A note on the norm-continuity for evolution families arising from non-autonomous forms

Abstract

We consider evolution equations of the form $$\begin{aligned} \dot{u}(t)+{\mathcal {A}}(t)u(t)=0,\ \ t\in [0,T],\ \ u(0)=u_0, \end{aligned}$$where $${\mathcal {A}}(t),\ t\in [0,T],$$ 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 Schrodinger operator with time dependent potentials.