Evolution equations without semigroups

Abstract Let { A ( t ):0⩽ t ⩽ T } be a family of linear closed operators defined on a dense set in a Banach space E . Evolution equations of the form d u /d t = A ( t ) u is studied in E , for a wide class of operators A ( t ), 0⩽ t ⩽ T , which in general have no resolvents. It will be proved that there exists a dense set S in E such that there exists a solution u ( t ) of the Cauchy problem for the considered equation with the initial condition u (0)∈ S . The correct formulation of the Cauchy problem is also studied in a certain class of solutions. Applications to general partial differential equations are given without any restrictions on the characteristic forms.