Semilinear evolution equations and fractional powers of a closed pair of operators

Abstract

SynopsisThe nonlinear evolution problem [Bu(t)]′ =A(t,Bu)u+f(t,Bu) withBa constant linear operator andA=A(t,Bu) a time-dependent nonlinear operator from one Banach space to another, is studied. Existence and uniqueness results are obtained by making use of the theory of B-evolutions and the fractional powers ofAandB. Two examples are presented in which the theory is applied to nonlinear equations with dynamic boundary conditions.