On the norm-continuity for evolution family arising from non-autonomous forms

Abstract

We consider evolution equations of the form \begin{equation*}\label{Abstract equation} \dot u(t)+ A(t)u(t)=0,\ \ t\in[0,T],\ \ u(0)=u_0, \end{equation*} where $A(t),\ t\in [0,T],$ are associated with a non-autonomous sesquilinear form $\mathfrak a(t,\cdot,\cdot)$ on a Hilbert space $H$ with constant domain $V\subset H.$ In this note we continue the study of fundamental operator theoretical properties of the solutions. We give a sufficient condition for norm-continuity of evolution families on each spaces $V, H$ and on the dual space $V'$ of $V.$ The abstract results are applied to a class of equations governed by time dependent Robin boundary conditions on exterior domains and by Schr\"odinger operator with time dependent potentials.