4]	Chafee, N.: Asymptotic behavior for solutions of a one-dimensional parabolic equation with homogeneous Neumann boundary conditions, J. Differential Equations, 18, 111-134 (1975). · Zbl 0304.35008
[5]	Grigorieff, R. D.: Diskret Kompakte Einbettungen in Sobolewschen Räumen, Math. Ann., 197, 71-85 (1972). · Zbl 0228.35009
[6]	Hale, J. K.: Stability and bifurcation in a parabolic equation, Dynamical Systems and Turbulence, Warwick, 1980 Lecture Notes in Mathematics, Springer, Berlin, Heidelberg, New York, 143-153 (1981).
[7]	Matano, H.: Asymptotic behavior and stability of solutions of semilinear diffusion equations, Publ. RIMS, Kyoto Univ. 15, 401-454 (1979). · Zbl 0445.35063
[8]	Ne?as, J.: Les méthodes directes en théorie des équations elliptiques, Academia, Editions de l’Academie Tchécoslovaque des Sciences, Prague, 1967.
[9]	Rauch, J., and Taylor, M.: Potential and Scattering theory on wildly perturbed domains, J. Functional Analysis, 18, 27-59 (1975). · Zbl 0293.35056
[10]	Stummel, F.: Perturbation theory for Sobolev spaces, Proc. Royal Soc. Edinburgh, 73A, 5-49 (1974/75). · Zbl 0358.46027
[11]	Stummel, F.: Perturbation of domains in elliptic boundary value problems. In Lecture Notes in Mathematics, 503, Springer, Berlin Heidelberg New York, 110-136 (1976). · Zbl 0355.65074
[12]	Stein, E.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton, New Jersey, 1979.