Global bifurcation of steady-state solutions (original) (raw)

1981, Journal of Differential Equations

In this paper, we study the bifurcation of steady-state solutions of a reaction-diffusion equation in one space variable. The steady-state solutions satisfy the equation a" +f(u) = 0 on the interval-L < x < L, where we take f(u) to be a cubic polynomial. The solution is assumed to satisfy either homogeneous Dirichlet, or homogeneous Neumann, or periodic boundary conditions. We take as bifurcation parameter the number L, and we obtain global bifurcation diagrams; that is, we count the exact number of solutions. These solutions can be viewed as the "rest points" of the equation U, = u,, + f(u). In order to determine the global flow of this latter equation, the precise knowledge of the number of rest points is a necessary first step. Our technique is a careful analysis of the so-called "time-map" S(a), a function defined by an elliptic integral, which measures the "time" an orbit takes to get from one boundary line to another. The relevant point is that we are able to count the exact number of critical points of S. This is done for both positive and negative solutions of the Dirichlet problem,' by proving estimates of the form S" + cS' > 0, or (0, for some (nonzero) function c = c(a). For the Neumann problem, we use entirely different techniques to prove that S is never critical, for any cubic polynomial J This implies at once that the Neumann problem can have at most one nonconstant solution (having a given number of maxima or minima). This solution is necessarily strongly nondegenerate, in the sense that zero is not contained in the spectrum of the linearized operator (see [2]). For the Dirichlet problem, the situation is far more complicated, and the bifurcation diagrams undergo qualitative changes, depending on the positions of the roots off: For example, Fig. 1 shows the bifurcation diagram (for the positive solutions), for two different cubic functions of the form f(u) = (a-u)(u-b)(u-c).