Linearization

Abstract

There is a well-known theorem in ordinary differential equations, going back to Poincaré, which states that the stability of a rest point can be inferred from “linearization.” More precisely, if one considers the ordinary differential equation in Rⁿ, u′ = f(u), and ū is a rest point (so that f(ū) = 0), then if the differential (matrix) df (ū) has all of its eigenvalues in the left-half plane, Re z < 0, it follows that ū is asymptotically stable; i.e., if u₀ is near ū, then the solution of the equation through u₀ tends to ū as t → + ∞. It is the main purpose of this chapter to prove an analogous theorem for partial differential equations, one which is sufficiently general to include systems of reaction-diffusion (parabolic) equations. In this context, the equation u′ = f(u) is replaced by an abstract equation of the form u _t = Au + f(u), where u takes values in a Banach space B; i.e., for each t, u(t) is in B and A is a linear operator. The main example is the case where A is a linear elliptic operator. The “rest points” ū, in this setting now are solutions of the equation Au + f(u) = 0, and the linearized operator becomes A + df (ū). We shall show that if the spectrum of this operator lies in the left-half plane, then again one can conclude that ū is asymptotically stable. This leads us quite naturally to a study of the spectrum of such linear operators. We shall undertake this study in §A, and in §B we shall prove a linearized stability theorem.