M 511

We consider non-autonomous parabolic equation of the form \begin{equation}\label{1} \dot u(t)+A(t)u(t)=0, \ t\in[0,T],\ u(0)=u_0. \end{equation} Here $A(t), t\in [0,T],$ are associated with a non-autonomous sesquilinear form $a(t,\cdot,\cdot)$ on a Hilbert space $H$ with constant domain $V\subset H.$ We give a brief introduction to $L^p$-maximal regularity for non-autonomous linear evolution equations of the form (\ref{1}). Furthermore, we study some fundamental theoretical properties of the associated evolution family. Recall that it is well known that, under suitable conditions, the solution of a non-autonomous linear evolution equation may be given by a strongly continuous evolution family. The later is in fact the non-autonomous counterpart of operator semigroup in the well-posedness theory of non-autonomous evolution equations.

Dr. Hafida Laasri

Hagen