Time-dependent nonlinear evolution equations

Abstract

Of concern is the nonlinear evolution equation \begin{align} \frac{du}{dt} & \in A(t)u,\ \ 0 < t < T \notag \\ u(0) & = u_{0} \notag \end{align} in a real Banach space $ X $, where $ A(t) : D(A(t)) \subset X \longrightarrow X $ is a time-dependent, nonlinear, multivalued operator acting on $ X $. It is shown that under certain assumptions on $ A(t) $, the equation has a strong solution. Applications to nonlinear parabolic boundary value problems with time-dependent boundary conditions are given.