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We prove $L^2$ $$\dot{u}\left(t\right)+A\left(t\right)u\left(t\right)=f\left(t\right)\text{for}\text{a.e.}t\in [0,T],u\left(0\right)=u_0,$$ where the operator $A\left(t\right)$ arises from a time depending sesquilinear form $𝔞(t,·,·)$ on a Hilbert space $H$ with constant domain $V.$ We prove the maximal regularity in $H$ when these forms are time Lipschitz continuous. We proceed by approximating the problem using the frozen coefficient method developed by El-Mennaoui, Keyantuo, Laasri (2011), El-Mennaoui, Laasri (2013), and Laasri (2012). As a consequence, we obtain an invariance criterion for convex and closed sets of...

We study stability and integrability of linear non-autonomous evolutionary Cauchy-problem $$\left(\mathrm{P}\right)\left\{\begin{array}{c}\dot{u}\left(t\right)+A\left(t\right)u\left(t\right)=f\left(t\right)t\text{-a.e.}\text{on}[0,\tau ],u\left(0\right)=0,\hfill \end{array}\right.$$ where $A:[0,\tau ]\to ℒ(X,D)$ is a bounded and strongly measurable function and $X$, $D$ are Banach spaces such that $D\underset{d}{\to }\hookrightarrow X$. Our main concern is to characterize $L^p$-maximal regularity and to give an explicit approximation of the problem (P).