The existence and attractivity of a local center manifold for fully nonlinear parabolic equation with infinite delay is proved with help of a solutions semigroup constructed on the space of initial conditions. The result is applied to the stability problem for a parabolic integrodifferential equation.

The object of this paper is to prove existence and regularity results for non-linear elliptic differential-functional equations of the form $\mathrm{div}a\left(\nabla u\right)+F\left[u\right]\left(x\right)=0,$ over the functions $u\in W^{1,1}\left(\Omega \right)$ that assume given boundary values ϕ on ∂Ω. The vector field $a:{ℝ}^n\to {ℝ}^n$ satisfies an ellipticity condition and for a fixed x, F[u](x) denotes a non-linear functional of u. In considering the same problem, Hartman and Stampacchia [Acta Math.115 (1966) 271–310] have obtained existence results in the space of uniformly Lipschitz continuous functions...