Stability and bifurcation of equilibrium solutions of reaction-diffusion equations on an infinite space interval
In: Journal of Differential Equations, Jg. 54 (1984-08-01), Heft 1, S. 19-59
Online
unknown
Zugriff:
Under the condition that f(x, y, z, α) and its partial derivatives decay sufficiently fast as ¦x¦ → ∞ we will study the (linear) stability and bifurcation of equilibrium solutions of the scalar problem ut = uxx + f(x, u, ux, α), ux(−∞, t) = ux(∞, t) = 0 (∗) where α is a real bifurcation parameter. After introducing appropriate function spaces X and Y the problem (∗) can be rewritten d dt u = G(u, α) , (∗∗) where G:X×R → Y is given by G(u, α)(x) = u″(x) + f(x, u(x), u′(x), α). It will be shown, for each (u, α)ϵX × R, that the Frechet derivative Gu(u,a): X → Y is not a Fredholm operator. This difficulty is due to the fact that the domain of the space variable x, is infinite and cannot be eliminated by making another choice of X and Y. Since Gu(u, α) is not Fredholm, the hypotheses of most of the general stability and bifurcation results are not satisfied. If (u0, α0 ϵ S = {(u, α): G(u, α) = 0}, (i.e., (u0,α0) is an equilibrium solution of (∗∗)), a necessary condition on the spectrum of Gu(u0, α0) for a change in the stability of points in S to occur at Gu(u0, α0) will be given. When this condition is met, the principle of exchange of stability which means, in a neighborhood of (u0, α0), that adjacent equilibrium solutions for the same α have opposite stability properties in a weakened sense will be established. Also, when Gu or its first order partial derivatives, evaluated at (u0, α0), are not too degenerate, the shape of S in a neighborhood of (u0, α0) will be described and a strenghtened form of the principle of exchange of stability will be obtained.
Titel: |
Stability and bifurcation of equilibrium solutions of reaction-diffusion equations on an infinite space interval
|
---|---|
Autor/in / Beteiligte Person: | Taliaferro, Steven D. |
Link: | |
Zeitschrift: | Journal of Differential Equations, Jg. 54 (1984-08-01), Heft 1, S. 19-59 |
Veröffentlichung: | Elsevier BV, 1984 |
Medientyp: | unknown |
ISSN: | 0022-0396 (print) |
DOI: | 10.1016/0022-0396(84)90141-4 |
Schlagwort: |
|
Sonstiges: |
|