THE C-REGULARIZED SEMIGROUP METHOD FOR PARTIAL DIFFERENTIAL EQUATIONS WITH DELAYS

摘要

This paper is devoted to study the abstract functional differential equation (FDE) of the following form ů(t) = Au(t) + Φu_t, where A generates a C-regularized semigroup, which is the generalization of Co-semigroup and can be applied to deal with many important differential operators that the Co-semigroup can not be used to. We first show that the C-well-posedness of a FDE is equivalent to the C-well-posedness of an abstract Cauchy problem in a product Banach space, where the operator C is related with the operator C and will be defined in the following text. Then, by making use of a perturbation result of C-regularized semigroup, a sufficient condition is provided for the C-well-posedness of FDEs. Moreover, an illustrative application to partial differential equation (PDE) with delay is given in the last section.