This article concerns the nonlinear evolution equation du(t)/dt is an element of A(t)u(t), 0 <= s < t < T, u(s) = u(0) in a real Banach space X, where the nonlinear, time-dependent, and multi-valued operator A(t) : D(A(t)) subset of X -> X has a time-dependent domain D(A(t)). It will be shown that, under certain assumptions on A(t), the equation has a strong solution. Illustrations are given of solving quasi-linear partial differential equations of parabolic type with time-dependent boundary conditions. Those partial differential equations are studied to a large extent.
