Existence and stability for partial functional differential equations

Abstract

Publisher Summary This chapter discusses the existence and stability for a class of partial functional differential equations. As a model for this class, one may take the equation wt(x, t) = wxx (x, t) + ƒ(t, w(x, t − r)), 0 ≤ x ≤ Π, t ≥ 0, w(0, t) = w(Π, t) = 0, t ≥ 0, w(x, t) = Φ (x, t), 0 ≤ x ≤ Π, −r ≤ t ≤ 0, where ƒ is a linear or nonlinear scalar-valued function, r a positive real number, and Φ a given initial function. The second derivative term corresponds to a strongly continuous semigroup of linear operators on a Banach space of functions determined by the boundary conditions. Accordingly, the approach relies primarily on semigroup methods and the treatment of equation as an ordinary functional differential equation in a Banach space. The chapter also describes semigroup and infinitesimal generator in the autonomous case.