Evolution equations without semigroups

摘要

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 du/dt=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.