Stability, Instability and Center Manifold Theorem for Fully Nonlinear Autonomous Parabolic Equations in Banach Space G. DA PRATO & A. LUNARDI Communicated by C. M. DAFERMOS Introduction In this paper we generalize some classical results on stability for autonomous semilinear evolution equations to the fully nonlinear case. The typical example we have in mind is the equation ut(t, x) = ~(u(t, x), Ux(t, x), Uxx(t, x)), t ~ 0, 0 --< x --< z~, (0.1) which will be studied by abstract methods, reducing it to the equation u'(t) = g(u(t)), t ~ 0 (0.2) where g : D -+ X is a regular function and D, X are Banach spaces with D con- tinuously embedded in 2(. Assuming that g(0) = 0, we will study stability, instability and saddle points of the zero solution of (0.2) by means of a linear approximation. To treat also some critical cases of stability, we will establish the existence of an attracting local center manifold for equation (0.2). Our treatment follows closely the methods used in the semilinear case (see for instance [1], [4]), but extending these ideas is not trivial because of technical difficulties due to the fully nonlinear character of problem (0.2). The main assumption on g is that the operator A = g'(0) : D ~ X generates an analytic semigroup e tA in X. The usual method for studying the initial value problem by linearization, namely u'(t) : g(u(t)) : Au(t) + ~p(u(t)), t ~ O, (0.3) u(0) = Uo, would be to solve the integral equation t u(t) = e ta Uo + f e (t-s)A W(u(s)) ds, t >= 0 0