Differentiability of C0-semigroups with respect to parameters and its application

In this paper we discuss some continuous dependency between a C0-semigroup and its parameter. We first get the following results: if the infinitesimal generator of a C0-semigroup is continuous in the generalized sense, strongly (or weakly) Fréchet continuously differentiable, or strongly (or weakly) Gâteaux continuously differentiable with respect to the parameter, respectively, so is the corresponding C0-semigroup. We then obtain corresponding expressions for the differential operators. Finally, the obtained results are applied to a mixed initial-boundary value problem of a semi-linear parabolic equation, and it is proved that the solution of the equation is Fréchet continuously differentiable with respect to the leading variable coefficient of the equation.  2003 Elsevier Science (USA). All rights reserved.