Evolution equations with dynamic boundary conditions

Abstract

SynopsisIn this paper, we study boundary problems with dynamic boundary conditions, that is, with boundary operators containing time derivatives. The equations under consideration are transformed into abstract Cauchy problems x – Cx = f and x(0) = x0. Abstract theoretical results concerning the operators C are obtained by the study of a naturally arising pseudodifferential operator. For existence and uniqueness theorems concerning solutions of parabolic and hyperbolic equations, we then apply the theory of semigroups in Banach spaces. Some examples of semilinear and quasilinear problems, to which our results apply, are given.