Stability and bifurcation of equilibrium solutions of reactiondiffusion equations on an infinite space interval
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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 dtu = 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)ε{lunate}X × R, that the Fréchet 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{lunate} 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. © 1984.
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